Science, Faculty of
Earth, Ocean and Atmospheric Sciences, Department of
DSpace
UBCV
Janssen, Joseph
Doctor of Philosophy - PhD
University of British Columbia
Since all ecological- and human-centric systems rely on a steady water supply, scientists must learn the true-underlying processes that force hydrological behavior from a few thousand catchments with available data in order to transfer that knowledge to the infinite number of ungauged catchments. While hydrologists have historically preferred to utilize physics-based equations to gain and transfer knowledge, data-driven methods for hydrological inference may provide more promising approaches. Due to the unique properties of hydrological data, such as extreme spatiotemporal dependence, high measurement uncertainty, threshold-like nonlinear processes, highly skewed distributions, non stationary relationships, uncommon causal pathway types, and high order emergent interactions, traditional statistical methods fail to learn important yet still undiscovered hydrological processes. These failures arise in a variety of hydrological studies including cross-catchment spatial analyses, single-catchment temporal analyses, and single-catchment spatiotemporal analyses, thus for each of these study types, we demonstrate previous methodological shortcomings and invalid assumptions, while developing and applying new data-driven approaches relating to information theory, causal inference, and statistical learning.
Diagnosing Spatiotemporal CatchmentHydrologic Behavior via CausalInference, Information Theory, andMachine LearningbyJoseph JanssenA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Geological Sciences)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2025© Joseph Janssen 2025The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the thesis entitled:Diagnosing Spatiotemporal Catchment Hydrologic Behavior via Causal Inference, In-formation Theory, and Machine Learningsubmitted by Joseph Janssen in partial fulfillment of the requirements for the degree of Doctorof Philosophy in Geological Sciences.Examining Committee:Ali Ameli, Assistant Professor, Department of Earth, Ocean and Atmospheric Sciences, UBCSupervisorElina Robeva, Associate Professor, Department of Mathematics, UBCCo-SupervisorRoger Beckie, Professor, Department of Earth, Ocean and Atmospheric Sciences, UBCSupervisory Committee MemberEldad Haber, Professor, Department of Earth, Ocean and Atmospheric Sciences, UBCUniversity ExaminerSteven Weijs, Associate Professor, Department of Civil Engineering, UBCUniversity ExaminerJasper Vrugt, Professor, Department of Civil and Environmental Engineering, University of Cali-fornia, IrvineExternal ExamineriiAbstractSince all ecological- and human-centric systems rely on a steady water supply, scientists must learnthe true-underlying processes that force hydrological behavior from a few thousand catchments withavailable data in order to transfer that knowledge to the infinite number of ungauged catchments.While hydrologists have historically preferred to utilize physics-based equations to gain and transferknowledge, data-driven methods for hydrological inference may provide more promising approaches.Due to the unique properties of hydrological data, such as extreme spatiotemporal dependence,high measurement uncertainty, threshold-like nonlinear processes, highly skewed distributions, non-stationary relationships, uncommon causal pathway types, and high order emergent interactions,traditional statistical methods fail to learn important yet still undiscovered hydrological processes.These failures arise in a variety of hydrological studies including cross-catchment spatial analyses,single-catchment temporal analyses, and single-catchment spatiotemporal analyses, thus for each ofthese study types, we demonstrate previous methodological shortcomings and invalid assumptions,while developing and applying new data-driven approaches relating to information theory, causalinference, and statistical learning.iiiLay SummaryEnsuring a steady water supply for all requires understanding how water systems truly work, espe-cially in unmonitored areas. Traditional methods have limitations, and water data itself is alreadytoo complex for standard analysis. This research develops and applies new, advanced data-driventools to overcome these challenges, helping us better learn the hidden behaviour governing the watercycle.ivPrefaceThis dissertation is the original work of Joseph Janssen. The research contained in this documentwas supervised by Dr. Ali Ameli and Dr. Elina Robeva. Several of the works were also supportedby various collaborators including Dr. Ardalan Tootchi, Vincent Guan, Majid Bayati, Shizhe Meng,Asad Haris, Stefan Schrunner, Jiguo Cao, William J Welch, and Nadja Kunz. This thesis consistsof 5 chapters. Each body chapter contains content from two different papers.The first part of Chapter 2 has been peer-reviewed and published as: Janssen, J., & Ameli, A. A.(2021). A hydrologic functional approach for improving large-sample hydrology performance in poorlygauged regions. Water Resources Research, 57(9), e2021WR030263. Water Resources Research(h5-index=87) is the top journal for hydrology-specific work. I performed the data collection andanalysis. Dr. Ali Ameli conceptualized of the work and both of us contributed to writing themanuscript.The second part of Chapter 2 has been peer-reviewed and published as: Janssen, J., Guan, V.,& Robeva, E. (2023, April). Ultra-marginal feature importance: Learning from data with causalguarantees. In International conference on artificial intelligence and statistics (pp. 10782-10814).PMLR. AISTATS (h5-index=100) is in the top five most prestigious venues for machine learningpapers. I conceptualized the work and performed all of the experiments and analysis. Vincent Guanand I contributed to writing the manuscript and all authors worked on the proofs.The first part of Chapter 3 has been peer-reviewed and published as: Janssen, J., Meng, S., Haris,A., Schrunner, S., Cao, J., Welch, W. J., ... & Ameli, A. A. (2025). Learning from limitedtemporal data: Dynamically sparse historical functional linear models with applications to Earthscience. Environmetrics, 36(4), e70018.. Environmetrics (h5-index=20) is a Q2 statistical journalfor environmental science related work. I performed the all of the analyses and wrote the paper.The other authors contributed to editing and idea generation.The second part of Chapter 3 is still in preparation for publication. I conceptualized and wrotemost of the paper, edited some of the code, performed some of the analyses, and did all the resultplotting.The first part of Chapter 4 has been peer-reviewed and published as: Janssen, J., Tootchi, A.,& Ameli, A. A. (2025). Tackling water table depth modeling via machine learning: From proxyobservations to verifiability. Advances in Water Resources, 104955. Advances in Water Resources(h5-index=55) is in the top five journals for hydrology related work. I performed most of the analysesand wrote the paper. Ardalan Tootchi gathered some of the data and did much of the GIS analysis.All authors contributed to editing and conceptualization.The second part of Chapter 4 has been published as a preprint as: Janssen, J., Guan, V., Ameli,A. A. A Causal Inference Framework for Catchment Boundary Delineation. Available at SSRNvPreface5389768.. I conceptualized of the work, performed the analysis and wrote the paper. The otherauthors contributed to editing and some writing.Gemini 2.5 and Grok 3 were used for some minor rephrasing and editing throughout this work. AItools were only used on works done in 2025.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research questions and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Dissertation structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Cross-catchment spatial analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Improving large-sample hydrology performance in poorly gauged regions . . . . . . 92.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3 Ultra-marginal feature importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.2 Axioms for explaining data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3.3 Ultra-marginal feature importance . . . . . . . . . . . . . . . . . . . . . . . . 422.3.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Within-catchment temporal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Dynamically sparse historical functional linear models with applications to Earthscience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52viiTable of Contents3.2.2 Historical functional linear model with dynamic sparsity (HFLM-DS) . . . . 553.2.3 Hydrology Data Application . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.4 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3 Self-Explaining LSTMs for hydrological inference and prediction . . . . . . . . . . . 693.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.3.2 Self-explaining Long Short-Term Memory (SE-LSTM) . . . . . . . . . . . . . 723.3.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 Within-catchment spatial analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2 Tackling water table depth modeling via machine learning . . . . . . . . . . . . . . 854.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3 A Causal Inference Framework for Catchment Boundary Delineation . . . . . . . . 1174.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.3.3 A proof of concept on virtual catchments: Experimental design . . . . . . . 1244.3.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3.5 A simulated case study comparing the efficiency of causal inference-based ver-sus topography-based delineations . . . . . . . . . . . . . . . . . . . . . . . . 1304.3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.2 Implications and significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139AppendicesA Appendix for âA Hydrologic Functional Approach for Improving Large-SampleHydrology Performance in Poorly Gauged Regionsâ . . . . . . . . . . . . . . . . 185A.1 Extra figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185B Appendix for âUltra-marginal Feature Importance: Learning from Data withCausal Guaranteesâ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189B.1 Mutual information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189B.1.1 Properties of mutual information . . . . . . . . . . . . . . . . . . . . . . . . 189viiiTable of ContentsB.1.2 Mutual information and feature importance . . . . . . . . . . . . . . . . . . 190B.1.3 Mutual information and machine learning evaluation functions . . . . . . . . 191B.2 Additional information about marginal contribution feature importance (MCI) . . . 192B.3 Additional information about ultra-marginal feature importance (UMFI) . . . . . . 193B.4 Additional information about other feature importance methods . . . . . . . . . . . 197B.5 Preprocessing methods for removing dependencies . . . . . . . . . . . . . . . . . . . 198B.5.1 Optimal transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198B.5.2 Linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200B.6 Experiments comparing linear regression and optimal transport . . . . . . . . . . . 201B.6.1 Removing dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201B.6.2 Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202B.7 Further feature importance experiments . . . . . . . . . . . . . . . . . . . . . . . . . 203B.7.1 Extra experiments on simulated data . . . . . . . . . . . . . . . . . . . . . . 203B.7.2 Extra BRCA experiments with known ground-truth feature importance . . . 206B.7.3 Experiments on unpermuted BRCA data . . . . . . . . . . . . . . . . . . . . 207B.7.4 Experiments on hydrology data . . . . . . . . . . . . . . . . . . . . . . . . . 208C Appendix for âDynamically sparse historical functional linear models with ap-plications to Earth scienceâ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217C.1 Additional Real Data Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217C.2 Additional Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217C.2.1 Selecting the Sparsity Parameter . . . . . . . . . . . . . . . . . . . . . . . . . 217C.2.2 Learned Dynamic Sparsity from Simulation Study . . . . . . . . . . . . . . . 217D Appendix for âSelf-Explaining LSTMs for hydrological inference and predictionâ 219E Appendix for âTackling water table depth modeling via machine learning: Fromproxy observations to verifiabilityâ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220E.2 Canadian well reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220E.2.1 Quebec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220E.2.2 Ontario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220E.2.3 Nova Scotia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220E.2.4 Alberta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221E.2.5 British Columbia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221E.2.6 Saskatchewan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221E.2.7 New Brunswick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221E.2.8 NewFoundland and Labrador . . . . . . . . . . . . . . . . . . . . . . . . . . . 221E.2.9 Manitoba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221E.2.10 Yukon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222F Appendix for âA Causal Inference Framework for Catchment Boundary Delin-eationâ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223F.1 Granger causality, predictive power, and mutual information . . . . . . . . . . . . . 223F.2 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224F.3 Extra Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224ixList of Tables2.1 Description of Catchment-Scale Individual Physical and Annual Climatic Attributes,Climatic Indices, Functional Indices, and Streamflow Signatures. Median (and in-terquartile ranges [Q1 & Q3]) for attributes, indices and signatures, across all catch-ments, are listed associated with non-transformed and non-scaled values. The trans-formation function for attributes, indices and signatures are listed (see Section 2.2.3for details). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Summary of the test set R2 values (on raw observed streamflow) across three differentmethods (columns) and two different catchments (rows). The best result in each rowis bolded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 Summary of the simulation study results. The average (standard deviation) acrossthe 100 replications is shown for each of the evaluation criteria defined in Section 3.2.4. 683.3 Summary of the simulation study results, comparing our old method (HFLM-DS)with the method developed in this paper (SE-LSTM). Inference is run for each of ourfocus catchments and three possible signal-to-noise levels. The inference quality isquantified by listing the correlation coefficient between the ground truth and inferredtime-varying unit hydrographs. The best result from each row is bolded. . . . . . . . 824.1 Mean and median climatic, topographic, and soil characteristics of ten ecoregionsalong which we evaluate model performance. The characteristics include long-termaverage Aridity index (-), Temperature (â¦C), Elevation (m), Depth to Bedrock (DTB,cm), Shallow Clay (%), and real observations of WTD (m). The ten ecoregions are (see4.2): Taiga (3), Northern Forests (5), Northwestern Forested Mtns (6), Marine WestCoast Forests (7), Eastern Temperate Forests (8), Great Plains (9), North AmericanDeserts (10), Mediterranean California (11), Temperate Sierras (13), Tropical HumidForests (15). Numbers in the parenthesis show the ecoregion class in the 15-classOmernik Ecoregion classification. Eco-region area is shown below the eco-regionâs class. 964.2 The performance of each model on âunseenâ real observations of water table depth(OBS) and unseen interior pixels of permanently wet water bodies (WB) across theentire study area (All) and within each ecoregion. The performance is evaluatedusing mean absolute error of simulated WTD at interior locations of permanentlywet surface water bodies wherein zero WTD is expected (MAE-WB), mean absoluteerror of simulated versus observed WTD (MAE-OBS), and the correlation betweensimulated and observed WTD (Corr-OBS). Ecoregions include: Taiga (3), NorthernForests (5), Northwestern Forested Mtns (6), Marine West Coast Forests (7), EasternTemperate Forests (8), Great Plains (9), North American Deserts (10), MediterraneanCalifornia (11), Temperate Sierras (13), Tropical Humid Forests (15). The best resultof each row is bolded and both MAE metrics are measured in meters. . . . . . . . . 109xList of Tables4.3 Various methods can be used to estimate catchment delineations from precipitationdata, including mutual information (MI), transfer entropy (TE), and causal effectestimation (CEE). To assess whether precipitation Xtâ1 at a given location influencesstreamflow Yt, each method conditions on an adjustment set of variables. Despite thepopularity of MI and TE, only CEE yields an unbiased causal estimate. We note thatthe backdoor criterion (5) can practically be satisfied by conditioning on the historicalprecipitation {Xtâ2, · · Â·Xtâl} as well as the historical precipitation of its neighbours,given sufficient lag l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.4 Classification Accuracy rates from our three considered methods for classifying pixelswithin the ground truth catchment (positive) versus pixels outside of the ground truthcatchment (negative). Higher is better across all metrics. Values in the parenthesesshow the standard deviation across our 7775 time series. . . . . . . . . . . . . . . . . 128B.1 The standardized interquartile range (SIQR), true positive rate (TPR), true negativerate (TNR), overall accuracy (OA), and the number of features for which feature im-portance can be calculated within 60 minutes are displayed after running the methodson the BRCA data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206xiList of Figures2.1 Overview of catchment locations and catchment-scale (a) constituent or individualclimatic, topographical and geological attributes. All attributes and indices werenormalized and transformed in this figure to simplify the spatial comparison (seeSection 2.2.3). The median and interquartile ranges of non-transformed and non-scaled attributes and indices, and their transformation functions, were reported inTable 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Overview of catchment locations and catchment-scale climatic and hydrologic func-tional indices. All attributes and indices were normalized and transformed in thisfigure to simplify the spatial comparison (see Section 2.2.3). The median and in-terquartile ranges of non-transformed and non-scaled attributes and indices, and theirtransformation functions, were reported in Table 2.1. . . . . . . . . . . . . . . . . . . 152.3 Pearson correlation (r) among (non-transformed) individual attributes and indices. . 182.4 Maps of hydrologic clustering (top panel) and the representative regime curve (meanregime curve) of catchments belonging to each cluster (bottom panel), for (a) frame-work 1, (b) framework 2, and (c) framework 3. Representative regime curve of eachcluster is colored according to the clusterâs color. . . . . . . . . . . . . . . . . . . . . 272.5 Tukeyâs honest significant difference test to determine whether values of streamflowsignatures between pairs of clusters, devolved by each framework, are statisticallydifferent (see Section 2.2.3 for more details). The dark blue shade (light blue shade)suggest a statistical difference under 1% (5%) significance level. Each panel showsTukeyâs p-values for one streamflow signature and one framework. The frameworkwith a larger number of dark or light blue squares is a stronger framework in iden-tifying catchments (dis)similarities in (a) shape-based streamflow signatures and (b)magnitude-based streamflow signatures. This figure shows the results for clusteringwith five clusters. Figure A.3 shows the results for clustering with six clusters. . . . 28xiiList of Figures2.6 Bayesian Information Criterion (BIC) values for regression-based predictions of (a)shape-based streamflow signatures (top panel) and (b) magnitude-based streamflowsignatures (bottom panel), using three frameworks and two scenarios of regressionnonlinearity (Equation 2.4-2.5). A lower BIC value refers to a larger extrapolationcapability. For each regression scenario and each signature, the color map quantita-tively shows the ratio between the BIC value of framework 2 or 3 to framework 1. Theoutperformance (lower BIC ratio) and the poorer performance (higher BIC ratio) offrameworks 2 or 3 compared to framework 1 are shown in blue and red, respectively.Darker blue refers to a larger outperformance compared to framework 1, and darkerred refers to a poorer performance compared to framework 1. Framework 1 employs10 individual climatic and physical attributes, framework 2 uses three climatic indices,and framework 3 considers six climatic indices and functional indices, as catchmentdescriptors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7 Goodness of fit R2 values and cross validation R2 values of random forests in predict-ing streamflow signatures using the three frameworks. (a) Shape-based streamflowsignatures (top panel). (b) Magnitude-based streamflow signatures (bottom panel).Framework 1 employs 10 individual climatic and physical attributes, framework 2uses three climatic indices, and framework 3 considers six climatic and functionalindices, as catchment descriptors. From left to right, three cross-validation scenar-ios (Semi-gauged 1, Semi-gauged 2, and Poorly gauged) systematically explore theinterpolation versus extrapolation capabilities of random forests predictions, wherethe Semi-gauged Scenario 1 mainly depicts the interpolation capability of the randomforests model and Poorly gauged Scenario mainly shows the extrapolation capabilityof the random forests. For each cross-validation scenario and each signature, the colormap quantitively shows the ratio between the R2 value of framework 2 or 3 to the R2value of framework 1. The outperformance (higher R2 ratio) and the poorer perfor-mance (lower R2 ratio) of frameworks 2 or 3 compared to framework 1 are shown inblue and red, respectively. Darker blue refers to a larger outperformance compared toframework 1, and darker red refers to a poorer performance compared to framework1, for each cross-validation scenario and each signature. Note that the values shown inthis figure for each cross-validation scenario and signature, are the mean of R2 valuesacross 200 random experiments (see Section 2.2.3 for more details). The maximumradii of the 99% confidence intervals of R2 values for all signatures are 0.001, 0.003,0.002, for semi-gauged 1, semi-gauged 2, and poorly gauged scenarios, respectively.These small radii of the 99% confidence intervals show the stability of R2 across therandom experimentations as well as the generalizability of our conclusions across thestudy site. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.8 Results for the experiments on simulated data from Subsection 2.3.4. Feature im-portance scores are shown as a percentage of the total for each of x1 to x4 from100 replications. Results are shown for marginal contribution feature importance(MCI), ultra-marginal feature importance with linear regression (UMFI LR), andultra-marginal feature importance with pairwise optimal transport (UMFI OT). . . 452.9 The full causal graph generating the data for the blood relation simulation experimentin Section 2.3.4. Blood related vertices to the response Y (blue) are coloured in red. Sand X4 are directly causally related to Y , whereas X3 is related to Y via the commonancestor S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46xiiiList of Figures2.10 Variable importance scores of CAMELS catchment attributes for explaining (a) meanannual runoff, (b) baseflow index, (c) slope of flow duration curve, and (d) halfflow date using accumulated local effects (ALE), conditional permutation importance(CPI), marginal contribution feature importance (MCI), permutation importance(PI), quadratic regression (QR), and ultra-marginal feature importance (UMFI). Thescores of each method are given as the percent of the total importance found for eachmethod following the simulation studies. The color of the square indicates the rankof that feature where darker blue indicates more importance while darker red indicateless importance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.11 Computation time for a single iteration of each method including: MCI (dark red),MCI with the soft 2-size-submodularity assumption (pink), UMFI OT (light blue),and UMFI LR (dark blue), plotted against the number of processed features. . . . . 493.1 A flow path diagram depicting how precipitation (red downward arrows) gets parti-tioned into evapotranspiration (ET), overlandflow (OF), shallow subsurface flow (SF)or deep groundwater flow (GF), with arrow width depicting illustrative relative effectsize. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Left: The daily streamflow (blue) and rainfall (black) for the Koksilah River (top)and the Withlacoochee River (bottom) for the year 1998. Right: The average dailystreamflow (blue) and rainfall (black) for the Koksilah River (top) and the Withla-coochee River (bottom) for 1979-2018. . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3 The relationships between the response y and the cause x is shown for D = 4 andT = 3. Causal connections are shown for replicate i at t = 2. . . . . . . . . . . . . . 563.4 The general structure of the discretized β(s, t) is shown for D = 2 and T = 3, wheres is lag and t is time. The bidirectional lines between nodes indicate that the nodesshould have similar values. The unidirectional lines indicate that there is periodicbehaviour with a period of T +1 (i.e., β(0, 3) and β(0, 1) are connected). The bottomrow of nodes represents the coefficients relating the current timeâs cause to the currenttimeâs response while the next rows represent the coefficients for consecutive past lags.The top row shows the zero boundary conditions. . . . . . . . . . . . . . . . . . . . . 583.5 The estimated function Î²Ë(s, t) for each lag s and time t for the Koksilah River locatedin British Columbia (left) and Withlachoochee River, located in Florida (right) fromthe whole data period (1979-2018). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.6 The architecture of the Self-explaining Long Short-Term Memory (SE-LSTM) modelfor simulating daily streamflow y(t) using Impulse Response Functions (IRFs). TheSE-LSTM transforms daily inputs A(t) into hidden states ht. The initial cell stateand hidden state, c0 and h0, are initialized as zero vectors. The final LSTM cellat time t illustrates the internal gates of the LSTM: the forget gate ft, input gate it,output gate ot, and candidate cell state gt. The final hidden state ht is passed througha dense layer fdense to produce the raw IRF coefficients Î²â. These coefficients are thenpassed through a sparsity and smoothness regularization layer (SPS), producing thefinal IRF β(s,A(t)), which is both smooth and sparse. The IRF is convolved withlagged precipitation x(tâs) over lags s = 0 to K, yielding the simulated streamflowy(t). The SE-LSTM is trained to minimize the mean squared error between y(t)and observed streamflow, allowing β(s,A(t)) to represent how lagged precipitationcontributes to streamflow generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74xivList of Figures3.7 The map of our 1110 catchments across the United States and Canada. The differentcolors show the clusters and the larger square points show our âfocusâ catchments. . 773.8 Left: Map of test set NSE (R2) values for each catchment based on SE-LSTM pre-dictions. Dark blue (1) is ideal, while white (0.5) is okay, and red (0 or negative) ispoor. Right: Comparison of several methods based on their predictive capabilities(NSE). Each line shows the cumulative fraction (y-axis) of catchments that have atleast x test set NSE (x-axis). We compare the results of Kirchner [2022], Janssen et al.[2025b], SE-LSTM (our current paper), and vanilla LSTM based on Bayati et al. [2025]. 793.9 The heatmaps show the average impulse response function along each lag (y-axis) foreach day of the year (x-axis). Darker blue indicates a higher streamflow sensitivity(on that day of the year) to previous rainfall, while white indicates zero sensitivity.There are very few red (negative sensitivity) results. The heatmap on the left displaysthe result from the Koksilah River (Vancouver Island) and the heatmap on the rightshows the result from the Withlacoochee River (Florida). . . . . . . . . . . . . . . . 803.10 The figures show the impulse response of streamflow across different days of precipi-tation for the Koksilah (left) and Withlacoochee (right) rivers. The blue bars indicateprecipitation occurrence and magnitude, while the lines denote the IRFs for a varietyof different days as denoted in the respective legends. . . . . . . . . . . . . . . . . . . 814.1 500 meter resolution pixels showing the locations of the available real observations ofWTD (red) and that of the delineated shorelines of surface water bodies (blue), acrossthe Prairie Pothole and Mississippi River Basin regions. Our real observations ofWTD and the delineation of the shorelines of surface water bodies extend throughoutthe entire United States and Canada. But for the sake of simplicity of visualization,we only show these two selected regions with fairly dense observations and delineatedshorelines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 The locations of ten ecoregions along which we compare and evaluate the performanceof the three machine learning-based and two physically-based WTD simulations. Thedashed red arch-shaped zone referred to focus zone and was explained in Section 4.2.3 954.3 Simulations of Degraafâs (right), Fanâs (middle) and our V3 (left) after the alignmentprocess in the lowland area north of Lake Saint Clair. All three simulations identifythe location of Lake Saint Clair and itâs shorelines (with close to zero WTD) at similarlocations. Shorelines are simulated at finer resolution using our V3 model than Fanâsand Degraafâs simulation, due to differences in grid resolution. . . . . . . . . . . . . 1024.4 Five different simulations of water table depths across the US and Canada usingmachine learning models (a-c) and physically-based models (d-e). . . . . . . . . . . . 1044.5 Pixel-by-pixel spatial correlations among the physically-based and machine learningWTD simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.6 Comparison of the distribution of areas with water table depths (WTD) less than 1 m(left) and less than 5 m (right) across three machine learning (ML) and two physically-based WTD simulations. The x-axis represents latitude in 1 degree increments, whilethe y-axis shows the total area of pixels with shallow WTDs within each 1 degreelatitude band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.7 Relative importance of each input variable within each machine learning model (seeSection 4.2.3 for the details on the derivation of importance score). The importancescores across a single model add to one. . . . . . . . . . . . . . . . . . . . . . . . . . 107xvList of Figures4.8 Distribution of residuals (Observed - Predicted WTD) across four quartiles of selectedglobally available inputs and across the five machine leaning and physically-basedmodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.9 The causal graph of a small hydrological system, which principally consists of stream-flow at one gauge, Qt, and three rain gauges, rg1, rg2, and rg3, from three times:tâ 2, tâ 1, t. We also place two unobserved variables Utâ2 and Utâ1, which denotelarger weather/atmospheric processes that cause rain to persist across several dayswithin the area. We know in advance that only the first two rain gauges (rg1 andrg2) have a causal effect on the streamflow Qt (shown by continuous arrows). rg3is outside of the true catchment, and therefore should not have a causal effect onstreamflow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.10 Maps describing the distribution of precipitation across the CAMELS-DE catchments. 1264.11 Example classifications of the true catchment along with each of the three methods forthe DEF13090 catchment (875 km2 catchment area) with 38 years of data and a max-imum NSE (or signal to noise ratio) of 0.8. Black line shows the catchmentâs groundtruth boundary polygon. Blue pixels indicate true (within the true catchment) forthe first panel, and indicate true (within the predicted catchment) for the remainingthree panels/methods. Salmon pixels indicate false (outside the true catchment) forthe first panel, and indicate false (outside the predicted catchment) for the remainingthree panels/methods. Missing pixels (white) may appear due to the pixels not beingfully contained within Germany. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.12 The plotted regression coefficients illustrate how much each factor impacts classifica-tion accuracy. The spread of the unit hydrograph in each catchment was calculatedas the average unit hydrograph variance across all contributing pixels. The lengthof time series coefficient denotes how much classification accuracy increases with 100years of extra data. The spatial autocorrelation of precipitation coefficient denoteshow much classification accuracy increases with a 0.1 increase in correlation. The â#of rainfall pixelsâ coefficient denotes how much classification accuracy increases withthe addition of 10 additional pixels (as catchment area increases). 99% confidenceintervals are shown (blue bars) around each regression coefficient (blue and whitecircle). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.13 The true DE811440 catchment is shown along with the estimated catchments by thetopographic method and the causal inference method, given 38 years of data and amaximum NSE of 0.9. Blue indicates true (within catchment) while salmon indicatesfalse (outside catchment). The unit hydrographs driving streamflow is shown in thetop left subfigure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131A.1 Pearson correlation (r) among non-transformed shape-based and magnitude-basedstreamflow signatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185A.2 The spatial maps of streamflow signatures. All signatures were normalized and trans-formed in this figure to simplify the spatial comparison. . . . . . . . . . . . . . . . . 186xviList of FiguresA.3 Tukeyâs honest significant difference test to determine whether values of streamflowsignatures between pairs of clusters, devolved by each framework, are statically dif-ferent. The dark blue shade (light blue shade) suggest a statistical difference under1% (5%) significance level. Each panel shows Tukeyâs p-values for one streamflowsignature estimated by the three frameworks. The framework with a larger num-ber of dark or light blue squares is a stronger framework in identifying catchments(dis)similarities in streamflow signatures. a) shape-based streamflow signatures, b)magnitude-based streamflow signatures. This figure shows the results for clusteringwith six clusters. Figure 2.4 (in the main text) shows the results for clustering withfive clusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187A.4 Maps of hydrologic clustering developed based on long-term streamflow observations(top panel) and the representative regime curve (mean regime curve) of catchmentsbelonging to each cluster (bottom panel). Representative regime curve of each clusteris colored according to the clusterâs color. This figure was reproduced from clusteringresults released by Brunner et al. [2020]. These results are referred to as âindepen-dent streamflow-based clusteringâ in the main manuscript. Brunner et al. [2020]labeled the representative regime curves belonging to the five clusters as: intermit-tent regimes (red), weak winter regimes (orange), strong winter regimes (grey), meltregimes (green) and New Yearâs regimes (blue). . . . . . . . . . . . . . . . . . . . . . 188B.1 PI-diagrams taken from Griffith and Koch [2014] for I(Y ;F ) when |F | = 2 (left)and |F | = 3 (right). Magenta represents unique information, redundant informationis colored with yellow, and synergistic information is in cyan. The starred regionsrepresent a single region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191B.2 The coefficient of determination (random forest OOB-R2) between the ith feature inthe BRCA dataset and all other features is plotted (black) for each i â {1, 2, ...50}.The R2 value between the ith feature and all other features after preprocessing withlinear regression (red) and optimal transport (blue) is also plotted. . . . . . . . . . . 202B.3 Cell (i, j) indicates how similar the jth variable in the BRCA dataset is comparedto its transformation via pairwise optimal transport or linear regression with respectto feature i. This is measured with the maximal information coefficient, which iscomparable to R2. To make the plots more clear and accessible, only the first 15features are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203B.4 Results for the experiments on simulated data from Subsection B.7.1. The resultsfor ablation, conditional permutation importance (CPI), and permutation importance(PI) were implemented with random forest (RF), and are shown in Figures B.4a, B.4b,B.4c, and B.4d . The results for MCI, UMFI LR, and UMFI OT were implementedwith extremely randomized trees (ET), and are shown in Figures B.4e, B.4f, B.4g,and B.4h. Feature importance scores are shown as a percentage of the total for eachof x1 to x4 from 100 replications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209B.5 Median feature importance scores provided by (a) MCI, (b) UMFI with linear re-gression, and (c) UMFI with pairwise optimal transport, for each gene in the BRCAdataset after 200 iterations. Genes colored in blue are known to be associated withbreast cancer while genes colored in grey are random permutations of randomly se-lected genes, which we assume to be unassociated with breast cancer. The first andthird quantiles of the scores are visualized as error bars for each gene. . . . . . . . . 210xviiList of FiguresB.6 Median feature importance scores provided by (a) UMFI with linear regression, and(b) UMFI with pairwise optimal transport, for each gene in the permuted BRCAdataset after 5000 iterations. Genes colored in blue are known to be associatedwith breast cancer while genes colored in grey are random permutations of randomlyselected genes, which we assume to be unassociated with breast cancer subtype. Thefirst and third quantiles of the scores are visualized for each gene. . . . . . . . . . . . 211B.7 Median feature importance scores provided by (a) ablation, (b) permutation impor-tance, and (c) conditional permutation importance, for each gene in the permutedBRCA dataset after 200 iterations. Genes colored in blue are known to be associatedwith breast cancer while genes colored in grey are random permutations of randomlyselected genes, which we assume to be unassociated with breast cancer subtype. Thefirst and third quantiles of the scores are visualized for each gene. . . . . . . . . . . . 212B.8 Median feature importance scores provided by (a) MCI, (b) UMFI with linear regres-sion, and (c) UMFI with pairwise optimal transport, for each gene in the unpermutedBRCA dataset after 100 iterations. Genes colored in blue are associated with breastcancer while genes colored in grey are randomly selected genes. The first and thirdquantiles of the scores are visualized for each gene. . . . . . . . . . . . . . . . . . . . 213B.9 Median feature importance scores provided by (a) ablation, (b) permutation impor-tance, and (c) conditional permutation importance, for each gene in the unpermutedBRCA dataset after 100 iterations. Genes colored in blue are associated with breastcancer while genes colored in grey are randomly selected genes. The first and thirdquantiles of the scores are visualized for each gene. . . . . . . . . . . . . . . . . . . . 214B.10 The average computation time for each method to process p features over 10 iterationsof the original BRCA data is plotted for each p â {5, 10, 15, 20, 25, 30, 35, 40, 45, 50}.We assume soft 2-size submodularity to run the MCI results. . . . . . . . . . . . . . 215B.11 The coefficient of determination (random forest OOB-R2) between the ith feature inthe CAMELS dataset and all other features is plotted (black) for each i â {1, 2, ...30}.The R2 value between the ith feature and all other features after preprocessing withlinear regression (red) and optimal transport (blue) is also plotted. . . . . . . . . . . 215B.12 Median feature importance scores provided by (a) MCI, (b) UMFI with linear regres-sion, and (c) UMFI with pairwise optimal transport, for each explanatory variable inthe CAMELS dataset, aggregated after 100 iterations. The first and third quantilesof the scores are visualized for each feature. . . . . . . . . . . . . . . . . . . . . . . . 216C.1 The estimated function Î²Ë(s, t) for each lag s for the Koksilah River located in BritishColumbia (left) andWithlachoochee River, located in Florida (right). The lines withineach plot show a single representative day within the âWetâ, âDryingâ, âDryâ, andâWettingâ periods that we delineated. For the Koksilah River, February 19th isâWetâ, June 9th is âDryingâ, August 28th is âDryâ, and October 17th is âWettingâ.For the Withlachoochee River November 16th is âWetâ, February 19th is âDryingâ,May 30th is âDryâ, and September 7th is âWettingâ. . . . . . . . . . . . . . . . . . . 217C.2 Each black point represents the fitted R2 after setting all coefficients within β(s, t)to zero which have a group norm less than the q parameter (standardized to be aquantile between 0 and 1). The red dot displays the chosen sparsity level. . . . . . . 218xviiiList of FiguresC.3 The estimated maximum time-varying lag Î´Ë(t) from 10 out of the 100 experiments inour simulation study from Section 4 (blue) versus the true δ(t) are visualized for twocases from our simulation study. For the Koksilah River located in British Columbia(left), the maximum R2 was 0.4 and the level of autocorrelation in the error wasmedium. For the Withlachoochee River, located in Florida (right), the maximum R2was 0.8 and the level of autocorrelation in the noise was set as high. These two casesrepresent the second worst (left) and second best (right) results from the eight caseswe tested in our simulation study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218D.1 Performance comparison maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219F.1 Example classifications of the true catchment along with each of the three methodsfor the DEA11040 catchment (1396 km2 catchment area) with 20 years of data anda maximum NSE (or signal to noise ratio) of 0.85. Black line shows the catchmentâsground truth boundary polygon. Blue pixels indicate true (within the true catch-ment) for the first panel, and indicate true (within the predicted catchment) for theremaining three panels/methods. Salmon pixels indicate false (outside the true catch-ment) for the first panel, and indicate false (outside the predicted catchment) for theremaining three panels/methods. Missing pixels (white) may appear due to the pixelsnot being fully contained within Germany. . . . . . . . . . . . . . . . . . . . . . . . . 225F.2 Plots showing the temporal autocorrelation of precipitation (top) and temporal par-tial autocorrelation of precipitation (bottom) of the first (left, DE110000) and last(right, DEG10620) catchment in the CAMELS-DE dataset. The dashed lines showsignificance at p=0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226xixChapter 1Introduction1.1 Background and motivationsGiven a precipitation event, how will a catchmentâs streamflow respond in the coming days ormonths? For hydrologists, ecologists, and engineers, this question is fundamental. Answering itwith certainty is possible if we have decades of high-quality, fine-scale data for a specific catchment.However, this ideal scenario is rare. Further, this problem intensifies when we consider a catchmentundergoing climate or land use change, or when we try to predict the behavior of one of the infi-nite neighboring, unmonitored catchments. Answering these questions accurately usually requiresknowledge transfer [Oruche et al., 2021, Ghosh et al., 2022a, Blo¨schl et al., 2013, Sun et al., 2021a].We must learn the true data-generating processes from available data and transfer this knowledgeto answer questions in regions where data is unavailable or to answer questions about future eventswhen data is not available. Three basic approaches for knowledge transfer exist. First, we may usephysics by experimentally deriving small-scale physical equations, then we may use these equationsto run mathematical simulations of streamflow given the globally-available catchment information.Alternatively, we may use expert knowledge which entails using a high-level understanding of thesystem to hypothesize how the system will behave. Finally, one may use statistics which purelyrelies on the data to tell the story of the system behavior and data-generating processes.Physically-based flow models are almost always based on the principle of mass balance and flowequations like Darcyâs law and Richardsâ equation [Beven, 1996b]. While these principles describethe results of laboratory experiments with high fidelity, extrapolating these principles to real worldscenarios at larger scales have proved troublesome [Flury et al., 1994, Hornberger et al., 1991]. In-deed, laboratory conditions, under which Darcyâs law has been derived and verified require that thesoils are uniform and well mixed in a small soil column [Beven, 1996b], and flow laws for undisturbedsoil columns with unknown and heterogeneous preferential flow paths remain elusive [Grayson et al.,1992]. Even if these equations were valid, accurate estimates of hydraulic conductivity at represen-tative scales do not and may never exist due to the destructive nature of measurement techniques[Beven, 1996b]. Instead, hydrologists tend to calibrate parameters which they purport to be phys-ically meaningful based on a model with incorrect assumptions (i.e., inexact boundary conditions,wrong equations, uncertain parameters). Relying on current expert knowledge is a great alternativetactic compared to physics-based modeling, but expert knowledge is obviously limited by what is al-ready known. In this dissertation we aim to expand our knowledge base through the final approach:statistics.With the rise of machine learning and ever growing hydrological datasets, hydrologists have begun tofocus their attention on learning hydrological processes via data-driven methods, but shortcomingswithin these studies are pervasive. Data-driven methods often have assumptions which are unknownto hydrologists or the methods were made for a specific purpose unrelated to hydrology. Making11.1. Background and motivationsvalid and robust inferences is not easy, regardless of whether its related to making predictions inungauged basins or learning a set of parameters that accurately describes a system. Due to theinherent difficulties often found in hydrological data such as sampling biases, highly confoundingspatiotemporal correlations, and nonlinear relationships, we must be skeptical of any predictions,inferences, and conclusions we make [Beven, 1989, Bergen et al., 2019, Klemes et al., 1982]. Inparticular, hydrologists, and probably all scientists more broadly need to constantly ask themselves:(1) What am I assuming by using this particular sequence of methods? (2) Are these modelingassumptions actually satisfied in my problem setting and what happens to my conclusions if theseassumptions are not satisfied? (3) Are these assumptions necessary or can the development of anew method remove them? (4) Was my method of choice even built to do the particular task I aminterested in? (5) Have I sufficiently evaluated my model? (6) How do I know that my model isgetting the right answer for the right reasons?Alongside Keith Beven, Vit KlemesË serves as the philosophical inspiration for this dissertation,though I should note that both of them may disagree with certain parts of this work. In hisexpansive academic works, Dr. KlemesË consistently argued against the use of empirical (or black-box or data-driven models) in favor of causal modeling due to the limited extrapolation capabilities ofempirical modeling. Alongside this, Klemes et al. [1982] warned against mathematical convenience,keeping with current fashion, unrealized arbitrariness, and reductionism. Since hydrologists oftenuse methods developed outside of hydrology, these violations happen unknowingly. However, evenwhen methods are developed within hydrology, they tend to have compounded assumptions thatresult in conceptual models that are sold as causal, though they remain unverified due to the possiblenonidentifiability of their physically-inspired parameters which in-turn lose their purported meaning.Instead, Klemes et al. [1982] advocates for causal methods as they can illuminate hydrologicalprocesses, extrapolate to unseen conditions, and detect better empirical models. I tend to agreewith these warnings and recommendations, thus in this work, we aim to develop new methods thatare causal, have limited assumptions, and/or have physically meaningful parameters.Contrary to my work, Klemes et al. [1982] simultaneously argues that data-driven approaches thatentail calibration or data fitting are not causal and instead looks to physically-based models foranswers. In the 1970s and 80s, this was indeed the predominant thinking, however, I think thatmore recent breakthroughs in causal inference, information theory, and machine learning allow fordata-driven approaches to remain causal in nature. In the early 1900s, statisticians were extremelyskeptical of the causal graph-based works of Seawall Wright [Wright, 1921], arguing against Wrightâsconclusions with the dominant philosophical and statistical world-views of the intellectual giants ofstatistics including, Ronald Fisher, Karl Pearson, and Francis Galton [Pearl and Mackenzie, 2018].Several years after Klemes et al. [1982], causal inference from observational data was establishedas a firm discipline and the goals of Wright (i.e., combining minor and reasonable domain-specificassumptions with empirical data to answer causal questions) were revived across the sciences [Pearland Mackenzie, 2018, Pearl, 1988]. For the remainder of this thesis, I hope to keep Dr. KlemesËhappy by ensuring that assumptions are kept to a minimum and causality is taken seriously, whilealso sticking to the flexibility of data-driven frameworks. Luckily, almost all questions in hydrologycan be framed as a causal question [Ombadi et al., 2020], as we will see later in this chapter.In a changing world, causal inference, and system identification more generally, will become evenmore central to scientific and hydrological research progress. Clearly the earth is constantly goingthrough changes, with the most rapid of these changes driven by us. Current IPCC projectionsshow that temperatures will continue to increase a further two degrees in the next 70 years while21.1. Background and motivationsevaporation and precipitation extremes continue to increase due to the increased energy in the Earthsystem [Allan et al., 2023]. We also must not forget about changes in land use, where wetland extenthas decreased by 25% in 50 years and desertification continues in drylands as water resources becomeoverextended [Masson-Delmotte et al., 2022]. Thus, we as hydrologists need to be more careful aboutthe assumptions we make via the methods we use. If the world were static, using simple data-drivenmethods is probably sufficient for consistently good inference and predictions. But, in a changingworld, we can no longer rely on simple correlative relationships and models that blindly fit the data.Correlative relationships and models/inferences built upon them will quickly break down since theyare vulnerable to distributions shifts. Instead, hydrologists need to focus on gaining a deeper andmore causally-oriented understanding of the systems they study. To gain such an understanding ofhydrological systems, we need to make inferences about the hidden process behind Earth systemdata through the lens of causal inference, information theory, and machine learning.Information theory is perhaps the most foundational mathematical discipline since it has been beau-tifully successful in describing theories in physics (second law of thermodynamics), human biology(consciousness), complex systems (emergent phenomena), and computing (digital communication)[Ito, 2016]. Recently, there have been some very basic yet exciting applications of information the-ory in hydrology [Moges et al., 2022, Tennant et al., 2020, Goodwell and Bassiouni, 2022, Farahaniet al., 2022, Franzen et al., 2020, Goodwell and Kumar, 2017], as well as some encouragement formore research in this interdisciplinary field [Goodwell et al., 2020, PerdigaËo et al., 2020, Kumar andGupta, 2020, Weijs and Ruddell, 2020, Gupta and Nearing, 2014]. Information theory and causalinference are intimately connected disciplines, with causal inference being a higher level interfacefor which scientific questions and relations can be posed, and information theory being a tool toexplicitly quantify these relations. Causal inference is great for hypothesis generation and acquiringobjective scientific knowledge [Janzing and Scho¨lkopf, 2010, Ombadi et al., 2020], however, normally,inferences about the causes of some phenomena cannot be made unless we intervene on the systemin a randomized control trail. Controlled trials are usually impossible in catchment-scale hydrologyKirchner [2022], so we must rely on data, prior knowledge, and temporal priority to infer causalrelations [Ombadi et al., 2020, Ghassami and Kiyavash, 2017]. Causal inference is important toinclude in our methods in contrast to pure machine learning and statistics since these methods canonly find associations. We will, however, still include statistical learning throughout our work due toits great flexibility. Statistical and machine learning has grown exponentially in hydrology preciselydue to their flexibility and we plan to continue this trend [Tyralis et al., 2019, Shen et al., 2023,Shen, 2018]. Flexibility is vital when we want to gain insights while using some model, especiallyin hydrology where unknown interactions and nonlinearities are prevalent [Young, 2002]. In theinductive approach to hydrology, theoretical preconceptions that may have flaws are disregardedand the model is specified using the data and as few assumptions as possible [Young, 2002, Guptaand Nearing, 2014]. If all theoretical preconceptions are automatically encoded into the model, in-formation loss due to preconceived biases are likely, and nothing new can be learned while old andperhaps incorrect assumptions are reinforced [Gupta and Nearing, 2014].Although the field is growing, causal and information theory-based analyses have rarely been donein hydrology due to a need for an extended knowledge beyond that which is received during stan-dard statistical training [Pearl, 2009]. Unlike causality and information theory, the use of machineand statistical learning has already exploded in popularity and is consistently used in high-impactjournal papers [Addor et al., 2018, Cheng et al., 2022, Anderson and Radic´, 2022, Stein et al., 2021,Araya and Ghezzehei, 2019]. Even when hydrologists attempt to use machine learning, causality, or31.2. Research questions and objectivesinformation theory in their works, they often misunderstand or misspecify assumptions or even useimproper methods. Although, researchers in statistics and machine learning constantly develop novelmethods which are then used by hydrologists in downstream tasks, these methods are not made withhydrology in mind so they are often incompatible for hydrology research. Indeed, analyzing datahas two primary purposes (1) make predictions and (2) extract information [Breiman, 2001c], andmachine learning researchers are often hyperfocused on obtaining better predictions [Pearl, 2021].Yet, hydrologists and other Earth scientists are more interested in extracting information from data.These misunderstandings, misspecifications, and incompatibilities will be a theme that we pursuethroughout this work. We echo the sentiment of KlemesË [1986]: âWhile this danger is serious anduniversal, it remains dormant until the models take the shape of easy-to-use software packages. Atthat stage the unholy trinity of good intentions, ignorance, and efficiency closes the circle and ahydrologic misconception becomes a virtually insurmountable obstacle to progress in hydrologyâ.1.2 Research questions and objectivesBroadly, this dissertation aims to increase our understanding of the data generating processes behindan array of hydrological data (i.e., learn what causes spatiotemporal variability in rainfall-runofffiltering functions). Current machine learning methods are developed with prediction as the primarygoal, however, in this thesis (and in the natural sciences more generally [Hogg and Villar, 2024,Grayson et al., 1992]), our goal is more about understanding, thus we aim to build machine learning-based tools that are useful for hydrological inference. These goals are easy to confuse, however,they are quite distinct. The questions we aim to answer, which we will describe in the followingparagraphs, require us to first develop novel inference methods that are also flexible enough to beapplicable on hydrological data.Blo¨schl et al. [2019] compiled a list of 23 unsolved problems in hydrology. The work in this thesis willtouch upon four of these important questions: (1) What causes spatial heterogeneity and homogene-ity in runoff, evaporation, subsurface water and material fluxes, and in their sensitivity to their con-trols, (2) How can hydrological models be adapted to be able to extrapolate to changing conditions,(3) How can we disentangle and reduce model structural/parameter/input uncertainty in hydrolog-ical prediction, and (4) What are the processes that control hillslopeâriparianâstreamâgroundwaterinteractions and when do the compartments connect. The overarching hypothesis of this work is thatdata-driven techniques have the potential to help hydrologists answer these questions by modelingand extracting the true data-generating processes behind hydrological complex systems, however themethods that are currently in use are insufficient for a variety of reasons. The data-driven techniquesbuilt in this thesis will derive from foundational work in information theory, causal inference, andmachine learning.All four of the unsolved problems in hydrology that we focus on in this work, as well as all workpertaining to predictions in ungauged basins or under climate change are fundamentally causal [Om-badi et al., 2020]. They require us to directly infer the causes of some phenomena, understand themechanistic reasons behind why a particular variable has a particular value, or extrapolate predic-tions under changing conditions, something that cannot be done with statistical association analysesalone [Pearl, 2009, Spirtes, 2010, Runge et al., 2019]. In hydrology we often ask important questionswhich are at each rung of the Ladder of Causation [Pearl and Mackenzie, 2018]. The foundationof causality lies in associations where hydrologists may ask which catchment characteristic is moststrongly related to a certain streamflow signature. The second rung of the ladder is intervention,41.2. Research questions and objectiveswhere hydrologists may ask what will happen to streamflow if land cover in particular part of thecatchment is changed. The final rung of the ladder is the counterfactual, where hydrologists mayask if a drought still would have happened if the climate were not warming [Pearl and Mackenzie,2018].The relationships we seek to uncover pertain to the first of our listed unsolved problems in hydrology:What causes spatial heterogeneity and homogeneity in runoff, evaporation, and subsurface water,and in their sensitivity to their controls [Blo¨schl et al., 2019]. Landscape characteristics on Earthare not evenly distributed across continents, and soils, topography, ecosystems, and climate allcoevolve together on a multitude of timescales [Troch et al., 2015]. Earth scientists understandthat interaction effects exist, but the scope of these interactions, under which conditions interactioneffects occur, and how many interactions exist is unclear [Massmann et al., 2021, Janssen and Ameli,2021]. Further, nonlinearities are commonly seen in the form of unknown threshold-like behaviors ornon-proportional input-output relationships due to antecedent or storage conditions [Fenicia et al.,2008]. Our aim in answering this question is thus to seek novel indices or novel patterns that explainthe heterogeneity of runoff and runoff-sensitivity both within and across catchments.The second and third questions ask us how to build models that are physically-meaningful andthus conducive to extrapolation while also reducing uncertainty. It has been recently argued byNearing et al. [2021] and Rahmani et al. [2025] that the vastly superior predictive performance ofneural networks over traditional hydrological models calls into question the usefulness of hydrologiststhemselves. This performance increase is certainly apparent in regions with extensive measurementsof streamflow, but in ungauged regions or under climate change scenarios, neural networks can getthe right answer for the wrong reasons, or even preform poorly all together [Wi and Steinschneider,2022, 2024, Bayati et al., 2025, Feng et al., 2024, 2023]. Still, there is a predominant mindsetthat there is an inherent trade-off between interpretability and predictive power. To alleviate theseissues, we argue that this trade-off does not necessarily exist. For example, in Chapter 3 we aimto build rainfall-runoff models that reduce the number of needed assumptions while keeping perfectinterpretability and inference capabilities. Further, in Chapters 2 and 4, we will help to decreasemodel uncertainty by increasing our knowledge base with respect to hydrologic (dis)similarity andconnectivity.Our final question relates to the connectivity and interactions between hydrologic compartments.We will investigate the spatial heterogeneity in runoff and subsurface flow at the sub-basin scale.In particular, we are interested in finding the areas of the catchment (either outside or inside thetopographic boundary) that provide disproportionate surface or subsurface flow (i.e., sensitivity toadditional water input). Furthermore, we would like to determine what makes a particular catchmentduring a particular time particularly sensitive to precipitation such that knowledge can be transferredto ungauged basins. These sensitivities are directly related to the second unanswered question as wellsince stream-groundwater interactions can form a contributing area and this contributing area canchange over time as the stream, groundwater, and hillslope all become saturated and hydrologicallyconnected. Connectivity is certainly driven by water table depth variability, so this will be studiedin detail. Finally, since most researchers in data-driven hydrology simply aggregate climate timeseries across the spatial scale of the topographically delineated catchment [Janssen et al., 2025b,Schrunner et al., 2025, Kratzert et al., 2018], inferring a more accurate extent of the catchmentcould help answer the second and third questions by reducing modeling uncertainties in gauged andungauged catchments [Blo¨schl et al., 2019].51.3. Dissertation structureWe have the following overarching objectives that we hope to consistently satisfy during the courseof this dissertation:1. Focus on learning the underlying processes that generate a variety of sources of hydrologicaldata instead of just trying to obtain better predictions2. Challenge the underlying thought processes that have cemented the consistent use of previousmethods in the hydrological literature3. Build methods that can be applied within hydrology (and perhaps across other domains) withas few assumptions as possible4. If some assumptions are necessary and/or violated, be open and honest about these shortcom-ings1.3 Dissertation structureThere are three primary types of hydrological data from which we can extract meaningful transferableknowledge. The first type of data comes from large sample hydrology where we treat each catchmentas an independent observation while comparing catchmentsâ hydrological responses given catchmentattributes. We will denote this type of study as cross-catchment spatial analysis. If we then zoominto a specific catchment, we can begin to analyze hydrological time series data, leading this to becalled within-catchment temporal analysis. After zooming in further on an individual catchment, wecan examine and compare specific catchment subcomponents for our final type of data analysis calledwithin-catchment spatial analysis. Through each of these three types of hydrological data, our aimis to take steps towards answering at least one of four unanswered questions above through learningtransferable knowledge that can be used for better hydrologic understanding and better predictionsin ungauged basins. Each of the next three chapters will cover one type of these analyses and willhave three sections. The first section will simply be a chapter summery. Then, the final two sectionswill comprise of two related works that have been or will be published.In the first paper of Chapter 2 [Janssen and Ameli, 2021], we use expert knowledge to derive threecatchment indices that describe long-term average partitioning, storage, and drainage characteristics.We showed that knowledge transfer from expert knowledge in the form of our indices can leadto better clustering and better predictions in ungauged basins since they eloquently express theinteractions amongst catchment characteristics in forming different types of flow. This motivatesus to build upon our knowledge of hydrological systems such that further advancements in processunderstanding, better predictions, and better regionalization (clustering), can be obtained. Our firstpaper shows the importance of soil and geology features, and we hypothesized that previous studiesmissed this importance due to methodological shortcomings related to the strong correlation betweenfeatures and their interactions. Therefore, in the second paper [Janssen et al., 2023], we propose,implement, and apply a new feature importance metric that can detect correlated interactions andlearn part of the structure of the causal graph.As we zoom into a single catchment at a time for Chapter 3, we begin to focus our efforts on timeseries analysis and learning an individual catchmentâs filtering function through time. In the firstpaper in the chapter [Janssen et al., 2025b], we propose a novel method called the dynamically sparsehistorical function linear model (DS-HFLM). This method can learn how, on average, precipitationis transferred into streamflow by estimating a distinct unit hydrograph for each day of the year.61.3. Dissertation structureImportantly, we impose smoothness across lags within each unit hydrograph and impose smoothnessacross unit hydrographs at consecutive days of the year. We show that DS-HFLM can obtain highlyinterpretable and accurate hydrological inferences while also displaying good predictive power intwo catchments in the US and Canada. The second paper of Chapter 3 greatly improves uponDS-HFLM by removing one of its key assumptions. Importantly, DS-HFLM assumes that day ofthe year acts as a good proxy for antecedent conditions. We remove this assumption by allowing aneural network to predict a unique unit hydrograph for each day based on antecedent precipitation,temperature and evapotranspiration. Using over 1000 catchments, we show that our new method(self-explainable LSTMs) can generate more accurate predictions compared to DS-HFLM while alsoallowing for improved inference capabilities.Finally, in the fourth chapter, we begin to look at the hydrological processes of individual pixelswithin catchments. Water table depth is the most important attribute in groundwater hydrologyas it can determine surface and subsurface runoff response, catchment boundaries, and sensitivityto climate/land use changes. Due to its vital importance we develop three US and Canada-widepredictions of WTD at 500 meter resolution using various amalgamations of proxy observations[Janssen et al., 2025c]. We compare the three machine learning models in addition to two physically-based models and conclude that due to sampling biases, assumption violations, and observationaluncertainty, no WTD model can verifiably predict WTD with acceptable certainty, especially in highelevation regions. One would have hoped that water table depth predictions are accurate enoughto delineate catchment boundaries, however, our previous work conclusively proves otherwise. Dueto this, we turn to causality by reformulating catchment delineation as a causal effect estimationproblem. As a proof of concept we show that standard causal effect estimation methods can reliablydelineate virtual catchments, while other non-causal methods like transfer entropy fail. We showthat causal inference may be applicable to these goals in the near future in regions where we havelots of low noise data and highly heterogeneous rainfall patterns.At a high level, the second chapter looks to see why filtering functions and streamflow generationmechanisms change across catchments. The third chapter aims to infer the filtering function andits changes through time. Then, the fourth chapter aims to look at the heterogeneity of spatially-distributed filtering functions. The fifth chapter simply consists of a dissertation summary, a list ofimplications, and a discussion about possible future work.7Chapter 2Cross-catchment spatial analysis2.1 Chapter summaryThis chapter explores the diverse hydrologic functions of catchments â specifically, why they par-tition, store, and drain rainfall and snowmelt in the way that they do. Large-Sample Hydrology(LSH) aims to derive generalizable conclusions about these behaviors by linking streamflow signa-tures to catchment characteristics. However, the descriptors of hydrologic functions necessary forsuch synthesis and extrapolation remain underdeveloped in LSH research. We hypothesize thatthis underdevelopment contributes to unexpectedly weak associations between catchmentsâ physi-cal features and streamflow characteristics, as well as poor predictability of streamflow hydrographshapes. We propose that the discrepancies observed between large-sample and field-based hydro-logical findings regarding the importance of physical attributes, and the limited predictive power ofLSH methodologies for shape-based signatures, stem from how catchmentsâ physical attributes areincorporated in LSH. While individual physical attributes may show little predictive power on theirown, their interaction can explain a substantial amount of variance. Recognizing the critical role ofthese interactions, this chapter explores this concept through two primary avenues: (1) the devel-opment of new indices and (2) the introduction of a novel, hydrology-inspired feature importancemetric capable of detecting interactions even amidst high correlations.In the first part of this chapter, we propose three dimensionless indicesâwhich directly quantifyhydrologic functionsâbased on how interactions between a catchmentâs climatic and physical at-tributes construct catchment functions. Using climatic and physical data as well as long-termstreamflow observations at hundreds of gauged catchments across the United States and Canada,our results depict that the use of functional indices as catchment descriptors improves the perfor-mance of LSH methodologies (hierarchical clustering, multivariate regression, and random forests)in identifying hydrologic similarities in shape-based streamflow signatures among catchments, andin predicting shape-based streamflow signatures in poorly gauged regions. This research highlightsthe importance of physical interpretability of LSH models and showcases the development of parsi-monious and process-based catchment-scale frameworks, allowing the analysis of globally availablecatchment data to progress the generalizable understanding of catchment hydrologic functions andstreamflow generation mechanisms.Hydrologists frequently prioritize learning from data rather than training the best possible model;however, research in machine learning often prioritizes the latter. Still, hydrologists interested inLSH rely on methods developed in the machine learning literature in order to quantify the relation-ship strengths between catchment characteristics and streamflow signatures. Marginal contributionfeature importance (MCI) was developed to break this trend by providing a useful framework forquantifying the relationships in data, however MCI is computationally expensive and struggles todetect interactions when the features are highly correlated, a common occurrence within LSH. In the82.2. Improving large-sample hydrology performance in poorly gauged regionssecond part of this chapter, we aim to improve upon the theoretical properties, performance, andruntime of MCI by introducing ultra-marginal feature importance (UMFI), which uses dependenceremoval techniques from the AI fairness literature as its foundation. We first propose axioms forfeature importance methods that seek to explain the causal and associative relationships in data,and we prove that UMFI satisfies these axioms under basic assumptions. We then show on simu-lated data that UMFI performs better than MCI, especially in the presence of correlated interactionsand unrelated features, while partially learning the structure of the causal graph and reducing theexponential runtime of MCI to super-linear. Finally, we compare the results of UMFI with othercommon feature importance methods, and, counterintuitively, we show that physical attributes areeven less important than previously thought, while land cover variables seem to be more importantthan previous studies showed. Both papers have been published at high impact venues (see Janssenand Ameli [2021] and Janssen et al. [2023]).2.2 A hydrologic functional approach for improving large-samplehydrology performance in poorly gauged regions2.2.1 IntroductionEncoded within catchment hydrology are complex interactions among catchmentsâ climatic andphysical (e.g., soil, geology, and topography) attributes. Large spatial-temporal variabilities of theseattributes, and diverse interactions among them, give rise to diverse catchment-scale streamflowgeneration mechanisms, in catchments around the world [Blo¨schl et al., 2013]. This diversity ofinteractions complicates the understanding of the dominant streamflow generation mechanisms in agiven catchment [Wu et al., 2021], which are central to obtaining the âright streamflow predictionsfor the right reasonsâ [Kirchner, 2006]. As qualitatively showed by Dunne [1978] and Dooge [1986],there could be a generalizable catchment-scale scientific framework that explains and synthesizes thebaffling diversity of streamflow generation mechanisms. Such a framework could allow the transferand regionalization of the mechanisms [McDonnell et al., 2007, Wagener et al., 2007], essentiallyrequired to predict streamflow responses in poorly gauged regions [McDonnell and Woods, 2004].There is, however, little in the way of a scientific framework to quantitatively explain why variationsin dominant streamflow generation mechanisms occur between catchments [Li et al., 2014a]. Thisquantitative framework, to be generalizable, should be developed based on globally available catch-ment data and should explain how interactions among catchment attributes influence the way acatchment transforms (or filters) climatic variability into streamflow variability [Troch et al., 2013].This study takes a small step toward developing and testing (the application of) one such general-izable quantitative framework.Several studies showed the efficiency of a small set of indices to describe a catchmentâs hydrologicbehavior and predict the signatures of the catchment water balance, within the context of large-sample hydrology (LSH). LSH uses (globally) available datasets of catchment physical and climaticattributes and explores hydrologic behaviors along a wide spectrum of climatic and physical settings.Thus, LSH has a great potential to produce a generalizable catchment-scale framework, with theability to define dominant streamflow generation mechanisms and ultimately predict hydrograph re-sponse with minimum parametrization and hydrologic data requirements [Addor et al., 2020]. Theindices, used in LSH (e.g., aridity index, snow fraction), are mostly derived from hydrologicallyrelevant interactions among climatic attributes that typically control overall catchment water bal-ance. For example, Knoben et al. [2018], Gnann et al. [2019] and more recently Meira Neto et al.92.2. Improving large-sample hydrology performance in poorly gauged regions[2020] illustrated the usage of climatic indices to estimate the magnitude and spatial variability ofmagnitude-based streamflow signatures (e.g., mean annual discharge), which explain (or summarize)the overall catchment water balance. LSH studies, however, showed that the shape-based streamflowsignatures, describing the hydrograph shape or rate of streamflow variabilities (e.g., slope of flowduration curve), are hard to predict using climatic indices, and the addition of catchment physi-cal constituent attributes, specifically soil and geology, could not improve these predictions [Addoret al., 2018, Beck et al., 2013]. Alternatively, several theoretical and field-based hydrology studieshave shown a large association between shape-based signatures and physical attributes, such as soil,geology, and topography, at the catchment scale [Carlier et al., 2019, Jencso and McGlynn, 2011,Mu¨ller et al., 2014, Pfister et al., 2017] and the hillslope scale [Ameli et al., 2015, 2018, Hopp andMcDonnell, 2009].We hypothesize that the disagreement between large-sample and field-based hydrology on the impor-tance of physical attributes, and the poor prediction capability of LSH methodologies for shape-basedsignatures, are associated with the ways catchmentsâ physical attributes are included in LSH. Phys-ical attributes, particularly soil and geology, if used individually in black-box statistical models,may not sufficiently represent the hydrologically relevant aspects of soil or geology. For example,the proxies used for storage potential in LSH studies (e.g., soil thickness and/or soil porosity) couldhave different impacts on streamflow characteristics at different levels of water availability. In twocatchments with similar storage potential, differences in magnitude and frequency of water availabil-ity could lead to distinctly different hydrologic connectivity patterns and thus different streamflowgeneration mechanisms [Hopp and McDonnell, 2009, Tromp-van Meerveld and McDonnell, 2006].In a statistical sense, alone, two predictors may have little predictive power, but together and in anformat, they may explain a large amount of variance. The importance of the interaction amongstcatchmentsâ climatic and physical attributes in controlling streamflow generation mechanisms andshape-based streamflow signatures were widely examined and emphasized in theoretical and field-based hydrology experiments [Ameli et al., 2015, Hopp and McDonnell, 2009, Kelleher et al., 2015],but not yet sufficiently acknowledged in LSH methodologies. In this context, Gnann et al. [2021]urged for the development of new process-based catchment descriptors to improve LSH method-ologies through sufficiently including the hydrologically relevant role of underground catchment at-tributes in different environmental settings. But how can new process-based catchment descriptorsbe created; preferably based on globally available catchment data?The concept of catchment hydrologic function has been suggested as a means to link catchment-scale hydrologic processes, or streamflow generation mechanisms, to streamflow responses [Black,1997]. In that spirit, we see catchments as non-linear filters, which perform a set of common hydro-logic functions broadly consisting of partitioning, storage, and drainage of incoming water (rainfall,snowmelt), in order to transform climatic variability into streamflow variability along a range ofstreamflow regimes [Black, 1997, Wagener et al., 2007]. These functionsâformed from complexinteractions among the catchment constituent climatic and physical attributesâtogether define thecatchmentâs dominant mechanisms underpinning streamflow signatures [Blo¨schl et al., 2013, Carrilloet al., 2011, McMillan, 2020]. Inspired by the Dunne qualitative diagram, Li et al. [2014a] theo-retically showed that the catchment-scale streamflow generation mechanisms could be explained bya small set of dimensionless indices, which were developed based on the interactions among cli-matic and physical attributes, and were capable of directly capturing the storage, partitioning, anddrainage functions. This theoretical analysis presented a holistic conceptual picture of the combinedimpacts of climatic and physical controls on various streamflow generation mechanisms. Jackson102.2. Improving large-sample hydrology performance in poorly gauged regionset al. [2016] and later Gabrielli and McDonnell [2020] also developed and used new indices, based onthe interactions among geological and topographical attributes, which were deemed valid to explaindrainage functions in a few extensively studied experimental hillslopes. These studies collectivelyshowed the premises of using functional indices to understand and quantify catchment-scale stream-flow generation mechanisms. Yet such indices, and in general the relationships between catchmentfunctions and streamflow signatures, have not been directly explored and tested within the contextof LSH and/or for predicting streamflow signatures.Here, we propose three functional indicesâwhich we develop based on recent findings on hydrolog-ically relevant interactions among catchment constituent climatic and physical attributesâto de-scribe catchment functional behaviors. Then, we create three frameworks of different types of catch-ment descriptors (or streamflow predictors) using globally available physical and climatic catchmentdata. We compare the three frameworks with widely used LSH methodologies including hierar-chical clustering, multivariate regression, and random forests. We explore whether a frameworkthat includes the proposed functional indices and standard climatic indices as catchment descriptors(framework 3) outperforms a framework with standard climatic indices alone (framework 2) or aframework with individual climatic and physical attributes (framework 1). This study comparesthe efficiencies of the three frameworks in explaining the degree of hydrologic (dis)similarity amonggroups of catchments, and at predicting shape-based and magnitude-based streamflow signaturesalong hypothetical scenarios of prediction in semi-gauged to poorly gauged regions. The efficienciesof the three frameworks are compared using a suite of independent streamflow signatures estimatedfrom observed streamflow hydrographs at hundreds of catchments, spanning a wide range of climatic,geological (surficial and bedrock), and topographical settings across the United States and Canada.Specifically, this study responds to calls made by Oudin et al. [2010], Troch et al. [2013], and morerecently by Knoben et al. [2018], Gnann et al. [2021] and Wu et al. [2021], to create new process-based descriptors of catchment functional behaviors for applications in large-sample or comparativehydrology, in order to improve the estimation of (dis)similarity in hydrologic behaviors among catch-ments and to improve the predictions of streamflow signatures in poorly gauged regions. In doingso, we introduce three functional indices and assess whether the addition of the functional indices toclimatic indices, or the use of the functional indices instead of their individual climatic and physicalcomponents, as catchment descriptors, could:1. Develop a process-based framework for explaining the degree of (dis)similarity among hydro-logically (dis)similar catchments.2. Enhance the capability of LSH predictive methodologies in extrapolating the prediction ofstreamflow signatures.The principle application we envision is to leverage the opportunities that field-based and LSH couldmake when they merge, in order to develop a parsimonious process-based framework, enablingthe analyses of globally available catchment data to progress the generalizable understanding ofcatchment hydrologic functions and streamflow generation mechanisms, as suggested by Troch et al.[2013] and Wu et al. [2021]. Such understanding is critically essential to (a) permit the transferand regionalization of streamflow generation mechanisms [Blo¨schl et al., 2013, McDonnell et al.,2007], (b) provide a general assessment of the hydrologic implications of climate and land usechanges [Berghuijs et al., 2014, Golden et al., 2017], (c) link catchment hydrology to highly relateddisciplines (e.g., catchment geochemistry) [Hrachowitz et al., 2016], and (d) explicitly representstreamflow generation mechanisms in earth system models in order to improve the prediction of112.2. Improving large-sample hydrology performance in poorly gauged regionsfuture water cycles which force and are being forced by global energy and biogeochemical cyclechanges [Clark et al., 2015].2.2.2 DataStreamflow Data and Catchment PolygonsWe started with daily streamflow observations at 671 and 1,602 stream gauges across the UnitedStates and Canada, respectively (a total of 2,273 gauges). Daily streamflow observations werethen used to calculate long-term values for a suite of streamflow signatures (see Section 2.2.3).Stream gauges in the United States were part of the CAMELS data set [Addor et al., 2017] wherea catchment boundary polygon for each gauge was developed and validated. The Canadian set ofgauges were derived from the HYDAT data set released by the Water Survey Canada. We derivedboundary polygons for the Canadian gauges using the Watershed Tool in ArcGIS (see Section 2.2.2).Catchments used in this study are asserted to have minimal development, stream diversions, andregulations.Climatic AttributesClimate data for the catchments in the United States and Canada were extracted from the ERA5-Land (ERA5-L) database [MunËoz-Sabater et al., 2021] for the period between 1981 and 2019. ERA5-L assimilates different sources of climate data to estimate global climate on land masses at 9 kmresolution and hourly and monthly timescales. Hourly data for temperature, total precipitation,rainfall, and snowmelt, along with monthly data for total precipitation, snowmelt, rainfall andactual evapotranspiration were extracted from the database. Due to large reported uncertainty ofassimilated potential evapotranspiration data in ERA5-L database (as reported in MunËoz-Sabateret al. [2021]), gridded monthly data of potential evapotranspiration released as the part of theTerraClimate database [Abatzoglou et al., 2018] were used. Catchment boundary polygons (Section2.2.2) were used to clip the gridded climatic data in order to calculate catchment-scale hourlyand monthly climatic attributes. The monthly catchment-scale data were used to calculate annual(hydrologic year; OctoberâSeptember) and ultimately long-term average annual snowmelt (SM) [m],total precipitation (P) [m], potential evapotranspiration (PET) [m] and actual evapotranspiration(AET) [m] (Table 2.1).There are five specialized hydrologically relevant climatic attributes calculated from ERA5-L dataand used in this study, including long-term average annual water surplus (WS), long-term medianwater input intensity (WII), long-term median of maximum seasonal water surplus (SWS), frequencyof high water surplus (N), and seasonality index (SI) (Table 2.1). Monthly water input (the sum ofmonthly rainfall and snowmelt) and monthly actual evapotranspiration data were used to calculatemonthly and annual water surplus, and ultimately to calculate long-term average annual watersurplus [m]. Water surplus refers to the net water which is potentially available for streamflow orgroundwater recharge. The sum of hourly rainfall and snowmelt data were used to calculate dailywater input intensity for events over the threshold of 0.5 mm/day. This threshold was used toremove small false or inconsequential rainfall or snowmelt events. These intensities were then usedover 1981â2019 to calculate long-term median water input intensity [m/day]. Seasonal water surpluswas calculated as the sum of monthly water surplus for each season. The median of the maximumseasonal value for each hydrologic year (OctoberâSeptember) was then taken over 1981â2019 tocalculate long-term median of maximum seasonal water surplus [m]. The frequency of high water122.2. Improving large-sample hydrology performance in poorly gauged regionsTable 2.1: Description of Catchment-Scale Individual Physical and Annual Climatic Attributes,Climatic Indices, Functional Indices, and Streamflow Signatures. Median (and interquartile ranges[Q1 & Q3]) for attributes, indices and signatures, across all catchments, are listed associated withnon-transformed and non-scaled values. The transformation function for attributes, indices andsignatures are listed (see Section 2.2.3 for details).Attribute Group Attribute/Index/Signature UnitMedianTransform(Q1-Q3)Climatic attributesPrecipitation (P) m 1.12 (0.95-1.27) LogSnowmelt (SM) m 0.20 (0.09â0.33) SqrtActual evapotranspiration (AET) m 0.60 (0.46â0.75) -Water surplus (WS) m 0.42 (0.31â0.65) LogPotential Evapotranspiration (PET) m 0.70 (0.59â0.94) LogWater input intensity (WII) m/day 0.004 (0.0â0.01) SqrtMax seasonal water surplus (SWS) m 0.24 (0.19â0.35) LogFrequency of high water surplus (N) - 0.55 (0.45-0.65) -Physical attributesSlope (S) m/m 0.04 (0.02-0.09) LogSoil hydraulic conductivity (Ks) m/day 0.41 (0.26â0.90) LogSoil porosity (Ï) - 0.40 (0.36â0.44) LogShallow Ks (Kss) m/day 0.45 (0.27â0.97) LogDepth to bedrock (D) m 14.7 (11.4â19.2) LogBedrock hydraulic conductivity (Kb) m/day 0.07 (0.01â0.29) LogClimatic indicesSeasonality index (SI) - 0.07 (-0.09-0.15) -Snow fraction (SF) - 0.19 (0.08â0.33) SqrtAridity index (AI) - 0.70 (0.52â0.88) LogFunctional indicesOverlandflow index (OI) - 0.01 (0.005â0.014) LogSubsurface storage index (SSI) - 21.6 (13.5â35) LogDrainage index (DI) - 7.16 (1.05â65.14) LogShape signaturesLow-FDC - 4.07 (2.77â6.14) SqrtMid-FDC - 2.99 (2.32â3.79) SqrtHigh-FDC - 5.45 (4.47â6.76) SqrtNormalized Q5 - 0.09 (0.04â0.16) SqrtNormalized Q95 - 3.53 (3.08â4.05) -Low flow frequency - 81.9 (34.3â123.5) SqrtHigh flow frequency - 12.95 (5.39â27.01) SqrtMagnitude signaturesQ5 mm/day 0.12 (0.04â0.28) LogQ95 mm/day 4.79 (3.16â7.86) LogQmean mm/day 1.34 (0.91â2.10) Log132.2. Improving large-sample hydrology performance in poorly gauged regionssurplus is the long-term average of the ratio between number of large events to number of all eventsoccurring during a hydrologic year. Large events are defined as the events with an event-scaleaverage water surplus larger than 1 mm/day. Hourly P and T data were used to calculate SIusing the equation introduced in Woods [2009]. Table 2.1 reports some statistics of these climaticattributes, and Figure 2.1 shows the spatial maps of these attributes across the study area.Figure 2.1: Overview of catchment locations and catchment-scale (a) constituent or individual cli-matic, topographical and geological attributes. All attributes and indices were normalized andtransformed in this figure to simplify the spatial comparison (see Section 2.2.3). The median andinterquartile ranges of non-transformed and non-scaled attributes and indices, and their transfor-mation functions, were reported in Table 2.1.142.2. Improving large-sample hydrology performance in poorly gauged regionsFigure 2.2: Overview of catchment locations and catchment-scale climatic and hydrologic functionalindices. All attributes and indices were normalized and transformed in this figure to simplify thespatial comparison (see Section 2.2.3). The median and interquartile ranges of non-transformed andnon-scaled attributes and indices, and their transformation functions, were reported in Table 2.1.Catchment Physical AttributesWe used globally available databases of gridded topographical and soil characteristics as well asshapefiles of geological properties to calculate the average and the coefficient of variation (over thewhole catchment) of physical attributes of each catchment using catchment polygon boundaries. Thedata sources are outlined in the next sub-sections. Table 2.1 and Figure 2.1 report some statisticsand spatial maps of these attributes.Topographical Attributes This study used the MERIT Hydro database [Yamazaki et al., 2019],which includes a 90-m resolution global-scale DEM, flow direction, and flow accumulation, in orderto calculate gridded slope (using the code developed in Addor et al. [2017]) as well as to generatestream orders and drainage networks. Flow direction and flow accumulation data were used to derivecatchment boundary polygons for Canadian gauges using Watershed Tool in ArcGIS. Calculated152.2. Improving large-sample hydrology performance in poorly gauged regionsareas of Canadian catchments were then validated against the areas computed based on topographicmaps by the Water Survey Canada (see Section 2.2.2). Boundary polygons of Canadian catchmentsas well as available polygons for the United States catchments were used to calculate catchment-scaleaverage slope (S) [m/m] and latitude.Soil Attributes We used a gridded (900-m resolution) global-scale database of hydraulic proper-ties developed by Dai et al. [2019]. These properties include depth to bedrock as well as saturatedsoil hydraulic conductivity and soil porosity at different depths down to 4 m. For the latter pa-rameters, we first calculated gridded vertically averaged values for the top 1 m of the soil, whichrepresent shallow saturated soil hydraulic behavior, and for the entire 4 m of the soil column, whichroughly represents saturated soil column hydraulic behavior. Catchment boundary polygons werethen used to clip the gridded data of soil properties in order to calculate the catchment-scale averagefor depth to bedrock (D) [m], shallow saturated soil hydraulic conductivity (Kss) [m/day], saturatedsoil hydraulic conductivity (Ks) [m/day], and soil porosity (Ï). For each catchment, the coefficientof variation over the catchment for these attributes was also calculated. Note that soil attributes arehighly uncertain [Dai et al., 2019], and spatial averaging may decrease their meaningfulness (due toscale problems, see Beven [1996b]) while largely keeping uncertainties constant (due to the spatialautocorrelation of their errors, see Wadoux and Heuvelink [2023]).Geological Attributes Here, global-scale shapefiles of bedrock permeability developed by Glee-son et al. [2014] were used. Permeability values were converted to bedrock hydraulic conductivityusing the equation suggested in Gleeson et al. [2014]. Catchment boundary polygons were then usedto clip these polygons in order to calculate catchment-scale average and coefficient of variation forbedrock hydraulic conductivity (Kb) [m/day].Catchment Selection CriteriaTo obtain a homogeneous catchment data set, we removed some of the catchments from our analysisaccording to the following criteria.1. Catchments with streamflow records less than 30 (acceptable) years during the hydrologicyears of 1981 through 2019, where each year should have fewer than 10% of days with missingdata to be considered as an acceptable year.2. Canadian catchments with a relative areal difference larger than 30% between our estimatedarea (Section 2.2.2) and computed area based on topographic maps by the Water SurveyCanada.3. Catchments with areas smaller than 50 km2 and larger than 1,000 km2. Smaller catchmentswere discarded because of the downscaling uncertainty in climatic attributes, while largercatchments were discarded owing to a potentially large river routing time delay impact onstreamflow response that was not considered in our indices/attributes. These cut-offs forcatchment size were adopted based on the experience of similar LSH studies [Wu et al., 2021,Sawicz et al., 2011].4. Catchments with a coefficient of variation (i.e., the relative variation around the averagecatchment-scale value) of soil and geological attributes larger than 50%.162.2. Improving large-sample hydrology performance in poorly gauged regions5. Catchments with relative surface water (e.g., wetland, pothole) area larger than 10% of thetotal catchment area. Here, we used the MODIS land cover database (500-m resolution) thatincludes 17 land cover types (e.g., surface water body, cropland) to identify catchments withthis characteristic.In the end, a total of 571 catchments (and stream gauges) out of the initial 2,273 catchments, wereused in our analyses (Figure 2.1).2.2.3 MethodsThe landscapes of 571 catchments were described using catchment-scale constituent (individual)climatic and physical attributes or dimensionless climatic and functional indices formed based onthose attributes, in three competing frameworks (see Table 2.1 for the list of attributes and indices).Framework 1 described each catchment using individual climatic and physical attributes (Section2.2.3). Framework 2 described each catchment with widely used climatic indices (Section 2.2.3).Framework 3 described each catchment using three functional indices (Section 2.2.3), in addition tothe three climatic indices used in the second framework. Functional indices represent hydrologicallyrelevant interactions amongst climatic, soil, geological and topographical attributes, and togetherthey capture some of the characteristics related to catchment partitioning, storage and drainagefunctions. The theoretical rationales of functional indices introduced in this study are also ex-plained in Section 2.2.3. The three frameworks of catchment descriptors (or streamflow predictors),composed of globally available catchment data, were then combined with (non)parametric statis-tical methodologies, including hierarchical clustering, multivariate regression, and random forests,to estimate the hydrologic clustering (and predict the magnitude) of a suite of streamflow signa-tures. Similar streamflow signatures, calculated based on real observations at the 571 stream gauges(Section 2.2.3), were used to assess and compare the three frameworksâ performances within eachstatistical analysis. Selecting the best framework out of the three competing frameworks was doneusing (a) Tukeyâs honest significant difference criterion that compares the performances of the threeframeworks in explaining catchments clustering or (dis)similarities in streamflow signatures (Section2.2.3), (b) Bayesian Information Criterion that selects the framework with the largest capability forextrapolating the regression-based predictions of streamflow signatures (Section 2.2.3), and (c) cross-validation criteria comparing the random forests-based predictive capability of the three frameworksin semi-gauged to poorly gauged regions (Section 2.2.3).Framework 1: Catchment-Scale Individual Climatic and Physical Attributes asCatchment DescriptorsCatchment-scale predictors of streamflow clustering and signatures in framework 1 include: long-term average annual water surplus (WS) [m], annual potential evapotranspiration (PET) [m], annualsnowmelt (SM) [m], frequency of high water surplus (N), precipitation seasonality or seasonality in-dex (SI), slope (S), saturated soil hydraulic conductivity (Ks) [m/day], soil porosity (Ï), depth tobedrock (D) [m], and bedrock hydraulic conductivity (Kb) [m/day]. Note that we excluded a coupleof attributes from our initial list of individual climatic and physical attributes, due to their largecorrelations with the other individual attributes included in framework 1, to avoid duplication of in-formation (in a manner similar to Addor et al. [2018]). This was particularly essential to fairly selectthe best framework using Bayesian Information Criterion and random forests which punish modelingframeworks with redundant input predictors. The following climatic attributes were excluded fromframework 1: long-term average annual precipitation, long-term median water input intensity, and172.2. Improving large-sample hydrology performance in poorly gauged regionslong-term median of maximum seasonal water surplus. These attributes are highly correlated withWS with the absolute values of correlation coefficient of 91%, 92%, and 95%, respectively (Figure2). Catchment latitude and shallow saturated soil hydraulic conductivity were also excluded giventheir high correlation with PET (|r| = 89%) and saturated soil hydraulic conductivity (|r| = 99%)(Figure 2), respectively. Individual attributes used in the first framework emulate major catchmentclimatic, geological, soil or topographical characteristics known to control catchment hydrologic be-haviors and streamflow signatures. These characteristics are also implemented directly (or usingproxies) in previous LSH studies (e.g., Addor et al. [2018]). Sections 2.2.2 and 2.2.2 explained thedetails of the derivation of catchment-scale individual climatic and physical attributes, respectively.Figure 2.3: Pearson correlation (r) among (non-transformed) individual attributes and indices.182.2. Improving large-sample hydrology performance in poorly gauged regionsFramework 2: Catchment-Scale Climatic Indices as Catchment DescriptorsPredictors of streamflow signatures in the second framework were dimensionless climatic indicescalculated based on the balance between available water and energy. These indices included long-term average annual snow fraction (SF), aridity index (AI) and seasonality index (SI). We calculatedsnow fraction index (and aridity index) as the ratio between long-term average annual snowmelt(and annual potential evapotranspiration) to long-term average annual precipitation. Seasonalityindex was obtained based on the implicit interaction between precipitation and temperature. Alarger SI refers to a larger synchronicity between precipitation and temperature (i.e., larger in-phaseseasonality). Section 2.2.2 explains the details of the derivation of individual components of theseindices. Knoben et al. [2018] showed that these three indices together could capture hydrologicallyrelevant interactions among climatic attributes and are able to explain hydrologic (dis)similarityamong catchments in magnitude-based streamflow signatures.Framework 3: Catchment-Scale Functional Indices as Catchment DescriptorsFramework 3 included the three dimensionless climatic indices used in the second framework as wellas three dimensionless hydrologic functional indices introduced in this section. Theories behind thederivation of the functional indices are presented in this section, whereas the evaluation of their powerto explain streamflow signatures is explored in Section 4. While the conceptual associations betweenthe functional indices and streamflow generation mechanisms (e.g., infiltration excess overland flow)are explained in this section, the quantitative relationships between the indices and mechanismsover space and time will be explored in our future works. All indices are computed as interactionsamong existing catchment-averaged attributes.Overlandflow Index The ability of a catchment to partition incoming water input across the landsurface may depend on the intensity of water input and infiltration capacity of shallow soil [Horton,1933, Woolhiser et al., 1996]. To represent this partitioning function, we used an Overlandflow Index(OI):OI =WIIKss(2.1)This index shows the catchmentâs potential to generate infiltration excess overland flow and refersto the ratio between long-term median water input intensity (WII) [m/day] and shallow saturatedsoil hydraulic conductivity (Kss) [m/day]. The former was calculated using hourly snowmelt andrainfall intensity data (Section 2.2.2), and the latter was calculated as the catchment-scale averagesaturated hydraulic conductivity of the uppermost 1 m of the soil (Section 2.2.2). A larger value ofOI for a catchment, comparatively, refers to a larger contribution of infiltration excess overland flowto the catchment stream, while a smaller OI favors a larger catchment-scale infiltration potential.We acknowledge that indices conceptually similar to the overland flow index (Equation 2.1) havebeen introduced before [Milly, 1994, Woods, 2003]. But to our knowledge, such indices have notbeen used in LSH, given the lack of soil property data and high (temporal) resolution rainfall andsnowmelt data at the spatial scale relevant for LSH studies (see Knoben et al. [2018]).Subsurface Storage Index Inspired by storage index [Woods, 2003] and the seasonality of watersurplus index [Wigington et al., 2013], our proposed Subsurface Storage Index (SSI) defines the sub-192.2. Improving large-sample hydrology performance in poorly gauged regionssurface storage functioning of catchments. This index shows the competition between the maximumsubsurface storage capacity of a catchment and total water surplus during the wettest season of atypical year when a large portion of streamflow could be generated. The former is defined as theproduct of the depth to bedrock (D) [m] and soil porosity (Ï), and the latter was calculated as thelong-term median of maximum seasonal water surplus (SWS) [m] (Section 2.2.2).SSI =DÃ ÏSWS(2.2)Catchment water surplus should be sufficiently large, above a certain threshold, to fill the subsurfacestorage water deficit as the first step to initiate (sub)surface hydrologic connectivity [McDonnell,2017] and generate considerable amounts of streamflow through mechanisms such as saturationexcess overland flow [Dunne and Black, 1970], transmissivity feedback [Dunne and Black, 1970] andsubsurface stormflow [Tromp-van Meerveld and McDonnell, 2006]. Below that certain thresholdof water surplus, hydrologic connectivity cannot be activated and no (or only a small amount of)streamflow can be generated. The threshold of water surplus that triggers hydrologic connectivitydepends partly on catchment antecedent wetness and subsurface storage capacity [Ameli et al., 2015,Hopp and McDonnell, 2009, Jencso et al., 2009] (other factors will be discussed in Sections 2.2.5).Indeed, a larger subsurface storage capacity may more strongly buffer large continuous seasonalevents, increasing the amount of water surplus required to initiate hydrologic connectivity. A largervalue of a catchment SSI refers to a larger buffering of water surplus during the wettest seasonand lessens the potential for hydrologic connectivity activation. A smaller SSI may favor quickstreamflow generation during the season when extensive precipitation events co-occur with littleenergy to drive evapotranspiration, and subsurface storage capacity is not sufficient to buffer suchlarge seasonal water surplus. Wlostowski et al. [2021], across 15 catchments of US critical zoneobservatory network, showed that large seasonal water surplus could dominate subsurface storagebuffer and contribute not only to stream high-flow, but also to streamflow recession.Drainage Index Streamflow generation mechanisms (and hydrologic connectivity) cannot be ex-plained thoroughly using overland flow index and subsurface storage index. As recently emphasizedthrough a statistical analysis conducted by Wu et al. [2021], the magnitudes of water input intensity,subsurface storage and water input volume are not sufficient to distinguish among four categoriesof streamflow generation mechanisms. Further distinguishing among saturation excess overland-flow, subsurface stormflow, transmissivity feedback, and groundwater flow requires the knowledge ofcatchment slope, soil-bedrock conductivity contrast and depth to bedrock (see Table 3 of Wu et al.[2021]). Theoretical and experimental studies also suggested that the dominance of these four mech-anisms depend on whether a catchment drains water surplus laterally or vertically in the subsurfacesystems. These studies collectively showed that catchments with larger slope (S), soil thickness (D)[m] and soil-bedrock conductivity contrast (Ks/Kb), may drain a larger portion of annual watersurplus (WS) [m] laterally within shallow porous media (e.g., Ameli et al. [2015], Ameli et al. [2018],Gabrielli and McDonnell [2020], Jackson et al. [2016], among others). Building on these statisticaland experimental studies, we introduce a new dimensionless index, Drainage Index (DI), as:DI =KsKbà SÃDÃNWS. (2.3)Here N refers to the frequency of the occurrence of high water surplus (Section 2.2.2), shown to have202.2. Improving large-sample hydrology performance in poorly gauged regionspositive association with the occurrence of shallow lateral drainage in field-based hydrology studies(e.g., Jencso et al. [2009]). Of course, there are other factors controlling how a catchment drainsannual water surplus as will be discussed in Section 2.2.5.Streamflow SignaturesTen magnitude-based and shape-based streamflow signatures at 571 stream gauges were calculated(Table 2.1). These signatures, known to identify (dis)similarities in hydrologic behaviors amongcatchments, were used as an independent tool to compare the performances of the three frameworksin explaining and extrapolating distinct catchment hydrologic behaviors. Magnitude-based signa-tures explain long-term average magnitudes of catchment water balance. Among several candidatessuggested in Olden and Poff [2003], magnitude-based signatures used in our study include: averagedaily streamflow (Qmean), the 5th percentile of daily streamflow (Q5), and the 95th percentile ofdaily streamflow (Q95). Shape-based signatures explain the shape and flashiness of streamflow hy-drographs without consideration for average streamflow magnitude. Note that a flashy hydrographis one that produces steep peeks soon after rainfall and then recedes sharply to stream low flowconditions [McMillan, 2020]. Shape-based signatures could reveal the rate of change of flow magni-tude as well as the frequency of extreme events such as floods or droughts [Richter et al., 1996], andcould, to some extent, explain streamflow resistance to external drivers (e.g., climate and land-usechanges) at different time-scales [Evenson et al., 2018].An example of a shape-based signature is the flow duration curve (FDC), which shows the distri-bution of probabilities of streamflow being smaller than or equal to a specified magnitude and istypically derived from daily streamflow data. The FDC aggregates streamflow values over time andshow the flashiness of hydrographs (or rate of temporal changes in flow) during high flows, averageflows and low flows [Vogel and Fennessey, 1995]. It has the potential to identify how a catchmentfunctions [McMillan et al., 2017] and has a large association with the degree of hydrologic connec-tivity at the catchment-scale [Jencso et al., 2009]. It, thus, is an appropriate signature to evaluatewhich framework could better explain catchment-scale hydrologic functions and processes. We usedthree slopes (or signatures) representing the flow duration curve, on a logarithmic scale, duringhigh flows (High-FDC), mean flows (Mid-FDC), and low flows (Low-FDC). These three slopes werecalculated as the slope between the 67th and 99th streamflow percentiles, between 66th and 33rdstreamflow percentiles, and between the 32nd and 1st streamflow percentiles, respectively. A largeslope value indicates a more variable streamflow, while a small slope value indicates a more dampedstreamflow variability, across a given flow regime. Although the lowermost and uppermost portionsof the FDC may have generally larger measurement uncertainties compared to the middle part (seeMcMillan et al. [2017]), Evenson et al. [2018] showed that there are certain streamflow generationmechanisms that can only be identified through these two extreme portions. Other studies alsoused these three slopes together as a means to determine the flashiness of a hydrograph and explainhydrologic functional behaviors across flow conditions (e.g., Berghuijs et al. [2016], Farmer et al.[2003], Oudin et al. [2010]).Other shape-based signatures used in this study include: low flow frequency and high flow frequency.We also used normalized Q5 (and normalized Q95) that reflect the ratio of the 5th percentile of dailystreamflow (and the 95th percentile of daily streamflow) to Qmean. The baseflow index was excludedfrom our initial list of streamflow signatures, due to its large correlation with the rest of shape-based signatures (|r| is larger than 80% between baseflow index and almost all other shape-basedsignatures; Figure A.1). To compute both magnitude-based and shape-based signatures, we used the212.2. Improving large-sample hydrology performance in poorly gauged regionsR package developed by Addor et al. [2017]. For each catchment, a signatureâs value per hydrologicyear (OctoberâSeptember) was calculated and then the long-term average of these annual signaturevalues, over acceptable years (see the definition of an acceptable year in Section 2.2.2), was computed.The spatial maps of the values of streamflow signatures used in this study are shown in Figure A.2and Table 2.1 reports some statistics of these signatures.Hydrologic Clustering and Tukeyâs Method to Compare the Performances of theThree FrameworksWardâs hierarchical clustering within the âhclustâ R package (Murtagh & Legendre, 2011) [Murtaghand Legendre, 2011] was used to group the 571 catchments using the three frameworks with threedifferent types of catchment descriptors (or streamflow predictors). Previous works showed that 5â6clusters may sufficiently group the general hydrologic behaviors of catchments across the UnitedStates (e.g., Brunner et al. [2020], Coopersmith et al. [2012]), although further refinement of group-ings could reveal some local hydrologic behaviors (e.g., Jehn et al. [2020]). Here, we show the resultsacross the United States and Canada corresponding to 5 clusters (for all three frameworks) in the re-sults section (Section 2.2.4) and the results corresponding to 6 clusters in the supporting information(Figure A.2). Note that the catchment clustering, associated with each framework, were originallyconstructed based on globally available climatic and physical data and without any information onstreamflow. To test the performances of the three frameworks in clustering hydrologically similarcatchments, we used two different streamflow-based lines of evidence.First, the spatial map of clustered catchments for the same set of catchments as our catchmentsacross the United States, developed by Brunner et al. [2020], was used as an independent streamflow-based clustering to visually assess which framework best identifies the map of catchments with(dis)similar streamflow regime behaviors. They clustered CAMELS catchments into five groupsbased on catchment streamflow regime behaviors, that is, the long-term average seasonal variationof streamflow within a typical year (Figure 2 of Brunner et al. [2020]). Their clustering resultwas reproduced in the supporting information (Figure A.4) and is referred to as âindependentstreamflow-based clusteringâ in the remainder of the study.Second, we compared the quality of hydrologic clustering developed by the three frameworks, in howstrongly they explain (dis)similarities in streamflow signatures among catchments. Once clusteringof catchments using each framework was completed, we placed the streamflow signatures of eachcatchment, estimated using streamflow observations, into its corresponding cluster. For a givensignature and for each framework, to statistically identify the extent of differences among the clustersof the streamflow signature, we used Tukeyâs honest significant difference test [Benjamini and Braun,2002], which allows multiple comparisons between pairs of clusters. For each framework and a givenstreamflow signature, our null hypothesis is that no statistically significant difference exists betweenthe streamflow signatures of the pair of clusters. The framework with the largest differences (orsmallest p-values) among the clusters of the given streamflow signature could best discriminateamong catchments with dissimilarity in that streamflow signature and could best group catchmentswith similarity in that streamflow signature.222.2. Improving large-sample hydrology performance in poorly gauged regionsRegression Method and Bayesian Information Criterion for Selecting the FrameworkWith the Highest Extrapolation CapabilityMultivariate regression has been widely used to predict signatures of streamflow quantity and quality(e.g., mean transit time) in (un)gauged catchments (e.g., Lull and Sopper [1966], Lyon et al. [2012],Shu and Ouarda [2012]). Based on the regional relationships obtained between catchment descriptorsand streamflow signatures in gauged catchments, the prediction of these signatures are extrapolatedto ungauged catchments. Blo¨schl et al. [2013] urged that in order for the regression method tobe realistic, extrapolatable, and provide the right prediction for the right reason, these black boxrelationships should include the knowledge of hydrologic functions. Here, we explored the extent towhich the inclusion of functional indices could improve the extrapolation capability of regression-based predictions of streamflow signatures. In doing so, we used the three frameworks in conjunctionwith two scenarios of regression. The first regression scenario (Equation 2.4) includes the 1st orderpolynomial and 1st order interaction terms. The second regression scenario (Equation 2.5) includesthe 1st and 2nd order polynomial terms and 1st order interaction terms.Yk = β0,k +Iâi=1(Î²i,kxi) +Iâ1âj=1Iâi=j+1(Î²i,j,kxixj) + ϵk (2.4)Yk = β0,k +Iâi=1(Î²i,kxi) +Iâi=1(Î²i,i,kx2i ) +Iâ1âj=1Iâi=j+1(Î²i,j,kxixj) + ϵk (2.5)Here, x refers to the catchment descriptors (or regression predictors). I is the total number ofpredictors, equal to 10, 3, 6 for frameworks 1, 2, and 3, respectively. Yk refers to the kth streamflowsignature, the βs refer to the regression coefficients, while ϵk refers to the normally distributederror term for the kth streamflow signature. Among all possible non-linear functions used withregression methods in hydrology (see Gan et al. [1991]), we selected polynomial terms given theirsimplicity of interpretation. Additionally, we included interaction terms to incorporate potentialbehaviors among the individual attributes in framework 1, allowing a fair comparison among thethree frameworks. See Wu et al. [2021] and Tyralis et al. [2021] for visualizations of the bivariaterelationships between individual catchment characteristics and individual signatures. Note that wedid not aim to develop a regression model that simulates streamflow signatures with the greatestaccuracy. Indeed, we compared the performance of the three frameworks in extrapolation of theregression-based predictions of streamflow signatures.Two approaches to select the most extrapolatable modeling framework are information criteria andcross-validation. Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC)are the most common information criteria approaches to select among competing predictive models[Laio et al., 2009, Lesaffre and Lawson, 2012]. These information criteria select the âbestâ frameworkby measuring the goodness of fit while punishing overly complex models, or by balancing betweenthe explanatory and extrapolation capabilities of the competing frameworks. BIC outperforms AIC(and standard cross-validation) when selecting the most extrapolatable parametric model with morefocus on choosing the scientifically correct model rather than the model with largest goodness offit [Aho et al., 2014]. For example, Bollen et al. [2014] found that BIC selected the true modelmore often than AIC when sample sizes are larger than 500. Most recently, Jung [2018] foundthat BIC consistently selects the true model much more often compared to AIC. In hydrology, Laio232.2. Improving large-sample hydrology performance in poorly gauged regionset al. [2009] showed promising performances for both AIC and BIC in selecting the best modelingframeworks, with BIC leaning, more than the AIC, toward selecting more parsimonious (i.e., lessparameterized) models with larger extrapolation capability. Here, with each scenario of multivariateregression, and for each streamflow signature, we used BIC in order to select the âbestâ frameworkamong the three frameworks, with an emphasis on selecting the most extrapolatable framework.BIC can be calculated as:BIC = pà ln(n)â 2 ln(LË) (2.6)For each framework, scenario, and signature, p represents the total number of parameters in themodel (number of β coefficients plus one for the variance of the error term), n represents the numberof samples (catchments), and 2 ln(LË) is the maximum of the likelihood function. The likelihood partof the equation measures the goodness of fit while the other term, pà ln(n), punishes overly complexmodels.Random Forests and Cross-Validation Criteria for Selecting the Strongest Frameworkin Predicting Streamflow Signatures in Semi-Gauged to Poorly Gauged RegionsRandom forests consist of an ensemble of decorrelated trees that allow high predictive accuracy, ro-bustness against overfitting, and adaptive learning inclusive of necessary interaction terms [Breiman,2001b]. With these desirable properties, several LSH studies used a list of catchment-scale (mostly)individual climatic and physical attributes as predictors in random forests, and relied on the randomforests algorithm to identify the functional relations and interactions among input predictors thatbest predict streamflow signatures [Addor et al., 2018, Tyralis et al., 2019, Prieto et al., 2019]. Giventhe black box nature of random forests and large uncertainties in catchment-scale data, it is notguaranteed that this method identifies âtrueâ and scientifically acknowledged causal relationshipsand interactions among individual attributes. This may negatively impact our confidence in randomforests predictions of streamflow signatures in poorly gauged regions, as will be further discussedin the discussion section. Here, we explored whether the inclusion of functional indices instead oftheir constituent climatic and physical components, as input predictors, could improve the capabil-ity of random forests in the prediction of streamflow signatures in semi-gauged to poorly gaugedregions. In doing so, we used the three frameworks in conjunction with three hypothetical scenariosof gauging pattern (or cross-validation) as will be explained in this section.Here, to simulate a streamflow signature, we used ârangerâ R package [Wright and Ziegler, 2017]with the three frameworks of streamflow predictors. First, the random forests models for the threeframeworks were trained on the entire data (i.e., 571 catchments) and then the trained models weretested (validated) on the same data set used for training. Indeed, this step showed the goodness offit of the model developed for each framework and each signature. Next, three different scenarios ofcross-validation criterion were used to compare the three frameworksâ capabilities in interpolatingversus extrapolating the random forests predictions of streamflow signatures. In the first scenario(Semi-gauged scenario 1), a leave one out cross-validation criterion was used, by predicting a catch-mentâs signature from trees in the trained random forests model that did not use the catchment ofinterest in training, then iterating through each catchment. In the second scenario (Semi-gaugedscenario 2), five-fold-cross-validation was conducted by randomly splitting the 571 catchments intofive âfolds,â then iteratively training on the four folds (80% of catchments) and testing on the242.2. Improving large-sample hydrology performance in poorly gauged regionsremaining fold (20% of catchments). In the third scenario (Poorly gauged scenario), Multiple Pre-dicting five-fold Cross-Validation (MPCV) [Jung, 2018] was used. In this scenario we randomlydivided the data set into five folds, then iteratively trained on one fold (20% of catchments), andtested the trained random forests models on remaining four folds (80% of catchments). Basically,MPCV could emulate a hypothetical scenario of random forests predicting streamflow signatures inpoorly gauged regions. Jung [2018], through the analysis of real and simulated datasets, showedthat MPCV could select the true models more often than standard cross-validation (e.g., scenarios1 & 2 above). Collectively, three different cross-validation criteria compared the random forestspredictive capabilities of the three frameworks, across a hypothetical spectrum of semi-gauged topoorly gauged regions.The goodness of fit and three cross-validation criteria were measured with the R2 (or NSE):R2 = 1ânâi=1(yËi â yi)2(yi â y¯)2 (2.7)For each framework, each cross-validation scenario (and goodness of fit), and each signature, yË isthe simulated streamflow signature in the validation phase, while y (and y¯) refer to the signaturevalue (and its average) estimated using streamflow observations (Section 2.2.3). n represents thenumber of samples (catchments) used in the validation or testing phase.Note that for each framework and signature, we first optimized the hyperparameters of each randomforests model using the most conservative cross-validation scenario, poorly gauged scenario (MPCV),with the minimum (maximum) number of catchments used for model training (validation). Thistreatment is essential to fairly compare the performances of the three frameworks in poorly gaugedregions. For each framework and signature, the same hyperparameters were then used for all threecross-validation scenarios and the goodness of fit scenario. The hyperparameters we chose to optimizeincluded replace (true or false), sample fraction (number between 0.5 and 1), minimum node size(integer between 1 and 10), number of trees (integer between 100 and 1,000), and fraction of variablesthat is available to use at each node in each tree (fraction between 0 and 1). Ultimately, theparameters of the random forest models, including the variables and thresholds used to split eachnode, were optimized independently within the training phase of each cross-validation and goodnessof fit scenarios. As explained above, random forests and creating the folds are stochastic. Wereplicated each experiment 200 times and recorded the mean value and the radii of the 99% confidenceintervals. A smaller radius of the 99% confidence interval shows more stability across the randomexperiments.Variable Scaling and TransformationBefore conducting our three statistical analyses, the data is transformed such that each attribute/indexis normally distributed, then scaled to have mean 0 and variance 1 (Figure 1 shows the scaledand transformed attributes/indices). For reference, Table 2.1 reports the median and interquartileranges, over the entire study area, of non-transformed and non-scaled attributes and indices as wellas the transformation function used for each attribute and index. Before performing hierarchicalclustering, data transformations are often done [Oliveira and Zaiane, 2010], and scaling the data isrequired to make each variable have equal weight when the clustering algorithm is applied [Hastie252.2. Improving large-sample hydrology performance in poorly gauged regionset al., 2009]. BIC is invariant to scaling measures but requires normally distributed residuals. Ran-dom forests are invariant under scaling and monotone transformations; thus, our random forestsmodels will be largely unaffected by these preprocessing steps [Hastie et al., 2009].2.2.4 ResultsThe Performances of the Three Frameworks in Identifying (Dis)Similarity AmongCatchments(Dis)Similarity in Regime Curves The three frameworks produce different spatial patternsof clusters of hydrologically similar catchments (Figure 2.4). Visual comparisons between Figures2.4 and A.4 suggest that the maps of (dis)similar catchments developed by framework 3 and 2 aregenerally more consistent, than framework 1, with the âindependent streamflow-based clusteringâ de-veloped based on the observations of long-term streamflow regime along the United States CAMELScatchments, particularly in the Western United States. The bottom panels of Figure 2.4 and FigureA.4 show the streamflow regimes, or the representative regime curves of catchments belonging toeach cluster, determined by our frameworks and the independent streamflow-based clustering, re-spectively. The five streamflow regimes, as suggested in independent streamflow-based clustering,include: intermittent regime (red), weak winter regime (orange), strong winter regime (gray), meltregime (green) and New Yearâs regime (blue). Comparing streamflow regimes determined by ourframeworks with the independent streamflow-based clustering shows that the framework 3, followedby framework 2, stronger than framework 1 reproduces five streamflow regimes. Note that we didnot expect to obtain an exact consistency between the representative regime curves developed byour frameworks and independent streamflow-based clustering; the representative regime curves inour frameworks were calculated using both USA and Canada catchments while these curves in theindependent streamflow-based clustering were calculated using only the USA catchments. However,our results highlight that using globally available climatic and physical data, and without usingstreamflow data, the representative regime curves developed by framework 3 can strongly replicatethe general patterns of seasonality of streamflow regimes calculated by independent streamflow-basedclustering.(Dis)Similarity in Shape-Based Streamflow Signatures The hydrologic clustering made byframework 3, stronger than the other two frameworks, identifies catchment (dis)similarities in allshape-based signatures (except high flow frequency) considered in this study, where most clusterpairs have statistically different values of shape-based signatures at a 1% significance level (darkblue shading) or at a 5% significance level (light blue shading) (Figure 2.5a). These signaturesdefine stream hydrograph shape and flashiness during low, mean and high flows. These results areindependent of the number of clusters used in the clustering approach; Tukeyâs honest significantdifference test for clustering with six clusters shows the outperformance of framework 3 over theother frameworks for all shape-based signatures except high flow frequency (Figure A.3a). Thesefindings generally highlight that the addition of catchment functional indices to climatic indices inframework 3, with a total of six catchment descriptors, could improve the efficiency of hydrologicclustering to identify catchments with (dis)similar hydrograph shape, compared to frameworks 2and 1 with 3 and 10 catchment descriptors, respectively.(Dis)Similarity in Magnitude-Based Streamflow Signatures Despite the outperformanceof frameworks 3 and 2 compared to framework 1 (particularly in clustering results with six clusters262.2. Improving large-sample hydrology performance in poorly gauged regionsFigure 2.4: Maps of hydrologic clustering (top panel) and the representative regime curve (meanregime curve) of catchments belonging to each cluster (bottom panel), for (a) framework 1, (b)framework 2, and (c) framework 3. Representative regime curve of each cluster is colored accordingto the clusterâs color.[Figure A.3b]), all three frameworks generally correctly distinguish between the values of Qmean, Q5and Q95 (Figure 2.5b). These signatures are primarily dependent on the overall water balance ofthe catchment and all three frameworks contain the required input predictors to emulate hydrologic(dis)similarity in long-term water balance.The Performances of the Three Frameworks in Extrapolation of Regression-BasedPredictions of Streamflow SignaturesWithin each regression scenario (Equation 2.4-2.5), the lowest BIC for the prediction of each sig-nature identifies the framework with the largest extrapolation capability of regression-based pre-dictions (darker blue shading; Figure 2.6). For both regression scenarios, framework 3 has thestrongest capability to extrapolate regression-based predictions of shape-based streamflow signa-tures. For magnitude-based streamflow signatures, framework 3 produces the most extrapolatableregression modeling for predicting Q5 and Q95. The outperformance of framework 3 in extrapo-lating regression-based predictions of both types of streamflow signatures is further improved asmore non-linear terms are added to regression scenario 2 (e.g., third order polynomial and highorder interaction terms; not shown here). Figure 2.6 also reflects, as the degree of nonlinearity ofregression increases (e.g., scenario 2), the extrapolation capability of regression-based predictions of272.2. Improving large-sample hydrology performance in poorly gauged regionsFigure 2.5: Tukeyâs honest significant difference test to determine whether values of streamflowsignatures between pairs of clusters, devolved by each framework, are statistically different (seeSection 2.2.3 for more details). The dark blue shade (light blue shade) suggest a statistical differenceunder 1% (5%) significance level. Each panel shows Tukeyâs p-values for one streamflow signatureand one framework. The framework with a larger number of dark or light blue squares is a strongerframework in identifying catchments (dis)similarities in (a) shape-based streamflow signatures and(b) magnitude-based streamflow signatures. This figure shows the results for clustering with fiveclusters. Figure A.3 shows the results for clustering with six clusters.282.2. Improving large-sample hydrology performance in poorly gauged regionsstreamflow signatures using framework 2, with only three climatic indices, outperforms framework1, with 10 individual catchment climatic and physical attributes.Figure 2.6: Bayesian Information Criterion (BIC) values for regression-based predictions of (a)shape-based streamflow signatures (top panel) and (b) magnitude-based streamflow signatures (bot-tom panel), using three frameworks and two scenarios of regression nonlinearity (Equation 2.4-2.5).A lower BIC value refers to a larger extrapolation capability. For each regression scenario and eachsignature, the color map quantitatively shows the ratio between the BIC value of framework 2 or3 to framework 1. The outperformance (lower BIC ratio) and the poorer performance (higher BICratio) of frameworks 2 or 3 compared to framework 1 are shown in blue and red, respectively. Darkerblue refers to a larger outperformance compared to framework 1, and darker red refers to a poorerperformance compared to framework 1. Framework 1 employs 10 individual climatic and physicalattributes, framework 2 uses three climatic indices, and framework 3 considers six climatic indicesand functional indices, as catchment descriptors.The Performances of the Three Frameworks in Random Forests-Based Predictions ofStreamflow Signatures in Semi-Gauged to Poorly Gauged RegionsThe accuracy of random forests, coupled with the three frameworks, are compared in predictingstreamflow signatures within hypothetical scenarios of gauging patterns (Figure 2.7). Goodness offit of random forests shows that the three frameworks could strongly estimate a suite of stream-flow signatures with R2 values larger than 86% for framework 2 and R2 values larger than 94% for292.2. Improving large-sample hydrology performance in poorly gauged regionsframeworks 1 and 3. Goodness of fit, however, cannot inform the extrapolation capabilities ofâoridentify the true model amongâthe three frameworks. The three cross-validation scenarios sys-tematically explore the interpolation versus extrapolation capabilities of the three frameworks, withthe last scenario (MPCV), or poorly gauged scenario, known to identify the most extrapolatableframework among competing frameworks (see Section 2.2.3). In the two semi-gauged scenarios (Fig-ure 2.7; Semi-gauged scenarios 1 & 2), framework 3 slightly outperforms framework 1 and stronglyoutperforms framework 2 in predicting magnitude-based streamflow signatures and the shape-basedsignatures describing the shape of hydrograph during low and mean flows.Figure 2.7: Goodness of fit R2 values and cross validation R2 values of random forests in predict-ing streamflow signatures using the three frameworks. (a) Shape-based streamflow signatures (toppanel). (b) Magnitude-based streamflow signatures (bottom panel). Framework 1 employs 10 in-dividual climatic and physical attributes, framework 2 uses three climatic indices, and framework3 considers six climatic and functional indices, as catchment descriptors. From left to right, threecross-validation scenarios (Semi-gauged 1, Semi-gauged 2, and Poorly gauged) systematically ex-plore the interpolation versus extrapolation capabilities of random forests predictions, where theSemi-gauged Scenario 1 mainly depicts the interpolation capability of the random forests model andPoorly gauged Scenario mainly shows the extrapolation capability of the random forests. For eachcross-validation scenario and each signature, the color map quantitively shows the ratio betweenthe R2 value of framework 2 or 3 to the R2 value of framework 1. The outperformance (higher R2ratio) and the poorer performance (lower R2 ratio) of frameworks 2 or 3 compared to framework1 are shown in blue and red, respectively. Darker blue refers to a larger outperformance comparedto framework 1, and darker red refers to a poorer performance compared to framework 1, for eachcross-validation scenario and each signature. Note that the values shown in this figure for eachcross-validation scenario and signature, are the mean of R2 values across 200 random experiments(see Section 2.2.3 for more details). The maximum radii of the 99% confidence intervals of R2 valuesfor all signatures are 0.001, 0.003, 0.002, for semi-gauged 1, semi-gauged 2, and poorly gauged sce-narios, respectively. These small radii of the 99% confidence intervals show the stability of R2 acrossthe random experimentations as well as the generalizability of our conclusions across the study site.302.2. Improving large-sample hydrology performance in poorly gauged regionsIn the poorly gauged scenario, cross-validation R2 values of framework 3 are larger than frameworks1 and 2 for streamflow signatures describing the shape of hydrograph during low, mean and high flows(Figure 2.7a; darker blue shading). Framework 2 performs relatively poorly (darker red shading),compared to the other two frameworks, in extrapolating the predictions of shape-based signaturesconsidered in this study. Acknowledging only a slight outperformance of framework 3 over framework1 for Low-FDC, High-FDC, and normalized Q95, these results suggest that framework 3 slightlyoutperforms framework 1 and strongly outperforms framework 2, in predicting the overall shape ofstreamflow hydrographs in poorly gauged regions. The performances of the three frameworks for therandom forests predictions of magnitude-based signatures (Qmean and Q95) in poorly gauged scenarioare strong, with cross validation R2 values larger than 76%, despite the slight outperformances offrameworks 1 and 3 over framework 2 (Figure 2.7b).The above findings highlight that, despite a lower number of catchment descriptors, there is aslight improvement in the extrapolation accuracy of random forests with climatic and functionalindices (in framework 3), rather than with their constituent components (in framework 1), in theprediction of shape-based signatures of low, mean and high streamflow conditions. Furthermore,the extrapolation accuracy of predictions of shape-based streamflow signatures is stronger with theapplication of functional and climatic indices together as input predictors (in framework 3), thanwith climatic indices alone (in framework 2). Further discussions on why framework 3 slightlyoutperforms framework 1, and why frameworks 1 and 3 strongly outperform framework 2, withinthe context of random forests extrapolation of shape-based streamflow signatures, are available inSection 2.2.5.2.2.5 DiscussionLarge-sample hydrology (LSH) may hold one of the keys to synthesizing, and ultimately transferringover space and time, the knowledge of catchment hydrologic functions, for understanding hydrologic(dis)similarity among catchments and predicting streamflow responses (and alterations) in poorlygauged regions [Hrachowitz et al., 2013]. However, the LSH potential in this regard has not beenthoroughly tapped into and tested, mainly due to a lack of fundamental catchment descriptorsthat emulate and quantify catchment hydrologic functions. This study proposed three indices thatdirectly quantify hydrologic functions to explore our overarching hypothesis that: the extrapolationcapability of LSHâs clustering and predictive methodologies improve with the inclusion of descriptorsof catchment hydrologic functions.Analyses of ResultsIdentifying Hydrologic (Dis)Similarity Using Clustering Method Hrachowitz et al. [2013]and later Barthold and Woods [2015] suggested that an effective hydrologic clustering scheme shouldbe strongly underpinned by the basic internal functions of a catchment. Despite some progress on de-veloping statistically driven catchment descriptors using principal component analysis (as reviewedby Guo et al. [2021]), Knoben et al. [2018] called for creating new process-motivated catchment de-scriptors, independent of streamflow data, in order to provide a comprehensive and extrapolatablepicture of the causes of (dis)similarities among catchments. Our results highlight that the inclu-sion of indices of hydrologic functions as catchment descriptors, developed independent of streamflowdata, pronouncedly improve the performance of hierarchical clustering in identifying catchments with(dis)similar shapes of streamflow hydrographs (Figure 2.3a). Results also show that the three cli-matic indices proposed in Knoben et al. [2018] might be deemed sufficient to cluster catchments with312.2. Improving large-sample hydrology performance in poorly gauged regions(dis)similar magnitude-based streamflow signatures (Figure 2.3b), where the addition of functionalindices only slightly improves the clustering performance. This is consistent with previous clusteringanalyses suggesting that the climatic indices are dominant drivers of hydrologic (dis)similarities inmagnitude-based streamflow signatures [Knoben et al., 2018].Prediction of Streamflow Signatures in Poorly Gauged Regions Buttle [2006] and morerecently Gnann et al. [2021] suggested that an accurate prediction of shape-based streamflow signa-tures in poorly gauged regions requires the inclusion of information on various catchment hydrologicprocesses underpinning streamflow generation mechanisms. The functional indices introduced inthis study could incorporate some of the hydrologic processes a priori in multivariate regression,and improve the extrapolation accuracy when predicting shape-based streamflow signatures at high,mean and low flow conditions (Figure 2.6a). These findings corroborate theoretical and field-basedhydrology studies that showed a large association between catchment functional behavior and sig-natures characterizing the shape of streamflow hydrographs [Jencso and McGlynn, 2011, Kelleheret al., 2015, Mu¨ller et al., 2014]. Our findings also corroborate LSH predictive studies that showedlittle impacts of catchment underground attributes and large influences of climatic indices in pre-dicting magnitude-based streamflow signatures (e.g., Addor et al. [2018], Meira Neto et al. [2020]).Indeed, multivariate regression, could sufficiently extrapolate the predictions of magnitude-based sig-natures with hydrologically relevant climatic information alone, and without a detailed knowledgeof hydrologic functions (Figure 2.6b).Random forests is considered as one of the strongest predictive methods for the data-driven iden-tification of functional and relations among input predictors [Tyralis et al., 2019]. The strong out-performance of random forests predictions of shape-based streamflow signatures using framework1âwith individual climatic and physical attributes as catchment descriptorsâover framework 2 withclimatic indices alone (Figure 2.7a), may highlight the strength of random forests in identifying some(but not all) of the process-based interactions among climatic and physical attributes underpinningstreamflow generation mechanisms. However, with a smaller number of predictors, we still see aslight improvement in the predictive accuracy of random forests for shape-based streamflow signa-tures once the process-based interactions among climatic and physical attributes are incorporateda priori through the use of indices of hydrologic functions in framework 3. We hypothesize thatthis outperformance enhances once more process-based information is incorporated in functionalindices. This will be further discussed in Sections 2.2.5 and 2.2.5. Our findings also showed thatrandom forests could sufficiently extrapolate the predictions of magnitude-based signatures with hy-drologically relevant climatic information alone (in framework 2), and without detailed knowledgeof hydrologic functions (Figure 2.7b).Conceptual Rational of Framework 3 With Climatic and Functional Indices Framework3 adds, in a simple way, the direct effect of climate variability (through climatic indices) to the âfilterâ(through functional indices) that transforms that climate variability into the streamflow variability.Small correlations among the functional and climatic indices (all |r| â¤ 46% and most |r| close tozero; Figure 2.3) imply that these indices complement each other, rather than repeat redundantconceptual (e.g., storage function) information, in order to explain (dis)similarities in streamflowgeneration mechanisms among catchments, and ultimately extrapolate the predictions of shape-based signatures to poorly gauged regions. The importance of the proposed functional frameworklies in the fact that there are very few catchment-scale formulations that use globally availablecatchment data to group catchments and extrapolate the predictions of shape-based streamflow322.2. Improving large-sample hydrology performance in poorly gauged regionssignatures, explicitly based on catchment basic functions. The prediction of shape-based signaturesis critically required (a) for a wide range of water resources management practices, (b) for explainingstreamflow vulnerability to external changes driven by land-use and climate change, and (c) fordeveloping mitigation and adaption strategies to manage the hydrologic impacts of such externalforces [Blo¨schl et al., 2013, Evenson et al., 2018].Why Could Indices Improve LSH Performance?Similarities of constituent attributes among catchments have been widely interpreted as a proxy forfunctional similarities (see Hrachowitz et al. [2013] for a detailed review). However, Oudin et al.[2010] showed that catchment individual attributes could not explain the similarities in catchmentfunctional behaviors in most landscapes. Our results further confirm that individual attributes per-form relatively poorly in explaining similarities in streamflow signatures. Framework 3 (with sixclimatic and functional indices) outperforms framework 1 (with 10 individual catchment physicaland climatic attributes) when identifying hydrologic similarity in a suite of shape-based signaturesas well as regime curves (Figures 2.4, 2.5 and A.3). This is true not only for identifying hydro-logic (dis)similarities among catchments using the clustering method, but also for extrapolationof regression-based predictions of streamflow signatures (Figure 2.6). This may suggest that, ifindividual catchment attributes are used as catchment descriptors, clustering and multivariate re-gression approaches are not able to sufficiently identify (and learn) the scientific interactions amongcatchment descriptors. Such interactions explain catchment functions and ultimately control hydro-graph shape. Employing scientifically accepted forms of catchment attributes as the predictors ofstreamflow could pronouncedly improve the clustering and regression extrapolation performances.Feature engineering, the process of creating new aggregative input predictors based on domainknowledge, is one of the most effective ways to improve the extrapolation capability of statisticalmethods, through the selection of scientifically meaningful predictors, interaction terms, casualfactors, and generally through the incorporation of natural processes that black-box models mayignore [Kuhn and Johnson, 2019, Roe et al., 2020]. For example, Tran et al. [2019] showed thatfeature engineering with domain knowledge could improve the classification accuracy of differentmachine learning algorithms in many benchmark datasets across several disciplines. The functionalindices introduced in this study bring hydrologic knowledge into statistical models and increasetheir physical interpretability by focusing not on individual controls but on conceptual controls (e.g.,storage) widely known to underline streamflow generation mechanisms and hydrologic response. Thesmall correlations between each functional index and its individual climatic component suggests thatthe functional index, and the scientific concept it conveys, are not disproportionately explained bythe climatic components but are explained as a result of the interaction amongst climatic and physicalcomponents. For example, the correlations between functional indices and their climatic componentsare 5%, 41%, and 6% for the overland flow index, subsurface storage index and drainage index,respectively (Figure 2.3), suggesting that climatic attributes are not enough to explain catchmentpartitioning, storage and drainage functions. Indeed, each component of a given index, particularly(surficial) geological attributes, is required for a robust explanation of the shape-based signatures,as it defines a unique dimension of the whole concept that the index conveys. Furthermore, theapplication of functional indices, rather than their individual components, reduces dimensionalitywhile preserving the important conceptual information within the data [Virgolin et al., 2020]. Inour study, the number of predictors in framework 1 with individual attributes are 10 compared tosix functional predictors in framework 3. These characteristics lead to reductions in variance, time332.2. Improving large-sample hydrology performance in poorly gauged regionscomplexity, storage capacity and multicollinearity, and help to mitigate overparameterization andenhance the extrapolation accuracy of statistical algorithms.One might argue that some of the modern machine learning methods (e.g., random forests) couldconsider all potential scenarios of âtrueâ interactions among individual attributes. However, thefunctional forms (e.g., interactions), calculated and derived statistically, are not always physicallyinterpretable. To further explore the validity of the relationships diagnosed by random forests, weunpacked the random forests black box and extracted the interactions among individual climatic andphysical attributes which random forests diagnosed in framework 1, using signed iterative randomforests algorithm [Kumbier et al., 2018]. This analysis revealed that the random forests was able toidentify a couple of interactions among attributes that we considered in our climatic and functionalindices. However, we were not able to interpret, and confirm with previous scientific literature,a large portion of relationships diagnosed by random forests in framework 1. There are severalpossible factors leading to the identification of potentially unscientific interactions among catchmentattributes by random forests. There is a possibility of compensations between functional forms (e.g.,interactions), leading to identification of (dis)similar functional forms between a group of catchmentsthat are not hydrologically (dis)similar in reality. Indeed, the functional forms obtained duringmodel training, may not reflect scientific causal relationships, but instead reflect the unscientificcorrelations dictated by biases and uncertainties in the data. Several practical and methodologicaluncertainties exist in LSH related to (a) input climatic and physical attributes [Addor et al., 2020],(b) the estimation of shape-based streamflow signatures [Almeida et al., 2016, McMillan et al., 2017,Westerberg et al., 2016], and (c) collinearity among catchment physical and climatic attributes[Troch et al., 2013], which could collectively hinder the ability of statistical methods to derivephysically meaningful functional forms. These problems affect all models but are more serious foroverparameterized models, with a large number of catchment descriptors, used for extrapolatinghydrologic behaviors to poorly gauged regions.Open Questions Regarding Functional IndicesOur intention with this study was to evaluate the usefulness of the three proposed functional in-dices as simple components of a broader complete picture of catchment hydrologic functions. Ofcourse, not all catchment functional characteristics, known to control streamflow generation mech-anisms, are included in our indices. Here, we did not aim to predict streamflow signatures with thegreatest fidelity, rather, we aimed to use globally available data and compare the clustering (andextrapolation) capabilities of functional indices with their individual components, to showcase theimportance of developing functional indices for improving LSH performance. The expected limitedpredictive capability of the three frameworks in poorly gauged regions opens new avenues for futureresearch. Such research could focus on improving our proposed functional indices, or developingnew functional indices, to further enhance the interpretability of catchment-scale descriptors usedin LSH methodologies. A critical step toward the improvement of functional indices will be a morerobust incorporation of nonlinearities associated with each hydrologic function and threshold-likeprocesses underlying these nonlinearities, as will be discussed in this section. Such nonlinearitiesand threshold-like processes stemmed from the impacts of catchment attributes, and individualattributes may not thoroughly reflect such nonlinearities in LSH methodologies. We expect thatthe absolute accuracy, and the outperformance of functional indices compared to their individualcomponents in the extrapolation of streamflow signatures will further improve as more (direct orproxy) data related to nonlinearities and threshold-like processes become available and are included342.2. Improving large-sample hydrology performance in poorly gauged regionsin functional indices.Partitioning Function The Overlandflow index (Equation 2.1) proposed here, leveraged therecent availability of rainfall and snowmelt data at high temporal resolution as well as data of shal-low saturated soil hydraulic conductivity. The index was able to enhance the performances of LSHmethodologies in explaining the shape of hydrograph, particularly during high streamflow. However,in some environments, shallow soil saturated hydraulic conductivity (Kss) can only roughly emulatesoil infiltration capacity, which controls threshold-like activation of surface hydrologic connectivityand infiltration excess overland flow [Appels et al., 2011]. Soil infiltration capacity may dependalso on the water table depth (or antecedent wetness), and intrinsic unsaturated soil characteristics[Parlange and Brutsaert, 1987]. This may raise the following questions to inform a more accuraterepresentation of soil infiltration capacity in LSH: What is an integrative low-dimension proxy thatcan emulate intrinsic unsaturated soil characteristics and antecedent wetness conditions, using glob-ally available data? Or, conversely, under which physical and climatic conditions, can linear Kss, asused in the denominator of overland flow index, sufficiently explain long-term catchment-scale soilinfiltration capacity?Storage Function Subsurface storage index (Equation 2.2) emulates maximum subsurface po-tential in storing the largest seasonal water surplus. Indeed, it explains the maximum efficiencyof the subsurface in delaying catchment response during wet seasons and in the threshold-like ac-tivation of hydrologic connectivity. In reality, however, the permanent groundwater table couldbe above the bedrock interface before the beginning of the wet season, and thus the subsurfacestorage (and delaying capability) might be lower (weaker) than what was considered here. Globalestimates of groundwater table depth could improve the numerator of the subsurface storage indexin order to more accurately estimate subsurface storage potential. Note that the the overestimationof subsurface storage function did not impact the general findings obtained in this study, given thecomparative nature of our analyses between two different forms (interaction vs. individual) of thesame components.Drainage Function Indices of hillslope-scale subsurface drainage have been widely developedprimarily based on analytical solutions of simplified versions of the Boussinesq equation (e.g., Ruppand Selker [2005], Troch et al. [2004]). These theoretical studies collectively emphasized the impor-tance of soil thickness, the vertical pattern of soil saturated hydraulic conductivity, water leakagealong the soil-bedrock interface, hillslope geometry (e.g., length, curvature), land surface slope, andwater input characteristics, in controlling how a catchment drains the stored water. Field-basedhydrology studies further depicted the importance of the slope of shallow weathered bedrock, micro-topography along the bedrock surface and the land surface, the conductivity contrast between soiland shallow bedrock, and the presence of sink-holes within deep aquifers and macro-pores withinshallow soil, in controlling hillslope-scale threshold-like behavior of hydrologic connectivity [Appelset al., 2015, Beven and Germann, 2013, Gnann et al., 2021, Hopp and McDonnell, 2009, Jacksonet al., 2016, McDonnell, 1990, Thompson et al., 2010]. However, globally available datasets do notexist for most of the features suggested above. Furthermore, it is not clear whether all the suggestedfeatures, explored during hillslope-scale studies, contribute to long-term catchment-scale subsurfacedrainage. For example, detailed micro-topographic features along the bedrock surface, or hillslope-geometry, might have a large impact on event-scale subsurface drainage of a given hillslope, but asmall impact on long-term subsurface drainage of the parent catchment.352.2. Improving large-sample hydrology performance in poorly gauged regionsOur drainage index uses globally available data and includes interactions amongst attributes knownto control annual catchment-scale lateral drainage in the shallow subsurface. However, we acknowl-edge that our index cannot provide a complete picture of subsurface drainage mechanisms controllingthe initiation of subsurface stormflow and transmissivity-feedback flow, which in some environmentsmight be dependent on soilsâ vertical hydraulic conductivity patterns [Ameli et al., 2016] or bedrockgeometry [Ameli et al., 2015]. Furthermore, macropore flow and pipe flow in the shallow and deepsubsurface are missing processes in our index. While the average hydraulic conductivity (whichwas used in our study) may represent catchment processes in some cases, according to percola-tion theory, the extremes (i.e., preferential flow paths), rather than the averages, may explain awider array of observed catchment processes at larger scales [Hunt et al., 2025, Hunt, 2005, Huntand Ghanbarian, 2016]. The above weaknesses of our drainage index may raise the following ques-tions for future research: Under which physical and climatic conditions do detailed geological andgeometrical characteristics of the bedrock surface and soil mantle control annual catchment-scalesubsurface drainage? What are simple and widely available geomorphological and hydrologic proxiesfor identifying the presence of sink-holes and macropores as well as for approximating soil verticalconductivity patterns and bedrock geometry?Spatial Heterogeneity and Spatial Scale Harman et al. [2009] theoretically showed thatcatchment-scale heterogeneities in subsurface storage and drainage functions could strongly de-fine a hydrographâs nonlinear behavior. Here, we explored the impacts of average catchment-scalefunctional behavior on streamflow signatures. Functional indices introduced in this study, couldefficiently incorporate the spatial heterogeneities of functional behaviors over the catchment withminimum parametrization and in a more conceptual way, compared to their individual components.Future research should quantify the contribution of the spatial heterogeneities of functional indicesto catchment-scale streamflow generation mechanisms and streamflow signatures, in different cli-matic and geological settings. Furthermore, in this study, we used medium size (50â1,000 km2)catchments. In relatively larger catchments, new research is needed to explore whether functionaland climatic indices still outperform over their constituent components, as well as over climaticindices alone, in explaining hydrologic (dis)similarity among catchments.Limitation to Large-Scale Applicability of the Functional Indices The usefulness of func-tional indices in improving LSH methodologies was evaluated for a large portion of North America.However, several types of landscapes were excluded from our analysis and thus our findings on theusefulness of functional indices may not be easily extrapolated to such landscapes.The storage index introduced in our study only considers shallow subsurface storage. Other typesof storage such as wetland storage, which could control streamflow generation mechanisms, werenot considered. Ameli and Creed [2017] showed that wetland storage is the dominant source ofstreamflow generation during mean and low flow conditions within wetland dominated catchmentsin the Prairie Pothole Region of North America. Wetland (or pothole) dominated catchments wereremoved from our analyses (Section 2.2.2) to reflect the lack of incorporation of such storages in ourfunctional indices. Therefore, the findings of this study cannot be extrapolated to a large portionof Prairie Pothole Region of North America (south-central Canada and the north-central region ofUSA) where a catchmentâs functions could be primarily controlled by wetland storage.Due to a lack of required terms and interactions which emulate the behaviors of macropore andpipe flow in our indices, our findings may not be generalizable to regions where macropore flow362.2. Improving large-sample hydrology performance in poorly gauged regionsand pipe flow dominate stormflow or baseflow generation. In this study, we also used medium size(50â1,000 km2) catchments and thus our findings are not generalizable to smaller and/or largercatchments. In addition, the statistical models used in this study, and the conclusions we made onthe usefulness of functional indices with LSH methodologies, are limited in their applicability to theranges of attributes considered in this study. Surficial and bedrock geological attributes consideredin this paper cover the global-scale ranges reported by Dai et al. [2019] and Gleeson et al. [2014],respectively. However, the global-scale ranges of climatic indices reported by Knoben et al. [2018]could diverge from the ranges of such indices in North America considered here. For example, snowfraction and the aridity index could be larger in Greenland and northern Asia, respectively, than theconsidered ranges in our study. A universal framework of catchment hydrologic functions should (a)incorporate different types of storages and processes (e.g., macropore flow) controlling streamflowgeneration, (b) incorporate potentially unique functions of very small and very large catchments,and (c) cover the global-scale ranges of catchment physical and climatic attributes.2.2.6 SummaryWe demonstrated the usefulness of three indices of hydrologic functions, developed from globallyavailable climatic and physical data, in order to explore whether including the knowledge of catch-ment hydrologic functions could improve LSH performance in explaining the degree of (dis)similarityamong catchments, and predicting a suite of streamflow signatures. In doing so, we used threecompeting frameworks of catchment descriptors (or streamflow predictors), where descriptors werecalculated using globally available catchment data. The performances of the three frameworks werecompared using observations of streamflow at hundreds of catchments across the United States andCanada. The results showed that individual catchment attributes (framework 1) and climatic indices(framework 2) performed relatively poorly in explaining (dis)similarities in shape-based streamflowsignatures among catchments, and in extrapolating the predictions of shape-based streamflow sig-natures. Once the functional indices were included as catchment descriptors (framework 3), theperformance of hierarchal clustering improved at identifying (dis)similarities in shape-based stream-flow signatures (and regime curves) among catchments. The performance of regression and randomforests also improved in extrapolating the predictions of streamflow signatures describing hydro-graph shape. Indeed, although LSH could sufficiently predict magnitude-based signatures (e.g.,Qmean) with climatic information alone, and without a detailed knowledge of hydrologic functions, arobust prediction ofâor the identification of (dis)similarity amongâhydrograph shape requires theknowledge of catchmentsâ hydrologic functions.Our findings, despite the simplifications employed in developing functional indices, provide somegeneral indication of the usefulness of functional indices in enhancing the interpretability of LSH, andimproving its extrapolation accuracy. These improvements are essential in order to more accuratelyaddress the challenges of predicting streamflow in poorly gauged regions, especially in light of climatechange, and in order to better represent hydrologic processes in earth system models. Future studieswill be needed to improve the incorporation of hydrologic nonlinearities, and the threshold-likeprocesses underlying these, in the functional indices, in order to further enhance the interpretabilityof functional indices and LSH methodologies.372.3. Ultra-marginal feature importance2.3 Ultra-marginal feature importance: Learning from data withcausal guarantees2.3.1 IntroductionScientists often seek to understand the relationships between a set of characteristics and some out-come of interest [Kruskal, 1984]. These relationships are ideally determined by performing carefullycontrolled experiments so that causality can be established. However, experiments can be difficultand costly to pursue, unethical to perform, or impossible to control [Wright, 1921, Vowels et al.,2021], leaving only observational data available. The relationships that are hidden within vastquantities of observational data are often difficult to determine, so statistical tools, such as featureimportance, have been explored.Recently, feature importance methods such as Shapley values [Shapley, 1953, Cohen et al., 2007,Lundberg and Lee, 2017], Shapley additive global importance (SAGE) [Covert et al., 2020], accu-mulated local effects (ALE) [Apley and Zhu, 2020], permutation importance (PI) [Breiman, 2001a],and conditional permutation importance (CPI) [Debeer and Strobl, 2020], have been used in high-impact journal papers by scientists who want to explain the mechanisms behind observational data[Addor et al., 2018, Bazaga et al., 2020, Stein et al., 2021, Johnsen et al., 2021, Schmidt et al.,2020, Gill et al., 2017, Janssen et al., 2022]. However, these methods are predominantly for modelexplanation or feature selection, so they have many shortcomings when used for other purposes suchas scientific inference [Freiesleben et al., 2022, Catav et al., 2021a]. ALE can nicely display howchanges in inputs lead to altered model predictions but important higher order effects are omitted[Molnar, 2020], and although CPI improves upon some limitations of PI, CPI gives zero importanceto perfectly correlated features even if they offer significant explanatory power towards the response[Covert et al., 2020]. Similarly, Shapley values diminish the importance of duplicated or highlycorrelated features [Catav et al., 2021a]. Further, only one model is trained in ALE, CPI, and PI.Thus, correlated features, which can alter the model assembly process, could be given artificially lowimportance if the goal is to explain the data [Hooker et al., 2021]. Due to the multiplicity of nearoptimal models with vastly different functioning, one cannot use a single model to explain the datagenerating processes [Marx et al., 2022, Molnar et al., 2021]. Instead of exploring a single model,the developers of SAGE, SPVIM, and marginal contribution feature importance (MCI) evaluate thedifference in accuracy between a model trained with the feature of interest and a model trainedwithout it, across all feature subsets [Catav et al., 2021a, Covert et al., 2020, Williamson and Feng,2020]. However, these methods have not been accepted by a wider scientific audience because oftheir high computational cost. In particular, we note that MCI is a recently developed method forexplaining data, and it was shown in extensive experiments to have better quality and robustnesswhen compared to Shapley values, SAGE, ablation, and bivariate methods [Catav et al., 2021a].Although MCI is a powerful and innovative method for explaining data, it has three key shortcom-ings. First, MCI demands an exponential number of model trainings, making it unsuitable even forsmall-to-medium-sized datasets. Second, although it can handle complex feature interactions anddata with correlated features, MCI underestimates the importance of correlated features that forminteraction effects because MCI usually ignores features that share information with the feature ofinterest, as explained further in Section 2.3.4. Third, MCI can give non-zero importance to featuresthat are completely unrelated to the response variable, as experimentally shown in Catav et al.[2021a, Figure S3] and theoretically shown in Harel et al. [2022]. We hypothesize that constructinginformation-preserving representations of the data that are independent of the feature of interest382.3. Ultra-marginal feature importancecould resolve these three issues. With this in mind, we introduce ultra-marginal feature importance(UMFI), a new variable importance method that can better describe data while drastically reducingruntime.The rest of this paper is organized as follows. Axioms for explaining data are proposed in Section2.3.2. The framework for UMFI is then formally presented in Section 2.3.3 along with its theoreticalproperties and its simple algorithm. In Section 2.3.4, we conduct experiments on simulated and realdata to assess the quality, robustness, and time complexity of UMFI compared to MCI. Finally, anoverview of the work, its limitations, and ideas for future work are discussed in Section 2.3.5.Related work This paper is greatly inspired by the development of marginal contribution featureimportance (MCI) by Catav et al. [2021a]. Let F = {x1, ..., xp} be the set of features used to predictthe response variable, y. The universal predictive power of a set of features S â F is given byν(S) = minfâG(â )E[l(f(â ), y)]â minfâG(S)E[l(f(S), y)], (2.8)where l is a specified loss function and G(S) is the set of all predictive models restricted to usingfeatures in S â F . ν is closely related to mutual information, with equality under ideal conditions[Covert et al., 2020], but in practice, ν is often approximated by machine learning evaluation func-tions [Covert et al., 2020, Catav et al., 2021a]. Using this, Catav et al. [2021a] defined the marginalcontribution feature importance (MCI) of a feature xi â F asIν(xi) = maxSâFÎ½(S âª {xi})â ν(S). (2.9)To achieve our goal of improving upon the shortcomings of MCI, we evaluate the importance of afeature of interest xi after preprocessing the data to remove dependencies on xi. Finding independentrepresentations of predictors for creating improved feature importance methods is a novel objective,though similar ideas have been alluded to as future work in Ko¨nig et al. [2021], Chen et al. [2020a],and Fan and Lv [2008]. The weaker concept of finding orthogonal representations of data has beendiscussed previously [Gibson, 1962], though the discussion has been limited to relative importancemeasures for multiple linear regression [Bi, 2012, Wurm and Fisicaro, 2014]. Methods which can notonly orthogonalize features, but also remove more general dependencies, have seen great progresswithin the domains of AI fairness and privacy. Some examples of these preprocessing techniquesinclude regression [Bird et al., 2020], optimal transport [Johndrow and Lum, 2019], neural networks[Song et al., 2019, Moyer et al., 2018, Gitiaux and Rangwala, 2021b,a], convex optimization [Calmonet al., 2017], kernels [Tan et al., 2020], and principal inertial components [Wang and Calmon, 2017].Linear regression and optimal transport are used in this paper. Gurushankar et al. [2022] studiesthe problem of using fairness techniques to perform partial information decomposition, which leadsto interesting connections with our work.2.3.2 Axioms for explaining dataBackground and notationWe use xi to denote an observed feature from the feature set F , and Xi to denote the randomvariable that xi is sampled from. Suppose that (F, y) = (x1, ..., xp, y) is sampled jointly from thejoint distribution (Xobs, Y ), where Xobs â Xfull = {X1, X2, ...Xr}, and p â¤ r. To accommodate392.3. Ultra-marginal feature importancedependency removal, we will require that the universal predictive power ν, defined in Equation (2.8),is also defined for transformations of the feature set. We therefore define the space of informationsubsets of a feature set F asI(F ) = {g(F ) : g is a deterministic function of F}. (2.10)We call these information subsets of F because I(Y ; g(F )) â¤ I(Y ;F ) holds for any deterministicfunction g by Theorem B.1.5.Given the data (F, y) and an evaluation function ν, the feature importance of xi â F is denoted byImpF,yν (xi) â Râ¥0. (2.11)In contrast to Xobs, we assume that Xfull is causally sufficient, meaning that there are no latentconfounders in the underlying data generating process [Yu et al., 2018]. Hence, we may consider thefull causal graphical model with graph G = (V,E), such that each vertex from V = {1, ..., r + 1}is associated to a random variable from Xfull âª Y := {V1, ..., Vr, Vr+1}, and the directed edge setE enables the graphical model to obey the global Markov property [Kang and Tian, 2009]. Thegraphical model can often be given by a structural causal model C = (G,U, E), where G is as definedabove, E is a set of mutually independent noise variables ϵ1, ..., ϵr+1, and U = {u1, ..., ur+1} is the setof functions relating the variables V1, ..., Vr+1 to their parents in G via the relation Vi = ui(Vpa(i), ϵi).AxiomsAny attempt to build a method that explains the data should begin by rigorously defining whatexplaining the data truly means. Different, but closely related definitions and goals have beenformulated by Chen et al. [2020a], Catav et al. [2021a], Gro¨mping [2009], Be´nard et al. [2022], andTolos¸i and Lengauer [2011]. Inspired by the above works, we provide three intuitive axioms forfeature importance methods intended for explaining data and scientific inference.1. Elimination axiom: Eliminating a feature xj from F can only decrease the importance ofother features:âxi â F {xj}, ImpF{xj},yν (xi) â¤ ImpF,yν (xi).2. Invariance under Redundant Information and Symmetry under Duplication (IRI& SD) axiom: Let xË be a redundant feature, i.e. xË â I(F ). Then adding xË to the featureset F to create F â² = F âª xË will not change the importance given to any preexisting feature inF . Also, in the case where the added feature xË is a duplicate of a preexisting feature xj â F(i.e., xË = h(xj) and h is bijective) both xË and xj will be given equal importance:xË â I(F ) =â ImpF,yν (xi) = ImpFâ²,yÎ½ (xi) âxi â FxË = h(xj) =â ImpF â²,yÎ½ (xË) = ImpFâ²,yÎ½ (xj).3. Blood relation axiom: A feature xi will be given non-zero and positive importance if andonly if it is blood related to the response Y in the full causal graphical model G = (V,E).Two vertices are said to be blood related if there is a directed path between them or if thereis a backdoor path between them via a common ancestor. Figure 2.9 illustrates an example.ImpF,yν (xi) > 0 ââ Xi â BRG(Y ).402.3. Ultra-marginal feature importanceThe elimination axiom comes directly from Catav et al. [2021a]. Once a feature is observed to berelated to the response, the relationship strength between the feature and response should not drop,regardless of the additional features added. In fact, the importance should often increase, sinceadding features could reveal further synergistic information about the response [Griffith and Koch,2014, Williams and Beer, 2010]. Hence, the elimination axiom suggests that a feature importancemethod intended for scientific inference should be expressive enough to capture complex featureinteractions.The invariance under redundant information (IRI) portion of our second axiom is a generalization ofthe duplication invariance property introduced in Catav et al. [2021a] and the group size invarianceproperty introduced in Tolos¸i and Lengauer [2011]. If a dataset contains two duplicate features,a model may use them equally often and therefore divide the importance equally between them(random forests), or only one of the features may be given importance (lasso) [Chen et al., 2020a].However, from the dataâs perspective, the original importance found before duplication should bemaintained [Tolos¸i and Lengauer, 2011, Catav et al., 2021a]. Further, after adding duplicate features,no additional interaction capability is available [Griffith and Koch, 2014], so the importance of allother features should remain the same. The same arguments can be made for features offering onlyredundant information, which arise much more frequently than exact duplicates, thus motivatingIRI. The symmetry under duplication (SD) aspect of our second axiom is functionally equivalentto the symmetry property presented in Catav et al. [2021a] and is similar to the correlated groupproperty in Tolos¸i and Lengauer [2011]. The purpose of this part of the axiom, as pointed out byTolos¸i and Lengauer [2011], is to prevent the feature importance method from dismissing featuresthat are not needed for better predictions, but which may be part of the true underlying causalmodel.We call the final axiom the blood relation axiom because it asserts that a feature should have non-zero importance with respect to the response y if and only if it is connected to y in the causal graphthrough a directed path or via a common ancestor, much like how blood relatives are connectedwithin a family tree. This axiom suggests that feature importance scores intended for data expla-nation should extract reliable knowledge about the underlying causal graph and data generatingprocess. When two features are blood related, or equivalently, when there is an open path betweenthem, the two features are said to be associated (see Greenland et al. [1999] and Williams et al.[2018]). Thus, a feature importance metric satisfying this axiom would give non-zero importanceto a feature if and only if there is a statistical association between that feature and the response.Statistical association is not only a quality of interest for many applications (e.g., genome-wideassociation studies), it also forms the foundation of Pearlâs causal hierarchy [Shpitser and Pearl,2008]. Some may argue that we should only assign importance to features with a directed path tothe response, but as pointed out by Gro¨mping [2009], even if this were desired, this is an impossiblegoal since X â Z â Y is Markov equivalent to X â Z â Y . It is argued by Gro¨mping [2009] thatany importance measure aimed at explanatory or causal purposes must give importance to X andZ in both scenarios.Additionally, a feature importance metric satisfying this axiom can partition the feature set intofeatures that are blood related to the response and features that are not blood related to the response.Although it does not enable us to immediately recover the full causal graph, this partitioning may bea helpful supplemental tool for other fast partial causal graph discovery methods [Soleymani et al.,2022]. This would be an especially relevant research direction since Reisach et al. [2021] recentlyshowed that classic causal discovery algorithms are more inaccurate than previously thought. This412.3. Ultra-marginal feature importancepartitioning could also be useful for sure independence screening [Fan and Lv, 2008, Schellhas et al.,2020].The axioms introduced in this section are only intended to define ideal properties for feature im-portance methods that are focused on scientific inference. We do not claim that a unique solutionexists since all three axioms are invariant to scalar multiplication, nor do we claim that a methodexists that satisfies all axioms in all scenarios.2.3.3 Ultra-marginal feature importanceLet F = {x1, ..., xp} be a set of p features of arbitrary type used to predict the response Y .Definition 1. We denote SFxi as a preprocessed feature set after dependencies on the feature ofinterest xi have been removed from F . An optimally preprocessed feature set is denoted by SËFxi, andwe say that a preprocessing SFxi is optimal if it obeys the following properties:1. SFxi â I(F )2. SFxi â¥â¥ Xi3. I(Y ;SFxi , Xi) = I(Y ;F )The first property ensures that SFxi = g(F ) for some deterministic function g, and hence no infor-mation from outside of F is gained during the transformation. The second property upholds thatSFxi is independent of Xi. The last property affirms that there is no unnecessary information lossor distortion incurred during preprocessing [Gitiaux and Rangwala, 2022]. For simplicity, we use Fand SËFxi to denote both their respective data and random vector instantiations.Provided that it exists, an optimal preprocessing SËFxi is not unique since scaling by a constant doesnot affect any of the optimality conditions. Therefore, given a feature of interest xi and a featureset F , we may consider the (possibly empty) equivalence class [SËFxi ] of optimal preprocessings. Inpractice, the last two properties of Definition 1 can be difficult to guarantee, but we later observe inSection 2.3.4 that non-optimal preprocessings can often be good enough to provide highly accuratefeature importance scores for scientific inference.Definition 2. Given an evaluation function ν : I(F )â Râ¥0, a feature set F , and a response y, wedefine the ultra-marginal feature importance (UMFI) of the feature of interest xi â F asUF,yν (xi) = ν(SFxi âª {xi})â ν(SFxi). (2.12)We note that this definition allows for the use of any preprocessing SFxi . When the utilized prepro-cessing SËFxi is optimal, we say that UMFI is optimally computed.Theorem 2.3.1. Suppose that the data (F, y) comes from a multivariate Gaussian distributionand that ν(·) is positively linearly related to I(Y ; Â·), then we can ensure that UF,yν satisfies (i) theelimination axiom and (ii) the redundant information invariance and duplication symmetry axiom,by using linear regression to optimally compute UF,yν . If we additionally assume that the jointdistribution of the variables (Xfull, Y ) is faithful to the causal graph G, then UF,yν also satisfies (iii)the blood relation axiom.The proof of this Theorem is in the Appendix. We note that the linear relation between ν(·) andI(Y ; Â·) is pivotal to the proof of Theorem 2.3.1. Under ideal conditions, this relationship holds[Covert et al., 2020], but in practice, the accuracy of the approximation depends on the quality422.3. Ultra-marginal feature importanceof the method, the loss function, and the response variableâs distribution. See Covert et al. [2021]and Supplement B.1.3 for a more thorough overview. Simulations have demonstrated that Gaussiangraphical models with sparse graphs are generally faithful to the graph; hence, assuming faithfulnessis often reasonable [Malouche and Sevestre-Ghalila, 2008].We remark that UMFI does not require Gaussianity to satisfy all axioms. Indeed, since the lastequivalence in Equation (B.1) can be relaxed to be an implication without Gaussianity, UMFIsatisfies one direction of the blood relation axiom, UF,yν (xi) = 0 =â xi Ì¸â BRG(Y ), for arbitrarydistributions. Additionally, we show that UMFI satisfies the blood relation axiom under alternateassumptions in Supplement B.3. We also note that the proof of Theorem 2.3.1(ii) does not requirethe Gaussian assumption, making the IRI & SD axiom true for arbitrary distributions.Algorithm 1: Algorithm for computing UMFI1: Let y be the response variable of the set of predictors F . Choose a feature xi â F .2: Obtain SFxi by using a technique that removes dependencies on xi from F .3: Specify a method f and a corresponding evaluation function νf .4: Estimate the predictive power, νf (SFxi), that SFxi has about y.5: Estimate the predictive power, νf (SFxi âª {xi}), that SFxi âª {xi} has about y.6: return UF,yνf (xi) = νf (SFxi âª {xi})â νf (SFxi)Since UMFI is model-agnostic, we provide a general algorithm, which can be applied using anypair of preprocessing functions and evaluation functions νf (Algorithm 1). We note that νf is notrestricted to the domain of machine learning models or even models in general. For example, onecould also implement UMFI with measures of dependence such as QMD [Griessenberger et al., 2022]or FOCI [Azadkia and Chatterjee, 2021].2.3.4 ExperimentsWe perform experiments to compare UMFI and MCI with respect to quality, robustness, andtime complexity. To implement UMFI, we consider optimal transport [Johndrow and Lum, 2019](UMFI OT) and linear regression [Bird et al., 2020] (UMFI LR) as methods to remove dependenciesfrom the data. A detailed overview of these implementations is shown in Supplement B.5 and exper-iments comparing their performance appear in Supplement B.6. For all experiments, we use randomforestsâ out-of-bag accuracy (cross-validation R2 for regression and cross-validation overall accuracyfor classification) as the evaluation metric νf since it can capture nonlinearities and interaction effects[Breiman, 2001a, Taufiq et al., 2023, Wright et al., 2016]. We use the ranger package to implementrandom forests with 100 trees and default hyperparameters [Wright and Ziegler, 2017]. All experi-ments were run in Microsoft R Open Version 4.0.2. Supplement B.7 contains additional experimentscomparing UMFI with other feature importance metrics including ablation, permutation importance,and conditional permutation importance. In the same section, we rerun the experiments comparingMCI and UMFI using extremely randomized trees instead of random forests and do an additionalcomparison on a real dataset from hydrology [Addor et al., 2017]. Code for all experiments in R,inlcuding basic MCI and UMFI functions, can be found at https://github.com/HydroML/UMFI.The analagous Python package can be found at https://github.com/guanton/umfi_python.432.3. Ultra-marginal feature importanceExperiments on simulated dataIn the following subsections, we compare UMFI and MCI on simulated data. The data in all sce-narios contains one response variable y, four explanatory features x1, x2, x3, x4, and 1000 randomlygenerated observations. Each study is repeated 100 times to test and ensure stability [Yu, 2013].Nonlinear interactions Interaction effects are common in many scientific disciplines where as-sessing feature importance is prevalent, including hydrology [Janssen and Ameli, 2021, Addor et al.,2018, Le et al., 2023, Li and Ameli, 2022], genomics [Catav et al., 2021a, Wang et al., 2021, Orlenkoand Moore, 2021, Wright et al., 2016], and glaciology [Edwards et al., 2021, Bach et al., 2018, Bren-ning and Azo´car, 2010, Sevestre and Benn, 2015]. So, as was done in Catav et al. [2021a], we assessthe ability of MCI and UMFI to detect nonlinear interaction effects in the data [Marx et al., 2021].We consider:x1, x2, x3, x4 â¼ N (0, 1)y = x1 + x2 + sign(x1 â x2) + x3 + x4.Feature importance metrics should ideally conclude that x1 and x2 have higher importance comparedto x3 and x4 because of the extra interaction term, sign(x1âx2). Figure 2.8a shows consistently goodperformance across all methods. Each method gave high relative importance scores to x1 and x2,while x3 and x4 received less, but still substantial importance. All methods show similar variability.Correlated interactions Interacting features are often correlated [Jakulin and Bratko, 2003,Janssen and Ameli, 2021]. So, this simulation study aims to repeat the nonlinear interactions study,except now x1 and x2 are highly correlated with eachother. In the same way, x3 and x4 are highlycorrelated with eachother. Let A,B,C,D,E,G â¼ N (0, 1). We consider:x1 = A+B, x2 = B + C, x3 = D + E, x4 = E +Gy = x1 + x2 + sign(x1 â x2) + x3 + x4.Just as with the interaction experiment with independent features, we would expect x1 and x2 to bemore important than x3 and x4 because of the extra interaction term, sign(x1 â x2). The results inFigure 2.8b show that UMFI provides substantially better feature importance scores compared toMCI when correlated interactions are present. MCI estimates that all features have approximatelythe same feature importance scores, while both UMFI methods appropriately give significantlygreater importance to x1 and x2 compared with x3 and x4. MCI fails in this experiment because itpenalizes feature subsets that share information with the feature of interest (Equation (2.9)). Forexample, if we are assessing the MCI score for x1, since x2 is strongly correlated with x1, thenthe predictive power offered by x1 on top of a subset S would be diminished by the presence ofx2 â S. Therefore, x2 is not utilized in the MCI score for x1, which prevents the detection ofthe interaction term sign(x1 â x2). UMFI is able to detect this interaction because it can extractthe information from x2 that interacts with x1 while keeping this extracted feature independentof x1. We suspect that similar results could be demonstrated in the presence of dependent, butuncorrelated interactions.442.3. Ultra-marginal feature importance(a) Nonlinear interactions (b) Correlated interactions(c) Correlation (d) Blood relationFigure 2.8: Results for the experiments on simulated data from Subsection 2.3.4. Feature importancescores are shown as a percentage of the total for each of x1 to x4 from 100 replications. Resultsare shown for marginal contribution feature importance (MCI), ultra-marginal feature importancewith linear regression (UMFI LR), and ultra-marginal feature importance with pairwise optimaltransport (UMFI OT).452.3. Ultra-marginal feature importanceCorrelation Feature importance methods that seek to explain data should not change the mea-sured importance of features in the presence of redundant variables according to the IRI axiom. Totest this, we implement a simulation study similar to the ones found in Catav et al. [2021a]. Letϵ â¼ N (0, 0.01). We consider:x1, x2, x4 â¼ N (0, 1), x3 = x1 + ϵy = x1 + x2.The addition of x3, which is approximately a duplicate of x1, should not alter the importance of x1,which should remain equally as important as x2 since they have the same influence on the responsey. The results shown in Figure 2.8c show that both MCI and UMFI work reasonably well. As withthe previous simulation experiment, the variability is consistent across methods. As was desired,UMFI with linear regression shows approximately equal relative importance scores for x1 and x2.The importance given to x2 was slightly greater than x1 according to MCI and UMFI with optimaltransport. Interestingly, MCI assigns some importance to x4, which was independent of the response,while both UMFI methods assign importance scores close to zero. Because of this, we conclude thatUMFI with linear regression performs the best in this simulated scenario.Blood relation To ensure that UMFI is suitable for scientific inference, and that it could be usedto learn part of the structure of causal graphs in theory as well as in practice, we implement theblood relation simulation experiment. In this study, data is generated from the causal graph inFigure 2.9, which was inspired by the collider causal graph found in Harel et al. [2022]. The featureS is unobserved, thus x3 and x4 are the only observed features that are blood related to the responsey. According to the blood relation axiom, x3 and x4 should be given high and positive importancewhile x1 and x2 should receive zero importance. In Section 2.3.3 and Supplement B.3, we prove thatin ideal scenarios, UMFI will satisfy the blood relation axiom. We hypothesize that we can extendthis to real-world scenarios where non-Gaussian features and interaction information appear. Totest this, we consider:δ â¼ U(â1, 1), γ â¼ Exp(1), ϵ â¼ U(â0.5, 0.5)x1, S â¼ N (0, 1), x2 = 3x1 + δ, x3 = x2 + Sy = S + ϵ, x4 = y + γ.SX1X2X3YX4Figure 2.9: The full causal graph generating the data for the blood relation simulation experimentin Section 2.3.4. Blood related vertices to the response Y (blue) are coloured in red. S and X4 aredirectly causally related to Y , whereas X3 is related to Y via the common ancestor S.462.3. Ultra-marginal feature importanceThe results shown in Figure 2.8d indicate that MCI fails to distinguish the blood related featuressince more than half of the total importance is shared between x1, x2 Ì¸â BR(Y ), while x3, x4 â BR(Y )together received less than half of the total importance. In contrast, UMFI LR and UMFI OT detectthat x1 and x2 should have approximately zero importance while giving most of the importance tox4 and the rest of the relative importance to x3.Hydrology experimentsAdditional results can be seen in Figure 2.10, where UMFI and MCI are compared to other popularfeature importance methods in hydrology across a range of hydrological signature. In general, wecan observe that UMFI gives more importance to vegetation indices such as forest fraction, leaf areaindex, and root depth, while giving less importance to geological features, compared to other moretraditional feature importance methods. This occurs because UMFI and QR are the only methodsnot affected by feature correlations and dependence. Since vegetation indices are correlated amongsteachother while also being extremely correlated with climate indices, they are given artificially lowimportance by other methods, but UMFI is invariant to these correlations, so it is able to keepthe importance high regardless of the strong dependence. Since geological features share littlecorrelation, the opposite effect can be observed with these features.Computational complexityMCI must train and evaluate a model for each element of the power set of the feature set, whichimplies O(2p) model trainings if there are p features. If the evaluation function ν obeys soft k-sizesubmodularity, then the maximizing subset has no more than k elements, which reduces the numberof model trainings to O(pk+1) [Catav et al., 2021a]. UMFI circumvents the exponential training timesince it can be evaluated after O(p) model trainings, though the extra step of dependency removal isrequired. To confirm the above statements, and to show that the extra model trainings required forMCI dominate the computation time for removing dependencies in UMFI, we ran a simple timedexperiment. For a range of dataset sizes from the BRCA data, we evaluate the computation timefor calculating the feature importance scores of all features using MCI and UMFI. We ran thisexperiment for a dataset with 5 features, and then slowly added features until our given time budgetof 1 hour ran out. Once all 50 BRCA features were used, more features were randomly generated.All datasets had 571 observations. These experiments were run using an Intel Core i9-9980HK CPU2.40GHz with 32GB of RAM. Code was parallelized in R, and 12 of the 16 available threads wereused.From Figure 2.11, we can observe that UMFI is approximately super-linear, with UMFI OT incur-ring more computational cost compared to UMFI LR. Giving each method one hour to run, MCIprocessed 19 features, MCI with the soft 2-size submodularity assumption processed 130 features,UMFI OT processed about 3000 features, and UMFI LR processed about 4000 features (Table B.1).2.3.5 ConclusionIn this study, we introduced three ideal axioms that feature importance measures should satisfy ifthey claim to be useful for learning from data. We then introduced ultra-marginal feature importance(UMFI), a new method that uses preprocessing techniques, originally developed in the domain of AIfairness, to provide fast and accurate feature importance scores for the purpose of explaining data.We proved that UMFI satisfies all three of the introduced axioms under certain assumptions. When472.3. Ultra-marginal feature importance(a) Mean annual runoff (b) Baseflow index(c) Slope of flow duration curve (d) Half flow date(e) Fraction of streamflow as low flow (f) Fraction of streamflow as high flow(g) Runoff ratio (h) Baseflow index lowFigure 2.10: Variable importance scores of CAMELS catchment attributes for explaining (a) meanannual runoff, (b) baseflow index, (c) slope of flow duration curve, and (d) half flow date usingaccumulated local effects (ALE), conditional permutation importance (CPI), marginal contributionfeature importance (MCI), permutation importance (PI), quadratic regression (QR), and ultra-marginal feature importance (UMFI). The scores of each method are given as the percent of thetotal importance found for each method following the simulation studies. The color of the squareindicates the rank of that feature where darker blue indicates more importance while darker redindicate less importance.compared with MCI, experimental results showed that UMFI, implemented with linear regressionand optimal transport as preprocessing techniques, was able to provide more accurate estimates offeature importance on real and simulated data, particularly in the presence of correlated interac-tions and unrelated features. Supplement B.7 shows that UMFI also compares favourably againstother baseline methods including permutation importance, conditional permutation importance, andablation.Throughout the work on this paper, several shortcomings appeared. First, we only considered twosimple methods for removing dependencies, linear regression and pairwise optimal transport. Other482.3. Ultra-marginal feature importanceFigure 2.11: Computation time for a single iteration of each method including: MCI (dark red),MCI with the soft 2-size-submodularity assumption (pink), UMFI OT (light blue), and UMFI LR(dark blue), plotted against the number of processed features.methods certainly exist in the literature, including optimal transport with chaining [Johndrow andLum, 2019], neural networks [Calmon et al., 2017, Song et al., 2019], or principal inertial components[Wang et al., 2019]. Though our two methods performed fairly well on the real and simulated datasetsin Section 2.3.4, optimal transport and linear regression failed to find representations of the datathat were independent of the protected attribute when we tested the methods on a hydrology datasetwith more shared information compared to BRCA [Addor et al., 2017] (Supplement B.7.4). However,neural nets, principal inertial components, or implementing optimal transport with better estimatesof the conditional CDF certainly could have given better results. Although dependencies were notremoved optimally for the hydrology dataset, the importance scores were still reasonably accurate.Second, UMFI scores are less robust than MCI since they have higher variability, however, because ofthe significantly lower computational cost, UMFI can be run multiple times and averaged to increaserobustness and compute confidence intervals. Third, it is not clear how closely random forests orother measures of multivariate dependence can approximate the universal predictive power ν andmutual information in practice. In the same vein, it is unclear if multivariate measures of dependenceusually satisfy properties such as monotonicity, redundancy invariance, and supermodularity, whichwe used UMFIâs adherence to our proposed axioms. Finally, though UMFI can work for any arbitraryfeature type, in this paper, we have only considered datasets with continuous explanatory variables.In future work, we would like to test how well other methods, such as neural networks, pair withUMFI while further testing on a wider variety of variable types (i.e., binary, categorical, and ordinal).We also believe that exploring the effectiveness of dependency removal techniques is worthwhile in itsown right. We would also like to test how well UMFI scales to large datasets. In these settings, UMFIwould benefit from a fast algorithm for computing confidence intervals and p-values to counteract itsvariability. Comparing different measures of multivariate dependence in their ability to approximateuniversal predictive power and satisfy common assumptions such as monotonicity would also be of492.3. Ultra-marginal feature importanceinterest to the broader feature importance community. Currently, UMFI is limited to providinga single measure of global variable importance. Extending our methods such that UMFI can bedecomposed into different orders of interaction effects or redundant information in a similar wayto functional ANOVA [Lengerich et al., 2020, Ma¨rtens and Yau, 2020, Hooker, 2007, Huang, 1998,Stone, 1994], commonality analysis [Amado, 1999, Daube et al., 2019, Seibold and McPHEE, 1979,Stoffel et al., 2021, Ray-Mukherjee et al., 2014], partial information decomposition [Rosas et al.,2020, Varley, 2023, Gutknecht et al., 2021, Mediano et al., 2022, Kolchinsky, 2022, Suzuki et al.,2022, Goodwell and Bassiouni, 2022, Goodwell et al., 2020, Gurushankar et al., 2022, Wollstadtet al., 2021], or functional decomposition [Bordt and von Luxburg, 2022, Hiabu et al., 2022] couldprovide interesting future research directions.To reiterate, UMFI is a powerful tool for detecting and explaining the relationships hidden withinobservational data. We emphasise that UMFI is just a framework. A variety of other methods canbe used to estimate the universal predictive power ν including, but not limited to, XGBoost, neuralnetworks, QMD, or FOCI [Griessenberger et al., 2022, Azadkia and Chatterjee, 2021]. Furthermore,new preprocessing techniques for dependence removal are still being developed in the AI fairnesscommunity, so these, in addition to other existing methods, can be used in future applications ofUMFI for additional improvements. We hope that UMFI will be a useful tool for learning from datain a variety of disciplines including bioinfomatics, earth sciences, psychology, and health science.50Chapter 3Within-catchment temporal analysis3.1 Chapter summaryThis chapter develops and applies two novel methods that can extract the time-varying filteringfunctions of catchments. These filtering functions relay how catchments partition, store, and drainrainfall and snowmelt. We are particularly interested in how these filtering functions (or the sensitiv-ities of streamflow to additional precipitation across lags) change in time and space due to differingcatchment characteristics or antecedent conditions. The explosive adoption of machine learningmodels within hydrology, due to their superior predictive capabilities, has been closely followed bya growing desire to interpret these black-box models. Methods developed to interpret black-boxmodels are frequently used to obtain compressed representations of why a model made a particularprediction (e.g., SHAP, LIME) or what features contributed to a modelâs overall performance (e.g.,permutation importance, SAGE). While the original black-box model may provide predictions ofstreamflow close to the limits of predictability, the extracted interpretations of the model will beimperfect, and accurate inferences about the data-generating processes cannot be made. We hypoth-esize that creating a model that is both flexible and inherently interpretable will be more popularand useful to the hydrological community. This chapter explores this hypothesis via the creation oftwo novel methods: (1) a purely statistical approach that can infer a unique unit hydrograph for eachday of the year using mild assumptions and (2) an assumptionless machine learning approach whichcan infer a unique unit hydrograph based a learned nonlinear function of antecedent conditions.In the first paper of this chapter, we develop a novel model inspired by the functional data analysisliterature. Recent work on sparse functional historical linear models confirms that they are promisingas a tool for obtaining complex and interpretable inferences, but several notable limitations exist.Most importantly, previous works have imposed sparsity on the historical coefficient function, buthave not allowed the sparsity, hence lag, to vary with time. We simplify the framework of sparsefunctional historical linear models by using a rectangular coefficient structure along with Whittakersmoothing, then reduce the assumptions of the previous frameworks by estimating the dynamic timelag from a hierarchical coefficient structure. We motivate our study by aiming to extract the physicalrainfall-runoff processes hidden within hydrological data. We show the promise and accuracy of ourmethod using eight simulation studies, further justified by two real sets of hydrological data.In the second paper in this chapter, we develop and apply a second method which utilizes theflexibility of machine learning to remove the primary assumption of our previous method, namelythat day of the year acts as a good proxy of antecedent conditions. To resolve this issue, we useLSTMs combined with time-varying unit hydrographs, which constrain the LSTM to be perfectlyinterpretable while not imposing any restrictive assumptions. We show that these self-explainingLSTMs can be used to generate state-of-the-art rainfall-runoff predictions along with immediatelyavailable interpretations. Most importantly, we demonstrate that the application of our method to513.2. Dynamically sparse historical functional linear models with applications to Earth sciencea large sample of catchments can be used to accurately infer the data-generating processes hiddenwithin hydrological time series, thereby leading to novel context-dependent catchment-scale laws.The first paper is already published as Janssen et al. [2025b], while the second paper is still inpreparation for submission.3.2 Learning from limited temporal data: Dynamically sparsehistorical functional linear models with applications to Earthscience3.2.1 IntroductionIn many complex systems, a causal variable is filtered into some response of interest via complicatedand unknown non-stationary processes. Exactly how the causal impact spreads and changes acrosstime is of great interest to scientists and policy-makers in a wide variety of disciplines. If a methodfor learning this filtering function in a data-driven manner existed, one would be able to extractuseful knowledge about the mechanisms that generate variability from a variety of time series datasources.More specifically, this research is motivated by the desire to learn the data generating processes be-hind streamflow time series. Observational units in hydrology called catchments, filter precipitationinto streamflow (Figure 3.1). This process may sound simple; however, slightly different geologic,climatic, or topographic features may produce vastly different filtering functions. Understanding thefiltering function for a given catchment is vital for a variety of reasons. For example, knowing that acatchment filters most rainfall slowly through subsurface and deep flow pathways would indicate thatthe stream is vulnerable to long-term soil and deep rock contamination from human activities suchas mining [Kunz, 2020]. Furthermore, inferring the geological or climatological factors that causeprocess variation can enable better predictions in regions where streamflow is not measured. Indeed,accurately estimating the functions that filter precipitation into streamflow can help hydrologistsunderstand the physical processes behind complex temporal hydrological data.As indicated by Botter et al. [2010], for a given catchment, water input and streamflow can becausally related via a historical filtering function. This filtering function evolves through timedue to changes in antecedent water input, evapotranspiration, temperature, and landscape variables[Mindham et al., 2023]. Evaporation or transpiration can alter the effect sizes of the filtering function,since a catchment with high evaporation only has a small portion of water available for filtering intostreamflow. On the other hand, the depth to which water flows or gets stored in/on the grounddictates the length of time between a precipitation event and a rise in streamflow [Jasechko et al.,2016, Somers and McKenzie, 2020]. Overland flow (OF) is the fastest and shallowest path thatprecipitation can follow, precipitation filtered into shallow subsurface flow (SF) travels down intothe soil and towards the stream at a slower pace, and deep groundwater flow (GF) travels deeplyinto the ground and is the slowest responding flow path (Figure 3.1).Streamflow and rainfall time series at the Koksilah River (British Columbia) and the WithlacoocheeRiver (Florida), illuminate the fact that the relationship between rainfall and streamflow is indeedcomplex and highly nonstationary (Figure 3.2). At the Koksilah River, very little winter rainfallis evaporated (days 1-50 and days 320-365), rather it quickly filters through the catchment asstreamflow. Conversely, in summer and early fall (days 100-310), any rainfall that may occur does523.2. Dynamically sparse historical functional linear models with applications to Earth scienceFigure 3.1: A flow path diagram depicting how precipitation (red downward arrows) gets partitionedinto evapotranspiration (ET), overlandflow (OF), shallow subsurface flow (SF) or deep groundwaterflow (GF), with arrow width depicting illustrative relative effect size.not result in notable streamflow intensification. At the Withlacoochee River, streamflow falls lessrapidly during periods of little-to-no rainfall compared to the Koksilah River (e.g., days 1-50).Evaporation is likely to be high in this catchment, therefore, streamflow may only increase in theevent of a large rainfall event as seen on day 50.The statistical model that most closely follows the causal filtering relation that we described in thepreceding paragraphs comes from Xun et al. [2022], who considered the historical functional linearmodel,yi(t) =â« tmax(0,tâÎ´)Î²(s, t)xi(s) ds+ ϵi(t). (3.1)Here, yi(t) is the outcome of interest at time t of replicate series i, xi(s) is a temporal explanatoryvariable at the past time s for the same replicate, and ϵi(t) is the residual term. The bivariatecoefficient function β(s, t) represents the effect of the explanatory variable at past time s on theoutcome at the current time t, where max(0, t â δ) â¤ s â¤ t. To estimate β(s, t), Xun et al. [2022]minimized an objective function containing three terms: (1) a least-squares loss, (2) a nested groupbridge penalty term which reveals the constant lag δ, after which β(s, t) is zero, and (3) a smoothnesspenalty term. This builds upon many previous methods including Malfait and Ramsay [2003] whooriginally developed the functional historical linear model, and Harezlak et al. [2007] who imposed adiscrete difference penalty on the coefficients similar to the P-spline framework [Eilers et al., 2015].While both Malfait and Ramsay [2003] and Harezlak et al. [2007] primarily focused on predictivecapability, Xun et al. [2022] were more interested in accurate parameter estimation. Parameterestimation will also be the focus of this paper.The method introduced by Xun et al. [2022] is compelling, though it has several notable downsides.First, it is overly complex. Its original formulation is non-convex, necessitating iterative lasso-equivalent optimization steps until convergence. The tent-like basis functions which parameterizeβ(s, t) add computational cost such that the number of basis functions is usually restricted (about230 basis functions were used in previous works). Second, the triangular finite element frameworkartificially imposes a constraint on the maximum time lag (i.e., the domain of integration cannotbe negative in Equation (3.1)). For example, if one were to predict streamflow on day one for theWithlacoochee River in Figure 3.2, only one value of rainfall (0mm) could be used, leading to aninaccurate prediction of 1998âs peak streamflow. In our application, however, except for the very first533.2. Dynamically sparse historical functional linear models with applications to Earth scienceFigure 3.2: Left: The daily streamflow (blue) and rainfall (black) for the Koksilah River (top) andthe Withlacoochee River (bottom) for the year 1998. Right: The average daily streamflow (blue)and rainfall (black) for the Koksilah River (top) and the Withlacoochee River (bottom) for 1979-2018.year, the model for streamflow on a day at the beginning of a year has rainfall data available fromthe last days of the previous year (see Section 3.2.2). Third, the previous frameworks do not allowfor a time dependent time-lag parameter δ(t). Although it is common to assume an arbitrary fixedwindow of time for which precipitation can affect streamflow, this is a severe limitation of previousworks [Janssen et al., 2021], so finding the true underlying δ(t) is of interest to the hydrologicalcommunity [Tennant et al., 2020, Chiu and Bittler, 1969]. Other similar methods, with similarissues, can be found in the distributed lag literature [Almon, 1965, Pesando, 1972, Rushworth et al.,2013, Asencio et al., 2014, Gupta, 1968, Liao et al., 2023].This article has three major contributions. First and foremost, to the best of our knowledge, thisis the first attempt to estimate a time-dependent time-lag parameter in historical functional linearmodels. Second, we simplify and increase the flexibility of previous historical functional linear modelmethods by replacing the tent-like basis functions with the tensor product Whittaker basis suchthat higher resolution coefficient and time-lag functions can be recovered. Finally, we successfullydemonstrate that our method can lead to novel research directions within hydrology by accuratelyestimating the ground-truth filtering function β(s, t) and time-dependent time-lag parameter δ(t).543.2. Dynamically sparse historical functional linear models with applications to Earth science3.2.2 Historical functional linear model with dynamic sparsity (HFLM-DS)Let yi(t) be the response at time t â [1, T ] for replicate i â 1, . . . , n, and let xi(t) be the correspondingcovariate. In our application, yi(t) is streamflow for time t within year i, and xi(t) is rainfall. Webegin by considering the exact causal streamflow relation, where V is the set of all (un)known and(un)measured antecedent variables which alter the way in which rainfall is filtered into streamflow,and s represents the lag (as apposed to Equation 3.1, where s represents some day in the past). Wethen separate the true filtering function h(s,V) into an identifiable component that is stable acrossyears due to consistent seasonal variations of antecedent conditions (β(s, t)) and an unidentifiablecomponent that changes from year to year (hâ²(s,V)):yi(t) =â« â0h(s,V)xi(tâ s) ds Exact filtering function model=â« â0[Î²(s, t) + hâ²(s,V)]xi(tâ s) ds Expand h into subcomponents=â« â0Î²(s, t)xi(tâ s) ds+â« â0hâ²(s,V)xi(tâ s) ds Distribute and separate terms=â« â0Î²(s, t)xi(tâ s) ds+ ϵi(t) Unparamaterized term as error.After separating terms, letting the unidentifiable part become the autocorrelated error term ϵi(t),and letting D â 1 be the maximum possible lag, our functional historical linear model is given byyi(t) =â« Dâ10Î²(s, t)xi(tâ s) ds+ ϵi(t), (3.2)and the discretized version (e.g., assuming daily measurements) is visualized in Figure 3.3. Onedetail that distinguishes our application from most applications in functional data analysis is thatin functional data, the replicates usually relate to independent units observed over a common timedomain (e.g., experimental subjects). While, in a sense, each year can be treated as a replicate,in reality, we have one data series for each of x and y over many years. Daily observations on xand y at the end of one year are followed by data on the first day of the next year. Due to this,in Equation (3.2), contrary to previous works, t â s can be < 1 (i.e., outside the original range oft â [1, T ]. When this occurs, we draw from the end of the previous replicate i â 1. An example ofthis is illustrated in the final two lagged causes in Figure 3.3, where yi(2) is being predicted by thefirst two observations of year i (xi(1) and xi(2)) as well as the last two observations of the previousyear (xiâ1(2) and xiâ1(3)).As with Xun et al. [2022] we assume without loss of generality that the response and covariate havehad their seasonal signals removed such that they have mean zero for each day of the year. Notonly does the usage of rainfall and streamflow anomalies remove the need for a functional interceptterm, it also removes any potential periodic confounding effects present in both signals [Moges et al.,2022].Let Ï1(s, t), ..., ÏK(s, t) denote a sequence of K known basis functions. Given these basis functionsand their corresponding basis coefficients b = [b1, ..., bK ]T , the coefficient function β(s, t) can bedenoted as553.2. Dynamically sparse historical functional linear models with applications to Earth scienceFigure 3.3: The relationships between the response y and the cause x is shown for D = 4 and T = 3.Causal connections are shown for replicate i at t = 2.β(s, t) =Kâk=1bkÏk(s, t).Using this basis expansion, we may reformulate Equation (3.2) asyi(t) =â« Dâ10Kâk=1bkÏk(s, t)xi(tâ s) ds+ Ïµi(t)=Kâk=1bkâ« Dâ10Ïk(s, t)xi(tâ s) ds+ Ïµi(t)=Kâk=1bkzik(t) + ϵi(t),wherezik(t) =â« Dâ10Ïk(s, t)xi(tâ s) ds.As stated in the introduction, one of our foremost goals is to infer the dynamic time lag δ(t) afterwhich the cause has no discernible effect on the response, i.e.,δ(t) = max{s} subject to β(s, t) Ì¸= 0. (3.3)The accuracy in determining δ(t) is directly related to the resolution of the basis functions thatparameterize β(s, t), but increasing the number of bases can be computationally prohibitive if wecontinue the use of the tent-like basis functions utilized in previous works [Xun et al., 2022]. Underthese circumstances, using the Whittaker basis (or equivalently zero-degree P-splines) is an appropri-ate solution as it is a fast exact interpolator with few notable shortcomings [Eilers, 2003, Whittaker,1922]. In this work, we parameterize the coefficient function β(s, t) with the finest resolution possible(one basis function per (lag, day of the year) pair) using tensor products of Whittaker basis functionsÏk(s, t) = wm(s)âwâ(t). If we have T observations in our time domain [1, T ], thenm â {0, ..., Dâ1},â â {1, ..., T}, and k â {1, ...,K}, where, for a specified k,m = kâ1â(âk/Dââ1)ÃD and â = âk/Dâ.563.2. Dynamically sparse historical functional linear models with applications to Earth scienceWe note that m denotes where we are in lag space, â denotes where we are in time space, and âxâ de-notes the ceiling of x (x rounded up to nearest integer). Here wm(s) and wâ(t) denote the Whittakerbasis functions, which are equivalent to zero-degree B-spline basis functions with a knot spacing ofone [Eilers, 2003]:wp(s) ={1 p â¤ s < p+ 10 otherwise.Since the Whittaker basis either has a value of 1 or 0, the functions take on the same values astheir discretized/observed versions. Due to this, in a slight abuse of notation, we continue to usefunctional notation (i.e., β(s, t) or yi(t)) even when we are discussing the computational/discretevalues of these functions.Suppose the maximum reasonable lagged influence between xi(t) and yi(t) for all t occurs at lags = D â 1, then we must estimate K = DT parameters {bk : k = 1, ...,K}. In general, the amountof available data is dense with few missing values but limited in temporal extent (i.e., we can haveT observations in [1, T ]), thus we expect that nT = N âª K, leaving us with an underdeterminedsystem. For example, in our real data study shown in Section 3.2.3, n â¤ 40, T = 365, and D = 150.This implies that N â¤ 40 Â·365 = 14600âª K = 150 Â·365 = 54750. This problem cannot be overcomewithout some assumptions about the structure of the coefficient function β(s, t), or equivalentlythe coefficients bk, and we therefore stipulate three major assumptions. First, for the same lag, thecoefficients at consecutive time points should have similar coefficient values (e.g., β(s, t) â β(s, t+1)).Second, for the same time point, coefficients at consecutive lags should have similar values (e.g.,β(s, t) â β(s+1, t)). Finally, we expect β(s, t) to be fairly sparse. Before fitting a model, the extentof the sparsity is unknown, though we expect the sparsity will be in regions with larger lags (i.e., foreach day there will exist a lag δ(t) after which the predictor stops contributing to the response). Aswith the coefficients themselves, we expect that δ(t) will smoothly vary with day of the year. Welimit the number of assumptions so that our methods are flexible and widely applicable [Clark et al.,2011], though we could have further assumed that the coefficient function should be non-negativefor our application since additional precipitation cannot reduce streamflow. Instead of adding thisconstraint, we keep our model flexible to allow an additional evaluation of the scientific validity ofour method.With these three assumptions in mind, we now impose regularization and sparsity constraints onthe coefficients of β(s, t). With the first assumption, we regularize the squared differences of thecoefficient function at consecutive points in time. This can be done with the horizontal first differencepenalty matrix (DH), which was first applied to functional historical linear models in Harezlak et al.[2007] and further used by Xun et al. [2022]. We further add periodic boundary conditions toDH in asimilar fashion as Garcia [2010], such that β(s, T ) â β(s, 1). Likewise, given the second assumption,we penalize the squared differences between coefficients with the same time but with consecutivelag values by introducing the vertical first difference penalty matrix (DV ). We additionally considera zero top boundary condition on the vertical penalties since lags past D â 1 are assumed to havezero influence. The structure of the coefficients and the penalties are visualized in Figure 3.4. As an573.2. Dynamically sparse historical functional linear models with applications to Earth scienceillustrative example, if we take D = 2, T = 3, and K = 6, then DH , DV , and β(s, t) are given by:DH =ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°â1 0 1 0 0 00 â1 0 1 0 00 0 â1 0 1 00 0 0 â1 0 11 0 0 0 â1 00 1 0 0 0 â1ï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£» DV =ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°â1 1 0 0 0 00 0 â1 1 0 00 0 0 0 â1 10 1 0 0 0 00 0 0 1 0 00 0 0 0 0 1ï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£» β(s, t) =ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°Î²(0, 1)β(1, 1)β(0, 2)β(1, 2)β(0, 3)β(1, 3)ï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£» .Figure 3.4: The general structure of the discretized β(s, t) is shown for D = 2 and T = 3, where s islag and t is time. The bidirectional lines between nodes indicate that the nodes should have similarvalues. The unidirectional lines indicate that there is periodic behaviour with a period of T +1 (i.e.,β(0, 3) and β(0, 1) are connected). The bottom row of nodes represents the coefficients relating thecurrent timeâs cause to the current timeâs response while the next rows represent the coefficients forconsecutive past lags. The top row shows the zero boundary conditions.In statistics, sparsity is usually induced via the lasso penalty [Tibshirani, 1996], however, it doesnot have the oracle and sign consistency properties, meaning it cannot distinguish which coefficientsshould be exactly zero according to the data-generating process. This was shown experimentally inthe original lasso paper [Tibshirani, 1996], and it was shown theoretically in Meinshausen and Yu[2009], Zhao and Yu [2006], Fan and Li [2001], and Leng et al. [2006]. The oracle and consistencyproperties of bridge penalties are better than lasso, though they still fail when the number ofparameters exceeds the number of observations and when there is extensive multicollinearity [Huanget al., 2008]. Both of these troublesome properties are expected in problems surrounding our intendedapplication. As an alternative, Zhao and Yu [2006] suggests using the L0 penalty, though thisposes a new problem of non-convexity and NP-hard computational complexity [Huo and Ni, 2007].The L0-norm has the oracle property and is selection consistent, meaning it will select the correctfeatures which are non-zero in the ground-truth model with a probability converging to 1 underweaker conditions compared to the lasso [Zhang and Zhang, 2012, Staerk et al., 2018]. Generallythe L0-norm problem is O(2K), but due to the strongly hierarchical nature of our model in whichwe assume we know the ordering of how the coefficients should be dropped to zero, optimizing583.2. Dynamically sparse historical functional linear models with applications to Earth sciencethe sparsity problem in our case changes to O(K). Thus, for our application it is computationallytractable and theoretically preferable for variable selection.Most previous works that have developed historical functional linear models have imposed sparsityin their model [Harezlak et al., 2007, Malfait and Ramsay, 2003, Xun et al., 2022], however, theyall assume that the lag after which the explanatory feature has no effect on the response is staticin time. This assumption is indeed too strong for many applications, especially in Earth science.Instead, we impose sparsity with a more flexible approach via nested group 2-norm thresholdingsuch that a dynamic time lag δ(t) can be reliably estimated. The nested group framework allowsus to place each coefficient in a series of groups according to a hierarchical structure [Zhao et al.,2009]. Let A0,t, A1,t, ..., ADâ1,t be a series of groups defined for each time point t â {1, ..., T}. Thegroup As,t contains the indices of the coefficients corresponding to β(s, t), ..., Î²(Dâ1, t). Notice thatthe groups are nested such that ADâ1,t â ADâ2,t â ... â A0,t, meaning a specific group contains allcoefficients with lags greater or equal to its s. Since As,t â Asâr,t for r â {1, ..., s}, we know that||bAs,t ||22 â¤ ||bAsâr,t ||22. In contrast to this, if we look at two different times t Ì¸= tâ², then As,tâ©As,tâ² = â .These properties of our groups and their norms allow us to ensure that (1) coefficients at later lagsmust be removed and set to zero before coefficients at earlier lags when they represent the sametime and (2) groups of coefficients at different times are formed independently such that the sparsitycan remain dynamic and δ(t) can be estimated accurately. We further note that coefficients onlyact locally, so if b3 = 0 (corresponding to β(0, 2), then the surrounding coefficients may not be zero(i.e., β(0, 1) Ì¸= 0 and β(0, 3) Ì¸= 0) unless the time is the same and the lag is greater (β(1, 2) = 0).These groups will be further discussed when our method and algorithm are detailed in Section 3.2.2.Autocorrelated errorsAs we illustrated in the derivation at the beginning of this section, our model is somewhat mis-specified due to replacing unmeasured dynamic components with the day of the year t [Grangerand Newbold, 1974]. While t serves as a good proxy for these dynamic antecedent conditions, itis imperfect due to climatic variations across various years [Wu et al., 2024]. This imperfection,as well as inherent day-to-day autocorrelation in streamflow and possible systematic measurementerrors [Horner et al., 2018, Sorooshian and Dracup, 1980, Schoups and Vrugt, 2010, Kim et al.,2023], can lead to strongly autocorrelated errors [Sun et al., 2021b, Thursby, 1987]. For example,suppose that for a given catchment, May is a dry month across most years, but for a particularyear, it is extremely wet. Our model will likely choose β(s, t) to be fairly small in May to fit thedata across most years, however, for the particularly wet May, our model will systematically underpredict streamflow (i.e., autocorrelated errors) since streamflow has persistent magnitudes and ourÎ²Ë(s, t) is too low for May in that particular year. If we ignore this property of the errors, we violatethe assumptions of linear regression (independent errors). Naturally, one could consider previouslags of the response to improve predictive capabilities and reduce autocorrelation, as is done inthe classical time series framework ARIMAX [Box et al., 2015], however, prioritizing forecastingaccuracy leads to several issues. First, previous streamflow does not cause future streamflow, thusincorporating it into the model leads us further from the true data-generating processes therebyinstigating incorrect causal effect estimates [Dafoe, 2018, Barnett et al., 2017, Runge, 2021, Achen,2000]. Further, while data for precipitation is globally available and accurate, other data such asstreamflow or evaporation is usually inaccurate or unavailable. Since hydrologists often require amodel that is easily transferable to all ungauged or data-limited regions [Janssen and Ameli, 2021],explicit use of previous streamflow values must be excluded from the model.593.2. Dynamically sparse historical functional linear models with applications to Earth scienceThe most common way to account for autocorrelated errors while not explicitly using lagged re-sponse values in the model, has been the Cochrane-Orcutt or Prais-Winsten correction method[Cochrane and Orcutt, 1949, Prais and Winsten, 1954]. Unfortunately, both Cochrane-Orcutt andPrais-Winsten corrections failed to outperform our method without autocorrelation correction forparameter estimation in our initial simulation studies. It is unclear exactly why these methods failed,but we have several suspicions. First, we noticed that artificially reducing the smoothness penaltyterm found from our validation set led to better results, but there was no consistent reduction wecould apply to produce optimal results. Second, we noticed that if we only consider a few lags (i.e.,D < 5), the corrective methods far outperformed default regression, however if D > 100, defaultregression far outperformed the corrective methods. Reviewing the causes of these and other issuescould be an interesting direction for future work [Dagenais, 1994, Sims, 1972, McGuirk and Spanos,2009, Mizon, 1995, Dafoe, 2018, Thursby, 1987].Instead, we found that using an autoregressive distributed lag model followed by deconvolutionachieved better results in the presence of autocorrelated errors. This method is comprehensivelyexplained in Kirchner [2022] but also appears in many earlier works [Baltagi and Baltagi, 2011,Pagano and Hartley, 1981, Tsay, 1985, Young, 2002, Wilkins, 2018]. As a simple illustration, considerthe equationsyt = Î±1ytâ1 + b0xt + b1xtâ1 + b2xtâ2 + ϵt (3.4)andyt = β0,txt + Î²1,txtâ1 + ut, (3.5)where the ϵt is a white noise process from N(0, Ï2) and u(t) = Î±1utâ1+ ϵt is an AR(1) process. Thetwo models are equivalent: we can subtract from (3.5) its lagged self multiplied by α1 to obtainyt = Î±1ytâ1 + β0,txt + (β1,t â Î±1Î²0,tâ1)xtâ1 â Î±1Î²1,tâ1xtâ2 + ϵt.We can therefore readily estimate the parameters of (3.4) for the simpler case of independent errors,then use deconvolution to obtain the β coefficients of interest in (3.5) from the modified b coefficients(see Kirchner [2022] for further details). The above simple example is easily applied to our morecomplex methodology and implemented in the algorithm below. Fortunately, this also reduces thecomputational cost compared to the Prais-Winsten procedure due to only requiring a single model fit,with a diagnostic test of no autocorrelation, followed by a simple deconvolution, instead of two modelfits. Note that all autocorrelated error correction approaches implicitly assume the common factorrestriction implying that the response does not cause the lagged explanatory variables [McGuirkand Spanos, 2009, Cook and Webb, 2021], a safe assumption for our application.To summarize, our problem setting incurs autocorrelated errors (which violates one of the coreassumptions of linear regression) due to slight model misspecification and the inherent properties ofour problem setting. The more traditional methods for autocorrelation correction from Cochrane-Orcutt and Prais-Winsten work poorly in our setting for reasons that should be explored in futurework. In the end, we use the deconvolution method suggested by Kirchner [2022] as it removes theautocorrelation, however, from our extensive testing, it does not significantly improve parameterestimation capabilities.AlgorithmLet the discretized response Y be defined as a column vector of length N â (Dâ 1). We then definethe sparse matrix Z â R(Nâ(Dâ1))ÃK which has D non-zero entries in each row corresponding to the603.2. Dynamically sparse historical functional linear models with applications to Earth sciencevalid entries of β(s, t) given the current time of yi(t) and the Whittaker basis. Using the Whittakerbasis saves significant computational effort when computing the design matrix Z. Instead of needingto evaluate NK integrals, no integration is needed as each entry of Z is either 0 or equal to the datapoint itself. Quantitatively, given similar amounts of data/parameters (about 10000 data points andabout 50000 parameters), in MATLAB using the code and triangular basis functions of Xun et al.[2022], computing Z takes about 10 minutes while in R with Whittaker basis functions, it takes lessthan half a second. This is the case even while R is known to be slower compared to MATLABaccording to the latest benchmarks done by the Julia team [Bezanson et al., 2017]. Here, b is thevector of bk âk â 1, ...,K. For example, suppose n = 2, D = 2, T = 3, and K = 6, thenY =ï£®ï£¯ï£¯ï£¯ï£¯ï£°y1(2)y1(3)y2(1)y2(2)y2(3)ï£¹ï£ºï£ºï£ºï£ºï£» Z =ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°: 0z1,1(2) : 0z1,2(2) z1,3(2) z1,4(2) : 0z1,5(2) : 0z1,6(2): 0z1,1(3) : 0z1,2(3) : 0z1,3(3) : 0z1,4(3) z1,5(3) z1,6(3)z2,1(1) z2,2(1) : 0z2,3(1) : 0z2,4(1) : 0z2,5(1) : 0z2,6(1): 0z2,1(2) : 0z2,2(2) z2,3(2) z2,4(2) : 0z2,5(2) : 0z2,6(2): 0z2,1(3) : 0z2,2(3) : 0z2,3(3) : 0z2,4(3) z2,5(3) z2,6(3)ï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£» b =ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°b1b2b3b4b5b6ï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£» .Our method first estimates a smooth coefficient function via:minb||Yâ Zb||22 + wh||DHb||22 + wv||DV b||22, (3.6)and the group 2-norms are computed using this initial coefficient function estimation (see the end ofSection 3.2.2). Then, to optimize for our time-varying maximum lag parameter δ(t), a sequence ofmodels are fit using coefficients with corresponding group norms greater than a sequence of q valuesinminb||Yâ Zb||22 + wh||DHb||22 + wv||DV b||22 s.t. bAs,t = 0 when ||bAs,t ||22 < q.Once the optimal q is chosen (selected via a userâs choice of methods), to account for autocorrelatederrors, previous response values are added as covariates and modified coefficients are computed.Finally, the coefficients are deconvolved back into their intended form. A detailed description of ouralgorithm is provided in Algorithm 2.3.2.3 Hydrology Data ApplicationWe apply our methodology to data from two diverse catchments to show its versatility for hydrology.For the real data experiments, the relationship between actual rainfall and streamflow is modelled.The simulation study also uses real rainfall data, to mimic actual patterns in the time series, butstreamflow is simulated according to known coefficient functions with known lag structures, so wecan assess estimation performance.The first catchment we explore is the Koksilah River which is located in Cowichan, British Columbia,Canada. The area of this catchment is 236km2 with an average elevation of 461 meters. About 11%of precipitation falls as snow and it is an extremely wet catchment with an aridity index (fractionof potential evapotranspiration over precipitation) of just 0.37. The second catchment we exploreis the Withlachoochee River which is located in Dade City, Florida, United States. The area ofthe catchment is 650km2, and it is low lying with an average elevation of only 42 meters above sealevel. It does not experience any snow and it is a fairly dry catchment with an aridity index of 1.07.613.2. Dynamically sparse historical functional linear models with applications to Earth scienceAlgorithm 2:Require: One response time series Y = [y1, ..., yN ] and one explanatory time seriesX = [x1, ..., xN ], split into training and validation sets (and test set if needed).1: Optimize b on the training set given a set of hyperparameter pairs (wh, wv), and find theoptimal hyperparameters such that some objective function (i.e, R2) is maximized on avalidation set.2: Find optimal b using the optimal weights wh and wv and the least squares criterion inEquation (3.6).3: Compute the group norms ||bAs,t ||22 for all (s, t).4: Find optimal sparsity threshold q to obtain δ(t).5: Given q and using Step 1, reoptimize the smoothing weights, wh and wv, and refit the modelby computing b using all data and ytâ1...ytâc as additional covariates.6: Deconvolve the optimized coefficients to obtain β(s, t).7: return β(s, t) and δ(t)To obtain rainfall for both these areas, we use EMDNA [Tang et al., 2021], a high-quality climatedataset that has complete daily precipitation and temperature values for 1979â2018 at 10km gridsquares across North America. From this data, daily rainfall values are generated for the Koksilahand Withlacoochee rivers by averaging precipitation and temperature values across the catchments,then using a temperature threshold of 0â¦C, we compute daily rainfall (Figure 3.2). Leap days areremoved, leaving us with 365Ã40 = 14, 600 observations. We decided that the maximum reasonablelag should be D = 150 days. Indeed, regardless of the catchment location, all precipitation inrain-dominated catchments should drain, become captured by deep groundwater systems, or beevaporated from the catchment within 150 days (5 months) [Jasechko et al., 2016, Tennant et al.,2020], leaving a total of 365Ã150 = 54, 750 parameters to be estimated to construct β(s, t) (Equation3.6). The choice of maximum lag also alters the total number of available observations since to predictthe response we require the previous 150 days of explanatory data. Therefore, the final number ofobservations is 14,451.Hyperparameters wh and wv are optimized with Bayesian optimization using the GPfit R package[MacDonald et al., 2015]. We start by evaluating 30 uniformly random initial hyperparameter setson the validation set and fitting a Gaussian process to said points. The following three steps arethen repeated 35 times to further optimize hyperparamters: (1) we choose the next point basedon the Gaussian process model by maximizing the expected increase in validation set R2, (2) weevaluate the new point via the validation set R2, and (3) we refit the Gaussian process model withall points that have thus far been evaluated. After experimenting on several catchments, we noticedthat the optimal wh is often much larger than the optimal wv, thus we set our search range as e[8,24]for wh and e[â5,15] for wv.Choosing the optimal threshold q is often the most difficult step in Algorithm 2. We do not specifi-cally define how this should be done in the methods section as we leave it up to the user and theirgoals, though we will give some guidance in the context of our hydrological application. Minimizingerrors on the validation set seemed to give high test set R2 scores, however, the final coefficientfunction indicated too little sparsity. The phenomena was also found in Rushworth et al. [2013]who observed optimizing the AIC leads to too little smoothness, pushing the authors to use theAIC-optimal smoothness parameter as a lower bound and artificially increasing this value. In all623.2. Dynamically sparse historical functional linear models with applications to Earth scienceexperiments below, we find the âknee-pointâ as outlined in Satopaa et al. [2011] to choose q since itgave consistently strong results (see Supplement).We let c = 2 for all experiments, meaning in step 5 of Algorithm 2, we add the previous two valuesof the response as additional covariates in an AR(2) distributed lag model. This is a conservativedecision since we will also be able to accurately model AR(1) errors [Schmidt, 1971]. All experimentswere run in R version 4.2.1.Applications to measured streamflowDaily streamflow data from the Koksilah River and the Withlachoochee River were gathered fromthe Environment Canada HYDAT database and from CAMELS [Addor et al., 2017], respectively.The average daily streamflow for both rivers is visualized in Figure 3.2. To improve predictive perfor-mance and remove patterns and non-Gaussianity from the residual plots, we transform streamflow yvia ynew = log(y+1). The estimates of our hyperparameters wh and wv are obtained after splittingthe data with 80% of the data for training and 20% of the data for validation. The final estimates ofβ(s, t), δ(t), and the whole dataset R2 are then obtained by training on all available data. In orderto quantify the overall confidence, certainty, and predictive performance of our model, we evaluateour model on a test set. To compute test set R2 values, we split the data with 60% for training,20% for optimizing the hyperparameters wh and wv, and 20% for testing.The estimates Î²Ë(s, t) for the Koksilah River after training on the entire dataset are shown in Figure3.5a. In the final model, the autocorrelated errors were modelled via an AR(2) process with fittedcoefficients (0.704, 0.122), showing substantial autocorrelation. In winter, rainfall frequency andmagnitude peaks after a rapid ramp up from September to October to November (days 260-320)(Figure 3.2), therefore, by December (day 335), the soil in the catchment is fully saturated andgroundwater is fully connected. When this happens, any additional rain either immediately transmitsto flood waters heading towards the stream, or the additional rainfall quickly pushes the wateralready in the soil towards the stream. Further, this behavior can be seen in Figure 3.2, whereDecember, January, and February (days 330â70) streamflow correlates strongly with rain falling theprevious day. Therefore, observing a δ(t) of around 3 days from day 350â60 empirically confirmsand supports existing expert knowledge. Two distinct peaks of maximum lag appear in summer andfall in Figure 3.5a. The peak in summer at around day 190, with a δ(t) of about 18 days, displayslong term recession behavior in the catchment. Starting in March, this catchment appears to enter adrying phase where evapotranspiration and runoff exceed water input (Figure 3.2). During the dryingphase (see Supplement), rainfall further back in the past begins to make a relatively larger impacton current streamflow, since the relative amount of antecedent dryness is a significant driver forstreamflow production. During August (around day 230), the soil is completely dry and temperaturesare high, therefore most rainfall contributes to soil wetness and evaporation instead of streamflow.Starting in mid-September (day 260), the Koksilah River catchment enters a rapid wetting phase asrainfall quickly becomes more frequent (see Supplement). Rainfall occurring up to around 40 daysin the past begins to have a significant impact on streamflow, since during this time if the previousdays were dry any additional precipitation will contribute to soil wetness, but if previous days werewet, additional precipitation will contribute to streamflow. The wetting phase ends in December(about day 350) when the soil becomes fully saturated regardless of the between-year variationsof precipitation in the wettest month, November. On the test set, R2 = 0.82 (in the transformedspace), indicating that we can have a high level of confidence in estimates of β(s, t) and δ(t) inFigure 3.5a. Clearly, the Koksilah River shows substantial nonstationarity in both the magnitude of633.2. Dynamically sparse historical functional linear models with applications to Earth sciencethe impulse response (β(s, t)) and the maximum lag (δ(t)), thus our methodological contributionsallow hydrologists to draw unprecedented and important inferences about this catchment.(a) Koksilah River (b) Withlachoochee RiverFigure 3.5: The estimated function Î²Ë(s, t) for each lag s and time t for the Koksilah River locatedin British Columbia (left) and Withlachoochee River, located in Florida (right) from the whole dataperiod (1979-2018).For the second catchment, the Withlachoochee River in Florida, the R2 was 0.55 on the test dataset(in the transformed space), indicating that we can have a moderate level of confidence in the coef-ficients plotted in Figure 3.5b. In the final model, the autocorrelated errors were modelled via anAR(2) process with fitted coefficients (1.389, -0.408), showing extreme autocorrelation. Though theWithlachoochee River results presented in Figure 3.5b look very different compared to the resultsfrom the Koksilah River (Figure 3.5a), they can be interpreted in a similar fashion. We first note thatthis catchment may not experience overland flow where rainfall bypasses the storage and stronglyaffects streamflow in a short period of time. This is reasonable since this is a dry catchment with anaridity index of greater than one, meaning that the potential for evaporation is greater than totalprecipitation. Starting in April (days 100-175), the catchment enters a dry phase. By this time ofthe year, the catchment has experienced about 6 months of relatively dry weather and temperatureshave begun to rise, therefore in May, most rainfall evaporates or contributes to soil moisture insteadof travelling as streamflow. During this dry period, both β(s, t) and δ(t) are near zero indicatingvery little streamflow response. By mid to late June (about day 180), a consistent amount of rainfallusually falls, outpacing evaporation (Figure 3.2), thus the catchment soils become wet enough toproduce streamflow. During this wetting period, streamflow responds slowly to water input (δ(t))rises to over 60 days), with strong and persistent recession behaviour (fairly constant β(s, t) acrosslags). As the catchment becomes more saturated, in the fall and early winter (day 275-365), webegin to see the largest magnitudes of β(s, t) and the response times δ(t) decrease from 60 to 30days, indicating a wet phase. From the beginning of the calendar year until day 100, rainfall fallsbelow evaporation and streamflow, leading to decreasing impulse response magnitudes (decreasingβ(s, t)) and larger lags becoming significant again (δ(t) rises to almost 50) (see Supplement FigureS1 for further comparisons).643.2. Dynamically sparse historical functional linear models with applications to Earth scienceBoth real-world studies lead to interesting and meaningful results with good R2 scores on the testset. In both cases, wetting, wet, drying, and dry periods could be delineated. Further, even thoughthe positivity of the coefficient function was not imposed as a constraint for our models, in bothcases, the model correctly identified that there is a strong positive relationship between rainfall andstreamflow at all times and all lags. This is a further indication that our method is reliable and cancorrectly identify dominant processes. We also note that the estimated δ(t) for each catchment variessmoothly with day of the year as expected. The test set R2 results for both catchments are similarto those seen from neural networks which are more complex and less interpretable [Hoedt et al.,2021, Anderson and Radic´, 2022, Kratzert et al., 2018]. Importantly, both catchments show theimportance of considering a time-varying maximum lag δ(t), which was our primary methodologicalcontribution. Without inferring the changes in these lags over the year, learning how to divide theyear into wet, drying, dry, and wetting periods for each catchment would have been substantiallymore difficult.Benchmark against previous hydrological inference methodsIn recent years, hydrologists and statisticians have developed several methods that are specificallymade for understanding the temporal dynamics of rainfall-runoff relationships [Schrunner et al.,2025, Kirchner, 2022]. Each method parametrizes their models differently, therefore the correctnessof their inferences about the ground-truth filtering function is difficult to compare, however, we cancompare their ability to fit and predict streamflow.In this subsection, we benchmark our model (HFLM-DS) against the most recent hydrologicalinference methods by comparing their predictive performances on the test sets of our two catchments.To the best of our knowledge, there are currently two primary methods that aim to infer the filteringfunction between rainfall and streamflow. The first method, developed by Kirchner [2022], extendsthe popular unit hydrograph model by allowing users to develop a series of unit hydrograph modelsthat change with precipitation intensity and antecedent conditions [Kirchner, 2024c]. This methodis similar to the method developed in this paper, however, instead of using day of the year toautomatically group data, the user must pre-specify how to break the full time series into groupsbased on antecedent conditions and rainfall intensity. In particular, to implement Ensemble Rainfall-Runoff Analysis (ERRA) models, we follow the modeling advice of Gao et al. [2025] by using theprevious 5 days of precipitation as an indication of antecedent wetness while using 4 thresholdsof precipitation intensity. The second method [Schrunner et al., 2025], called Gaussian slidingwindow regression (GSWR), uses a series of Gaussian kernels to infer the timing and magnitudesof the different flow paths as shown in Figure 3.1 (i.e., overland flow, shallow subsurface flow, andgroundwater flow). We follow the implementation details described in Schrunner et al. [2025] duringour experiment.Table 3.1: Summary of the test set R2 values (on raw observed streamflow) across three differentmethods (columns) and two different catchments (rows). The best result in each row is bolded.Location GSWR ERRA HFLM-DS (ours)Koksilah River 0.70 0.78 0.81Withlacoochee River 0.39 0.50 0.67From the results summarized in Table 3.1, it is clear that HFLM-DS (average test set R2 =0.74)653.2. Dynamically sparse historical functional linear models with applications to Earth sciencerepresents an important improvement over ERRA (average test set R2 =0.64) since it is moreinterpretable (due to its ability to infer the dynamics of the largest important lag) while havingsignificantly better predictive performance. Further, HFLM-DS (average test set R2 =0.74) vastlyimproves over GSWR (average test set R2 =0.55) in terms of the ability of each model to representthe complexities in streamflow data. This difference in predictive capability on unseen data is likelydue to the ability of HFLM-DS to change its filtering function with time (day of the year), whereasGSWR assumes that the rainfall-runoff relationship is stationary in time. We limited our evaluationto measuring the R2 (or NSE) statistic here, however, in future work, one could evaluate how eachmethod performs in predicting individual streamflow signatures which may point to more specificmodel deficiencies.3.2.4 Simulation StudiesThe aim of our simulation study is to quantify and understand our ability to estimate β(s, t) andδ(t) in a variety of noisy environments.Evaluation criteriaXun et al. [2022] compared methods by computing the root mean squared error of δ, the percentbias of δ, and the mean integrated squared error of β(s, t). The percent bias of δ does not fullycapture the accuracy of δ estimates when δ varies over time, and the other metrics, such as the rootintegrated squared error, are less interpretable compared to R2, thus we introduce three evaluationcriteria for our simulation studies. The first criterion, R2(β, Î²Ë), aims to evaluate the accuracy inestimating the true coefficient function in an interpretable fashion:R2(β, Î²Ë) = 1ââ« â« {β(s, t)â Î²Ë(s, t)}2 ds dtâ« â« {β(s, t)â β¯}2 ds dt (3.7)The second criterion, δ(t)-bias, is the average of Î´Ë(t) across time minus the average of δ(t) acrosstime. Our third criterion, δ(t)-correlation evaluates how well we can estimate the inter-periodvariability in δ(t) across time. This metric is simply calculated as the correlation between theground-truth δ(t) and the estimated Î´Ë(t). While biases could cancel over time, and strong correlationdoes not imply that Î´Ë(t) has the correct scale, together these evaluation metrics account for eachotherâs shortcomings while independently providing information about the accuracy in which we areestimating the scale and temporal dynamics of δ(t). Ideally, β-R2 and δ(t)-correlation are close toone, while δ(t)-bias is close to zero.Simulation scenariosIn this section, we run eight simulation studies that aim to quantify our methodâs ability to estimateβ(s, t) and δ(t) in noisy environments. We use the same real rainfall data used in the previous sectionto form the covariate matrix Z, then we define eight scenarios with different values for β(s, t), δ(t),noise levels, and noise autocorrelation coefficients, such that known ground-truth values for theresponse yi(t) can be simulated. Streamflow is notoriously difficult to predict. Depending on thelocation of the catchment and its dynamics, even when using powerful black-box methods such aslong short-term memory (LSTM) neural networks [Hochreiter and Schmidhuber, 1997a], combinedwith more predictor variables, the test set R2 can range all the way from zero to one [Ayzel andHeistermann, 2021, Hoedt et al., 2021, Kratzert et al., 2018]. In future scenarios for which our663.2. Dynamically sparse historical functional linear models with applications to Earth sciencemethod may be applied, we hypothesize that inferences from models with R2 < 0.4 will be difficult.Further, from previous works we know that interpretable models with R2 > 0.8 will be rare inhydrology. Thus, we simulate the response vector with additional noise such that if we calculatethe true response ytrue(t) from the ground-truth β(s, t) we would produce R2(y, ytrue) = 0.8 orR2(y, ytrue) = 0.4, whereR2(y, ytrue) = 1ââ« {y(t)â ytrue(t)}2 dtâ« {y(t)â y¯}2 dt , (3.8)y(t) is the simulated data after adding noise to ytrue(t), and y¯ is the average of y(t) over t. Further,we specify two possible levels of autocorrelation for our simulated noise. From our results in theprevious section, we know that the noise can be highly autocorrelated, thus we specify mediumAR(0.6,0.1) and high AR(1.5,-0.52) autocorrelation scenarios.The eight scenarios emulate the patterns obtained in the real hydrology study in Section 3.2.3.The estimated β(s, t) values from Figures 3.5a and 3.5b for the two catchments, along with theirrespective rainfall values, allow us to simulate two sets of noiseless ground-truth response time series.With two noise levels and two autocorrelation levels for the additive error, there are a total of eightsimulation scenarios. For each scenario, we repeat data generation 100 times such that stable resultsand uncertainty levels can be obtained. Each simulation iteration takes less than 10 minutes on anIntel i9-9980HK 2.4 GHz processor with 32 GB of RAM, so our methods are computationally feasiblefor any modern hardware setup.Simulation resultsTable 3.2 shows that the introduced method is highly promising and robust. We consistently observethat regardless of location, the data with less noise allowed for more accurate and stable inferencesof β(s, t) and δ(t). We also consistently observed that higher autocorrelation leads to poorer δ(t)-corr results. The average results displayed in Table 3.2 as well as the individual simulation runsconsistently resulted in negative δ(t)-biases, revealing systematic errors. This indicates that onaverage δ(t) > Î´Ë(t), meaning that the estimated coefficient function is often too sparse and thechosen q should decrease, suggesting that our method for choosing the threshold q is not optimal.Although our chosen sparsity levels are too large due to wrongfully removing some small but non-zero coefficients, we note that the biases seen in Table 3.2 are relatively small compared to themagnitudes of δ(t) shown in Figure 3.5a and 3.5b, thus we believe our results are satisfactory (seeFigure S3 for a visualization).For the Koksilah River, R2(β, Î²Ë) was consistently above 0.9 for the high noise scenarios and above0.96 for the low noise scenarios. The high noise scenarios also gave consistently worse estimates ofδ(t), with higher autocorrelation also having an impact.For the Withlachoochee River, the estimates for β(s, t) are consistently accurate with R2 rangingfrom 0.888 to 0.927 in the high noise scenario and from 0.934 to 0.968 in the low noise scenario.For both noise scenarios, the bias was about -4 days for the high correlation scenarios days while itwas about -2.5 days for the low correlation scenarios. Considering δ(t) can range from 2 to over 60(Figure 3.5b), these bias results are quite promising. Further, our method could recover δ(t) with acorrelation above 0.98 in both low noise scenarios. The δ(t) correlation was slightly lower for the highnoise low autocorrelation scenario (0.943) and much lower for the high noise high autocorrelationscenario (0.848).673.2. Dynamically sparse historical functional linear models with applications to Earth scienceTable 3.2: Summary of the simulation study results. The average (standard deviation) across the100 replications is shown for each of the evaluation criteria defined in Section 3.2.4.Location R2(y, ytrue) AR(2) coefs R2(β, Î²Ë) δ(t)-bias δ(t)-corrKoksilah River 0.4 0.6, 0.1 0.926 (0.016) -2.699 (1.806) 0.772 (0.126)Koksilah River 0.8 0.6, 0.1 0.971 (0.005) -1.773 (0.769) 0.925 (0.043)Koksilah River 0.4 1.5, -0.52 0.904 (0.030) -4.133 (2.60) 0.534 (0.280)Koksilah River 0.8 1.5, -0.52 0.961 (0.009) -2.42 (0.944) 0.850 (0.145)Withlacoochee River 0.4 0.6, 0.1 0.927 (0.014) -2.773 (1.684) 0.943 (0.039)Withlacoochee River 0.8 0.6, 0.1 0.968 (0.005) -2.391 (0.571) 0.992 (0.003)Withlacoochee River 0.4 1.5, -0.52 0.888 (0.037) -4.912 (4.155) 0.848 (0.097)Withlacoochee River 0.8 1.5, -0.52 0.934 (0.015) -3.736 (1.045) 0.982 (0.009)3.2.5 Discussion and conclusionsIn this work, we build off the iterations of the functional historical linear model introduced byMalfait and Ramsay [2003], Harezlak et al. [2007], Rushworth et al. [2013], and Xun et al. [2022].We greatly simplify their formulations while allowing for dynamic sparsity and high resolutionestimates of the coefficient function using the Whittaker basis. In Section 3.2.2, we illustrate ouralgorithm with several diagrams and introduce the three minor assumptions we make to reduce thehighly overparameterized problem to one that is tractable. We only assume horizontal smoothness,vertical smoothness, and that sparsity is concentrated at larger lags to keep our methods flexibleand applicable to a wide variety of applications outside of our main hydrologically-centered goal.Throughout the work, we are strongly driven by the goal of accurately estimating rainfall-runoffrelationships. After gathering rainfall and streamflow data from Vancouver Island and Florida,we estimated the coefficient functions that filter rainfall into streamflow at both locations andcompared our conclusions with expert knowledge. We found that the functions have an interestinghydrological interpretation where catchments go through four distinct phases: (1) wetting, (2) wet,(3) drying, (4) dry. Because the Koksilah River (Vancouver Island) has high seasonal variability intemperature and rainfall, the four phases can be distinctly parsed. On the other hand, we foundthat the Withlachoochee River (Florida) is an extremely dry catchment, therefore it never entersthe wet phase, but all other phases can be inferred. Indeed, none of these inferences could havebeen made without the methodological improvements pertaining to the dynamic sparsity. Finally,in an extensive simulation study, we found that our simple methodology can successfully recoverthe ground-truth coefficient function β(s, t) as well as the dynamic time lag δ(t) with high accuracy.Though the accuracy for which we can recover the coefficient function and time-lag clearly dependson the level of noise and autocorrelation, we found that even when the R2 is less than or equal to 0.4and the errors are extremely autocorrelated, accurate conclusions can be made from our inferences.Several shortcomings of our methods and experiments can be identified. First, though our algorithmcan potentially include multiple features, these additional features may not follow our sparsity andsmoothness assumptions. For example, if we included snowfall as a predictor of streamflow in asnow-rich catchment, we may expect snowfall to take days or weeks or even months to melt andbegin to contribute to streamflow. This would contradict our sparsity assumption since very low andvery high lags would be expected to have zero coefficients. Second, we assume throughout the workthat the behavior of the catchment does not change across different years. Although our method683.3. Self-Explaining LSTMs for hydrological inference and predictiondoes not capture year-to-year relationship variability, it can be used to extract multiple coefficientfunctions before and after some known change. For example, one could estimate the rainfall-runoffrelationship before and after a large change in land use or before and after a period of severe climatechange. Focusing now on the limitations of our experiments, we note that our experiments are onlylimited to a specific application in hydrology, though we hypothesize that our methods can work wellin other domains. Further, our simulation study is limited since we provided ground-truths that weknow can be reached from the algorithm. This limitation points back to the second methodologicallimitation since our method may only be able to recover certain types of smoothly varying coefficientfunctions.In future work, one could solve one or more of the above shortcomings. Perhaps the flexibility of ourmethods could also be improved by adding adaptive smoothness in a similar fashion as the adap-tive lasso, since different days or lags could have different smoothness [Zou, 2006, Centofanti et al.,2022, Yang and Hong, 2017, Martinez and Carroll, 2010, Ballout et al., 2023]. Adding time-varyingAR terms to the autoregressive distributed lag model or accounting for heterscedasticity may alsoimprove the extraction of filtering functions [Kim et al., 2023, Rao, 2004]. Further, although forour purposes we adequately quantify our certainty about the estimated coefficients through a testset evaluation, this could be extended by creating confidence intervals [Chatterjee and Lahiri, 2011,Lahiri, 1999, Paparoditis and Politis, 2003]. Specifically, we suggest using the block residual boot-strap as described in Asencio et al. [2014] or Paparoditis and Politis [2003]. Extending our methodto be appropriate for spatial analysis could also be interesting. With smooth spatial attribution,one could understand how impactful a specific grid cell of a cause is on the response. Further,with sparsity, one could delineate the spatial area that is impactful on the response. Our currentwork takes the perspective of penalized least squares and sparsity thresholding, but another inter-esting perspective leading to different methods could come from a causal inference and conditionalindependence framework [Laumann et al., 2023]. Better methods for choosing our hyperparameterscould also be explored [Bach, 2008, Hall et al., 2009, Chatterjee and Lahiri, 2011].We hope our method can serve as a reliable tool for learning from time series data across multipleareas of science or serve as an inspiration for further methodological development. This area ofresearch is certainly rich with potential scientific discoveries.3.3 Self-Explaining LSTMs for hydrological inference andprediction3.3.1 IntroductionOne of the primary goals within hydrology is streamflow prediction. In operational settings, hy-drological conceptual models such as HBV have been a popular choice for managers in governmentand industry positions due to its reliability and simplicity [Seibert and Bergstro¨m, 2022]. Suchconceptual models have seen operational use cases in a large swath of around 100 countries includ-ing Norway, Sweden, Switzerland, Canada, Finland, Germany, and Austria [Seibert and Bergstro¨m,2022]. HBV and other conceptual models (i.e., Xinanjiang, GR4J, VIC, and Sacramento) are alsopopular within the hydrological research community [Zhao, 1992, Liang et al., 1994, Bergstro¨m,1976, Perrin et al., 2003, Burnash, 1973], however there has been a recent explosion of researchinterest in deep learning for streamflow prediction in gauged and ungauged regions following thepublication of Kratzert et al. [2018]. After many follow-up benchmarking studies in a variety of re-693.3. Self-Explaining LSTMs for hydrological inference and predictiongions [Lees et al., 2021, Loritz et al., 2024, AcunËa Espinoza et al., 2024, He et al., 2025, Izadi et al.,2024, Kratzert et al., 2019b, Feng et al., 2022, Wang and Gupta, 2024b, Acuna Espinoza et al., 2024,Reichert et al., 2023, Mai et al., 2022, Wi and Steinschneider, 2022], it is generally believed thatdeep learning models (especially LSTMs) can consistently outperform physical-conceptual modelsin terms of predictive accuracy. Predictive accuracy, however, is not always enough. Hydrologicscientists aim to gain a better understanding of complex hydrological systems [Kirchner, 2024a],policy makers concerned with climate change need to know that our hydrologic models are behavingcorrectly under distribution shifts (i.e., getting the right answer for the right reasons), and watermanagers often want to know why their operational model is making a particular prediction. Deeplearning models, in their present form, do not generally satisfy all of these concerns at once, butsome progress has been made in these directions.Researchers generally believe that there is a tradeoff between predictive accuracy and interpretability[Alvarez Melis and Jaakkola, 2018, Wang and Gupta, 2024a, Rudin, 2019], as we often need complexmodels to accurately predict complex phenomena (such as streamflow), and complexity usually leadsto a lack of interpretability. The desire for good predictions is so tempting that interpretability isusually sacrificed and complex uninterpretable models are trained. However, the overall desire tointerpret a complex modelâs predictions is still present, thus, researchers in methods-based fieldslike machine learning and statistics along with applied researchers in hydrology have continuedto create methods that enable us interpret complex models post-training. Explainable AI (XAI)techniques, such as Integrated Gradients [Sundararajan et al., 2017], SHapley Additive exPlanation[Lundberg and Lee, 2017], Expected Gradients [Erion et al., 2021], and surrogate modeling [Bayatiet al., 2025], are widely used to interpret deep learning models. Post-training interpretability isextremely common, but severe limitations exist. Indeed, there will always be a gap between themodel and its interpretation due to the need to simplify the model into interpretable components[Rudin, 2019]. Instead of constructing often misleading interpretations about an uninterpretablemodel, many researchers argue that we should focus on constructing models that are themselvesinterpretable such that the explanations are exactly equivalent to the model they explain [Maieret al., 2024, Rudin, 2019, Richman and Wu¨thrich, 2023, Thompson et al., 2023].Models that can make important inferences about hydrological processes usually poorly fit the datadue to the overall complexity of hydrological processes. Machine learning models are developed bycomputer scientists and statisticians who are normally solely interested in maximizing predictivecapabilities [Maier et al., 2024, Miller et al., 2021]. Meanwhile, hydrologists are more interestedin gaining an understanding of the system. These cultural differences between applications-basedscientists and machine learning engineers or methods-based scientists mean that machine learningmethods are not suitable for drawing inferences and hydrological models have suboptimal predictiveperformance [Maier et al., 2024, Janssen et al., 2023, Gro¨mping, 2009, Miller et al., 2021].Large uninterpretable models result in problems of equifinality where many models with differentcharacteristics produce the same predictive performances. While, equally predictive models mightimply that the models are equally good to some users, hydrologists want their models to be right forthe right reasons [Kirchner, 2006]. Indeed, without this, drawing inferences from the model underclimate or land use changes or predicting in ungauged regions may result in misleading conclusions[Maier et al., 2024]. LSTMs without physical constraints were found to be unreliable under climatechange conditions [Wi and Steinschneider, 2024, 2022], however adding constraints can sometimesworsen the nonphysical behavior of a different part of the model [Yang and Chui, 2021]. For thesereasons, it is important for any novel method with claims of physical realism to verify that the703.3. Self-Explaining LSTMs for hydrological inference and predictionlearned parameters or structure are indeed physically meaningful [Young, 2003], as the data maynot fully constrain the system [McIntyre et al., 2011]. This verification may be enough to supportreliable extrapolation to unseen regions on the data manifold.With the recent introduction of differentiable learning to the academic literature [Feng et al., 2022,Miller et al., 2020], some progress has been made towards building complex and accurate modelsthat are also interpretable and follow known processes [Gu¨ven et al., 2017]. Feng et al. [2022]introduces a conceptual model (HBV) with dynamic parameters, but using a fixed conceptual modellike this presupposes that we know the number of storages and flow paths. The mass conservingLSTM was also introduced to increase interpretability and model realism, but its interpretability islimited and it was not made for inference [Hoedt et al., 2021]. On the other hand, a few statisticalmethods have been recently developed for hydrological inference [Kirchner, 2022, Janssen et al.,2025b, Schrunner et al., 2025, Yoo and Park, 2008]. For example, Kirchner [2022] introduced aregression and deconvolution-based model for extracting the rainfall-runoff response. This methodassumes, however, that we know how to divide events into groups, such that the runoff responseis constant within each group. This assumption is often violated in real-world use cases. As animprovement to the method introduced by Kirchner [2022], Janssen et al. [2025b] developed andapplied the dynamically-sparse functional historical linear model. Janssen et al. [2025b] showed thattheir method could outperform the methods of Kirchner [2022] and Schrunner et al. [2025] whilealso demonstrating their methodâs capability of recovering ground-truth parameters. The methodintroduced in Janssen et al. [2025b] importantly rests on the assumption that the rainfall runoffresponse (or unit hydrograph) only depends on day of the year (i.e., the day of the year representsa perfect representation of antecedent conditions).The primary focus of this work is knowledge discovery from data with minimal assumptions, enablingus to refine and improve existing hydrological models in a changing climate. Conceptual models arebuilt on assumptions that may not always hold under nonstationary conditions, necessitating a data-driven approach to uncover hydrologic functioning. Minimal assumptions are particularly beneficialin knowledge discovery, as they reduce the risk of biasing interpretations with predefined modelstructures [McIntyre et al., 2011, Young, 2003, Dubova et al., 2025, Miller et al., 2021]. Nearing et al.[2019] argues that hydrologic data represent manifestations of hydrologic functioning, meaning thatappropriate analytical tools can help unravel these underlying processes. The method introduced inthis work, called self-explaining long short-term memory (SE-LSTM), serves as such a tool, allowingus to extract physically meaningful relationships from data without imposing rigid model constraints.In line with Kirchner [2022] and Janssen et al. [2025b], we exploit the interpretability of the unithydrograph and the strong predictive capabilities of LSTMs in order to form SE-LSTM.A key concern may be why we introduce SE-LSTM instead of using a hybrid approach like LSTM-HBV. The rationale lies in the validity of assumptions and its simplicity. If a set of assumptions donot hold under changing climate conditions, then neither the models nor their interpretations remainscientifically sound [Dubova et al., 2025]. SE-LSTM, by contrast, enables an adaptive, assumption-minimal exploration of hydrologic behavior. Furthermore, the model demonstrates the causal celerityphenomenon, reinforcing its scientific grounding. In addition to this, the unit hydrograph is thesimplest and most interpretable representation of hydrogeological processes, making it a suitablecandidate for closing the interpretability, predictive power trade-off [Hrachowitz and Clark, 2017,Rudin, 2019, Dubova et al., 2025]. This work, therefore, aligns with the broader pursuit of improvinghydrologic models not by rigidly adhering to existing assumptions but by systematically uncoveringthe processes that govern hydrologic functioning.713.3. Self-Explaining LSTMs for hydrological inference and predictionVery little has been done to construct purely data-driven models that are themselves interpretablewith strong predictive power along with proven inference capabilities. To resolve these shortcom-ings found in previous works, we reduce the assumptions of the statistical models built by Kirchner[2022] and Janssen et al. [2025b] while building on the self-explaining neural network frameworkintroduced by Alvarez Melis and Jaakkola [2018]. Similar work has also been done by Yang et al.[2022], Thompson et al. [2023], and Richman and Wu¨thrich [2023] in the context of tabular dataand Munkhdalai et al. [2022] in the context of time series. Other similar works include Yoshikawaand Iwata [2022], Yoshikawa and Iwata [2021], Park et al. [2015], Yamada et al. [2017], Al-Shedivatet al. [2020]. In particular, Richman and Wu¨thrich [2023] and Yang et al. [2022] show that these self-explaining neural nets have better inference capabilities compared to their simpler counterparts (i.e.,generalized least squares). In this paper, as our core contribution, we use differentiable modeling tolearn a unique unit hydrograph for each day and for each catchment based on catchment character-istics and antecedent precipitation, temperature, and potential evapotranspiration. Fortunately forhydrologists, we hypothesize that due to the rainfall-runoff relationship being, in a sense, simple,we can create an interpretable model without any cost to expressiveness [Thompson et al., 2023].Indeed, since all catchments act as filtering functions that transfer precipitation into streamflow,all catchment dynamics can be expressed (without any assumptions) as a time-varying convolution.Thus, as with more general neural network-based methods, our self-explainable LSTM (SE-LSTM)does not impose any violatable constraints on the learned function, giving it perfect flexibility, in-terpretability, and applicability. To evaluate our novel method, we compare its predictive capabilityacross over 1000 catchments in Canada and the United States. Lastly, and most importantly, weshow that it can act as a inference tool in several simulation studies.3.3.2 Self-explaining Long Short-Term Memory (SE-LSTM)SE-LSTM is an interpretable deep learning model that leverages Long Short-Term Memory (LSTM)networks [Hochreiter and Schmidhuber, 1997b] to represent how past precipitation influences stream-flow at time t. Central to this framework is the Impulse Response Function (IRF) kernel, whichquantifies the contribution of past precipitation to current streamflow [Kirchner, 2022, 2024a]. Inreal world hydrological systems, this IRF, or streamflowâs sensitivity to water input, changes basedon the state of the catchment. To model and learn the state of the system, our method SE-LSTMincorporates both dynamic climate inputs and static catchment attributes at each time step, collec-tively denoted as A(t). These inputs are used to generate a dynamic IRF kernel, β(s,A(t)), via anLSTM:β(s,A(t)) = LSTM(A(t)). (3.9)Here, s is the lag time, representing how many days in the past a precipitation input occurred relativeto time t. Streamflow at time t, denoted as y(t), can then be modeled exactly as a convolution ofthe precipitation x(tâ s) with the IRF kernel that is a function of all antecedent conditions (A):y(t) =â« â0Î²(s,A(t))x(tâ s) ds. (3.10)Simplifying this to the discrete setting and assuming that the celerity wave is faster than K daysgives us:723.3. Self-Explaining LSTMs for hydrological inference and predictionyt =Kâs=0Î²(s,A(t))xtâs. (3.11)The primary objective of this methodology is to solve for the time-varying and non-stationarycoefficient β(s,A(t)) in Equation 3.11, such that y(t) best represents the observed streamflow. Thisis an inverse optimization problem, which is considered ill-posed because small changes in the inputdata can lead to large variations in the solution, or multiple solutions may exist for the samedata. This issue is common in hydrological modeling when attempting to infer dynamic systemresponses from limited or noisy observational data. Solving such problems remains challengingusing traditional regression and statistical tools, which often require strong assumptions and lack theflexibility to capture dynamic, nonlinear behavior. Recent literature has demonstrated the efficacy ofdeep learning models in efficiently solving complex and underdetermined inverse problems [Genzelet al., 2023, Karniadakis et al., 2021]. Motivated by the demonstrated power of deep learningmodels in this domain, we employed an LSTM network to estimate the β-coefficients in the SE-LSTM framework. Beyond this, LSTM offers the flexibility to learn IRFs as functions of inputfeatures A(t), without requiring a predefined functional form. In contrast, traditional statisticalmethods typically rely on fixed or parametric formulations and are unable to capture such dynamic,input-dependent relationships without strong a priori assumptions.In this paper, we limit A(t) to include daily precipitation (P), temperature (T), and potentialevapotranspiration (PET), as well as 20 time-invariant catchment attributes described in the datasection, but in future applications A(t) can easily be extended to any set of antecedent conditionsand catchment characteristics. These inputs to the LSTM allow the learned IRF kernel β(s,A(t))to vary over time and depend on a learned current state and climatic conditions. As with mostdeep learning applications in hydrology, we assume that K = 365. Figure 3.6 shows the conceptualstructure of SE-LSTM.SE-LSTM captures the inherent memory and nonlinearity of hydrological systems through its learnedIRFs, where the effect of precipitation on streamflow is both delayed and state-dependent. Tradi-tional hydrological models, such as the unit hydrograph, have long used IRFs to characterize thedelayed impact of precipitation on runoff. The unit hydrograph models streamflow as a linear convo-lution of effective rainfall with a fixed, time-invariant response function. This is hydrologistsâ mostinterpretable model [Hrachowitz and Clark, 2017], as it allows us to exactly map how the input(precipitation) is transformed into outputs (streamflow), however, its use is limited by its overlysimplistic (time-invariant) view of hydrological processes.More recent studies have advanced the use of IRF kernels in hydrology, recognizing their inter-pretability and physical relevance. For instance, Kirchner [2022] introduced a statistical methodfor estimating a time-invariant IRF kernel, effectively modeling streamflow as a fixed weighted sumof past precipitation. However, this time-invariant assumption fails to capture the variability inhydrologic responses caused by changing antecedent moisture conditions. To address this, Kirchner[2022] proposed a modified approach where the IRF kernel varies with the intensity of antecedentprecipitation, requiring predefined thresholds for changes in response behavior. Janssen et al. [2025b]further extended this concept by developing Historical Functional Linear Model with Dynamic Spar-sity (HFLM-DS). This statistical method allows the IRF kernel to vary with the day of the year.This study showed that catchments can âresetâ due to long periods of drought (empty storage) orlong periods of precipitation (full storage). Following such resets, streamflow becomes dependent733.3. Self-Explaining LSTMs for hydrological inference and predictionFigure 3.6: The architecture of the Self-explaining Long Short-Term Memory (SE-LSTM) modelfor simulating daily streamflow y(t) using Impulse Response Functions (IRFs). The SE-LSTMtransforms daily inputs A(t) into hidden states ht. The initial cell state and hidden state, c0 andh0, are initialized as zero vectors. The final LSTM cell at time t illustrates the internal gatesof the LSTM: the forget gate ft, input gate it, output gate ot, and candidate cell state gt. Thefinal hidden state ht is passed through a dense layer fdense to produce the raw IRF coefficients Î²â.These coefficients are then passed through a sparsity and smoothness regularization layer (SPS),producing the final IRF β(s,A(t)), which is both smooth and sparse. The IRF is convolved withlagged precipitation x(tâs) over lags s = 0 to K, yielding the simulated streamflow y(t). TheSE-LSTM is trained to minimize the mean squared error between y(t) and observed streamflow,allowing β(s,A(t)) to represent how lagged precipitation contributes to streamflow generation.on only a limited window of recent precipitation, thus assuming a finite maximum lag K is withinreason.Despite these advancements, previous methods are constrained by simplifying assumptions. Most743.3. Self-Explaining LSTMs for hydrological inference and predictionnotably, they do not support a fully time-variant representation of hydrological systems. In con-trast, the SE-LSTM framework allows for a fully time-variant IRF kernel without requiring suchstatistical assumptions. By employing LSTMs to dynamically generate the IRF kernel based onthe evolving catchment state, SE-LSTM offers a flexible and physically interpretable approach tomodeling streamflow. This design better reflects the nonlinear, state-dependent nature of hydrologicsystems and enables improved understanding and prediction of streamflow dynamics.This work aligns with a growing body of research advocating for self-explanatory neural networksover traditional post hoc interpretation of black-box deep learning models. Our approach shares keyprinciples with recent state-of-the-art methods in computer scienceâsuch as those proposed by Al-varez Melis and Jaakkola [2018]âwhich enable interpretation through a learnable coefficient matrix.Notably, the IRF coefficients in our model can be viewed as an interpretable attention mechanism,similar in spirit to the approach described by Richman and Wu¨thrich [2023]. While sharing simi-larities in theory and purpose, our method introduces fundamental differences that distinguishes itfrom its computer science counterparts. Specifically, it is tailored to represent rainfall-runoff rela-tionships in a manner consistent with established hydrologic concepts. In addition, we incorporateregularization techniques (detailed in the next section) that are essential for promoting physicallymeaningful and hydrologically consistent behavior.Regularization techniquesRegularization techniques are a fundamental component of DL, designed primarily to (i) preventoverfitting by discouraging overly complex models that memorize training data, and (ii) incorpo-rate prior knowledge into the learning process to guide the model toward more generalizable andphysically consistent solutions. In this study, we applied several regularization techniques, includingdropout, grouped sparsity, and smoothness. Dropout is a widely used regularization method thatwas applied to the hidden state ht prior to feeding it forward. It functions by randomly mask-ing a fraction of the elements in ht, a fraction known as the dropout ratio, thereby preventing themodel from memorizing noise or learning spurious patterns during training. The dropout ratio is auser-defined hyperparameter and must be selected based on model performance and overfitting risk.In addition to dropout, in SE-LSTM, sparsity and smoothness are applied as two forms of regular-ization that shape the learned IRFs to be physically meaningful and interpretable. The motivationand rationale of these two methods are the same as those explained for HFLM-DS (see Janssenet al. [2025b]). Sparsity is enforced through a gating mechanism that controls which parts of theIRF are allowed to remain active. The key idea is that not all time lags in the input sequence (e.g.,precipitation) should influence the output (e.g., streamflow); instead, the model is encouraged toidentify and retain only the most relevant lags. This is implemented by computing the cumulativesquared values of lag contributions and applying a threshold based on a sparsity hyperparameter.This threshold acts as a cutoff, beyond which lag contributions are suppressed.To make this work within a deep learning framework, where all operations must be differentiable,a binary (Boolean) mask is first constructed based on whether the cumulative energy at each lagexceeds the threshold. However, binary operations are non-differentiable and thus incompatible withgradient-based optimization. To resolve this, SE-LSEM uses a soft gating mechanism inspired bythe Bernoulli gate: a continuous approximation using the sigmoid function is applied to the samethresholding operation, producing soft values between 0 and 1. These soft values are combined withthe hard binary mask in a way that ensures the gradient flows through the soft part during training,753.3. Self-Explaining LSTMs for hydrological inference and predictionallowing the model to learn a sparse IRF in an end-to-end differentiable manner. The target sparsitylevel is controlled by a user-defined hyperparameter, which defines the energy threshold percentile.This quantile determines how many lagged contributions are retained and is tuned as part of themodelâs hyperparameter optimization.In addition to sparsity, smoothness is also applied to the IRF to encourage physically realisticbehavior. Hydrologically, it is unreasonable for adjacent lagged contributions to fluctuate wildlyfrom one day to the next. To discourage this, SE-LSEM includes a smoothness loss that penalizesrapid changes between consecutive IRF weights. Specifically, the second-order differences of the IRFvaluesâi.e., the differences between successive first differencesâare computed, and their squaredvalues are averaged to form the smoothness penalty. This acts like an L2 penalty on the curvature ofthe IRF, encouraging gradual transitions across time lags. The strength of this penalty is controlledby a user-defined smoothness coefficient, which is another hyperparameter in the model and isselected through tuning. Together, the sparsity and smoothness mechanisms help ensure that theIRF remains interpretable, parsimonious, and aligned with known hydrological behavior.3.3.3 ExperimentsIn order to meet our goals and compare our newly devised method SE-LSTM with similar methodsin the hydrological literature, we propose two primary experiments. In the first experiment, we usereal climatic and streamflow data in order to compare the predictive power across our entire set ofcatchments of SE-LSTM, other unit hydrograph-based inference methods, and the standard deeplearning LSTM model (referred to as Vanilla LSTM throughout this text). This experimental setupwill also allow us to demonstrate the unmatched interpretability provided by the outputs of ourSE-LSTM models while we focus on our two primary study catchments. In the second experiment,we use real climatic and synthetic streamflow data in order to assess SE-LSTMâs ability to infer thehidden data generating processes behind noisy time series.DataThis study uses daily catchment-averaged time series (1979-2018) and static data from 1110 catch-ments across the United States and Canada (CAMELS and HYDAT databases). The catchmentselection process, dataset sources, and selected variables match the work of Bayati et al. [2025]. Thedynamic inputs fed into the LSTM include precipitation, temperature, and potential evapotranspi-ration. Daily precipitation and temperature time series was derived from the EMDNA dataset for1979â2018 [Tang et al., 2021]. The daily potential evapotranspiration data was calculated from theFAO Penman-Monteith equation [Allen et al., 1998, Singer et al., 2021] using data from the ERA5-Land dataset [MunËoz-Sabater et al., 2021]. Alongside the three dynamic inputs, we also include20 static catchment attributes including catchment area, mean slope, mean elevation, shallow soilhydraulic conductivity, soil hydraulic conductivity, soil porosity, depth to bedrock, maximum sub-surface water storage, bedrock hydraulic conductivity, ratio of soil to bedrock conductivity, averageprecipitation, average potential evapotranspiration, aridity index, snow fraction, seasonality index,high precipitation frequency, low precipitation frequency, high precipitation duration, low precipita-tion duration, and forest fraction. The sources of all data can be found within Bayati et al. [2025],Janssen and Ameli [2021], or Addor et al. [2017]. Figure 3.7 illustrates the spatial distribution ofthese 1110 catchments. For some of our experiments and visualizations below, we focus on the twocatchments that are are also focused on in Janssen et al. [2025b]. The large square on Vancouver is-land (Figure 3.7) denotes the Koksilah River catchment. It is an extremely wet and fairly small (236763.3. Self-Explaining LSTMs for hydrological inference and predictionkm2) catchment with a low amount of snow (< 12% snow fraction). Our second focus catchment isdenoted by the large square in Florida (Figure 3.7). This is the Withlachoochee River catchment,and it is fairly dry and fairly large (650 km2) catchment with no snow. The clusters shown in Figure3.7 are formed based on catchment averaged indices and are used to separate training in order toreduce memory consumption as will be explained in the next subsection.Figure 3.7: The map of our 1110 catchments across the United States and Canada. The differentcolors show the clusters and the larger square points show our âfocusâ catchments.Incorporating static climatic and physical catchment attributes as inputs to deep learning models is awidely adopted approach in rainfall-runoff modeling [Feng et al., 2022, Kratzert et al., 2019a]. Theseattributes allow deep learning models to leverage similarities and differences among catchments,which is essential for developing a unified model capable of generalizing across diverse catchments.By integrating key information about catchment characteristics, these attributes enhance modeltransferability, enabling effective hydrological simulations in both gauged and ungauged catchments[Kratzert et al., 2019a].Model trainingTo train the neural network-based models for each catchment (SE-LSTM and vanilla-LSTM), wefirst train a regional model based on all catchments within a cluster (see clusters in Figure 3.7).We then fine tune on the training set of the catchment of interest. In all cases, the training set isthe first 40% of the data, then the validation set is the next 20%, and the test set is the final 40%.The validation set is used for hyperparamter tuning and early stopping. For each of the 8 regionalmodels, hyperparameters are not tuned. Instead, we use default hyperparameters for these modelswhich are smoothness factor=100.0, dropout=0.4, sparsity=0.0, scale=1000, LSTM Units=128, and773.3. Self-Explaining LSTMs for hydrological inference and predictionregl2=0.0. The regional model is run with a maximum of 100 epochs, a batch size of 2048, and earlystopping patience of 20 (based on validation set R2). Unlike with the regional model, we do optimizehyperparameters for the fine-tuned models. Hyperparameters are optimized with the GPSampler inthe Optuna Python package. Following 40 randomized trials, the Gaussian process begins to pick thenext best trail for an additional 10 iterations. Based on some initial experimentation and explorationwe set the hyperparamters ranges as: lambda smooth â [0.0, 2000.0], sparsity â [0.0, 0.99], regl2â [0.0, 1.2], dropout â [0.1, 0.8]. After choosing the optimal set of hyperparameters, the model isfine-tuned model is retrained with a maximum of 70 epochs, a patience of 5, and a learning rate of5Ã 10â4.The same training/validation/test splits are used to train both statistical models (HFLM-DS andERRA). In the case of ERRA, we use precipitation intensity and antecedent wetness (the averageprecipitation across the last 10 days is used as a proxy) as indicators to divide hydrological conditionsinto groups with different representative unit hydrographs as shown in Kirchner [2024a] and as wasrecommended in the ERRA tutorial [Kirchner, 2024b]. The antecedent wetness conditions are brokenup into four groups based on the quantile breakpoints of 50, 75, and 90. Precipitation intensity isused to break the hydrological conditions into six evenly-spaced groups, and we used a maximumlag of 40 days. As for HFLM-DS, we use a maximum lag of 210 days (7 months) and refrain fromcalculating the dynamic sparsity as this part of the modeling framework increases the computationalcost without a strong effect on the predictive performance. The horizontal smoothness wh â e[10,20]and vertical smoothness wv â e[â5,15] hyperparameters are optimized by maximizing the validationset R2. In contrast to the LSTM-based methods, these methods are run in R. All code is availableat https://github.com/HydroML/SE-LSTM.Demonstration of predictive powerIn our first set of experiments, we simply compare the methods of Kirchner [2022], Janssen et al.[2025b], and Kratzert et al. [2018], with our method (SE-LSTM). For this experiment, we use all1110 catchments, and compare the test set NSEs across methods.The results for SE-LSTM are illustrated on the left side of Figure 3.8. Overall, SE-LSTM performsextremely well in catchments along the west coast of North America and in the mountainous Rockies(NSE > 0.75. Meanwhile, in the central United States and Canada, the streamflow predictions bySE-LSTM are quite poor (< 0.5 NSE). The eastern half of the United States and Canada also showfairly strong predictive accuracies (0.5 < NSE < 0.9).Looking to the right side of Figure 3.8, we can begin the compare the predictive performancesof the methods. While Kirchner [2022] showed that his ERRA method is highly interpretable,the predictive performance is often low, likely due to the method over-simplifying the catchmentprocesses. The HFLM-DS method introduced by Janssen et al. [2025b] increases the allowed modelcomplexity, and this results in a substantial increase in predictive power. Indeed, the median test setNSE for ERRA is around 0.2, while the median NSE for HFLM-DS is around 0.5. After removingthe assumption that the day of the year is a perfect proxy for antecedent conditions, we can seea substantial increase the the NSE at all levels of uncertainty. With this assumption removed, weobserve a median NSE increase of over 0.1. By comparing SE-LSTM and HFLM-DS across theUnited States and Canada (Figure D.1), we can conclude that this assumption is likely to not bevalid anywhere as there are not any noticeable areas where SE-LSTM AND HFLM-DS have similarpredictive capabilities.783.3. Self-Explaining LSTMs for hydrological inference and predictionFigure 3.8: Left: Map of test set NSE (R2) values for each catchment based on SE-LSTM predictions.Dark blue (1) is ideal, while white (0.5) is okay, and red (0 or negative) is poor. Right: Comparisonof several methods based on their predictive capabilities (NSE). Each line shows the cumulativefraction (y-axis) of catchments that have at least x test set NSE (x-axis). We compare the resultsof Kirchner [2022], Janssen et al. [2025b], SE-LSTM (our current paper), and vanilla LSTM basedon Bayati et al. [2025].These clear improvements over previous unit-hydrograph-based inference methods are exciting, how-ever, there is still a large gap between the performance of the Vanilla LSTM (median NSE= 0.754)and the SE-LSTM (median NSE = 0.655). This gap is not equally distributed across North Amer-ica. According to Figure D.1, the performance of SE-LSTM and Vanilla LSTM are quite similaralong the west coast of North America. Further, SE-LSTM actually outperforms Vanilla LSTM inmany catchments across the deep south (cluster 2). This is consistent with the results of Kratzertet al. [2018], who showed that LSTMs can sometimes underperform traditional models, especiallyin Florida. There seem to be two possible reasons for the continued underperformace of SE-LSTMcompared to Vanilla LSTM: (1) we actually made one or more assumptions on the filtering functionthat turns precipitation into streamflow that is not valid across most North American catchments,or (2) SE-LSTM is more sensitive to the optimization scene and this was not done in the optimalway. Though it is outside the scope of this current work, we hypothesize that the blame can fallon the latter reason for two reasons. First, we cannot conceive of any filtering function that couldbe excluded by the model structure imposed by SE-LSTM. Second, it is certainly possible that theLSTM architecture is not optimal for predicting time-varying unit hydrographs and a sequence-to-sequence architecture may be more suitable as this would allow for smoothness constraints acrossconsecutive days, an important addition to the model developed by Janssen et al. [2025b].Demonstration of perfect interpretabilityThe outputs of SE-LSTM can be interpreted and displayed in a variety of ways, making this methodchiefly interpretable compared to its differentiable modeling and deep learning counterparts. Twodifferent interpretation strategies for our two primary catchments of interest are displayed in Figures3.9 and 3.10.In the first interpretation strategy, we can match the output of Janssen et al. [2025b] by plottinga heatmap of the average IRF for each day of the year. This displays how, on average, the IRFs793.3. Self-Explaining LSTMs for hydrological inference and predictionchange throughout the year. For example on the left side of Figure 3.9 we can note that streamflowis always rapidly responding to precipitation input, but the response in much more attenuated inthe summer months. Further, on the right side of Figure 3.9, we notice that this catchment hasmuch longer recession behavior. These conclusions match nicely with those of Janssen et al. [2025b],who studied the same catchments.Figure 3.9: The heatmaps show the average impulse response function along each lag (y-axis) foreach day of the year (x-axis). Darker blue indicates a higher streamflow sensitivity (on that day ofthe year) to previous rainfall, while white indicates zero sensitivity. There are very few red (negativesensitivity) results. The heatmap on the left displays the result from the Koksilah River (VancouverIsland) and the heatmap on the right shows the result from the Withlacoochee River (Florida).If one were more interested in specific events rather than long-term average catchment functioning,the illustrations in Figure 3.10 which display our second interpretation strategy may be of greaterinterest. In these figures, we can infer the response of the catchment due to individual precipitationevents. For example, on the left side of Figure 3.10, we can see that any rainfall happening onOctober 14th 2003, will have a relatively small response with a two day delay due to dry antecedentconditions and low water input intensity. In contrast to this, just two days later, the antecedentconditions are wet and the water input intensity is high and in response to this we have a large andrapidly responding (one day delay) IRF. For the Koksilah river, we chose to focus on the streamflowresponse before and during a large precipitation event, so for the Withlacoochee river, we chooseto look at the streamflow responses during and after a large precipitation event. During the largeprecipitation event (September 9th to 11th, 2017), we note that the IRF magnitudes are fairly large,with their peaks happening about four days after the corresponding water input and a continuedlarge response for another two weeks. This is indeed starkly in contrast to what we observed forthe Koksilah river, illustrating their vastly different catchment functioning. A few days after thelarge precipitation event, there is barely any response to smaller precipitation events as illustratedby the flatter IRFs on September 15th and 16th. Note that these IRFs are illustrated as forwardIRFs in contrast to the backward IRFs used in the first interpretation strategy. Both IRF typesare mathemtically equivalent, but they differ in their point of reference. For the IRFs displayed in803.3. Self-Explaining LSTMs for hydrological inference and predictionFigure 3.9, the âday of the yearâ represents the day of the year of streamflow and the IRFs representthe weights of all past rainfall events (in terms of celerity not velocity). For the forward IRFs inFigure 3.10, the reference day is the day of precipitation, so each IRF represents the correspondingstreamflow in response to that dayâs precipitation. To train our SE-LSTM, we use the backwardstrategy, however, this can be easily transformed (post training) into the forward IRFs as was donefor Figure 3.10.The interpretations that we have discussed so far have followed basic hydrological principals, whichis encouraging as this shows that the LSTM is probably forming a good representation of theunderlying processes. However, the LSTM clearly does not always follow hydrological principals,and the beauty of SE-LSTM is that these shortcomings are automatically shown, enabling possiblefuture resolutions. In Figure 3.9, although almost all IRF weights are positive, there are some smallnegative weights in the average IRFs from both catchments. This would imply that the LSTM thinksthat more precipitation may cause less streamflow. Clearly, this is extremely unlikely to be true.Further, the forward IRFs during the large precipitation event in the Withlacoochee basin showsan inverse relationship between antecedent wetness/precipitation intensity and IRF magnitude. Forsome reason, the LSTM gave more weight to the small event on September 9th in comparison tothe much larger event on September 11th. This may show that the LSTM cannot differentiatebetween quick successions of precipitation events, therefore, points for future improvements areeasily ascertained due to this unmatched interpretability.Figure 3.10: The figures show the impulse response of streamflow across different days of precipita-tion for the Koksilah (left) and Withlacoochee (right) rivers. The blue bars indicate precipitationoccurrence and magnitude, while the lines denote the IRFs for a variety of different days as denotedin the respective legends.Demonstration of inference capabilitiesThe aim of this simulation study is to quantify how well time-varying unit hydrographs can berecovered from noisy observational data. In order to obtain a set of ground truth unit hydrographsfor each of our study catchments, we first let our fine-tuned SE-LSTMs generate all unit hydrographsbased on the complete climate and catchment attribute data. We then use real precipitation data togenerate a complete time series of noiseless streamflow data. Following this, we add some amount ofGaussian noise such that the NSE (or R2) between the noised time series and noiseless time series iseither 0.5 (high noise scenario), 0.7 (medium noise scenario), or 0.9 (high noise scenario). With this,we now have real climate time series paired with noisy streamflow time series that was generated813.3. Self-Explaining LSTMs for hydrological inference and predictionfrom a known set of unit hydrographs. Using this noisy streamflow time series, we run and comparethe inferences made by HFLM-DS and SE-LSTM.The results of this study are shown in Table 3.3. It should be clear from the table that the inferencesmade by SE-LSTM are extremely accurate. In all scenarios, SE-LSTM was able to extract the hiddenground truth set of unit hydrographs with a correlation above 0.85. Overall, the inference resultsfrom HFLM-DS are much weaker. Although the correlations are fairly high (> 0.55), there wasconsiderable bias in the inference results (R2 â 0, not shown), indicating that HFLM-DS is notrobust to model misspecification. Interestingly, with both models, the level of noise did not seem tohave much of an effect on inference accuracy.Table 3.3: Summary of the simulation study results, comparing our old method (HFLM-DS) withthe method developed in this paper (SE-LSTM). Inference is run for each of our focus catchmentsand three possible signal-to-noise levels. The inference quality is quantified by listing the correlationcoefficient between the ground truth and inferred time-varying unit hydrographs. The best resultfrom each row is bolded.Location R2(ynoisy, ynoiseless) HFLM-DS SE-LSTMKoksilah River 0.5 0.72 0.86Koksilah River 0.7 0.72 0.89Koksilah River 0.9 0.74 0.94Withlacoochee River 0.5 0.65 0.91Withlacoochee River 0.7 0.58 0.89Withlacoochee River 0.9 0.63 0.893.3.4 ConclusionsThis paper introduces the self-explaining LSTM (SE-LSTM), which is a novel method designedby and for hydrologists who want to have a powerful and interpretable predictive model that canalso make valid inferences. We aimed to close the gap in predictive performance between simplehighly interpretable models and high complexity low interpretability models, a consequence of the socalled interpretability-explainability tradeoff, while also enabling improved learning capabilities andstaying assumption-free. While the gap has not been fully closed, we were able to take importantsteps in this direction by introducing the SE-LSTM which employs LSTMs to learn the time-varyingimpulse response function (or the function that catchments represent when filtering precipitationinto streamflow) via differentiable modeling. Through both real world and synthetic experiments,we showed that the SE-LSTM clearly outperforms previous unit hydrograph-based methods in bothinference and predictive performance. Although we think this represents a significant step forwardin machine learning-based hydrology, several limitations and directions for future work clearly exist.The first notable limitation of SE-LSTM is that there are no intermediate outputs. Learning inter-pretable latent states/fluxes such as groundwater storage, evapotranspiration, and flow paths in oneof the most important goals of differentiable modeling within hydrology, as these latent factors canhelp with interpretability, calibration, and validation [Feng et al., 2024]. This could be resolved infuture work by integrating SE-LSTM with the intermediate storage and water table depth frame-work from Gao et al. [2025]. Second, although we argue that SE-LSTM remains perfectly flexiblesince it does not inherently assume any structure with respect to the rainfall-runoff response, we see823.3. Self-Explaining LSTMs for hydrological inference and predictiona clear performance gap between the Vanilla LSTM and our SE-LSTM. This shortcoming should beable to be resolved by improving optimization tactics in future work.83Chapter 4Within-catchment spatial analysis4.1 Chapter SummaryWhile James Dooge expressed the central importance of solving the water balance in hydrology over30 years ago, closing the water balance in catchment-scale systems, a seemingly simple problem,still remains elusive today. When studying water balance, the changes in groundwater storageand the validity of the boundary conditions are scarcely discussed. Water table depth (WTD)is perhaps the central attribute that can cleanly describe the status of the groundwater storagesystem. Generally, a large-scale (e.g., continental or global) continuous map of static WTD can besimulated using either physically-based (PB) or machine learning-based (ML) models, though thedevelopment of physically-based models has often been the focus. These predictions mark importantdevelopments as spatial patterns of water table depth play a crucial role in assessing ecologicalresilience, shaping urban/agricultural development, and modeling the entire Earth system. WTDcan also help hydrologists determine the hydrologic connectivity and the sensitivity of differentparts of a catchment to water input due to WTD determining unsaturated flow length and the theproduction of different flow types (i.e., saturation excess overland flow). This concept of connectivity(centrally related to the boundary conditions we impose for water balance studies) is usually assumedto be driven by topography-based catchment delineation, though this can lead to misrepresentationsof the hydrologic system due to complex underground drainage systems, unknown karst pathways,fault systems, and slow cross-boundary groundwater movement. This chapter aims to illuminatewithin-catchment processes by exploring the uncertainty of WTD predictions and the ability ofcausal methods to determine connectivity and catchment boundaries.In Chapter 4.2, we construct three fine-resolution (500 m) ML simulations of WTD, using theXGBoost algorithm and more than 20 million real and proxy observations of WTD, across the UnitedStates and Canada. The three ML models were constrained using known physical relations betweenWTDâs drivers and WTD and were trained by sequentially adding real and proxy observations ofWTD. Through an extensive (pixel-by-pixel) evaluation across the study region and within ten majorecoregions of North America, we demonstrate that our models (corr=0.6-0.75) can more accuratelypredict unseen real and proxy observations of WTD compared to two available PB simulationsof WTD (corr=0.21-0.40). However, we still argue that currently-available large-scale simulationsof static WTD could be uncertain within data-scarce regions such as steep mountainous regions.We reason that biased observational data mainly collected from low-elevation floodplains and theover-flexibility of available models can negatively affect the verifiability of large-scale simulations ofWTD. Ultimately, we thoroughly discuss future directions that may help hydrogeologists decide howto improve machine learning-based WTD estimations. In particular, we advocate for the use of proxysatellite data, the incorporation of physical laws, the implementation of better model verificationstandards, the development of novel globally-available emergent indices, and the collection of more844.2. Tackling water table depth modeling via machine learningreliable observations.In Chapter 4.3, we introduce a causality-based paradigm for solving the catchment delineationproblem. We first assert and argue that rain at a particular location causes streamflow if and onlyif the particular location is within the true catchment boundary. Next, we show and explain whythe purportedly âcausalâ methods usually utilized in hydrology, such as transfer entropy, cannotbe applied to such a problem. Finally, we explore if, where, and when such causal reasoning cansolve hydrologistsâ catchment delineation problem by experimenting on a large sample of virtualcatchments in Germany. The paper in Chapter 4.2 has been peer-reviewed and published as Janssenet al. [2025c], while the paper in Chapter 4.3 has been published as a preprint as Janssen et al.[2025a].4.2 Tackling water table depth modeling via machine learning:From proxy observations to verifiability4.2.1 IntroductionGroundwater is the most abundant source of liquid freshwater on Earth, with an important influenceon above-ground processes [Gleeson et al., 2016]. Groundwater affects water quantity and qualityin streams [Miguez-Macho and Fan, 2012, Ali et al., 2011], lakes [Vaheddoost and Aksoy, 2018, Xuet al., 2021, Ameli and Craig, 2014], and wetlands [Ameli and Creed, 2019, 2017], and regulatesthe behaviour of ecological and human-centric systems [Siebert et al., 2010, Huggins et al., 2023].In riverine ecosystems, groundwater provides critical water supplies to plants in arid riparian andnon-riparian areas, without which trees and plants would quickly deteriorate in drought conditions[Kibler et al., 2021, Xu et al., 2022]. The central property of groundwater at a particular locationand the particular aquifer is the water table depth (WTD), defined as the depth below which theground becomes fully saturated with water. Static WTD, which is the main focus of this paper, isdefined as the long-term average (non-transient) water table depth at a given location. This quantitycan vary by several orders of magnitude from one location to another. Although transient WTDsimulations are often needed to form robust conclusions about hydrological processes, large-scale(e.g., continental and global scales) fine-resolution spatial estimates of static WTD, can be used as agood first-order approximation in order to inform (1) the direction of groundwater movement [Freezeand Cherry, 1979], including whether or not the roots of trees in a given location have access to water[Tai et al., 2018, Cooper et al., 2003, Schook et al., 2022], (2) the spatial heterogeneity of hydrologicprocesses within and across catchments as well as the celerity of the rainfall-streamflow response[Scaini et al., 2018], and the nonlinearity of stormflow [Kidron, 2021] and baseflow generation [Aliet al., 2011, Detty and McGuire, 2010, Kim et al., 2004, Li and Ameli, 2023], (3) the overall gainingor losing status of surface water bodies connected to groundwater systems [Jasechko et al., 2021], (4)the delineation of catchment boundaries [Hinton et al., 1993], and (5) the spatial extent of droughtand flood vulnerabilities [Brooks et al., 2015, Smith and Majumdar, 2020, Gleeson et al., 2021].Previous studies have attempted to simulate static WTD at local, regional, national, and globalscales using various methods [Zell and Sanford, 2020, Maxwell et al., 2015, Fan et al., 2013, Kochet al., 2019]. For example, in an aquifer near Toronto, Canada, Desbarats et al. [2002] used ageostatistical method (kriging with external drift) to estimate the WTD at 90m resolution usingthe information of elevation and topographic wetness index. Kollet and Maxwell [2008] and Tranet al. [2022] used ParFlow, a physically-based groundwater model, to simulate WTD in a catchment854.2. Tackling water table depth modeling via machine learningin Oklahoma and Colorado, respectively. Further, national-scale simulations of WTD have beenmade by Maxwell et al. [2015] using ParFlow (across a large portion of the United States with 1km resolution), by Koch et al. [2021] using a machine learning algorithm called catboost (acrossDenmark with 10 m resolution), by Ben-Salem et al. [2023] using kriging (across Spain and Portugalwith 4 km resolution), and by Koch et al. [2019] using random forests to simulate winter-time staticWTD in Denmark with 50 m resolution. One of the most prominent examples of a global simulationof WTD was conducted by Fan et al. [2013], who used a physically-based model to develop a 1 kmgrid of WTD for the entire planet. Another estimation of global WTD was made by de Graaf et al.[2017] using PCR-GLOBWB and MODFLOW at a 10 km resolution, as an extension of their earlierwork [de Graaf et al., 2015]. As an improvement to de Graaf et al. [2017], Verkaik et al. [2024]developed the PCR-GLOBWB-MODFLOW global groundwater model to simulate WTD at 1 kmresolution. Other global physically-based models of WTD include simulations from Zeng et al. [2018](1 km resolution), Reinecke et al. [2019] (9 km resolution), Niu et al. [2007] (100 km resolution),and Lo and Famiglietti [2011] (300 km resolution).Attempts to simulate national or global-scale WTD have typically fallen into two categories: (1)physically-based models and (2) machine learning models, with each containing notable shortcom-ings. Physically-based numerical methods are flexible enough to incorporate spatiotemporal varia-tions in recharge, heterogeneous soil properties, and time-varying boundary conditions. However,they may only make valid predictions for locations that perfectly follow a long list of assumptionsunder which the model was built (i.e., laboratory scale equations, laboratory conditions, homo-geneity, laminar flow, etc...), with the scale being one of the most important and often contradictedassumptions [Herrera et al., 2023, Beven, 1996b]. Darcyâs law and Richardsâ equation form the back-bone of physically-based models, but they may lose their validity when applied beyond laboratoryscales and conditions (e.g., the scale of 1-hectare hillslope) [McDonnell et al., 2007]. This impliesthat, due to nonlinearity and heterogeneity, a model setup with a fine grid resolution and importantparameters such as effective hydraulic conductivity, which is accurately measured at the same fineresolution, is required to reliably and robustly apply physically-based models [Wildemeersch et al.,2014, Vogel and Ippisch, 2008, Beven, 1995, Beven*, 2001]. See Beven [1996b] for a critique ofdistributed models that violate these scale assumptions. Attempting to run physically-based modelsat much smaller scales would be challenging due to dramatically increased computational costs andthe lack of accurate forcing/input data at such scales. Without major breakthroughs in computing,running a physically-based model at such a fine resolution for an entire continent may continue to becomputationally infeasible [Koch et al., 2021]. This, in addition to the unavailability of parameterdatasets at such fine scales, may partially explain why global-scale physically-based simulations ofWTD generally have coarse resolutions (e.g., 1 km in Fan et al. [2013] and 10 km in de Graaf et al.[2015]). The vertical extent of the simulation domain in physically-based models can also affect theaccuracy with which WTD can be estimated. Maxwell et al. [2015] stated that one of the mainlimitations of their physically-based model developed to estimate WTD across a large portion of theUSA was specifying the lower model boundary at 102 m. The lack of knowledge on the proper lo-cation of a modelâs lower boundary condition leads to possible underestimation of WTD magnitudein dry and mountainous regions, where it is known that actively circulating groundwater can be asdeep as 200 meters [Condon et al., 2020]. Moreover, physically-based simulations strongly dependon parameters such as hydraulic conductivity, which are challenging to estimate accurately at globalscales with fine resolution [de Graaf et al., 2015, Maxwell et al., 2015, Ofterdinger et al., 2014]. Whileparameter uncertainty also exists in machine learning simulations â as will be discussed below âmachine learning models can manage this uncertainty by down-weighting the influence of uncertain864.2. Tackling water table depth modeling via machine learningparameters.Machine learning methods may resolve some of the previously mentioned shortcomings of physically-based models. Machine learning models (1) do not rely on uncertain and scale-dependent physically-based assumptions, (2) are pronouncedly less computationally intensive, (3) will down-weight im-portant variables with high measurement uncertainty such as hydraulic conductivity, and (4) haveshown to provide appropriate predictive accuracy once compared against WTD observational data.The latter is particularly important, as large-scale physically-based simulations of WTD have beenmostly verified against estimated hydraulic heads and not against direct observations of WTD (seethe discussions in Reinecke et al. [2020] and Yang et al. [2023] on how such model validation canhinder the development of trustworthy models). We especially emphasize this point because, evenafter the clear discussion of Reinecke et al. [2020], several works have continued to validate theirlarge-scale groundwater models against the estimated hydraulic head [de Graaf et al., 2020, Rahmanet al., 2023, Callaghan et al., 2024]. In contrast, when machine learning models were used to simu-late WTD, such as in Koch et al. [2019], Koch et al. [2021], and Ma et al. [2024], the performanceof the model was verified by comparing the modelled versus observed WTD observations.Despite the advantages of machine learning in modelling large-scale static WTD, such modelsstrongly rely on observational data, which, in groundwater hydrology, have considerable bias, uncer-tainty and sparsity issues. The lack of physical realism, together with sparse, biased, and uncertainobservational data, reduces the chance that machine learning models will learn the true functional(or structural) behavior of groundwater systems. This is critically required to generalize and ro-bustly extrapolate WTD estimates beyond available sparse observations. Poor measurement quality,sparse well drilling in certain areas, unknown impacts of external and non-natural drivers such aspumping, vertical gradients due to having an unknown mix of measurements from confined, uncon-fined, and perched aquifers may lead to spurious relationships in WTD datasets that the machinelearning model will hopelessly follow [Koch et al., 2021]. One other notable downside of currentML models in hydrology is that they only aim to predict a single variable (i.e., WTD), however,models such as ParFlow can simulate/validate both WTD and surface flow simultaneously [Maxwellet al., 2015]. Still, the main objective of some of the studies that use physically-based models is thelarge-scale simulation of WTD [Fan et al., 2013, 2007, de Graaf et al., 2015], and we continue withthis trend. Of course, the limitations of ML models are not limited to those noted above. Morelimitations and sources for uncertainty will be extensively discussed in Section 4.2.5.With the advent of technologies such as geophysical surveys and remote sensing products, we arenow amassing fine-resolution globally available data that can (in)directly inform WTD. One notableexample includes a fine-resolution (e.g., 30 m) satellite-based map of the probability of surface wateroccurrence [Pekel et al., 2016], obtained through merging time-varying satellite information. Thesetypes of fine-resolution proxy data have rarely been used in developing (or validating) machine learn-ing or physically-based groundwater models. Furthermore, machine learning models are becomingmore flexible in incorporating physical-based information, which can be used to increase the physicalrealism of such models.In this paper, we estimate three fine-resolution (500 m) spatially continuous maps of (static) WTD inunconfined aquifers across the United States and Canada. We leverage the computational efficiencyand flexibility of machine learning while incorporating physical boundary conditions, known toconstrain groundwater systems, and physical (functional) relationships known to control WTD. Indoing so, we use (1) more than 9 million real observations of WTD collected at more than 900,000874.2. Tackling water table depth modeling via machine learningobservational wells across the United States and Canada, (2) a physically-constrained XGBoostmachine learning model, and (3) more than 12 million satellite-based proxy observations. Usingthe XGBoost machine learning model and subsets of real and proxy observations, we simulate threesets of simulated WTD across the study region. To obtain these three sets of predictions, thetraining data used to develop each machine learning model increases from model one to modelthree by sequentially adding satellite-based proxy observations to real observations of WTD. Indoing so, we showcase the consequences of using proxy observations in simulating WTD. Ultimately,using independent/unseen real and proxy observations of WTD, we evaluate and compare our threemachine-learning simulations of WTD across ten ecoregions of North America. Additionally, inthis comparison, we include two available physically-based simulations of static WTD, which coverour study regions and our study period. These inter-model and model-observation comparisonsindicate where different machine learning and physically-based simulations corroborate each otherâspredictions of WTD and where each simulation follows (or does not follow) currently availableobservations. Finally, focusing on structural drawbacks, prediction verifiability, and observationaluncertainties, we critically discuss the challenges of machine learning-based modelling of WTD whileaiming to inform future model development, data collection, and experimental analysis. In doing so,this paper goes beyond the previous machine learning-based studies of WTD [Ma et al., 2024, Kochet al., 2019, 2021], by (1) training three physically-constrained machine learning models based onmultiple sources of (proxy) observations and systematically evaluating how the addition of new proxyobservations affect performance and reliability (2) comparing the predictive performance of thesethree machine learning models and two available physically-based simulations across ten eco-regionsof North America, and (3) discussing the verifiability of these models.4.2.2 DataInput Variables Used to Develop Machine Learning ModelsClimate data Climatic attributes and conditions influence recharge intensity and timing, whichare among the main drivers of regional water table depth [de Graaf et al., 2017, Maxwell et al., 2015,Moeck et al., 2020]. Here, we calculated the long-term average values of climatic attributes, includingprecipitation, temperature, temperature of the month of January, rainfall intensity, maximum snowwater equivalent (SWE), snow fraction, aridity index, and precipitation excess for the period between1979-2009, which covers the time from which the bulk of real and proxy WTD observations areavailable. January temperature was used in addition to average temperature, as previous studiesshowed that winter temperature affects soil frost and, ultimately, WTD properties [Fan et al.,2013]. To quantify long-term average precipitation, temperature, January temperature, rainfallintensity, and snow fraction, we used the EMDNA dataset [Tang et al., 2021], which provides dailyprecipitation and temperature at a resolution of 0.1 degrees. The long-term average snow fractionwas calculated as the sum of snowfall (precipitation during days with below zero temperature) overthe sum of precipitation. Rainfall intensity was calculated as the average rainfall intensity (mm/day)across days with over 1 mm of rain.Long-term average values of potential evapotranspiration, actual evapotranspiration, and maximumsnow water equivalent were estimated using the TerraClimate dataset [Abatzoglou et al., 2018] ata resolution of 0.04 degrees. After converting the spatial resolution of all climate datasets to thereference 15 arc-sec cell size through geospatial resampling as discussed in subsection 4.2.3, we com-bined gridded precipitation from EMDNA and potential evapotranspiration from TerraClimate tocalculate the gridded continent-wide aridity index values as the ratio between potential evapotran-884.2. Tackling water table depth modeling via machine learningspiration and precipitation [Addor et al., 2017]. Gridded precipitation excess was also calculated forNorth America as precipitation from EMDNA minus actual evapotranspiration from TerraClimate.The long-term average maximum snow water equivalent (SWE) was calculated as yearly maximumSWE from TerraClimate, averaged across all years. All climatic attributes were downscaled to ourtarget resolution of 500 m (see Section 4.2.3).Topography Conceptually, in most mountainous landscapes, the water table usually follows anextremely damped version of topography [Desbarats et al., 2002, Moreno et al., 2016]. Indeed, inmountainous regions, water tables are often observed and simulated to be deep [de Graaf et al.,2017]. This is not only due to lateral groundwater flow seeking equilibrium with gravity-drivengradients but also due to decreased recharge at higher elevations [Moeck et al., 2020]. Further, flattopography encourages discharge formations such as rivers, wetlands, and lakes, while mountainousterrains usually correspond to low-storage areas [Moeck et al., 2020]. At the same time, precipitationincreases with elevation while evapotranspiration tends to decrease. Hence, topography is one ofthe primary drivers of WTD, with the complex relationship between topography and groundwaterrecharge (and WTD) depending on how topography interacts with precipitation, evapotranspiration,and geological formations [Moeck et al., 2020]. Machine learning may provide a strong foundationfor exploring and simulating such a complicated relationship. Here, we used different topographicalattributes which are known to control recharge and WTD. These attributes include elevation, slope,and topographic index. To cover the entire extent of the United States and Canada, we mergedtwo elevation datasets. SRTM15+ [Tozer et al., 2019] was used for zones below 60 degrees North,while GTOPO30 [Daac, 2004] was used for a small portion of our study region with latitudes largerthan 60 degrees. The former dataset is available at our target resolution (500 m), while GTOPO30is a global digital elevation model (DEM) with a grid spacing of 30 arc seconds (approximately 1km), which was converted to our target resolution using ArcGISâs spatial analysis toolbox, simplyby splitting each 30 arcsecond pixel into four 15 arcsecond ones. Using the merged elevation datasetand GIS spatial analysis tools, terrainâs slope was calculated in degrees. Marthews et al. [2015]created a global 500 m resolution dataset of Topographic Index (TI), the index that was introducedby Beven and Kirkby [1979], to characterize the topographic control on the spatial distribution ofwater table. TI dataset was directly used in our model without any preprocessing.Geology Gleeson et al. [2014] developed global maps of bedrock permeability and porosity, twokey properties for understanding groundwater hydraulics [Moeck et al., 2020]. In our study, bedrockpermeability and porosity were initially used as input data to the machine learning models toinform the geological characteristics of the shallow subsurface. However, our results showed closeto zero importance for these two attributes, meaning that our machine learning models did not usethe information of these two attributes to simulate the spatial pattern of static WTD at local toregional scales.Our machine learning models do not choose deep geology data as important factors potentiallydue to (1) low spatial variability of this data across the North America, (2) high uncertainties indeep geological data (as opposed to climate/topographic data, which are more certain) and (3)the coevolution of climate, soil, land cover and geology, leading to much of the information ofdeep geology already being encoded in other features. As will be introduced in the next section,incorporating hydraulic characteristics of deep soil could, to a large extent, emulate deeper subsurfacehydraulic properties. In fact, due to the minimal variability of geology data, the information contentthat helps to predict static WTD seems to only be informed by correlated attributes such as (deep894.2. Tackling water table depth modeling via machine learningand shallow) soil hydraulic properties. This does not mean that bedrock hydraulic properties areunimportant, but it implies that the machine learning algorithm, which learns through the variabilityof predictors, opts to learn geologically relevant information/processes from highly variable andrelevant soil hydraulic properties rather than almost invariant bedrock properties. Soil hydraulicproperties (and even climate attributes) can be highly correlated with bedrock permeability inmany environments, carrying important information about bedrock geology the currently availabledatasets of bedrock permeability cannot convey. We refer the readers to Heudorfer et al. [2024],Ghosh et al. [2022b], and Li et al. [2022] for further details on how machine learning models mayleverage the correlation/coevolution among attributes and learn the relevant information of a givenuncertain attribute via other more certain attributes.Soil data While climate is the driver of overall catchment recharge, soil hydraulic propertiescontrol the partitioning between subsurface vertical infiltration (and recharge) and lateral flows[Maxwell et al., 2015, Moeck et al., 2020]. Low soil hydraulic conductivity prevents water fromentering the subsurface, encouraging the input water to exit the catchment via overland flow insteadof contributing to soil moisture or aquifer recharge [Maxwell et al., 2015, Moeck et al., 2020]. de Graafet al. [2015] argues that the soil saturated hydraulic conductivity is one of the most dominant controlsof WTD, where higher conductivity lowers the water table and enhances regional flow. Soil thickness(and soil porosity) may also impact groundwater placement [de Graaf et al., 2015].The Soilgrids dataset [Poggio et al., 2021] is a comprehensive global-scale information system pro-viding high-resolution soil property maps for various depths and soil layers. The dataset covers awide range of soil properties, including sand, silt, and clay fractions, organic carbon content, bulkdensity, depth to bedrock (or soil thickness), and more at a spatial resolution of approximately 250meters. To characterize hydraulic properties of shallow portion of the earth surface, clay, silt, andsand fractions within the top 15 cm depth were aggregated to a resolution of 500 m. Similarly, theavailable data of clay, silt, and sand fractions below 1 m were aggregated to a resolution of 500 mto present the hydraulic properties of a relatively deeper portion of the Earthâs surface material.These aggregated fractions at two different vertical positions were then utilized as input parametersfor the machine learning groundwater models. Furthermore, the gridded depth to bedrock data wasgathered from Dai et al. [2019], converted to 500 m resolution, and used as an additional soil-relatedinput to our machine learning models.Land cover Vegetation plays a critical role in determining water input partitioning and ultimatelygroundwater recharge [Allen et al., 1998, Moeck et al., 2020]. Different types of vegetation havevarying rates of transpiration, which is the process by which plants take up and release water intothe atmosphere through the stomata on leaves. For example, a forested area with tall trees mayhave a higher rate of transpiration than a grassland, leading to a higher rate of evapotranspiration.This variation in transpiration and evaporation rates will affect the amount of water available torecharge the groundwater system, ultimately affecting the water tableâs depth. Changes in land usecan impact recharge rates even if the climatic conditions remain the same, as noted by DeFries andEshleman [2004], Minnig et al. [2018].The Collection of Moderate Resolution Imaging Spectroradiometer (MODIS) Land Cover DynamicsProduct, called MCD12Q, provides global land cover and phenology metrics at 500-meter spatialresolution. MCD12Q dataset includes annual land cover maps with 17 classes, including 11 naturalvegetation classes, three human-altered classes, and three non-vegetated classes. In this study,904.2. Tackling water table depth modeling via machine learningMCD12Qâs land cover map of the year 2000 was acquired and reclassed. Different forested land coverclasses were combined to form one unique forested land cover class. Similarly, different shrublandclasses and grassland classes were combined to form one unique shrubland class and grassland class,respectively. This led to a total of 7 land cover classes which were used as categorical inputs tothe machine learning-based groundwater models. The classes selected for this analysis were chosenbased on established findings on the relationships between land cover and groundwater recharge rate(and ultimately WTD) [Owuor et al., 2016, Scanlon et al., 2005, Mohan et al., 2018, Zhang andSchilling, 2006].Real Observations of Water Table DepthWater well observational information in the United States and Canada was compiled from na-tional organizations and provincial databases. Water well observational information in the UnitedStates was downloaded from USGSâs database on âGroundwater levels for the Nationâ (https://nwis.waterdata.usgs.gov/nwis/gwlevels). In Canada, historic well reports are documentedand made available for public use in most provinces. Details of provincial databases, data format,and web links for each province are provided in the Supplementary material document. A total of9.6 million observations collected between 1852 and 2022 (with the majority collected between 1979-2009) were compiled at 905,371 observational wells across the USA and Canada. When multipleobservations over time in a given well were present, the minimum value of WTD was taken to obtainone representative observation for the given well. This treatment enhances the chance of obtainingrepresentative static WTD observations that are minimally influenced by pumping. Moreover, whenconverting point-based WTD observations to the target resolution of our study (i.e. 500 m pixels),there were instances, especially across flat agricultural lands, where two or more observational wellswere situated within a single pixel. In such cases, the median WTD was designated as the pixelâsrepresentative observation of WTD. Figure 4.1 shows (in red) the 500 m resolution pixels with esti-mated representative observations of WTD across the Prairie Pothole region of North America andthe Mississippi River Basin. Note that the mentioned processes in converting temporal point-scaleobservations to static pixel-scale WTD are in line with the previous large-scale (e.g., national orglobal) studies conducted to quantify static WTD.Proxy Observations of Water Table DepthOccurrence of surface water inundation connected to groundwater systems Persistentwater inundation on the Earthâs surface is a strong indicator of hydraulic connection between surfacewater and the surrounding groundwater system [de Graaf et al., 2017, Fan et al., 2013, Maxwellet al., 2015, Koch et al., 2019]. Indeed, the static water table depth will often be zero or near zeroat the interior or shoreline of persistent rivers, streams, wetlands, potholes, and lakes [de Graafet al., 2017]. Pekel et al. [2016] used more than three million Landsat satellite images to recordthe months (including summer) when water was inundated on the ground, between 1984 and 2015.They developed a probabilistic metric, Water Occurrence Percentage (WOP), which reflects thepercentage of time at a given 30 m x 30 m pixel when water was present on the ground surface. Themetric varies between 0 (no water) and 100 (permanent water), with values in between reflectingthe intermittent presence of water. This milestone database was used in our study to develop twosets of WTD proxy observations. The first set of proxy observations reflects the permanently wetinterior pixels of (typically small) surface water bodies and is used in our study as one line ofindependent evidence to evaluate and compare the performance of three machine learning and two914.2. Tackling water table depth modeling via machine learningphysically-based simulations of WTD. The second set of proxy observations reflects the shorelinesof surface water bodies of different sizes and is used in our study to train two (out of three) of ourmachine learning models. Note that there is no overlap between the first and second sets of pixelsutilized for the first and second objectives. Consequently, the pixels that evaluate the performanceof five machine learning and physically-based models are mutually exclusive from those used fortraining the two ML models. To preprocess the data, we first aggregated the 30 m resolution surfacewater inundation database to our target resolution (500 m). During this aggregation process, therepresentative WOP at each 500 m resolution pixel was calculated as the maximum WOP amongall WOPs of the original 30 m resolution pixels. Similar to the original 30 m resolution database,the pixelâs representative WOP at our target resolution ranged from 0 to 100.The first set of proxy data, aimed at delineating the permanently wet interior pixels of surfacewater bodies, was compiled by identifying pixels where surface water inundation was consistentlydetected over a specific occurrence threshold. Pixels experiencing water inundation for more than95% of the time (i.e. WOP>95%) were delineated and merged. Then, we used the HydroLAKESdataset [Messager et al., 2016], which provides polygonal representations of water bodies, to selectthose pixels which were fully inside water body polygons. 7,920,475 pixels representing the interiorof permanently wet surface water bodies were extracted and used to evaluate the two physically-based and three machine learning modelsâ performances in locating pixels where WTD is expectedto be zero (this will be further explained in Section 4.2.3). HydroLAKES appears to have missedsome water features with WOP>95%. Notable examples are thousands of long and stretched waterbodies, which may have been missed in surveying conducted to develop the HydroLAKES dataset,due to their narrow widths, restricted access, or other reasons. In such cases, merged pixels withWOP>95% were considered as permanently wet surface water bodiesâ pixels. Regarding the selectionof 95% threshold of WOP, we have to clarify that visible and near-infrared range images used inPekel et al. [2016], could have missed some inundated pixels in some months, because the light atthese frequencies could be impeded by cloud cover, canopy, and ice. To account for this uncertainty,we incorporated an arbitrary error margin of 5%, or the 95% threshold of WOP, rather than using100% threshold of WOP to delineate permanently wet pixels. Opting for smaller error margins (e.g.,1-2%) would misrepresent significant lakes, while larger than 5% error margins would pose a risk ofoverestimating the permanently wet surface water bodies. Therefore, a WOP of 95% was deemed themost appropriate threshold for this part of the study. Furthermore, we only relied on surface waterbodies with surface areas between 30 and 80,000 hectares to ensure that a disproportionately largenumber of pixels were not selected from the interior of very large surface water bodies (e.g. the GreatLakes of North America). Indeed, excluding interior pixels of large water bodies enhanced the spatialuniformity of pixels, allowing for a fair evaluation and comparison of the modelsâ performance.The second set of proxy data, aimed at delineating the semi-permanently wet shorelines of surfacewater bodies, was compiled again by pixel-based WOP values. Pixels meeting the criterion ofsemi-permanent to permanent inundation (WOP>75%) were delineated and merged. Then, theperipheral, non-interior pixels around the groups of merged pixels were identified as the pixelsof surface water bodiesâ shorelines. The selection of the 75% threshold ensured the inclusion ofareas beyond the interior of permanently wet surface water bodies that are still experiencing semi-persistent surface water inundation. Peripheral pixels with WOP>75% indicate prolonged surfacewater inundation with potentially stable soil saturation, consistent with our definition of staticWTD defined as the long-term non-transient WTD. Our preliminary analysis showed that choosinga larger threshold of WOP, such as 85-90%, could have excluded a large portion of surface water924.2. Tackling water table depth modeling via machine learningbodiesâ shorelines delineated using other datasets (e.g., HydroLakes). On the other hand, optingfor smaller thresholds (e.g., 60-50%) risked including ephemerally inundated areas like wave run-upand overland runoff, which have no stable connection to surrounding groundwater systems. The30 m resolution and the temporal information on surface water inundation provided by Pekel et al.[2016] dataset facilitated the identification of the shorelines of numerous small and/or locally (andsemi-permanently) inundated areas, which were not delineated by other available datasets such asHydroLAKES. In the end, 12,167,332 pixels were delineated as the pixels of shorelines of surfacewater bodies across the study region at a resolution of 500 m. Proxy pixels also include riverchannels and flooded riparian zones where the watercourse is wide enough to result in WOP>75%.These pixels were considered as the locations with static WTD equal to zero (i.e. the water tableis typically to be at the land surface). They will be utilized as proxy observations of WTD to traintwo of our machine learning models (see Section 4.2.3). Figure 4.1 shows (in blue) the delineatedlocations of shorelines of surface water bodies across the Prairie Pothole region of North Americaand the Mississippi River Basin at 500 m resolution.(a) Prairie Pothole Region, Canada (b) Mississippi, USAFigure 4.1: 500 meter resolution pixels showing the locations of the available real observations ofWTD (red) and that of the delineated shorelines of surface water bodies (blue), across the PrairiePothole and Mississippi River Basin regions. Our real observations of WTD and the delineation ofthe shorelines of surface water bodies extend throughout the entire United States and Canada. Butfor the sake of simplicity of visualization, we only show these two selected regions with fairly denseobservations and delineated shorelines.Height Above Nearest Drainage Digital Elevation Models (DEMs) inform the development ofnumerical descriptors relevant to identifying hydrological processes and WTD [Sivapalan and Kalma,1995]. One of WTDâs most relevant DEM-based descriptors is the Height Above Nearest Drainage(HAND) [Nobre et al., 2011, Koch et al., 2019]. Using a hydrologically coherent DEM, HAND valueswere calculated by (1) finding the drainage pixels using DEM-based upslope accumulated area.Drainage pixels are those with above some threshold value of upslope accumulated area. (2) Thenearest drainage pixel was found for a given pixel, and their elevational differences were computed toobtain the HAND value for the given pixel. An extensive experimental campaign in northern Brazil934.2. Tackling water table depth modeling via machine learningby Nobre et al. [2011] showed a strongly positive linkage between the static (non-transient) WTD andHAND values along steep mountainous regions. Indeed, WTD at a given pixel can strongly followthe surface elevation of its local drainage network. The global HAND dataset, originally calculatedat a resolution of 3 arc-seconds [Yamazaki et al., 2019], was utilized and resampled to match ourstudyâs reference resolution (500 m). The use of HAND data as additional proxy observations ofWTD in our third machine learning model will be further explained in Section 4.2.3.Global-scale Physically-based Simulations of WTDWe compare our three machine learning-based simulations of WTD (see Section 4.2.3) with physically-based simulations developed by Fan et al. [2013], and de Graaf et al. [2015] (data was gathered fromDe Graaf and Stahl [2022]). These two studies used historical data (1960-2010) to estimate staticwater table depth at the global scale. Fan et al. [2013] used a simple 1 km gridded Darcy-basednumerical scheme to simulate WTD. For the top 1 meter of soil, they used available datasets ofhydraulic conductivity. They assumed an exponential decay of hydraulic conductivity for deepergeology and calibrated three parameters using water table depth observations in North Americantemperate zones. They also calibrated permafrost parameters using observations of wetland ex-tent in Northern USA and Canada. de Graaf et al. [2015]âs global 11 km resolution simulationof WTD was produced by coupling the land-surface model PCR-GLOBWB with the groundwatermodel MODFLOW. Each pixel in their coupled model has one canopy layer, two soil layers, andone groundwater layer. Groundwater flux is forced using temperature, potential evapotranspiration,and precipitation, while land cover, soil, and topography conditions are set using globally avail-able datasets. Boundary conditions are set at large lakes and coastlines. Some hydrogeologicalparameters within the model of de Graaf et al. [2015] were chosen based on how well the simulatedhydraulic heads fit the hydraulic heads estimated from piezometer observations. In the remainderof this paper, we refer to studies conducted by de Graaf et al. [2015] as de Graafâs simulation andFan et al. [2013] as Fanâs simulation.EcoregionsEcoregions are defined as areas of the Earthâs surface that contain distinct assemblages of naturalcommunities and environmental conditions. They are a way to divide the planet into relativelyhomogeneous areas based on similarities in climate, geology, topography, vegetation, and wildlife.The United States Environmental Protection Agency (EPA) divides North America into 15 differentecoregions based on the Omernik Ecoregion Framework [Omernik, 1987]. The Omernik EcoregionFramework is widely used by scientists and resource managers to understand and manage diversenatural ecosystems [Gallant, 1989]. These 15 ecoregions are: Arctic Cordillera, Tundra, Taiga, Ma-rine West Coast Forest, Eastern Temperate Forest, Great Plains, North American Deserts, Mediter-ranean California, Southern Semi-Arid Highlands, Temperate Sierras, Tropical Dry Forest, TropicalRainforest, Western Mountains, Valleys, and Coast Arctic Plains and Mountains. In each ecoregion,we compare and evaluate (using different lines of evidence) the performance of the three machinelearning-based and two physically-based simulations of WTD. Note that the real observations ofWTD were only available within ten ecoregions, and hence in the remainder of the paper, we onlyuse the information of these ten ecoregions (Figure 4.2). Table 4.1 reports some statistics regardingthese ten ecoregionsâ climatic, topographic, and soil characteristics.944.2. Tackling water table depth modeling via machine learningFigure 4.2: The locations of ten ecoregions along which we compare and evaluate the performanceof the three machine learning-based and two physically-based WTD simulations. The dashed redarch-shaped zone referred to focus zone and was explained in Section 4.2.34.2.3 MethodsThis paper aims to produce three sets of machine learning-based simulations of WTD (at 500 mresolution) in shallow unconfined aquifers across the USA and Canada. Section 4.2.3 explains ouralgorithm to filter out real WTD observations collected in confined aquifers. We then explain theresampling algorithm used to downscale coarse climatic input attributes to our target resolution(Section 4.2.3). We explain the machine learning methodology, the physical constraints used toconstrain the machine learning models, and the calibration procedures used to develop the threemachine learning models (Section 4.2.3). We also explain how the three models were developedby gradually adding WTD proxy data to our training set (Section 4.2.3). The evaluation criteriaused to evaluate and compare the performances of the three machine learning simulations and twophysically-based simulations of WTD are discussed in Section 4.2.3.Filtering of Real WTD ObservationsThis paper particularly focuses on the simulations of WTD within unconfined aquifers. Hence,we excluded the real well observations potentially collected within confined aquifers. Here, weleveraged the observations of WTD in the United States and British Columbia (Canada) flaggedwith information on confined versus unconfined status of the aquifers where the observations werecollected. We explored whether well depth and geologic and climatic characteristics could explainthe confined versus unconfined status of the observational data. We could then extrapolate thisinformation to the other observational data where the confined versus unconfined status is unknown,but other characteristics are known. Since the ultimate aim was extrapolation, a powerful yetinterpretable model was required. Hence, we employed a non-greedy decision tree algorithm, whichprovides accurate and interpretable extrapolation, using the evtree R package [Grubinger et al.,2014]. Inputs to the algorithm include climatic attributes, such as aridity index, precipitation, and954.2. Tackling water table depth modeling via machine learningTable 4.1: Mean and median climatic, topographic, and soil characteristics of ten ecoregions alongwhich we evaluate model performance. The characteristics include long-term average Aridity index(-), Temperature (â¦C), Elevation (m), Depth to Bedrock (DTB, cm), Shallow Clay (%), and realobservations of WTD (m). The ten ecoregions are (see 4.2): Taiga (3), Northern Forests (5),Northwestern Forested Mtns (6), Marine West Coast Forests (7), Eastern Temperate Forests (8),Great Plains (9), North American Deserts (10), Mediterranean California (11), Temperate Sierras(13), Tropical Humid Forests (15). Numbers in the parenthesis show the ecoregion class in the15-class Omernik Ecoregion classification. Eco-region area is shown below the eco-regionâs class.Region Statistic Aridity Temperature Elevation DTB Shallow Clay WTD(3) Mean 0.75 -4.7 405 2331 17.1 4.12483km2 Median 0.74 -4.65 331 2382 18.1 2.47(5) Mean 0.78 1.22 404.97 2039 18.2 6.7548140km2 Median 0.76 0.98 370 1835 18.4 4.57(6) Mean 0.93 0.89 1366 1641 15.9 19.127406km2 Median 0.73 0.51 1233 1525 15.7 8.60(7) Mean 0.26 3.53 605 1723 12.4 7.9221301km2 Median 0.21 3.47 406 1628 11.9 4.57(8) Mean 0.88 13.15 205 1705 20.3 5.31122212km2 Median 0.89 13.3 181 1453 21.0 3.96(9) Mean 2.26 10 684 3908 28.1 17.0228570km2 Median 2.16 8.88 616 3755 28.3 9.14(10) Mean 6.44 13.4 1320 3659 19.5 27.776231km2 Median 5.13 12.0 1398 2376 19.7 15.0(11) Mean 3.86 16.1 433.2 1282 21.6 19.616278km2 Median 3.42 16.3 283 801 21.9 14.3(13) Mean 2.31 14.9 1935 529 23.7 40.72858km2 Median 2.01 14.5 2019 188 23.1 17.9(15) Mean 1.14 25.1 98.0 361 35.0 1.46857km2 Median 1.16 25.4 29 248 36.8 1.19snow fraction; geological attributes, such as depth to bedrock; topographical attributes, such aselevation; and well information, such as well depth. We found that aridity index (AI) and welldepth were the strongest predictors of aquifer type. For dry regions (AI > 1.48), wells at depths lessthan 241 meters are usually unconfined, while wells at depths greater than 241 meters are usuallyplaced in confined aquifers. In more wet regions (AI < 1.48), wells at depths less than 19 meters areusually in unconfined aquifers, while greater than 19 meters usually indicate a confined region. Thecross-validation metric showed that this simple yet interpretable model could identify unconfinedreal observations of WTD with over 85% accuracy, while the overall accuracy in identifying the classof real observation (confined versus unconfined) was over 73%. Although our overall accuracy isdecent and our model is both simple and interpretable, we want to note that extrapolating to areasoutside of the training set is always troublesome for data-driven models, but we believe this is thebest currently available option. This model was applied to the entire study region to label real WTDobservations (and their corresponding wells) as confined versus unconfined. Finally, we excluded allobservations labelled as confined. In the end, this led to 541,418 pixels (at 500 m resolution) of real964.2. Tackling water table depth modeling via machine learningobservations of WTD. Other criteria for aggregating point-scale to pixel-scale observations can befound in Section 4.2.2.Resampling of Climate DataThis study aimed to estimate water table depths at a resolution of 15 arcseconds (500 m at theequator). However, the continental-scale climatic input variables explained in Section 4.2.2 wereonly available at coarser resolutions, such as 360 arcseconds. The coarse-resolution climate datasetsin GIS raster formats were resampled to the 15-arcsecond target resolution using the GeographicallyWeighted Regression method (GWR). In this technique, the independent variable of the down-scaling was elevation. We used the GWR for Raster Down-scaling in the SAGA software package(Spatial and Geostatistics menu).Machine Learning Modeling of WTDMachine learning algorithm: XGBoost Decision trees can be simple yet powerful and in-terpretable models of complex data. Methods that combine a large number of decision trees intoan ensemble exhibit strong predictive capabilities and computational efficiency on tabular data[Shwartz-Ziv and Armon, 2022]. Tree-based ensemble methods include random forests [Breiman,2001b] and gradient-boosted decision trees such as XGBoost [Chen and Guestrin, 2016], CatBoost[Prokhorenkova et al., 2018], and LightGBM [Ke et al., 2017]. While random forests is the most pop-ular tree-based ensemble method in hydrology [Tyralis et al., 2019], our preliminary results showed astronger performance of XGBoost on our dataset. Other studies also showed the strong performanceof XGBoost in the simulation of groundwater systems [Osman et al., 2021]. XGBoost, the machinelearning algorithm used in our paper to simulate WTD, is centred around the sequential optimiza-tion of shallow decision trees [James et al., 2013]. A single shallow decision tree is a weak learner asit captures the broad relationships in the data with a minimal chance of overfitting. The sequentialnature of the algorithm allows XGBoost to learn slowly while correcting itself during the learningprocess. This is captured by iteratively regressing the residuals of the previously built set of shallowtrees. The learning rate and level of stochasticity can be controlled via various hyperparameters.Building an XGBoost algorithm requires robust calibration and validation of regular XGBoost pa-rameters and optimization of hyperparameters [Probst et al., 2019, Bilolikar et al., 2023]. In devel-oping each model, the (real and proxy) WTD observations are split into a training, validation, andtest set. The regular XGBoost parameters are calibrated using the training set, and hyperparam-eters are calibrated against the validation set. 20 iterations of random search were used to selecthyperparameters based on the modelâs performance (R2) on the validation set. Once hyperparam-eters are chosen, the model is retrained on both the training and validation sets and tested on thetest set. We report the performance of the three machine learning models against (real and proxy)WTD observations of the test set. The final simulation of each machine learning model is then madebased on the full model trained on all data.Training, validating and testing sets For training, validation, and testing our machine learn-ing models, we divided the observations into two groups. The first group includes pixels within anarc-shaped focus zone across the United States (Figure 4.2 shows this zone in red), and the secondgroup includes the remainder of the pixels across our entire study domain (USA and Canada). Thearc-shaped focus area encompasses a large portion of the USA and small parts of Canada, including974.2. Tackling water table depth modeling via machine learningthe most of the Eastern Temperate Forests, Great Plains, North American Deserts, and significantportions of the Northern Forests and Northwestern Forested Mountains, including the USAâs Rock-ies. It excludes the Eastern and Western coasts, large portions of Florida, Texas, California, Oregon,and most parts of Canada. This zone has been used as the study domain in large-scale WTD sim-ulations by Maxwell and Condon [2016], Maxwell et al. [2015], Ma et al. [2024], and Yang et al.[2023]. This focus zone contains a large portion (about one-third) of our real observations of WTDand includes USAâs most important groundwater-dependent ecosystems and resources developmentlandscapes while excluding some highly exploited aquifers (e.g., Californiaâs Central Valley Aquifer).Hence, accurate prediction (and extrapolation) of static WTD is both plausible (given dense WTDdata availability) and necessary within the focus zone. In our study, the training set includes 1/3(randomly selected) of the real and proxy observations within the focus-zone domain and 2/3 ofthe (real and proxy) observations from the rest of study region. The validation set includes 1/3 ofthe (real and proxy) observations within the focus-zone + 1/3 of observations from the rest of thestudy region. The test set includes the remaining 1/3 of (real and proxy) observations within thefocus-zone domain.Implementing physical constraints Groundwater systems are inherently complex. This com-plexity is influenced by various factors, including climate, land use, topography and geology, andhow they interact together, which need to be included in numerical models to effectively representgroundwater dynamics [Ojha et al., 2015]. While machine learning techniques have the potentialto capture these complexities to some extent, their flexibility can lead to models that may becomeunwieldy or difficult to interpret. To avoid this, some constraints should be forced on the internalrelationships of machine learning models to ensure logical consistency in their predictions [Bierkensand Wada, 2019]. Such constraints can also enhance the generalization and extrapolation capabilitiesof machine learning models [Janssen and Ameli, 2021].One standard course of action in machine learning modeling is to use domain knowledge to constrainthe model via physically meaningful monotonic relations among input variables and the response[Fallah Tehrani et al., 2012, Ben-David et al., 2009, Bartley et al., 2019, Cano et al., 2019]. Amonotone increasing (decreasing) relationship between a variable xj and the response y is defined asfollows: given an increase in xj , and that all other variables are held constant, the response shouldnot decrease (increase) in value [Bartley et al., 2019, Fallah Tehrani et al., 2012]. Utilizing monotonicconstraints can decrease the size of the search space of good models, ensure sensible model behaviorand ultimately increase generalizability [Gutie´rrez and Garc´ıa, 2016, Bartley et al., 2019, Cano et al.,2019]. While monotonicity is a strong assumption, it is clearly applicable across many areas of science[Liu et al., 2020a]. One prominent example of this, as suggested by Fallah Tehrani et al. [2012], isthat doctors would not trust a model where increased cigarette consumption leads to a predicteddecrease in lung cancer risk. Likewise, groundwater hydrologists should not trust a model in whichincreased precipitation from one pixel to the other, while keeping all other variables constant, leads toincreased WTD predictions. XGBoost can easily incorporate such physically meaningful monotonicconstraints [Dong et al., 2022, Bartley et al., 2019, Ovchinnik et al., 2019, Yang and Chui, 2021]. Indoing so, XGBoost rejects any split that violates the direction of monotonicity [Dong et al., 2022].Further, the mean values of a split are passed down to its children such that monotonicity cannotbe violated further down in the tree [Dong et al., 2022, Bartley et al., 2019]. This simple frameworkallows for guaranteed global monotonicity while not dramatically increasing computational cost[Bartley et al., 2019, Ovchinnik et al., 2019]. While this approach has been used extensively inother areas of science, we are not aware of any previous applications of monotonic constraints in984.2. Tackling water table depth modeling via machine learninggroundwater hydrology.Groundwater levels are influenced by various terrestrial characteristics and environmental factorsthat govern recharge processes and groundwater dynamics [Delin et al., 2007, Jan et al., 2007,Nolan et al., 2007]. A comprehensive overview of the climatic, geological and topographical factorsaffecting groundwater level and recharge rate is provided by Moeck et al. [2020]. The location of thewater table is a function of net groundwater recharge and how fast the system retains or laterallydrains groundwater [Maxwell et al., 2015]. For instance, a higher aridity index, all else being equal,cannot lead to a higher recharge rate [Brutsaert and Stricker, 1979, Moeck et al., 2020]. We knowfrom the simple mass-balance equation that spatial increases in long-term average precipitation anddecreases in evapotranspiration, all else being equal, cannot lead to a decrease in recharge rate andgroundwater storage. Similarly, a larger topographic index at a given pixel, all else being equal,implies a larger amount of water flowing toward the pixel and should not allow for deeper WTD.A larger sand fraction, all else being equal, suggests faster vertical/lateral drainage of groundwater,leading to deeper WTD. Conversely, a larger clay fraction suggests a slower groundwater drainage,leading to shallower WTD. Physically-based models also satisfy these monotonic relationships ifthey converge and meet the constraints imposed by mass balance and momentum equations.Building on the above scientific rationales, we imposed the following constraints to all three set-upsof machine learning models developed in our paper: (1) WTD must monotonically increase with alarger aridity index, and with larger deep/shallow sand fractions, and (2) WTD must monotonicallydecrease with precipitation, precipitation excess, deep/shallow clay fractions, and topographic index.Again, it is important to note that within each monotonic constraint, we technically assume thatall other variables remain unchanged. Within the XGBoost model, this implies that between twopixels with identical geology, topography and land cover, the one with a larger aridity can nothave shallower simulated WTD. Alternatively, between two pixels with identical climate, land cover,and topography, the one with a larger deep/shallow sand fraction and smaller deep/shallow clayfraction can not show shallower WTD. While such constraints can be precisely imposed on machinelearning models, the situation in which all climatic and biophysical factors are identical except forone factor may not always happen in reality. However, the imposed constraints intuitively followbasic principles from groundwater hydrology and, as explained before, are the ones that physically-based models should also satisfy. Despite their simplicity, these constraints can mitigate the risk ofequifinality of machine learning models by significantly reducing the size of the search space of goodmodel.With further advances in the algorithms of machine learning models and with the continued iden-tification of increasingly complex (nonlinear) relationships among hydro-geological factors, futureworks should be able to train more robust physically-constrained machine learning-based ground-water models. Note that the (nonlinear) functional relationships among hydro-geological factors inshaping water table depth can be derived using fine-resolution physically-based models or advancedcausal inference methodologies. While the former is a plausible near-future target, advanced causalinference methodologies able to identify complex nonlinear relationships from uncertain observa-tional data have not been developed yet. Until then, the simple yet intuitive constraints used inour paper can be considered the first attempt in groundwater hydrology to improve the physicalrealism of machine learning models. Note that during our initial experimentation, we found that themodel fit and test set R2 decreased slightly due to adding the above constraints. Ben-David et al.[2009] and Bartley et al. [2019] reported a similar condition and finding. Indeed, the unconstrainedXGBoost model may overfit to the noise and biases in the observational data; obviously these noise994.2. Tackling water table depth modeling via machine learningand biases, as will be discussed in Section 4.2.5, do not obey any expert knowledge-based monotonicconstraints [Ben-David et al., 2009].Note that to evaluate the need for the use of constraints in our models, we conducted metamorphic (orstress) testing, which is a widely-used analysis to evaluate the presence of poor extrapolation behaviorin machine learning models [Yang and Chui, 2021, Reichert et al., 2024, Wi and Steinschneider,2024, 2022]. In this test, we compare the reliability of our ML model with and without physicalconstraints, under the hypothetical scenario of increased temperature (+4 degrees) and decreasedprecipitation (-20%). A reliable model should not show shallower simulated WTD under these muchdrier conditions. The ML model without physical constraints showed shallower simulated WTDin about half of the study region pixels, while in the model with constraints, none of the pixelscan become shallower with increased temperature and decreased precipitation. Considering thispreliminary stress testing analysis (results not shown), it is clear that there is a need to incorporatephysical constraints to help the model better understand the relationships between physical/climaticattributes and WTD.Three machine learning model setups Three different sets of (real and proxy) observationsof WTD were used to optimize XGBoostâs (hyper)parameters, leading to the development of threemachine learning models and WTD simulations. In the first model (V1), we only used 541,418pixels of real observations of WTD from unconfined aquifers, with each real observation receiving aweight equal to 1. As we will see in the results section, this model could accurately simulate WTDalong the observational locations while missing excessively shallow WTD close to permanent surfacewater bodies. Hence, we introduced the second model (V2), in which 12,167,332 pixels of WTDproxy data along the semi-permanently wet shorelines of surface water bodies were added to thereal observations of WTD, in order to optimize the (hyper)parameters of the XGBoost algorithm.WTD was assumed to be zero at these pixels. These WTD proxy data received a weight equal to themagnitude of water occurrence probability (WOP) at the given pixel, which should be larger than75%. Indeed, more persistent water inundations are more likely to have a zero WTD. See Section4.2.2 for more details on delineating the semi-permanently wet shorelines of surface water bodiesand the quantification of WOP.As we will see in the results section, despite very accurate performance of V2 along and closeto (real and proxy) observational locations, V2 led to excessively shallow WTD along some high-elevation steep mountainous landscapes. This contradicts the findings of local-scale studies, whichhave consistently observed a deep WTD along high-elevation mountainous regions [Manning et al.,2013, Smerdon et al., 2009, Ofterdinger et al., 2014, Somers et al., 2019]. Indeed, due to a lack of realand proxy observations along high-elevation steep mountainous landscapes, V2 may not properlylearn groundwater processes occurring along these landscapes. Height above the nearest water body(HAND) was shown to approximate WTD [Koch et al., 2019, Nobre et al., 2011], especially in dryregions with mountainous topography [Gleeson et al., 2011, Desbarats et al., 2002]. Hence, HANDvalues in the mountainous regions of North America may offer a fair approximation of WTD inthese regions. Thus, we introduced the third model (V3), in which 2% of pixels, with HAND valueslarger than 30 meters (a total of 766,814 pixels), were randomly sampled and added to the (realand proxy) observations used in V2 (see Section 4.2.2 for more details on HAND data). Alongthese pixels, WTDs were assumed to be equal to HAND values. Note that the 30-meter thresholdrepresents the third quartile of HAND values, corresponding to the steepest and highest lands acrossthe study region. HAND data is a more uncertain proxy of WTD than the wet shorelines of surface1004.2. Tackling water table depth modeling via machine learningwater bodies, thus, a smaller weight of 0.75 was used for HAND data in V3 model development andoptimization.Interpretation of machine learning models The same climatic and physical input variables,and the same physical constraints, were used for all three models (input variables were explainedin Section 4.2.2). To interpret what the machine learning algorithm learnt, we can quantify andrank the extent to which each XGBoost model relies on each input variable. To achieve this foreach model, we computed each input variableâs gain feature importance. The split-level gain metricof a given input variable was computed as the gain in accuracy after each split belonging to theinput variable. The split-level gains were then averaged across all splits in the ensemble of trees toform the final gain metric of each input variable for each model [Chen et al., 2019]. Such a metricaims to reflect the structure that the machine learning model learnt during model development (oroptimization) and can be used to infer the extent to which such structure follows available scientificliterature relevant to WTD prediction.Model Evaluation CriteriaWe compared the performances of our three machine learning-based and the two physically-basedsimulations of WTD within individual ecoregions and across the entire study domain. The unseendata against which the modelsâ performances are evaluated include (1) unseen real WTD obser-vations (OBS) and (2) proxy data on the permanently wet interior pixels of surface water bodies(WB), wherein we expect WTD to be zero (see Section 4.2.2 for more details on the delineationof these pixels). As explained in Section 4.2.3, across the entire study region, unseen or test setreal WTD observations for the three machine learning simulations include randomly selected onethird of available real WTD observations within arc-shaped focus zone (Figure 4.2 shows this zone),which were not used in machine learning models development. Within each ecoregion, unseen realWTD observations were identified through 10-fold cross-validation. Proxy data from permanentlywet interior pixels of surface water bodies were not used in machine learning model developmentand were technically unseen for the three machine learning simulations. The two physically-basedsimulations, to some extent, relied on real observations of WTD and/or the location of surface waterbodies to obtain more optimal model parameters (see Section 4.2.2). However, to be fair to thesesimulations, in our paper, we assume that all real WTD observations and the interior pixels of sur-face water bodies were unseen by these physically-based models. Hence, for the two physically-basedsimulations, all real and proxy WTD observations were considered unseen across the entire studyregion and each ecoregion.For all machine learning and physically-based simulations, at pixels where unseen real observationswere located, we computed the mean absolute error (MAE) between simulations and real observa-tions of WTD (MAE-OBS). Similarly, along the interior pixels of water bodies wherein zero WTDis expected, we computed the mean absolute error or difference from zero (MAE-WB). The authorsof some of physically-based simulations of WTD usually argue that their simulations of WTD arebiased replicas (and not accurate simulations) of the observed WTD, as some of physically-basedmodels do not account for human intervention [Fan et al., 2013, Maxwell et al., 2015, Yang et al.,2024, 2023, Rahman et al., 2023, de Graaf et al., 2015]. Human intervention due to pumping couldsignificantly bias the observations of WTD. Hence, for all 5 different simulations, we also estimatedthe correlation between simulations and real observations of WTD (Corr-OBS) along pixels whereunseen real observations were located. In doing so, we evaluated how much each model captures1014.2. Tackling water table depth modeling via machine learningthe spatial variability of WTD without harshly punishing physically-based simulations for possiblepumping biases. In addition to comparing the performances of machine learning and physically-based models model simulations against unseen real and proxy observations of WTD, we exploredthe pixel-by-pixel statistical association among the five simulations, across the study region. We dothis by computing the correlation between each pair of simulations and visualizing it as a correla-tion matrix. Such inter-model correlational analysis can reveal similarities between two simulations,while helping us to determine output trustworthiness. Although the similarity of two model out-puts is not a direct measure of correctness, two independent labs coming to the same conclusionsis a good measure of replicability, and two models trained with different subsets of data leading tosimilar outputs shows stability. Stability and replicability are indirect measures of trustworthinessand are a lower bound on the requirements of scientific discovery/usefulness [Yu, 2013, Fonio et al.,2012, Barnett et al., 2016, Open Science Collaboration, 2015].We also evaluate the residual distribution of all five models across different quartiles of 6 inputattributes, namely aridity index, elevation, precipitation, depth to bedrock, shallow sand fraction,and deep silt fraction. This analysis can reveal how bias and variance in WTD estimation changeacross models at different levels of climate, soil, and topographic attributes.In order to accurately integrate different datasets of input variables and compare different simu-lations, we should ensure that all pixels and geographical points of datasets and simulations areperfectly aligned. Different datasets can sometimes be slightly offset due to differences in the un-derlying reference systems or grid resolution. In this study, we adjusted the alignment of all inputdatasets to machine learning models as well as two physically-based simulations, in a manner similarto Tootchi et al. [2019]. Pixel locations and alignments along the Pacific and Atlantic Oceans shore-lines as depicted in the SRTM+ dataset (geographic coordinate system: World Geodetic System 84- WGS 84 - EPSG:4326) were considered as the reference spatial pixels and system. Then, if needed,we aligned the oceansâ shoreline locations in all datasets of input variables and two physically-basedsimulations. For instance, Figure 4.3 depicts that after re-alignment, our machine-learning simula-tion and two physically-based simulations similarly identified the location of a small lake as well asits shoreline (pixels with close to zero WTD).Figure 4.3: Simulations of Degraafâs (right), Fanâs (middle) and our V3 (left) after the alignmentprocess in the lowland area north of Lake Saint Clair. All three simulations identify the locationof Lake Saint Clair and itâs shorelines (with close to zero WTD) at similar locations. Shorelinesare simulated at finer resolution using our V3 model than Fanâs and Degraafâs simulation, due todifferences in grid resolution.1024.2. Tackling water table depth modeling via machine learning4.2.4 ResultsDescription and Comparison of the Spatial Patterns of WTD SimulationsOverall, the three machine learning simulations consistently show shallower WTD on the easternportion of the continent of North America than the western portion (Figures 4.4a to 4.4c). However,the WTD simulations of the three machine learning models exhibit significant differences at localscales. V1 could not reasonably predict WTD close to (and within) lakes and rivers, particularlyalong lakes in Canadaâs North and Prairie Pothole Region, where WTD was predicted to be around10m. In contrast, V2 accurately predicts that WTD should be near zero close to (and within) lakeand river locations. However, V2 also leads to excessively shallow (less than 0.1 m) WTD along alarge portion of Canada (with the exception of the Canadian Prairies, South-eastern Canada andcentral British Columbia), while V1 is more conservative with predictions of 1-5 m in such locations.Regardless, the overall spatial patterns in V1 and V2 are similar, and these two simulations are highlyspatially correlated (0.78), while V3 has a negligible spatial correlation with two other XGBoostmodels (0.11-0.16) (Figure 4.5).One notable distinction across the three machine learning models occurs along the Canadian RockyMountains, where V2 shows very shallow groundwater simulations, while V3 estimates the depthto be deeper than 50m along high-elevation steep lands, and V1 predicts a WTD of up to 10m.Furthermore, across the Canadian Rocky Mountains, WTD is highly spatially variable in V3, fol-lowing the large spatial variations in elevation and slope, while V1 and V2 predict fairly spatiallyuniform WTD. Unlike V1 and V2, V3 follows the general patterns and magnitudes of Fan and deGraafâs simulations across the Canadian Rocky Mountains. Similarly, across the USAâs Appalachia,V3 clearly distinguishes between the Appalachian Mountains of the eastern USA and the rest ofthe eastern USA. While V1 (V2) shows a fairly uniform distribution of WTDs in this region withno clear spatial pattern, V3 follows the general patterns and magnitudes of Fan and de Graafâssimulations with highly locally variable WTD. Across the USAâs Appalachia, V3âs WTD is deep(> 50m) along steep, high-elevation landscapes, while the rest of the eastern coast has distinctlyshallower WTDs at 0-10m. Another major distinction occurs in the large portions of Canadaâs Northsurrounding Hudson Bay. In V1, this region has WTDs ranging from 1 to 20 meters, similar to deGraafâs physical-based simulation, whereas V2 predicts almost the entire region to have around zeroWTD. While V3 predicts a large portion of this region with nearly zero WTD, some local areas havemedium to high WTDs. V3 in this region follows Fanâs simulation in terms of the spatial patternof WTD. Finally, all three machine learning models, in line with Fan and de Graafâs simulations,predict that the southwestern deserts of the USA have extremely deep water table depths (> 70 m).The overall spatial patterns of de Graaf and Fanâs simulations are similar, although Fanâs simulationhas much greater local-scale spatial variability. This weakens the spatial correlation between deGraaf and Fanâs simulations (0.33). In terms of the magnitude of WTD at the regional scale, Fanâssimulation, in line with V3, shows that the lower Mississippi River basin, the coastal region fromsouthern Florida to New Jersey, the midwest United States, and northeastern Canadian wetlandshave generally shallow WTDs (< 5m). At the same time, local areas of deep WTDs also exist alongmountainous ridges of the Rockies and Appalachia for both Fan and V3 simulations. Generally,Fan and V3 exhibit high local variability, which arose from simulating the phreatic surface as anextremely damped version of local topography, particularly in mountainous regions (i.e., BritishColumbia). This pattern matches with most conceptual diagrams of the water table along a hillslopecross section (i.e., see Figure 1 of Moreno et al. [2016], Figure 1 of Zang et al. [2018], and Figure1034.2. Tackling water table depth modeling via machine learning1 of Rahman et al. [2023]). The equilibrium seeking physical equations of Fan and the addition ofHAND observations in V3 leads to these models predicting shallow WTD< 5m in the valleys anddeep WTD> 100m on the mountain ridges. Indeed, WTD strongly follows local-scale elevationaldifferences in both Fan and V3 simulations.(a) XGBoost - V1(b) XGBoost - V2(c) XGBoost - V3(d) de Graafâs Simulation(e) Fanâs SimulationFigure 4.4: Five different simulations of water table depths across the US and Canada using machinelearning models (a-c) and physically-based models (d-e).Comparison among five simulations Analysis of pixel-by-pixel spatial correlation betweeneach pair of WTD simulations, using all pixels of the study region, depicts there is generally aweak spatial correlation among the two physically-based simulations of WTD (r â 0.33) across theUSA and Canada (Figure 4.5). There is also a weak spatial correlation between the two physically-based simulations of WTD and the V1 and V2 machine learning-based simulations (0.11 < r <0.24). Interestingly, our V3 simulation is strongly correlated with both physically-based simulations(0.52 < r < 0.61). V1 and V2 are highly correlated with each other (r = 0.78), but are lesscorrelated with V3 (r = 0.16, 0.11). Such correlations are supported by the continental-scale spatialdistribution of simulated WTDs shown in Figure 4.4 and the corresponding comparisons made inthe previous subsections.Shallow Groundwater Areas (SGAs) in North America exhibit uneven distribution across latitudes,driven by complex interactions among climatic, hydrologic, and geologic factors (Figure 4.6). Astrong concordance is observed between the results of V3 and Fan, particularly in sub-polar latitudes.Among the simulations, V1 and DeGraaf display the smallest SGAs, while V2 shows the largestextent.1044.2. Tackling water table depth modeling via machine learningFigure 4.5: Pixel-by-pixel spatial correlations among the physically-based and machine learningWTD simulations.Figure 4.6: Comparison of the distribution of areas with water table depths (WTD) less than 1 m(left) and less than 5 m (right) across three machine learning (ML) and two physically-based WTDsimulations. The x-axis represents latitude in 1 degree increments, while the y-axis shows the totalarea of pixels with shallow WTDs within each 1 degree latitude band.Interpretation of Machine Learning ModelsThis section interprets the importance of the input variables within each of our three machinelearning models (Figure 4.7). V1 strongly relies on climatic features such as temperature andaridity index to explain the variability of real observations of WTD, with both being responsiblefor over 10% of the model performance. In total, climatic features have a part in over 60% of V1âsperformance, and topographic features explain more than 25%, with close to 15% of the impactcoming from elevation. V1âs simulation and the derived variable importance are highly consistentwith Ma et al. [2024] machine learning simulation of WTD across a large portion of the USA. Landcover and soil properties make up the rest of the model. While land cover and sand (and clay)1054.2. Tackling water table depth modeling via machine learningrelated fractions contribute the least (< 5%), shallow and deep silt fractions and depth to bedrockplay moderately strong roles (5â10%). In V2, the aridity index greatly impacts model performance(> 10%), while almost all other climatic, topographic and soil-related properties have uniformlymoderate (5â 10%) effects on V2 model performance.V3 is almost exclusively reliant on topographic variables (â 70% in total), such as topographic index(â 37%), slope (â 15%), and elevation (â 17%). In total, climatic features explain only 15% of themodel, with January temperature having the largest impact (â 5%), while V3 has little (less than5%) reliance on snow fraction, precipitation, precipitation excess, maximum snow water equivalenton the ground, and rainfall intensity. V3 only slightly relies on sand/silt/clay fractions (less than5%), while it is moderately reliant on depth to bedrock (â 5%) (Figure 4.7).Overall, V1 and V2 rely on topographic variables (topographic index, slope, and elevation) at aboutthe same level, while V3 is over twice as dependent on these variables. V1 (V3) is strongly (weakly)reliant on climatic information, while V2 gives almost all input variables a more or less uniformscore (Figure 4.7). Land cover does not seem to be a useful variable across all models. Soil-relatedfeatures such as sand and clay fractions are only given relatively high importance by V2. The starkdifferences in simulating WTD in mountainous areas between V1 (or V2) and V3, as explained inSection 4.2.4, likely have to do with V3 being strongly driven by local topography instead of climate,while V1 (or V2) is strongly driven by climate. Note that the variable importance metric used in thisstudy is only meant to interpret each model, not the underlying physical processes. Thus, from thisanalysis, we can only conclude how heavily each model uses each variable to explain the variability oftarget observations, not how strongly each variable is related to WTD in general (see Janssen et al.[2023] for details). This important dichotomy is illustrated by the fact that each model gives vastlydifferent importance scores and the ground-truth importance scores are currently unknown. Further,it is important to note that these importance scores are only related to the spatial heterogeneity ofWTD, and it does not consider temporal variation.Relative Comparisons of Models Predictive PerformancesThis section compares each modelâs ability to predict WTD at pixels wherein unseen real and proxyobservations are located within each ecoregion and throughout the study region. Mean absolute error(MAE) and Pearson correlation were used as performance metrics to evaluate model performance inpredicting observed or expected WTDs at unseen locations. A summary of predictive performanceresults for each ecoregion and the entire study region can be found in Table 4.2.Along the Temperate Sierras (13), MAE-WB values are highest, meaning all models perform rela-tively poorly (MAE-WB > 2) in predicting permanent water body locations. This ecoregion exhibitsthe largest elevation across the study region (median of 2019m). All models, except V2, performpoorly in predicting permanent water bodies along the North American Desert (10), which has thehighest aridity among all ecoregions and the second-highest elevation. Again, all models, except V2,perform poorly in predicting permanent water bodies in Mediterranean California (11), which hasthe second highest aridity. In the Eastern Temperate Forest (8), with the second lowest average ele-vation, V2 and V3 perform exceptionally well (MAE-WBâ 0.1), and Fan (MAE-WBâ 0.3) performsvery well, in predicting small water bodies. Similarly, in the Great Plains (9) and Northern Forest(5), V2 and V3 perform exceptionally well and Fan performs very well. Along the NorthwesternForested Mountains (6) and Marine West Coast Forest (7), V2 is the only simulation with a strongperformance in predicting permanent water bodies (MAE-WBâ 0.1). Unsurprisingly, the region1064.2. Tackling water table depth modeling via machine learningFigure 4.7: Relative importance of each input variable within each machine learning model (seeSection 4.2.3 for the details on the derivation of importance score). The importance scores across asingle model add to one.with the lowest average observed WTD (i.e. the tropical humid forests of Florida (15)) also hasthe lowest error in locating permanent water bodies across all models, with V2 and V3 performingexceptionally well (MAE-WB â 0). Throughout the entire study region, our V2 (MAE-WB=0.1)simulation best predicts the unseen interior locations of permanently wet surface water bodies.V3 (MAE-WB=2.42) and Fan (MAE-WB=2.54) simulations also perform relatively well, while V11074.2. Tackling water table depth modeling via machine learning(MAE-WB=5.38) and de Graaf (MAE-WB=15.36) simulations are relatively poor in this respect.Table 4.2 clearly indicates that the three machine learning-based simulations provide stronger core-lation between simulated and observed WTD (Corr-OBS between 0.6-0.75) compared to the twophysically-based simulations (Corr-OBS between 0.21-0.4). Within ecoregions, both MAE and corre-lation metrics depict that the three machine learning-based simulations perform better in predictingunseen real observations of WTD, except in Taiga (3) with a small number of observations, wherethe Fan simulation performs best. Although they are never better than all ML models, except inthe case of Taiga above, the simulations of de Graaf et al. [2015] and Fan et al. [2013] sometimesperform better than one set-up of machine learning model. For example, in terms of MAE of waterbody locations, Fan simulations outperform V1 in almost all regions and Fan outperforms V3 forthis metric in Taiga, Marine West Coast Forests, and Eastern Temperate Forests. Further, in thesparsely observed region of Taiga, both de Graaf and Fan outperform V3 in terms of correlation withobservations. In predicting WTD at unseen wells across the entire study region, V1 performs thebest among all models with a correlation of 0.75 and MAE of 7.81. This is not surprising since it isthe only simulation exclusively driven by real observations of WTD. Although most of V2âs trainingdata comes from proxy observations (and not real observations of WTD), its performance whenpredicting unseen real observations degrades very little, with a correlation of 0.71 and an MAE of8.18. The errors and accuracies among these two models and within each ecoregion also correspondclosely. V3, generally, performs worst among the three machine learning models in predicting realobservation of water table depth while still performing better than the two physically-based models.The residuals (i.e. Observed - Predicted WTD) from de Graaf et al. [2015] are substantially largerin magnitude and also more heavily biased (predictions are too deep) compared to those of Fan, V1,V2, and V3 (Figure 4.8). Across most quartiles of input attributes, there are very few differences inthe residuals between V1 and V2. V3 generally has slightly higher error compared to V1 and V2,while Fan generally has higher error compared to V3.In terms of specific predictor variables, Figure 4.8 shows that aridity index, elevation, and precipi-tation are strong predictors of residual magnitudes. In particular, WTD errors tend to be larger asaridity or elevation increase. Across all models, as regions become drier (from the fourth quartileQ4 to the first quartile Q1 of precipitation), the errors between modelled WTD and real well obser-vations tend to increase. Higher depth to bedrock (comparing Q1 with Q4) seems to be associatedwith higher errors in the machine learning-based models and Fanâs simulation. Higher shallow sandfraction tends to lead to slightly higher errors in all models.4.2.5 DiscussionThis paper presented three physically constrained machine learning (XGBoost) simulations of WTD.The three models (V1, V2, and V3) were built by sequentially adding proxy observations relevantto WTD (see Section 4.2.3). Unseen real and proxy observations of WTD, which were not usedfor training and validation of machine learning models, were then employed to explore and com-pare the performances of the three machine learning simulations and two physically-based availablesimulations of WTD across ten ecoregions of North America. Overall, all five models poorly pre-dict WTD across three ecoregions with the highest aridity and elevation (Temperate Sierras, NorthAmerican Desert, Mediterranean California). In the rest of the USA and Canada, along with thelocations with available proxy data on surface water bodies V2, V3, and (in some eco-regions) Fanâssimulation perform relatively strongly. Where real observations of WTD exist, V1, V2, and V31084.2. Tackling water table depth modeling via machine learningTable 4.2: The performance of each model on âunseenâ real observations of water table depth(OBS) and unseen interior pixels of permanently wet water bodies (WB) across the entire studyarea (All) and within each ecoregion. The performance is evaluated using mean absolute error ofsimulated WTD at interior locations of permanently wet surface water bodies wherein zero WTDis expected (MAE-WB), mean absolute error of simulated versus observed WTD (MAE-OBS), andthe correlation between simulated and observed WTD (Corr-OBS). Ecoregions include: Taiga (3),Northern Forests (5), Northwestern Forested Mtns (6), Marine West Coast Forests (7), EasternTemperate Forests (8), Great Plains (9), North American Deserts (10), Mediterranean California(11), Temperate Sierras (13), Tropical Humid Forests (15). The best result of each row is boldedand both MAE metrics are measured in meters.Region Real Obs Eval Fan de Graaf V1 V2 V3(3) 483 MAE-WB 2.53 49.23 5.58 0.04 2.83MAE-OBS 3.93 29.19 7.38 6.69 28.71Corr-OBS 0.18 0.31 0.50 0.43 0.05(5) 48140 MAE-WB 0.91 17.07 3.90 0.06 0.32MAE-OBS 8.54 30.62 3.96 4.64 6.10Corr-OBS 0.03 0.09 0.54 0.51 0.29(6) 27406 MAE-WB 4.56 91.04 6.62 0.09 5.52MAE-OBS 21.87 152.07 11.83 12.20 16.26Corr-OBS 0.18 -0.07 0.73 0.71 0.47(7) 21301 MAE-WB 9.90 194.36 5.09 0.10 11.97MAE-OBS 11.97 65.69 5.02 5.02 6.95Corr-OBS 0.13 0.05 0.60 0.60 0.38(8) 122212 MAE-WB 0.30 3.66 3.04 0.10 0.10MAE-OBS 7.16 23.02 2.73 3.25 4.22Corr-OBS 0.18 0.09 0.62 0.52 0.28(9) 228570 MAE-WB 0.79 15.24 4.96 0.12 0.38MAE-OBS 13.33 39.20 8.62 8.47 9.22Corr-OBS 0.38 0.20 0.70 0.69 0.64(10) 76231 MAE-WB 8.12 139.18 9.30 0.53 4.67MAE-OBS 23.36 94.55 14.28 15.32 17.25Corr-OBS 0.37 0.14 0.73 0.69 0.55(11) 16278 MAE-WB 7.76 116.09 9.55 0.65 4.57MAE-OBS 14.74 72.41 9.10 8.68 12.14Corr-OBS 0.24 0.08 0.72 0.71 0.49(13) 2858 MAE-WB 9.96 202.93 22.33 2.19 9.70MAE-OBS 39.20 479.42 29.21 29.21 29.76Corr-OBS 0.10 -0.02 0.57 0.57 0.19(15) 857 MAE-WB 0.29 1.93 1.67 0.05 0.03MAE-OBS 1.46 1.36 0.96 1.31 1.31Corr-OBS 0.09 0.10 0.12 0.26 0.09All 541418 MAE-WB 2.54 40.78 5.38 0.10 2.42MAE-OBS 13.47 52.90 7.81 8.18 9.61Corr-OBS 0.40 0.21 0.75 0.71 0.601094.2. Tackling water table depth modeling via machine learning(a) Aridity Index(b) Elevation(c) Precipitation(d) Depth to Bedrock(e) Shallow Sand Fraction(f) Deep Silt FractionFigure 4.8: Distribution of residuals (Observed - Predicted WTD) across four quartiles of selectedglobally available inputs and across the five machine leaning and physically-based models.perform relatively stronger than physically-based models in following the variability or magnitudesof well-based observations. The interpretation of the three machine learning models suggested thatthe V3 model strongly used the topographic wetness index (TWI) information to simulate WTD,while V1 strongly relies on climatic information. In this regard, V1 resembles Ma et al. [2024]âsmachine learning simulations of US WTD, which used random forests trained on only well-basedobservations of WTD.One may argue that the information that V3 learnt from the data and used to predict WTD1104.2. Tackling water table depth modeling via machine learningcorroborates well with classical physically-based models (e.g., TOPMODEL [Beven et al., 2021]),in which TWI dominantly controls the spatial pattern of WTD. Recent hillslope-scale experimentalworks in both wet and arid mountainous landscapes also suggested that TWI dominantly controls thelocation of WTD from the hillslope ridge line (where TWI is small and WTD is deep) to the hillslopetoe (where TWI is relatively large and WTD is shallow) (e.g., van Meerveld [2024], Karlstrom et al.[2023]). Among the three machine learning simulations, V3 not only somewhat strongly follows WTDobservations and predicts interior wet pixels of surface water bodies, in most of North Americaâsecoregions, it also generates predictions which are highly spatially correlated with simulations fromFanâs and de Graafâs physically-based models, particularly along mountainous regions. Indeed, V3provides a good balance between what (real and proxy) data dictates and what physically-basedmodels suggest. However, both data and model structures, are uncertain and difficult to verify.These uncertainties will be critically discussed below, guiding future works on data collection andcompilation as well as model development and verification in groundwater hydrology.Uncertainty in The Observations of WTDTo assess the reliability of different WTD simulations, we must rely on historical observations ofwell-based water levels. However, these real observations may have associated errors and, undersome circumstances, may not represent the water table depth meaningfully [Taucare et al., 2024].This is particularly important as the model evaluation statistics can depend strongly on the sampleof observations on which one chooses to evaluate the model (see Reinecke et al. [2020] and Table4.2). In the remainder of this subsection, we explore uncertainties related to the real observationsof WTD and how they can affect the simulations of WTD and the verifiability of those simulations.Observations of WTD are susceptible to systematic errors due to vertical hydraulic gradients [Jasechkoand Perrone, 2020]. It is fairly well known that water tapped within confining layers can experienceadditional pressure heads, leading to extremely shallow measurements of well water levels. In thesesystems, the vertical gradient would be upwards, and the true water table would be far deeper thanwhat the readings from well observations convey [Vincent et al., 2013, Hilton and Jasechko, 2023].Very few well observations have indicators (and labels) of whether or not a confined or unconfinedaquifer is tapped, further adding to the uncertainty of observations of WTD. The upward verticalgradient issue can even arise in unconfined aquifers, especially in the valley bottoms of steep terrains[Hilton and Jasechko, 2023]. Furthermore, downward hydraulic gradients could be common in somelandscapes [Elc¸i et al., 2003, Jasechko and Perrone, 2020, Gabrielli et al., 2018], where the WTDcould be measured far below the true water table. This can happen due to fairly independent causessuch as long well screens [Elc¸i et al., 2003]. Theoretical and numerical simulations conducted byElc¸i et al. [2003], as well as an experimental study conducted in California by Jasechko and Perrone[2020] depicted this phenomenon. Indeed, well levels are only a good representation of the water ta-ble when the well screen is placed at (or close to) the true water table or where perfectly hydrostaticconditions exist [Vincent et al., 2013, Jasechko and Perrone, 2020].Human error could also cause some types of data uncertainty. Across our study region, severalreal WTD observations are unreliable. Over 100 USGS observations showing a water level deeperthan the deepest well ever drilled in North America and thousands of USGS observations showingnegative WTDs. Various other obvious errors have been detected in other works as well [Jasechkoand Perrone, 2020]. While these detected observations are wrong and should be removed beforetraining or validating a model (as was done in our paper), other observations may be equally wrong,but impossible to detect and remove. In addition to large human errors, which may be present in a1114.2. Tackling water table depth modeling via machine learningminority of observations, random observational errors can occur everywhere [Vincent et al., 2013].For example, Silliman and Mantz [2000] found that noise in water level measurements is high enoughto prevent the estimation of the vertical hydraulic gradient.Pumping has greatly affected groundwater levels across the US, especially in the last hundred years[Hilton and Jasechko, 2023, Luczaj et al., 2017]. Groundwater pumping can have an outsized im-pact on lowering the water table in warm, low-elevation landscapes due to the higher likelihood ofagricultural uses [Fan et al., 2013, Gleeson et al., 2021]. Pumping can affect WTD observations nearand in the surrounding areas where pumping occurs. The observations collected near pumping areasdo not necessarily reflect the natural and permanent WTD [Jasechko and Perrone, 2020]. Withdata from California, Jasechko and Perrone [2020] found that pumping can cause drawdowns of 10sof meters, and groundwater level recovery after overexploitation often takes decades or centuries inarid regions and may even be permanent [Van Camp and Walraevens, 2009, Sommer and Froend,2011]. Even after the use of a well has ceased, an artesian well may continue to extract groundwaterand affect observations of WTD [Hilton and Jasechko, 2023].Temporal variability of the water table may hinder our ability to identify static WTD at a particularlocation [Mole´nat et al., 2005]. A large portion of real observations in the USA and Canada (â¼ 81%)were taken at one time, with no follow-up observations at that location. Water levels can fluctuatedue to long-term or oscillatory changes in climate, tides, runoff, barometric pressure, mass loading,or pumping [Condon et al., 2021, Rust et al., 2019, Rasmussen and Crawford, 1997, Flickinger andMitchell, 2020, Mole´nat et al., 2005, Hayashi et al., 2016, Jasechko et al., 2024]. Appraisal of USGSdata depict, for example, in a well near Seco Creek, Texas, the WTD fluctuated by over 50 meters inless than two decades of observations. For a USGS groundwater well in San Bernardino, California,yearly oscillations occur with an amplitude of nearly 10 meters, with an additional decreasing trendfrom about 30 meters in 2005 to 60 meters in 2020. Such temporal variation of WTD could alsohappen in regions with typically shallow WTD, such as the prairie pothole region of central USAand Canada. In this region, the water table may fluctuate by over 6 meters in less than a decade,even without pumping influences, as exemplified by Hayashi et al. [2016]. In aquifers worldwide,long-term trends of 10 m to over 100m in observed WTD can be caused by excessive pumping[Jasechko et al., 2024, Lopes et al., 2006]. Even when we exclude pumping effects, natural seasonalvariations were shown in Nevada to vary WTD by about 6m [Lopes et al., 2006].Finally, and perhaps most importantly, large-scale sampling biases across the USA and Canadahave affected the overall spatial pattern of available observations of WTD [Reinecke et al., 2023].Hydrologists and engineers typically dig wells where there is likely water (i.e., river valleys andproductive aquifers). If a well is dug to some maximum acceptable depth and no water is found, noWTD measurement can be taken. Hence, deep WTD observations are not widely available acrossNorth America. Furthermore, the WTD observation dataset can be biased towards observationsin low-elevation areas with large populations and moderate climates instead of dry (or wet) high-elevation areas [Boerman, 2022, Lopes et al., 2006]. Therefore, the overall distribution of observationsof WTD, available for modeling studies, may be shallower than the actual distribution of WTD acrossthe USA and Canada [de Graaf et al., 2015].Note that the need for high quality data is not limited to development of machine learning mod-els, as the mentioned data uncertainties can equally affect the verifiability of physically-based andmachine learning models. We further discuss the future needs and directions to access high qualityobservational or proxy data of WTD in Section 4.2.5.1124.2. Tackling water table depth modeling via machine learningUncertainty in Machine Learning-based simulations of WTDThe rapid adoption of machine learning within hydrology has led to several important advancementsdue to these algorithmsâ ability to easily model and predict hydrologyâs complex and non-linearprocesses. Yet, several shortcomings still remain for groundwater modelling using machine learningframeworks, as evidenced in our paper by the lack of strong fit among models as well as the lackof strong fit between simulations and observations of WTD. The three primary shortcomings arethe overreliance on observational data, equifinality of model structure, and the inability to directlymodel physical processes [Istalkar et al., 2023, Beven, 2002]. While all these shortcoming are alsopartially relevant to large-scale physically-based simulations of WTD, here we mainly focus on theireffects on machine learning model development in groundwater hydrology.The three machine learning-based groundwater models analyzed in our work are based on the samepowerful machine learning algorithm, XGBoost, and the same set of physically-based constraints.Yet each model provides locally distinct simulations of WTD (Figure 4.4). Furthermore, the modelperformance against unseen real observations of WTD is not as strong as the performance obtainedin other areas of hydrology (e.g., streamflow prediction). Using similar tree-based machine learningmethods, Koch et al. [2019] and Koch et al. [2021] found that the highest R2 against unseen obser-vations they could achieve across Denmark was about 0.55. Further, Ma et al. [2024] showed thatrandom forests, could obtain 0.65 test set NSE with respect to log-transformed WTD along a largeportion of the USA . Our test set R2 (NSE) against unseen real observations of WTD across theentire USA and Canada was quite similar for V1 (0.56) and V2 (0.52).Typically, stronger performance against observational data is preferred, and systematic biases andlarge random errors in predictions suggests that the developed model misrepresented the ground-water system. However, due to the considerable uncertainties of the observational data (see Section4.2.5), it is unclear if the models that make the best predictions against real observations of WTD(V1 and V2) are truly the best. Experimental and modeling studies revealed WTD is usually ex-tremely deep across steep (and wet) portions of the Rocky Mountains of Canada, in line with thesimulations made by V3 [Chen et al., 2020b, Fan et al., 2013, de Graaf et al., 2015, Smerdon et al.,2009]. Yet in the corresponding ecoregions (6 and 7), the performance of V3 is worse than V1 andV2 (but still better than the physically-based simulations), if one considers the real observationsof WTD as ground truth (Table 4.2). Without well-regulated observations that reflect the trueWTD, we, as groundwater hydrologists, will forever be blind to the ground truth. With many pa-rameterization (or model structure) options available with machine learning algorithms (similar tophysically-based models), only reliable observations can help us choose the correct path. If we knewthe exact physical equations driving the water table location at a coarse scale (e.g., 500 m), knewaccurate subsurface parameters at all locations, and could engineer a model that perfectly incor-porates these equations and parameters without simplification, the reliance on observational datawould be less, and one could ignore biased training data. But clearly, these coarse-scale ground-truthequations and parameters have yet to be found.Equifinality can have a profound negative effect on the trustworthiness of machine learning simula-tions. During hyperparameter tuning and training, we noticed that there are likely many XGBoostparameter sets spanning wide ranges of the parameter and hyperparameter space, with satisfactoryperformance on the validation/test set. Indeed, as other studies showed, the most complex and leastinterpretable models often have the best predictive capability [Nearing et al., 2021, Yang and Chui,2021]. To manage equifinality, one solution is to add physical realism by adding physical constraints1134.2. Tackling water table depth modeling via machine learningto machine learning algorithms, as we did in our paper. However, this practice reduced our modelsâpredictive capability, as noted at the beginning of this section and as shown in other studies aswell [Yang and Chui, 2021]. One possible explanation is that adding physical realism to part of amachine learning model, in the form of monotonically constraining relationships of some variables,could diminish the physical realism of other more uncertain parts of the model as exemplified inYang and Chui [2021]. To properly constrain machine learning models, a better understanding ofall physical (potentially complex and nonlinear) relationships among inputs and output data and/ormore diverse observations relevant to different processes connecting the input to output is needed[Beven, 2002].Machine learning models can far âoutperformâ physically-based models on test set data (as shownin our paper and other hydrology related disciplines , e.g., Nearing et al. [2021]). However, giventhe high sampling biases of our real observations of WTD and observational data uncertainties (seeSection 4.2.5), across unmeasured or unmeasurable locations, machine learning is still not guaran-teed to âphysically outperformâ fine-resolution physically-based models. Although we attempted toenhance the physical realism of our machine learning models through the use of additional proxyobservations and physics-based constraints, we still expect future works could further enhance thephysical realism of machine learning models in groundwater hydrology. We will discuss these pointsfurther in the next section.Future DirectionsCurrently, our knowledge about water table locations is highly uncertain and stored in hearsay andnewspaper reports [Jasechko and Perrone, 2020]. This is mainly because potentially uncertain andequifinal machine learning (or even physically-based) models cannot be verified with potentiallyuncertain (and non-representative) observations of WTD. Here, we discuss future directions for im-proving the physical realism and verifiability of machine learning models in groundwater hydrology.The physical realism of machine learning models could be enhanced by incorporating known physicallaws of groundwater hydrology (e.g., Darcyâs Law). However, there is no guarantee that includingthese laws would verifiably increase the physical realism of models at the scale of interest. This is alsotrue for the physically-based models where laboratory-scale physical laws may not control pixel-scalegroundwater processes. How climate, topography, and geology interact at local and regional scalescould have an outsized effect on the pixel-scale WTD. Indeed, groundwater flow is not only drivenby the landscape attributes at a given location but also by that of surrounding areas [Grabs et al.,2009]. For example, blowing snow, runoff over frozen ground, and lateral groundwater movementhave pronounced causal effects on groundwater levels several kilometers away [Hayashi et al., 2016].Yet, such complex and spatially-varying interactions and the functional relationships connectinglandscape attributes (locally and regionally) to WTD are understudied and remain unincluded incurrent machine learning models. Once such interactions and functional relationships are defined,they could be used to select among many candidate models or incorporated a priori in machinelearning models.Reliable verification among competing machine learning models could be made possible by testinghow competing groundwater models simulate known causal relationships between the drivers ofWTD (e.g., precipitation, topographic index) and WTD (or hydraulic gradient). This can be doneusing metamorphic testing if we know a priori the function and form of causal relationships amongthe drivers and WTD [Yang and Chui, 2021]. The monotonic physical constraints used in our paper1144.2. Tackling water table depth modeling via machine learning(see Section 4.2.3) are a simple form of such causal relationships. However, in reality, the functionalforms of such relationships at local and regional scales can be more complex than what we usedin our paper. Hence, future experimental or statistical inference studies could focus on exploringthe functional form of such causal relationships at different scales. Another alternative to verifymodels could be the use of virtual experimentation wherein the competing machine learning modelsvirtually simulate land use and climate alterations to choose which model more reliably replicatesknown behaviour [Kirchner, 2006].Process-oriented indices, data, and markers with information relevant to WTD can improve machinelearning-based groundwater models through different pathways. They can be used as (1) input vari-ables to the model, as done for the topographic index in our paper, (2) proxy observations for modeltraining, as done for the HAND and the probability of surface inundation in our paper, (3) modelconstraints or boundary conditions, or (4) as an (in)direct verifier to select the model structurethat makes good predictions for the right reasons. For example, previous works showed that theincorporation of information on the landscapeâs hydrologic functioning (i.e. how landscapes storeand transmit groundwater), through novel scientific indices, could enhance the generalizability ofmachine learning-based groundwater-surface water interaction models [Janssen and Ameli, 2021].Topography-based indices and satellite data, reflecting how landscapes have evolved, can be derivedat fine resolution and ultimately can be used to enhance the capabilities of machine learning mod-els. Indices such as height above the nearest drainage or water body (HAND), horizontal distancefrom the water body, and topographic index turned out to be the most important topography-baseddrivers of WTD in previous studies [AËgren et al., 2014, Gabrielli et al., 2018, Koch et al., 2019,2021, Grabs et al., 2009]. Distance to hydrogeologically significant features such as alluvial-fanor consolidated rock may also be important [Lopes et al., 2006]. One could also go beyond oursimple monotonic constraints by more closely coupling physically-based and machine learning mod-els. Perhaps there is a way to incorporate differentiable modelling framework [Shen et al., 2023]within groundwater modelling by increasing the flexibility of rigid groundwater flow laws, or byconstraining data-driven methods to only allow for information/groundwater to flow according topressure gradients. Future works may also test the reliability of laboratory-scale physical laws atcoarse pixel-scale using differentiable modelling by comparing these laws and hypotheses with therelationships learned by physics-informed machine learning models [Wang et al., 2022]. Finally, newlaws that govern groundwater flow within undisturbed soils at larger scales may be discovered usinggeophysical imaging and system identification techniques [Mitchell et al., 2011, Terpin et al., 2024,Guan et al., 2024].Alternative topography-based indices such as upgradient recharge [Lopes et al., 2006], regionalgeomorphic indices [Ali et al., 2014], convergence index, and river density [Naghibi et al., 2020],could be tried in future machine learning models since they can have emergent process-orientedcausal relationships with WTD. For example, Forster and Smith [1988] showed that the WTDpattern is highly sensitive to the concavity or convexity of hillslopes. Certain ecological markerscould also be associated with WTD and constraining groundwater models [Sommer and Froend,2011]. For example, in Nevada, Lopes et al. [2006] noted that the Greasewood shrub species cannotgrow unless the water table is less than 15 meters deep. Further studies on how different speciesuptake water and in-depth remotely sensed plant location data could revolutionize our knowledgeof WTD. With the massive increase in the availability of satellite data, which can capture thegroundwater-dependent features on the land surface at a fine spatial-temporal resolution, futureworks could further leverage these data to enhance the physical realism and overall strength of1154.2. Tackling water table depth modeling via machine learninggroundwater models. While the use of a suite of input indices, millions of pixels of data, andemergent constraints could be straightforward for machine learning models (as done in our paper),such information may be difficult to utilize in physically-based models.The approaches stated in this section could help us inch closer to reliable WTD estimations, butthe bottom line is that we need more reliable and trustworthy observations of WTD which areprecise and truly representative of the water table at a given place. Alternatively, some additionalinformation which could guide us on the usefulness of the real observations, such as labelling confinedversus unconfined observational wells as was used in our paper (see Section 4.2.3), may increasecertainty. Furthermore, future works could incorporate water usage datasets to help predictionsfollow the natural drivers of WTD while also accounting for or removing pumping biases [Huanget al., 2018, Wada et al., 2014, Hoekstra et al., 2012, Khan et al., 2023, Hofste et al., 2019, Janssenet al., 2021]. More importantly, the groundwater modelling community urgently requires extensiveglobal observations of WTD at aquifers with different geology located along non-floodplains andparticularly steep mountainous regions. Before such data becomes available, all water table depthpredictions along steep regions should be regarded, at best, as educated guesses.4.2.6 ConclusionsIn this paper, three physically constrained machine learning models were developed to simulate 500m resolution static WTD across the United States and Canada, by sequentially adding more than 12million proxy observations of WTD to a large set of real well observations of WTD. In the first model(V1), only real observations of WTD were used to train the XGBoost machine learning algorithm.In V2, in addition to real observations of WTD, close to 12 million proxy observations along theshorelines of surface water bodies were used to train the XGBoost algorithm. Finally, in the thirdmodel (V3), more than 700,000 DEM-based proxy observations, which can roughly reflect WTD inmountainous landscapes, were added to the previous set of real and proxy observations in order totrain the XGBoost algorithm. The spatial pattern of the three machine learning-based simulationsof WTD, as well as two existing physically-based simulations of WTD, were compared against eachother and evaluated against unseen (i.e. not used for model training) real and proxy observations.These comparisons and evaluations were made across the entire study region and within ten majorecoregions of North America to determine where these five simulations corroborate each otherâsconclusions and where each simulation followed currently available observations. The novelty ofour work is to incorporate multiple sources of proxy observations to predict WTD using physicallyconstrained machine learning models. Through a systematic development of three machine learningmodels and by comparing their simulations with two available physically-based models, we alsoevaluate the opportunities and challenges with data-driven WTD modeling.We found that V1, V2 and V3, stronger than physically-based simulations, replicate unseen real ob-servations of WTD, while V2 and V3 reasonably simulates unseen locations of permanently wet sur-face water bodies, in most of North Americaâs ecoregions. All five machine-learning and physically-based simulations performed poorly in predicting WTD along the most arid and/or high-elevationecoregions. Pixel-by-pixel comparisons of all five simulations showed that the two physically-basedmodels provided vastly different simulations of WTD when compared to each other and V1/V2.V3, on the other hand, closely followed the overall patterns of the two physically-based simulationswhile reasonably matching proxy and real observations of WTD. Opening the black box of our threemachine learning models showed that V3 strongly followed classical groundwater-surface water in-teraction models due to topographic index strongly controlling the spatial pattern of V3 predictions,1164.3. A Causal Inference Framework for Catchment Boundary Delineationparticularly at the local scale.Regardless of our extensive evaluation against unseen real and proxy observations, and our fairlypromising results, as compared to available literature, we remain skeptical of available large-scalepredictions of WTD, particularity along steep mountainous regions. Crucially, the verifiability ofnew, potentially promising models remains limited by our incomplete grasp of groundwater processesat large pixel scales and the pervasive influence of deeply ingrained observational biases. Yet, weremain optimistic given several future directions, including developing novel ecological or emergentindices, incorporating more carefully taken/processed observations, gaining new insights on thenonlinear functional relationships among climatological, topographical, and geological variables inshaping static WTD, and expanding the ways in which such insights can be incorporated into newmodels.4.3 A Causal Inference Framework for Catchment BoundaryDelineation4.3.1 IntroductionVirtually all investigations in catchment hydrology depend on an initial, accurate delineation of thestudy areaâs boundary. Precise delineations are essential for understanding and predicting hydro-logical responses across both gauged and ungauged basins, with direct implications for flood anddrought forecasting, climate-change impact assessment, closing the water balance equation, water-resources planning, and the protection of drinking-water supplies from upstream pollution [Mengistuand Spence, 2016, Spence and Mengistu, 2019, Liu et al., 2020b, Wang and Gupta, 2024a, Beven,2006, Wohl et al., 2019]. Land-use practices, such as forestry, mining, and urban development, canmarkedly alter streamflow generation and quality, but without a reliable delineation of the truecatchment boundary, quantifying their impacts remains highly uncertain [Mengistu and Spence,2016]. Indeed, in physical, conceptual, and machine learning modelling frameworks, enforcing massbalance across poorly defined boundaries biases parameter estimates and water-balance compo-nents, reducing model realism and undermining conceptual understanding within these frameworks[Kayembe and Mitchell, 2018, Bouaziz et al., 2018, Zheng et al., 2024, Kirchner, 2006, Goswami andOâConnor, 2010]. As delineating catchment boundaries is often the first step in hydrology-relatedworkflows, uncertain delineations will have an immediate negative effect on all subsequent analyses,which in turn affects all scientific insights, practical management, and political decision-making.Before proceeding, we clarify what we mean by âcatchmentâ. In hydrology, two complementary def-initions coexist. The velocity-based perspective treats the catchment as the source area from whichindividual water molecules originate, typically identified via conservative or non-conservative tracers[Ameli et al., 2018, Spence and Mengistu, 2019, Ameli et al., 2021]. In contrast, the celerity-baseddefinition regards a land parcel as part of the catchment if rainfall there âtriggersâ a measurablestreamflow response at the outlet within a specified time window. Although these concepts haverarely been distinguished in practice, making the distinction is critical: velocity-based delineation in-forms contaminant transport studies, whereas celerity-based delineation underlies flood and droughtforecasting, climate-impact assessment, and water-resources management. In this paper, we adoptthe celerity-based definition, inferring the catchment as the set of areas whose rainfall time seriesinduce a detectable streamflow response at least once per year.One of the most widely used methods for delineating catchment boundaries relies on surface to-1174.3. A Causal Inference Framework for Catchment Boundary Delineationpography, which often misrepresents the true contributing areas. Slow cross-boundary groundwatertransfers may mask gains and losses of water, while fast preferential flow paths through fractures,ditches, or impermeable urban infrastructure can escape detection [Gordon et al., 2022]. Evenstate-of-the-art machine-learning rainfallârunoff modelsâwhile outperforming conceptual modelson global benchmarksâoften see degraded performance when mass-balance constraints are imposedgiven traditional boundary polygons. Relaxing these constraints can improve predictive accuracy,underscoring the fact that boundary errors, rather than model structure, may drive these failures[Pokharel et al., 2023, Hoedt et al., 2021, Frame et al., 2023, Wang and Gupta, 2024b, Beven,2020]. Furthermore, global and regional mass-balance analyses reveal that >50% of topographicallydelineated catchments exhibit discharge-to-recharge ratios that deviate substantially from unity, in-dicating open systems rather than closed basins [Schaller and Fan, 2009, Fan, 2019, Liu et al., 2020b,Gordon et al., 2022, Huang et al., 2023]. Focused case studies confirm these discrepancies. For ex-ample, in a New Zealand headwater catchment, >50% of extracted recharge reemerged outside thetopographic divide [Ameli et al., 2018]. Further, in a New Hampshire catchment, the subsurfacecontributing area exceeded the surface basin by >50% [Tiedeman et al., 1998]. Meanwhile, in anarid California catchment, cross-basin transfers supplied >300% of nominal inputs [Thyne et al.,1999]. Hidden conduits and fault zones in agricultural, rocky, or fractured settings can significantlyinfluence flow pathways [Thyne et al., 1999, Fan, 2019], and karst systems may route subsurface flowsquickly and unpredictably. Beyond mass-balance failures, DEM-based delineation is highly sensitiveto the choice of elevation product, resolution, and sink-filling algorithm [Erdbru¨gger et al., 2021,Datta et al., 2022, Moges et al., 2023, Ray, 2018]. Paradoxically, increasing DEM resolution can ac-tually exacerbate errors by revealing micro-topography that distorts DEM-based divides [Charrierand Li, 2012, Moges et al., 2023], thus future DEM improvements may not lead to better delin-eations. Together, these uncertainties underscore the limitations of purely topographic approachesfor reliably identifying true catchment areas.Groundwater equipotential surfaces (i.e., the contours of equal hydraulic head) provide a theo-retically sound basis for delineating true catchment boundaries, since streamlines (flow paths) runperpendicular to these contours and define the extent of subsurface contributing areas [Hinton et al.,1993, Erdbru¨gger et al., 2021]. In practice, equipotential maps are derived either from physicallybased groundwater models or from data-driven inference of water-table depth (WTD). However,no verifiable, high-resolution WTD product exists. Even when forced with similar input datasets,different modeling approaches produce starkly inconsistent head distributions and thus divergentequipotential delineations [Janssen et al., 2025c]. Moreover, verified groundwater boundary condi-tions still may miss unobserved fast flow paths in karst or urban aquifers [Pang et al., 2023, Bre-dehoeft, 2005]. As a result, the extreme uncertainty in WTD estimation undermines any attemptto map equipotential surfaces reliably, rendering groundwater-equipotential delineation impracticalwith current data and methods.Recent works have applied data-driven connectivity metrics to infer contributing area sensitivitiesand groundwater connections under the banner of causal inference [Franzen et al., 2020, Giani et al.,2022, Deng et al., 2023, Cappelli et al., 2023, Rinderer et al., 2018, Ahamed et al., 2025]. Forexample, Franzen et al. [2020] computed the maximum mutual information between streamflowand lagged precipitation at different grid cells to map the spatiotemporal changes in rainfall-runoffrelationship strength in a Colorado basin. Ahamed et al. [2025] used transfer entropy to quan-tify the information flow between spatially distributed rainfall/snowmelt and baseflow. Rindereret al. [2018] systematically evaluated a suite of âcausalâ (effective) and âassociativeâ (functional)1184.3. A Causal Inference Framework for Catchment Boundary Delineationmeasuresâincluding cross correlation, mutual information variants, transfer entropy, and Grangercausalityâfor subsurface connectivity, finding high detection error rates when compared againsttopographic ground truth. Similar assessments in karst terrains [Delforge et al., 2022], and recentextensions using attention-based and feature importance-based frameworks [Longyang et al., 2024,Deng et al., 2023, Cappelli et al., 2023], have appeared in the literature.Despite decades of advances, no existing method reliably delineates true catchment boundarieswithout unrealistic assumptions and great uncertainty. Topography-driven approaches routinelymisrepresent contributing areas as evidenced by their failure to provide closed mass-balances, whilegroundwater equipotential delineations remain impractical due to extreme uncertainties in watertable depth estimation [Schaller and Fan, 2009, Erdbru¨gger et al., 2021, Janssen et al., 2025c].Data-driven methods from information theory and machine learning have derived subsurface con-nectivity and illuminated the spatial variation in streamflowâs sensitivity to water input, but havenot been directly applied to the catchment delineation problem. Furthermore, as shown in otherfields, metrics like transfer entropy and mutual information may not be well-suited for causal in-ference due to their inability to remove the effect of confounders [Rinderer et al., 2018, Zhou andRavishankar, 2022, Friston, 2011]. To resolve these gaps, we recast celerity-based catchment delin-eation as a formal causal-inference problem. We introduce the theory of causal effect estimation tothe hydrological community and evaluate and compare the efficiency of traditional metrics (mutualinformation, transfer entropy) and rigorous causal estimators (obtained via causal effect estimationtheory) in solving the problem of catchment delineation. We evaluate whether these approachescorrectly identify catchment boundaries by attempting to differentiate pixels that are inside ver-sus outside known catchment boundaries. Then by systematically varying noise levels, length oftime-series, and streamflow flashiness (or different extent of groundwater contribution to stream)across a wide range of catchments of different sizes and with different precipitation spatial syn-chronicity, we identify where and under what circumstances causal reasoning can accurately infercelerity-based catchment area. Our results lay the groundwork for a data-driven and assumption-free delineation framework that will converge to the true catchment boundary as data and methodsimprove. Although our focus is on celerity-based delineation using rainfall and streamflow timeseries, we believe our methodology and discussion will be similarly useful for velocity-based catch-ment delineation when precipitation/streamflow tracer or isotope data become available, offering apromising direction for future research.4.3.2 MethodsDefining catchment delineation as a causal problemWe aim to delineate the catchment area given a streamflow time series and a set of gauge-basedor pixel-based precipitation time series. In particular, we want to know which areas, defined by arain gauge or a pixel, trigger streamflow at the stream gauge over short and long lags. Catchmentdelineation, therefore, amounts to identifying the rain gauges (or rain pixels) that have a causaleffect on streamflow. The total causal effect (TCE) of some input (X) on some output (Y ) hasbeen rigorously defined in the causal inference literature as [Henckel et al., 2022, Liu et al., 2020c,Schamberg et al., 2020, Yu and Li, 2021]:TCE(X â Y ) = limvâ0E[Y (x+ v)â Y (x)]v. (4.1)1194.3. A Causal Inference Framework for Catchment Boundary DelineationIn words, the total causal effect of X on Y is the expected sensitivity in Y after intervening on X bysome infinitesimal amount v [Schamberg et al., 2020]. Note that X and Y can be any random vari-ables (even random variables in time, i.e., Xt and Yt). We assert that this is an exact mathematicalreformulation for the problem of delineating celerity-based catchments. The total causal effect ofa rainfall grid on catchment streamflow represents the integrated sensitivity of current streamflowcaused by historical water input perturbations over a defined antecedent period (e.g., 50 days). Ifthe total causal effect is zero, then rainfall at that location during the defined antecedent period hasnot affected (or triggered) streamflow. Otherwise, if the total causal effect is nonzero, then rainfall atthat location will eventually lead to an increase in streamflow. Notably, the magnitude of the causaleffect also gives information about the strength of the connectivity or the streamflowâs sensitivityto rainfall at a given location. Thus, solving (4.1) aligns exactly with what most hydrologists meanwhen they refer to determining a contributing area or a catchment. Indeed, a catchment shouldincorporate all locations along which rainfall-induced celerity can push stored water (either in sub-surface or surface micro-topographies) toward the stream, at a given time-scale (or series of timelags). Equivalently, equation (1) can be used with rainfall and stream tracer time series to identifythe source locations (or velocity-based catchment areas) that affect stream tracer concentration.Previous MethodsIn this section, we will provide an overview of the two primary methods that several hydrologystudies used to estimate the effect of precipitation in specific areas on streamflow (though they werenot used to delineate catchment area directly): mutual information and transfer entropy. Then in thenext subsection, after introducing the basics of causal effect estimation theory, we will explain whythese methods may poorly estimate the causal effects required for accurate catchment delineation,as well as how we can estimate total causal effects accurately.Correlation and mutual information The correlation between two random variables X,Y isdefined ascorr(X,Y ) =E[(X â µX)(Y â µY )]ÏXÏY. (4.2)Here, µX and µY represent the means of X and Y , while ÏX and ÏY represent their standarddeviations. Unlike correlation, mutual information works for any kind of relationship, linear ornonlinear. Even if two random variables are uncorrelated, they can still have high mutual informationif one helps explain the other in complex, nonlinear, and non-monotonic ways. Mutual informationgeneralizes correlation beyond linear dependence. It is defined asI(X;Y ) =âyâYâxâXPX,Y (x, y) log(PX,Y (x, y)PX(x)PY (y)). (4.3)Mutual information quantifies how much uncertainty is reduced about one variable when we knowthe value of another variable. Mutual information between two random variables X and Y isnon-negative, and is zero if and only if the variables are independent, i.e. X â¥â¥ Y . Mutual infor-mation can therefore be interpreted as the distance between the true joint distribution of X and Y ,PX,Y (x, y), and the joint distribution presuming independence between X and Y , PX(x)PY (y).Transfer entropy and Granger causality While mutual information quantifies the overall sta-tistical dependence between two variables, it does not account for the directionality or temporal1204.3. A Causal Inference Framework for Catchment Boundary Delineationordering of information flow (i.e., I(Y ;X) = I(X;Y ), so we cannot distinguish between informa-tion flowing from X to Y versus Y to X). To address these limitations, methods such as trans-fer entropy TXâY have been developed to capture directional interactions that relate time series[Schreiber, 2000]. It does so by leveraging the histories of the two time series, across different timelags Xtâl, · · · , Xtâ1 and Ytâk, · · · , Yt. Whereas mutual information measures the joint distributionâsdeviation from an independent prior (X â¥â¥ Y ), transfer entropy considers a conditionally indepen-dent prior, such that Xtâ1...Xtâl does not provide any extra information about Yt. This is formallygiven by the following generalized Markov property:P (Yt = yt|Ytâ1 = ytâ1, · · · , Ytâk = ytâk)= P (Yt = yt|Ytâ1 = ytâ1, · · · , Ytâk = ytâk, Xtâ1 = xtâ1, · · Â·Xtâl = xtâl).(4.4)Transfer entropy measures the joint distributionâs deviation from this prior (4.4), and can thereforebe interpreted as conditional mutual information. Formally, the transfer entropy TXâY for thestandard case l = 1, is defined asTXâY =âxâXP (Yt:tâk, Xtâ1) logP (Yt|Ytâ1:tâk, Xtâ1)P (Yt|Ytâ1:tâk) = I(Yt;Xtâ1|Ytâ1:tâk). (4.5)In our case, the value of transfer entropy explains how much additional predictive power one variable(say, rainfall) and its histories have for another variable (say, streamflow), over and above what thatsecond variable can predict about itself. A high transfer entropy value from rainfall to streamflowwould suggest that current and historical rainfall drive much of the streamflow variability, even afteraccounting for streamflowâs own memory. We note that transfer entropy is closely related, and oftenequated, to Granger causality (see [Granger, 1969, Barnett et al., 2009, Hlava´ckova´-Schindler, 2011,Shojaie and Fox, 2022] and Theorem F.1.1 in the appendix).Causal effect estimationHere, we theoretically demonstrate that mutual information and transfer entropy are inadequatefor solving the catchment delineation problem, as they fail to estimate causal effects in an unbiasedmanner. To do this, we first introduce and overview the required background knowledge for esti-mating the total causal effect. As many of these terms will be new to hydrologists, we will explainthem through a hydrologically-motivated causal graph (Figure 4.9).Definition 3 (Adjustment set). Suppose that we wish to estimate the total causal effect of a set ofinput variables (also known as treatment variables in the causality literature) X on a response (oroutput) variable Y . An adjustment set Z is the set of variables that is conditioned on (or adjusted)in order to estimate the causal effect of X on Y .For example, we may set the input variable X to be precipitation data from rain gauge 1 at aprevious time (tâ 2), e.g. rg1tâ2, and the response variable Y , to be the gauged streamflow at timet, e.g. Qt. If we estimate the causal effect from X to Y via correlation or mutual information, whichare purely marginal and associative metrics, then the adjustment set is the empty set. In the caseof transfer entropy, we condition on the past k values of streamflow, therefore the adjustment setconsists of these k past values, i.e. Z = {Qtâ1, · · · , Qtâk}. This is summarized in Table 4.3.Definition 4 (Valid adjustment set [Runge, 2021]). Given an input variable X and a responsevariable Y , an adjustment set Z is said to be valid if conditioning on Z provides an unbiased totalcausal effect estimate for X on Y (see Equation 4.1).1214.3. A Causal Inference Framework for Catchment Boundary Delineationrg1tâ2 rg1tâ1 rg1tQtUtâ2 Utâ1rg2tâ2 rg2tâ1 rg2trg3tâ2 rg3tâ1 rg3tFigure 4.9: The causal graph of a small hydrological system, which principally consists of streamflowat one gauge, Qt, and three rain gauges, rg1, rg2, and rg3, from three times: tâ 2, tâ 1, t. We alsoplace two unobserved variables Utâ2 and Utâ1, which denote larger weather/atmospheric processesthat cause rain to persist across several days within the area. We know in advance that only thefirst two rain gauges (rg1 and rg2) have a causal effect on the streamflow Qt (shown by continuousarrows). rg3 is outside of the true catchment, and therefore should not have a causal effect onstreamflow.It follows that we need to find a valid adjustment set in order to verify if a rain gauge at a given lag ishaving a causal effect on streamflow. If one were to use an invalid adjustment set, the causal effectsthat we estimate may be biased, leading to false negatives or false positives when determining theinclusion of a rain gauge within the catchment boundary. The classical way to ensure the validityof an adjustment set is via Pearlâs backdoor criterion [Pearl, 1993].Definition 5 (Backdoor Criterion [Pearl, 1993]). A set of variables Z is said to satisfy the backdoorcriterion for input variable X and response variable Y if (1) Z â© des(X) = â , and (2) Z blocks allpaths between X and Y that contains an arrow into X (i.e., backdoor paths into X).The first condition implies that we cannot condition on any children, grandchildren, etc. (i.e.,descendants) of X. For example, to quantify the causal effect of Utâ2 on Qt we cannot conditionon rg2tâ1 since it is a child of Utâ2 (Figure 4.9). This condition should be satisfied to ensure thatsome portion of the total causal effect is not incorrectly removed. The second condition means thatwe must block pathways that increase the noncausal association between X and Y by adjusting for1224.3. A Causal Inference Framework for Catchment Boundary Delineationvariables on such paths. For example, suppose one wants to measure the causal effect of rg3tâ2 onQt which must be zero. Based on Figure 4.9 and Definition 5, the set {rg2tâ1, rg1tâ2} would notsatisfy the backdoor criterion since there is still at least one open path between rg3tâ2 and Qt (i.e.,rg3tâ2 â Utâ2 â rg2tâ2 â Qt). Hence, for quantifying the total causal effect of rg3tâ2 on Qt, whilethe first criteria is met and rg3tâ2 has no descendants, the set of all other observed variables (i.e.,{rg2t, rg2tâ1, rg2tâ2, rg1t, rg1tâ1, rg1tâ2, rg3t, rg3tâ1}) should be considered as a valid adjustmentset. Note that conditioning on rg3tâ1 and rg3t is optional since these do not have a directed pathto Qt (for more details, see Pearl [1993]). Once we apply this valid adjustment set, all paths fromrg3tâ2 to Qt would be blocked by one of the variables in the adjustment set, as such the total causaleffect of rg3tâ2 to Qt should be estimated as zero.Although the backdoor criterion ensures the validity of adjustment sets, there are some valid ad-justment sets that do not satisfy the backdoor criterion [Shpitser and VanderWeele, 2011, Shpitseret al., 2010]. A complete characterization of all valid adjustment sets is provided by the adjustmentcriterion developed by Shpitser et al. [2010] and Van der Zander et al. [2014] (see Section F.2 of theSupplement, Perkovi et al. [2018] and Runge [2021] for a clear explanation).Table 4.3: Various methods can be used to estimate catchment delineations from precipitation data,including mutual information (MI), transfer entropy (TE), and causal effect estimation (CEE).To assess whether precipitation Xtâ1 at a given location influences streamflow Yt, each methodconditions on an adjustment set of variables. Despite the popularity of MI and TE, only CEE yieldsan unbiased causal estimate. We note that the backdoor criterion (5) can practically be satisfied byconditioning on the historical precipitation {Xtâ2, · · Â·Xtâl} as well as the historical precipitation ofits neighbours, given sufficient lag l.Method Adjustment set used in estimationMutual information (MI) â Transfer entropy (TE) Previous streamflow: {Ytâ1, . . . , Ytâk}Causal effect estimation (CEE) Any set Z satisfying Definition 4,e.g. satisfying the backdoor criterion (Definition 5)While it has been recently used in some hydrology studies, the above definitions imply that mu-tual information and transfer entropy are not sufficient for determining causal effects. Mutualinformation and correlation are purely associative metrics, and it is fairly well known, even amonghydrologists, that associative metrics like these cannot measure causality since correlation does notimply causation. But the specific reason behind this is less widely understood. This explanationcan come directly from considering the backdoor criterion. For example, in Figure 4.9, there willbe a statistically significant association between rg3tâ2 and Qt due to the common cause Utâ2.Mutual information has an empty conditioning set, so the backdoor path from Qt into rg3tâ2 isnot blocked and therefore the backdoor criterion and the adjustment criterion are not satisfied. Asimilar argument can be made regarding the use of transfer entropy as a method that has beenused to measure causal effects in hydrology. Note that the analogous causal graph for Qtâ1 can bemade from Figure 4.9 by simply shifting all variables one day back in time. Thus, since transferentropy only conditions on previous streamflow, it would fail to satisfy the adjustment criterion formeasuring the causal effect of rg2tâ2 on Qt in two respects. First, Qtâ1 should not be conditionedon since it is a descendent of the input rg2tâ2, therefore imposing this conditioning factor would1234.3. A Causal Inference Framework for Catchment Boundary Delineationimmediately bias any causal inferences. Second, as with associative metrics like mutual information,several open backdoor paths still remain (i.e., the path rg2tâ2 â Utâ2 â rg1tâ2 â Qt). Thesepoints will be evaluated experimentally in the results section, where we will compare a causal effectestimation method that uses the adjustment criteria with mutual information and transfer entropy.4.3.3 A proof of concept on virtual catchments: Experimental designIn the following experiments, we aim to compare the efficacy of mutual information, transfer entropy,and causal inference in determining causal effects through the study of catchment delineation. Wealso aim to study the effects of data quality (or the presence of noise in streamflow time-series), datalength, streamflow flashiness (or percent base/groundwater flow), and precipitation characteristics onour ability to delineate catchments with causal effect estimation. Since the goal of these experimentsis to evaluate where and under what circumstances causal inference is able to solve catchmentdelineation under realistic data settings, we require a set of known ground-truth catchments. Virtualcatchments are a popular tool in this setting since we can realistically simulate streamflow whilefully knowing the underlying data generating processes. These types of case studies with virtualstreamflow have been used throughout the hydrological literature [Macdonald et al., 2025, Li et al.,2014b, Ameli et al., 2015].DataWhile producing a large-sample hydrology dataset for Germany (CAMELS-DE), Loritz et al. [2024]noticed large discrepancies between Merit Hydro, HydroSHEDs, and German government-reportedcatchment areas. In fact, 800 stations were removed from the CAMELS Germany dataset justbecause catchment delineations were too uncertain (â¥ 20% uncertainty in area). The availability oflarge diverse sets of high quality hydroclimate data is extremely important, especially in the age ofmachine learning [Kratzert et al., 2024], thus these large uncertainties pose a significant constrainton data availability and therefore model generalizability. Further, the calculation of the catchment-scale catchment attributes and meteorological forcing depend significantly on reliable delineationof catchment boundaries, so any errors in catchment delineations lead to misleading catchmentattributes and ultimately misleading inference and assessment [Heudorfer et al., 2025, Ghosh et al.,2022b, Li et al., 2022]. Due to the potential to fix such issues with a novel and independent catchmentdelineation approach, we decided to focus on Germany for our experiments.The CAMELS-DE dataset was downloaded from Dolich et al. [2024], though we only used their listof catchments (1555 total) as well as their derived catchment boundary delineations (via MERITHydro) [Yamazaki et al., 2019] to use as our hypothetical ground truth. With having known ground-truth catchment boundaries, we used spatially-distributed precipitation data within each catchmentboundary to generate synthetic streamflow data. The details of both sets of data are laid out below.Precipitation data Although we are using virtual catchments in our experiments, we use realprecipitation data in order to maximize the experimental realism. Since it is globally available withequally spaced measurements (in contrast to gauge-based products), we use the PERSIANN-CDRsatellite-based precipitation product [Nguyen et al., 2018, Hsu et al., 1997, Ashouri et al., 2015]. Welimit ourselves to satellite precipitation data because interpolated products based on limited ground-based observations may have strange dependency structures that may not be easy to account for,so we leave this for future work. Precipitation data was downloaded via the CHRS data portal1244.3. A Causal Inference Framework for Catchment Boundary Delineation(https://chrsdata.eng.uci.edu/) as daily values across Germany at 0.25 degree resolution for1983-2020 [Nguyen et al., 2019]. For each catchment boundary in the CAMELS-DE dataset, weextracted and saved the precipitation time series corresponding to PERSIANN pixels both withinthe catchment boundary and in the 0.5 degree radius of the catchment boundary. Pixels thatwere in the catchment (IC) boundary were classified as being part of the ground-truth catchment(IC= 1), while pixels outside of the boundary were classified as outside the boundary (IC= 0). Ifthe catchment was too small for any pixels to lie within the boundary, the closest pixel was chosento be part of the ground-truth catchment and classified as IC= 1.The precipitation data and catchment boundaries are visualized in Figure 4.10. As can be seen fromthe figures, the catchment areas vary widely from 5 to 14,000 km2. Long-term average precipitationvaries from around 2mm per day in northeastern Germany to about 3.5mm per day in the south ofGermany (Figure 4.10c). As can be seen from Figures 4.10a and 4.10b, precipitation events are morelocalized in northern Germany compared to southern Germany. We hypothesize that less spatialautocorrelation (or synchronicity) may lead to a greater ability to separate signals and recover theground-truth catchment. This hypothesis will be explored in Section 4.3.4.Virtual streamflow data Given the ground-truth catchment delineation and a set of remotely-sensed precipitation data, we generate streamflow data for our experiments. For each pixel, stream-flow is generated at the outlet via a convolution between that pixelâs rainfall and a randomly gener-ated unit hydrograph of length 50 days. If the catchment consists of multiple pixels, the streamflowgenerated by each pixel in the catchment (i.e., IC= 1) are summed to produce the total streamflow.For our first experiment (method comparison), we require diverse data settings, so we generatestreamflow based on the spatially distributed rainfall data from each of the 1555 German catch-ments. The catchment area (or the number of pixels within the true catchment boundary) andthe spatial autocorrelation of rainfall vary across these catchments, setting the stage for evaluat-ing their effects on the efficacy of causal inference in delineating the true catchment boundary. Inaddition, to increase the data diversity and evaluate the effect of more factors on causal inferenceefficiency, we systematically vary three additional factors: unit hydrograph spread, the signal-to-noise ratio, and the number of available years of data. For each study catchment, we generate fivedifferent streamflow time series by systematically varying these three additional factors, for a totalof 5 Ã 1555 = 7775 synthetic streamflow time series. In short, the five primary factors that varyacross these 7775 streamflow time series include:â¢ Number of pixels: The size of the catchment (5 - 14000 km2) alters the number of pixelsboth within and outside of the catchment (see Figure 4.10). We suspect that larger catchmentswill be more difficult for data-driven catchment delineation.â¢ Precipitation spatial autocorrelation: The 1555 catchments we consider have a range ofautocorrelation between 0.91 and 0.96. More similar (or more correlated) precipitation timeseries may make the signals harder to differentiate.â¢ Signal-to-noise ratio: For each of our 7775 streamflow time series, we add random Gaussiannoise to the streamflow time series such that the NSE between the noiseless and noised timeseries is between 0.5 and 0.9 (sometimes referred to as max NSE). The level of noise, which israndomly chosen for each time series, will likely hurt our ability to recover the ground truthsystem to varying degrees.â¢ Spread of unit hydrograph: All unit hydrographs follow a gamma distribution with shape1254.3. A Causal Inference Framework for Catchment Boundary Delineation(a) Correlation with neighbours (b) Mutual information with neighbours(c) Mean daily precipitation (mm) (d) Precipitation autocorrelationFigure 4.10: Maps describing the distribution of precipitation across the CAMELS-DE catchments.and scale parameters being randomly generated between 1 and 2. A unit hydrograph witha larger shape/scale, or equivalently, a larger spread, would reflect the contribution of moredelayed flow, similar to groundwater contribution.1264.3. A Causal Inference Framework for Catchment Boundary Delineationâ¢ Length of time series: For each of the 7775 streamflow time series, we select between 5 and38 years of synthetic streamflow data for our analysis.Our first two experiments (method comparison and analysis of future data requirements) use thesame set of data. While the five factors enable us to have a diverse set of data for which tocompare methods, they also enable us to understand where and under what circumstances causaleffect estimation may be a viable option for catchment delineation. For example, by comparingthe accuracies of pixel classification (i.e., within or outside of the true boundary) across differentmaximum NSE levels and numbers of available years, we can determine the quality and quantityof data needed for accurate catchment delineations. Further, by comparing delineation accuraciesacross different catchments and different levels of groundwater contributions (determined by theunit hydrograph parameters), we can illustrate where (within Germany) causal effect estimationwill initially become useful.Method implementationInformation theoretic methods While mutual information and transfer entropy have been usedfor catchment delineation, it has been shown that previous implementations may be unstable, sincethey are dependent on coarse-graining specifications [AÂ´lvarez Chaves et al., 2024]. For example,in order to compute mutual information, Franzen et al. [2020] discretized their precipitation timeseries into a binary series (i.e., they only differentiated between wet and dry days), resulting ina large reduction in available information. In order to implement the mutual information andtransfer entropy baselines, we instead leverage the fact that they can be approximated by usinga performance metric from a well calibrated machine learning model. Indeed, this is theoreticallyjustified since mutual information is equivalent to the universal predictive power (see Covert et al.[2020]) and transfer entropy is equivalent to conditional mutual information. We further note thattransfer entropy is equivalent to Granger causality, as shown in Theorem F.1.1 in the appendix, sinceGranger causality is defined as the drop in universal predictive power when removing a feature.We therefore implement these information theoretic baselines by computing the out of bag R2 ofextremely randomized trees, in order to approximate the universal predictive power [Geurts et al.,2006], as done in Janssen et al. [2023]. Specifically, we use the âextratreesâ option in the rangerpackage with 40 trees [Wright and Ziegler, 2017]. This ML model effectively captures nonlinearitiesand interactions, and should therefore capture the optimal performance of both baseline methods.Standard errors are then computed by repeatedly (20Ã) permuting the rows of each column in thedata set (i.e., removing all information) and measuring either the predictive power of all features (formutual information) or the drop in predictive performance when removing the features belongingto a pixel (for transfer entropy). Hence, for both approaches a pixel can be considered as inside thecatchment if the measured quantity is greater than three times the standard error and out of thecatchment boundary if the quantity is less than three times the standard error.Causal Effect Estimation For all experiments, we infer causal effects of precipitation in a pixelvia regression estimates. As our adjustment set for each precipitation pixel, we condition on itsprevious 10 lags and the 10 lags of all other pixels that are located both within and outside theunknown true boundary, in order to ensure that we satisfy the backdoor criterion (Definition 5).The first condition is satisfied since past precipitation values cannot be descendants. The secondcondition is satisfied since precipitation data for a given day only depends on its recent history,rather than across all time lags. Although a more optimal choice may be possible [Runge, 2021], it is1274.3. A Causal Inference Framework for Catchment Boundary Delineationreasonable that by conditioning on 10 lags, all biasing paths would be blocked. Indeed, the temporalautocorrelation of precipitation only lasts about three days (see Figure F.2 in the Appendix), andthe most influential lag on streamflow can be assumed to be between one and eight (see Section4.3.3 and Janssen et al. [2025b]). Statistical significance (i.e., a coefficient implies that a pixel iswithin the catchment (IC= 1)) is determined by calculating the coefficient confidence intervals withwidths of three times the standard errors.Investigation into future data requirements To assess where (i.e., under what hydrologi-cal/climatological conditions) and under what circumstances (i.e., which data and noise settings)causal inference may become a competitive method for delineating catchments, we regress the fivefactors that vary across our 7775 virtual datasets (see Section 4.3.3) against classification accuracy.We will investigate the regression coefficients to determine the effects of each of the five factors onimproving (i.e., positive effect) or worsening (i.e., negative effect) the classification accuracy.4.3.4 Experimental resultsMethod comparisonTable 4.4 summarizes the results of our comparative analysis. As can be seen from the table, causalinference is the only method that is able to extract the catchment delineation with high classificationaccuracy (0.992). While both mutual information and transfer entropy have high true positive rates,they both have a zero true negative rate, therefore, their overall classification accuracy is extremelylow (0.047). This is perfectly illustrated in Figure 4.11 where the ground truth catchment has twopixels that contribute to streamflow. Both mutual information and transfer entropy infer that allneighbouring catchments/pixels contribute water to the outlet, while causal inference can correctlydetect the two pixels that are part of the âtrueâ catchment. Another example for a larger catchmentcan be found in the Supplementary material.Table 4.4: Classification Accuracy rates from our three considered methods for classifying pixelswithin the ground truth catchment (positive) versus pixels outside of the ground truth catchment(negative). Higher is better across all metrics. Values in the parentheses show the standard deviationacross our 7775 time series.True Positive Rate True Negative Rate Classification AccMutual Information 1 (0.0) 0 (0.0) 0.047 (0.024)Transfer Entropy 1 (0.0) 0 (0.0) 0.047 (0.024)Causal Effect Estimation 0.962 (0.170) 0.999 (0.007) 0.992 (0.020)Analysis of future data requirementsThe results of our analysis of future data requirements are shown in Figure 4.12. A higher signal-to-noise ratio and a larger set of data leads to higher classification accuracies (Figure 4.12). Moreattenuated streamflow responses (more spread out unit hydrographs and therefore higher groundwa-ter contributions) slightly (but statistically significantly) worsens the classification accuracy, thoughthe regression coefficient is close to zero. Higher spatial autocorrelation (or synchronicity) of precip-itation, and considering more pixels (reflective of larger catchments), leads to worsening accuracy.All regression coefficients are statistically significant (p < 0.01). For the case of spatial autocor-1284.3. A Causal Inference Framework for Catchment Boundary DelineationFigure 4.11: Example classifications of the true catchment along with each of the three methods forthe DEF13090 catchment (875 km2 catchment area) with 38 years of data and a maximum NSE (orsignal to noise ratio) of 0.8. Black line shows the catchmentâs ground truth boundary polygon. Bluepixels indicate true (within the true catchment) for the first panel, and indicate true (within thepredicted catchment) for the remaining three panels/methods. Salmon pixels indicate false (outsidethe true catchment) for the first panel, and indicate false (outside the predicted catchment) for theremaining three panels/methods. Missing pixels (white) may appear due to the pixels not beingfully contained within Germany.relation of precipitation, due to the low variability, only ranging from 0.91 to 0.96, its confidenceintervals are much wider than the other factors.1294.3. A Causal Inference Framework for Catchment Boundary DelineationFigure 4.12: The plotted regression coefficients illustrate how much each factor impacts classificationaccuracy. The spread of the unit hydrograph in each catchment was calculated as the average unithydrograph variance across all contributing pixels. The length of time series coefficient denotes howmuch classification accuracy increases with 100 years of extra data. The spatial autocorrelation ofprecipitation coefficient denotes how much classification accuracy increases with a 0.1 increase incorrelation. The â# of rainfall pixelsâ coefficient denotes how much classification accuracy increaseswith the addition of 10 additional pixels (as catchment area increases). 99% confidence intervals areshown (blue bars) around each regression coefficient (blue and white circle).4.3.5 A simulated case study comparing the efficiency of causalinference-based versus topography-based delineationsCase study setupBased on the previous results section, we can surmise that causal inference has the potential tobe a powerful method for catchment delineation in catchments that are not too large, and whereprecipitation is highly localized while both the length of the available time series data and the signal-to-noise ratio are large. In light of this finding, we test causal effect estimation against topography-based delineation on a medium sized catchment in northeastern Germany (where autocorrelationis low, see Figure 4.10b). For this case study, we assume that the runoff response within thetopographic catchment is rapid, as reflected by the black kernel (Figure 4.13). We further assumethat there exists a large area outside of the topographic catchment (three pixels) which contributesslower groundwater flow, as reflected by red kernel (top left of Figure 4.13). The full 38 years ofdata is used, and noise is added to the streamflow time series until the NSE between the noised andnoiseless versions is 0.9.Case study resultsThe results of this simple case study, shown in Figure 4.13, illustrate that causal inference canindeed detect hidden and cross-boundary. While topography-based delineations will always misspurely groundwater-driven recharge zones, causal inference removes all assumptions and is therefore1304.3. A Causal Inference Framework for Catchment Boundary Delineationable to detect these zones to a large extent. This particular case study shows that causal inferencestill misclassifies one pixel, but this performance is still substantially better than the topographiccatchment (missing three pixels). One should note that due to hidden karst or urban pathways,topographic catchments may even miss fast flow paths, so its failure modes are not limited to thiscase study where only missing delayed groundwater contributions were highlighted.Figure 4.13: The true DE811440 catchment is shown along with the estimated catchments by thetopographic method and the causal inference method, given 38 years of data and a maximum NSEof 0.9. Blue indicates true (within catchment) while salmon indicates false (outside catchment). Theunit hydrographs driving streamflow is shown in the top left subfigure.1314.3. A Causal Inference Framework for Catchment Boundary Delineation4.3.6 DiscussionLimitations of Information-Theoretic Metrics for Causal InferenceOur theoretical analysis of the backdoor criterion, together with our comparative experiments,demonstrate that neither mutual information nor transfer entropy reliably estimate causal effects,especially in the context of catchment delineation. Although these measures quantify statistical de-pendencies and directed associations, they were not designed to disentangle confounding influencesor to satisfy the conditions required for causal identification. In hydrology, it is common to interprettransfer entropy as a proxy for causation. For instance, Rinderer et al. [2018] asserts that transferentropy extracts causal information when quantifying groundwaterâstream connectivity. Similarly,Bennett et al. [2019] describes transfer entropy outputs as indicating how inputs âcontributeâ andâinfluenceâ model outputs, and Tennant et al. [2020] employ transfer entropy to infer streamflowgeneration mechanisms and critical timescales for predictability. Additional studies including Good-well et al. [2020], Konapala et al. [2020], Larsen et al. [2017], and Sendrowski and Passalacqua [2017]also frame transfer entropy as capturing causal influence.In contrast, other disciplines have long recognized the distinction between directed associations andtrue causal effects. In computer vision, Zhou and Ravishankar [2022] groups Granger causalityand transfer entropy with purely associative metrics (e.g., correlation), distinguishing them fromstructural causal models. Within neuroscience, Friston et al. [2013] explicitly characterizes transferentropy and Granger causality as measures of directed functional connectivity âwithout concern forcausal relationsâ, and warns that conflating these with causal inference can impede scientific progress[Friston, 2011]. We acknowledge that transfer entropy is straightforward to compute and has yieldedvaluable insights into inferring hydrologic connectivity. However, as Devito et al. [2005] cautions,relying solely on the simplest available tools when addressing fundamentally causal questions maybe problematic. Instead, formal causal effect estimators, grounded in do-calculus and backed byclear identification criteria [Pearl, 1993], are necessary to ensure robust and scientifically meaningfuldelineations of catchment boundaries. Carefully following causal theory lets us determine whetherand how we can estimate a causal effect (e.g., the effect of rainfall in a given pixel on streamflow)using only observational time series and our assumed causal diagram, without needing to performactual physical experiments.Prospects for Causality-Driven Catchment Delineation Under Improving DataOur experiment on future data requirements (Section 4.3.4) demonstrates that causal inferencemethods can offer a competitive, and largely assumption-free, alternative to topographic delin-eations in moderately-sized catchments characterized by localized precipitation. Regions such asAustraliaâs eastern coast, which exhibit low spatial autocorrelation in rainfall and possess extensivelong-term records [Han et al., 2020], are particularly well suited for this approach. Similarly, tropicalzones in Brazil and southeast Asia, as well as high-elevation areas like the Tibetan Plateau and theRocky Mountains, display substantial spatial variability in extreme rainfall events [Ghanghas et al.,2023, Tan et al., 2021, Zhang and Wang, 2023]. Climate-change projections indicate that as tem-peratures increase, the spatial heterogeneity of precipitation will intensify [Ghanghas et al., 2023].Concurrently, the ongoing expansion of high-quality hydrometeorological observations and improvedrainfallârunoff modeling (furthering our ability to explain more hydrograph variability) will furtherenhance the power of causal effect estimators. With longer, more reliable precipitation time seriesand more accurate discharge measurements, the fidelity of causality-based delineation should only1324.3. A Causal Inference Framework for Catchment Boundary Delineationimprove. In this context, causality-driven methods are poised to become an increasingly robust toolfor catchment boundary identification as data quality and availability continue to advance, thoughtheir utility may by limited to certain regions at first.Limitations of the current study and future workTo demonstrate its practicality on real-world delineation or boundary verification problems, severalshortcomings must be addressed. Here we assume that all of the noise or measurement error isGaussian and belongs purely to the streamflow variable. We suspect that our results from causalinference will show much worse behaviour under scenarios with non-Gaussian streamflow measure-ment errors or substantial precipitation measurement errors. Model misspecification in the causalgraph generation step or in the regression step may further hinder the applicability. Note that onlymultivariate linear regression was used for causal effect estimation in this paper, as it is the simplestpossible method to illustrate our proof of concept. Even more accurate results would be expected bysimply adding feature selection, group penalties, or greater flexibility via machine learning methods[Nouraie et al., 2025, Janssen et al., 2025b, Longyang et al., 2024]. It is also unclear how applicablecausal inference would be if the pixels were not measured independently (i.e., interpolated precip-itation products based on point-based observation stations). Note that this is the reason we stuckwith a course-resolution purely satellite driven product. If future work shows that interpolatedproducts are theoretically and experimentally sound, this would open up the possibility of usingfiner resolution and more accurate precipitation datasets. Furthermore, the dynamic nature of thecontributing area is a vital aspect to consider. Future work could address this by using time-varyingunit-hydrograph estimations [Janssen et al., 2025b]. Future work can extend this sort of study toBrazil which may have severe water balance and catchment delineation issues [Huang et al., 2023,Schwamback et al., 2022, UchoËa et al., 2024, Tan et al., 2022].4.3.7 ConclusionsIn this paper, we argue that causal inference, implemented by conditioning on valid adjustment sets,can be useful as an assumption free and independent paradigm for catchment delineation. We showthat causal inference is at least conceptually possible to use in this domain via experiments on virtualcatchments. In particular, our results demonstrate that causal inference may already be applicablefor catchment delineation in moderately-sized catchments with spatially heterogeneous rainfall. Wealso illustrate and explain the importance of considering the correct adjustment set and dispel someof the confusion surrounding the causal or non-causal nature of common tools like transfer entropyand mutual information. We hope that this paper will influence catchment hydrologists by makingthem carefully consider the uncertainties surrounding their most fundamental assumption (that thetopography-driven boundary is accurate). Although causal inference currently has some limitationsthat need to be resolved, we remain optimistic that our proof of concept paves the way for practicaluses of causal effect estimation within this domain in the near future.133Chapter 5Conclusion5.1 SummaryIn this dissertation, we have explored how hydrologists currently extract insights from data-drivenspatiotemporal analyses, while developing novel methods. These novel methods contain far fewershortcomings compared to the previous state-of-the-art since the development process consideredthe unique obstacles inherent within hydrological data. After introducing the research motivationsand objectives in Chapter 1, Chapter 2 explored how to improve cross-catchment spatial analyses.In the first part of the chapter, we developed three novel hydrological indices which we showedto be powerful representations of independent streamflow generation mechanisms as they allowfor improved regionalization and better predictions in poorly gauged basins. Since these indiceswere created by considering the interactions or synergistic information that arises when consideringmultiple catchment characteristics (i.e., soil and bedrock hydraulic conductivity or water inputintensity and shallow soil hydraulic conductivity), our results demonstrated the importance andprevalence of interactions within cross-catchment spatial analyses. After noticing that the methodsused in most cross-catchment spatial analyses (i.e., feature importance) were insufficient when theexplanatory variables are highly correlated and/or possess interactions, we sought to develop anovel feature importance metric that resolves these shortcomings in the second part of Chapter2. We surpassed this goal with the development of ultra-marginal feature importance since thisnovel method can not only detect correlated interactions, it can also guarantee the detection ofblood relatives in the causal graph with sub-quadratic computational complexity. By applyingultra-marginal feature importance on a large sample hydrology dataset, we showed that previousliterature had underestimated the importance of land cover while overestimating the importance ofsoil and geological attributes.Starting with Chapter 3, the focus of this dissertation shifted to learning how catchment filteringfunctions change with time. In the first part of that chapter, we began by relating streamflow toprecipitation convolved with the time-varying filtering function we aimed to learn. We know thatthis filtering function (or unit hydrograph) changes based on some complex and unknown function ofcurrent and antecedent conditions. Due to its complexity, in the first part of Chapter 3, we assumedthat day of the year can act as a good proxy for these antecedent conditions (i.e., January 10thhas the same/similar antecedent conditions across different years). With this key assumption, wedeveloped the historical functional linear model with dynamic sparsity (HFLM-DS). The HFLM-DSmethod aims to estimate a smooth coefficient function (β(s, t)) which changes for each lag s and foreach day of the year t. We ran HFLM-DS on two catchments and were able to infer when and howeach catchment transitions between wet, drying, dry, and wetting periods, while also quantifyingthe shape and magnitude of these unit hydrographs in each period. Importantly, we also showedthat HFLM-DS is good at recovering ground-truth parameters while also showing strong predictive1345.2. Implications and significancecapabilities. In the second part of this chapter, we returned to our key assumption in the firstpart and remove this assumption using differentiable modeling and neural networks. In particular,we used LSTMs to predict the filtering function for each day based on antecedent precipitation,temperature, and potential evapotranspiration. This new âself-explainableâ LSTM (SE-LSTM) wasshown to outperform previous unit-hydrograph based methods on held-out streamflow time serieswhile also providing perfect interpretability and improved inferential capabilities.In the third and final body chapter of this dissertation (Chapter 4), we ended by exploring howeach pixel within a catchment can differentially impact streamflow. Groundwater connectivity andsaturation levels are known to have large impacts on spatial heterogeneity in streamflow generation,thus in the first part of Chapter 4, we studied one of groundwaterâs fundamental properties, watertable depth (WTD). In the last 15 years, there has been considerable effort put fourth to simulatewater table depth at regional scales with fine resolution using physically-based models. In ourwork, we showed that these physically-based models lack any reasonable correspondence with wellobservations, so we aimed to resolve this by developing a machine learning model to predict WTDacross North America. Using real well observations as well as a large set of proxy observations, weended up building three machine learning models for WTD prediction. However, due to samplingbiases, observation unreliability, and the lack of physical realism within machine learning models, weremain skeptical of all water table depth predictions, especially those in high elevation areas. Givensome common hydrological assumptions, more accurate catchment delineations could, in theory, bemade using high resolution estimates of WTD, however, due to our continued skepticism of theseproducts, we turned to causality as a possible resolution in the second part of Chapter 4. In this finalpart, we first argued that an area can be considered part of a catchment if and only if precipitationfalling in that area causes streamflow at the measured gauge. Then, by bringing in the backdoorcriterion and adjustment set criterion into the hydrological literature, we explained why previousmetrics used for causal purposes (i.e., transfer entropy) are in fact not causal. Following this, wedemonstrated to hydrologists via synthetic experiments that by using the backdoor criterion ina regression setting, one can accurately delineate catchments, and potentially demonstrate cross-boundary connectivity using causality and widely available spatially distributed time series data.Finally, we closed this chapter by analyzing and discussing where and when causal effect estimationcould become a valuable tool for learning more accurate catchment delineations.5.2 Implications and significanceThe most interesting and important scientific knowledge is not a set of dry facts, instead it is acomplex network of quantifiable and predictive cause and effect relationships [Pearl and Mackenzie,2018]. In this thesis, our aim was to take steps towards enabling hydrologists to expand theirunderstanding of these complex hydrological relationships. Unfortunately, learning from hydrologicaldata poses unique challenges to existing statistical and machine learning methods, thus we developednew data-driven methods directly inspired by hydrological problems in order to help hydrologistsperform more robust analyses that need less assumptions.Since these methods have now been developed and extensively tested, we hope that future studieswill apply them to spatial, temporal, and spatiotemporal hydrological datasets to increase our knowl-edge of ungauged basins and to answer several unanswered questions in hydrology. We hope thesemethodologies will be applicable to areas within and beyond hydrology, and we hope these hydro-logical insights will be useful for answering global problems in hydrology, climatology, geochemistry,1355.3. Future Workand Earth science.Based on the applications we have already demonstrated in this thesis, we can begin to observesome novel insights. Our second chapter has demonstrated the importance of interactions withinlarge sample hydrological data. We saw that incorporating expert knowledge-derived interactionswithin prediction and regionalization workflows can result in clear improvements. We then showedthat traditional machine learning methods fail to capture these interaction effects within featureimportance scores, especially when there is significant correlations. Our feature importance metriccan be used in any scenario where the goal is to advance science, and using our method insteadof others will lead to a better understanding of the underlying causal graph as well as a betterunderstanding of relationship strength in the presence of interaction and high correlations. Basedon our work in Chapter 3 relating to building complex but interpretable models via time-varyingunit hydrographs, we have observed diverse hydrologic behavior in time and space. More specifically,we began to answer our first three unsolved problems in hydrology (see Chapter 1.2). We foundthat antecedent conditions can almost fully determine the hydrograph response to water input. Forexample, in dry periods, we consistently saw that the runoff response is slight and finishes quickly,and in wet periods, the response is large but drains quickly. The importance of removing assumptionsand the pervasive issue of inference after model misspecification was also demonstrated. Indeed, inthe second part of Chapter 3, SE-LSTM shows much stronger inference capabilities compared toHFLM-DS. However, despite the lack of assumptions within SE-LSTM, it still fails to match thepredictive power of vanilla LSTM, suggesting that the optimization is not optimal. These conclusionswill encourage future researchers in hydrology to carefully consider their assumptions and challengetheir confidence in their conclusions under possible misspecification. They may also be encouragedto improve optimization procedures for SE-LSTM. Finally, Chapter 4 holds significant work and hasimportant implications surrounding strongly-held beliefs in surface and groundwater hydrology. Inthe first work of that chapter, we showed that previous estimates of water table depth across NorthAmerica are extremely poor and we argued that the observations and machine learning simulationsmay also be unreliable. Since previous works displayed great confidence in their predictions, wehope that our more robust (and negative) evaluations will lead to a greater focus on groundwaterresearch, especially in high elevation regions. In our final work of this dissertation, we challengedand dispensed with two pervasive assumptions within surface water hydrology. First, we arguedthough an extensive literature review that catchment delineations are less reliable than most large-sample hydrologists would like to admit, then we showed and explained (for the first time within thehydrological literature) why common methods like transfer entropy cannot show causal relationshipsin hydrology. As with many of our other works, this work hopes to force hydrologists to considerthe most basic assumptions they are making and the intended purpose of the methods they use.Most importantly, when many decades of accurate satellite-based precipitation products are globallyavailable, we believe that this work will revolutionize the way we think of and determine hydrologicconnectivity and catchment delineation.5.3 Future WorkThroughout the completion of this thesis, more and more questions have arisen, with the number ofnew questions far outnumbering the number of answers. The questions I am left with relate to fourprimary topics: (1) the impact of data quality/quantity on hydrological inference and prediction, (2)the role of optimization in choosing hydrological models, (3) the exploration of causal representations1365.3. Future Workin hydrological models, (4) the learning of new hydrological laws that govern flow in undisturbedhillslopes. We will explore each of these topics a bit further in the following paragraphs.It is clear from an inference and prediction perspective, getting more and higher quality data willlead to great progress in the field of hydrology [Janssen et al., 2025c, Kratzert et al., 2024], but thereare still underexplored questions within this topic. In the second chapter of this thesis, we studiedthe spatial patterns of runoff by comparing hydrological signatures across catchments. We exploredmany of these hydrological signatures including baseflow index, average streamflow, different slopesof the flow duration curve, though the list extends far beyond just these. This raises the questionof whether or not there are yet to be discovered signatures that describe a portion of hydrographvariation left unexplained by the current set of commonly used signatures. More specifically, Iwonder how well we can predict the shape of hydrographs from streamflow signatures and howwe can create new interpretable signatures that increase this predictive capability [Weijs et al.,2013]. In the third chapter, we turn to time series data where the data quality can be questionable.After reading Keith Bevenâs work on disinformative events [Beven and Westerberg, 2011, Beven,2024], I wonder how pervasive this disinformative data is within modern large sample hydrologydatasets like CAMELS. Further, future work should build and test novel methods (specifically madefor hydrological data) that can identify and remove disinformative data in order to ensure betterhydrological inferences. Currently, CAMELS-like datasets are limited to lumped climatology data atdaily time steps, but future work should determine the utility of gridded and hourly catchment dataas well as how to incorporate these into modern (and traditionally lumped) hydrological models.Pushing the boundaries of data accuracy and granularity can inform where and when our models,rather than our data, are at fault and need to be improved. In our fourth chapter, we discuss thepervasive issue of sampling bias in groundwater studies. Though it is more subtle in surface waterhydrology, data from rain and stream gauges could incur missing values at nonrandom instances dueto flood events causing logger malfunctions or due to the coevolution of Earth systems. The impactof this and its possible resolution through counterfactually generated data augmentation strategiesshould be explored further [Xie et al., 2021].One key part of hydrological model development discussed very little in this thesis is optimization.Keith Beven often discusses the space of âbehaviouralâ models, which include those that surpasssome expected threshold of predictive accuracy [Gupta et al., 2024, Beven, 1996a]. Although itis usually hidden from us behind easy-to-use software packages, optimization methods pick one ofthese âbehaviouralâ models in a arbitrary and sometimes even biased way. Exploring the entirespace of behavioural models via special types of optimization methods may be useful to determineparameter identifiability, physical realism, and methodological shortcomings [Maus et al., 2023]. Forexample, in the case of our rainfall-runoff models presented in Chapter 3, we should explore the rangeof smoothness and sparsity hyperparameters that lead to acceptable solutions and identify betteroptimization schemes that lead to either better predictions or better inferences (as these goalsmay be meaningfully different). Due to the potentially large space of âbehaviouralâ models onemay experience parameter nonidentifiability in the worst case scenario, meaning that a potentiallyinfinite set of parameters are equally valid [Guan et al., 2024, 2025]. In this case, a particularoptimization method may be biased towards a specific explanation due to noise structure or theproperties of the loss landscape. For example, Tsai et al. [2021] showed that an LSTM recoversparameters that are smoothly distributed across space, but it is questionable whether this is alwaysthe case, which would indicate a possible smoothness bias. Likewise, Wang and Gupta [2024b] builta physically interpretable machine learning-based model, but they found that the model massively1375.3. Future Workprefers to generate baseflow rather than overland or shallow subsurface flow, indicating a possiblebias towards complexity. This strongly relates to our next topic of future work: ensuring thatdata-driven hydrological models are causal.While causality seems like a lofty goal for a discipline as messy as hydrology, causality, and moregenerally system identification, can ensure we get the right answer for the right reasons, our modelscan extrapolate to unseen conditions, our inferred parameters are physically-meaningful, and thelearned latent variable are representative of the true system. Thus, future work in this area is ofextreme importance. In particular, a substantial amount of future attention should be paid to re-solving the potential biases in all models (conceptual, machine learning, hybrid) due to optimizationissues discussed in the previous paragraph. This is fundamentally a causal problem as one needsto force these models to learn the true data generating processes instead of getting distracted byeasy-to-use correlations. By learning the data generating processes, one would hope that we can addcomplexity to the conceptual part of hybrid models while reducing the complexity and increasing theinterpretability of the data-driven portions. This perhaps can be done in the future by combiningdifferentiable modeling with hypothesis testing [Wang et al., 2025], in order to learn new and morecomplex functional relationships between latent drivers of streamflow. One should worry however,that imposing a true physical constraint on one part of the model may make another part of themodel less physical. This phenomenon has been observed in Yang and Chui [2021] and discussed inLi et al. [2023], but it is extremely relevant to progress in causal hydrology, so much more researchis needed in this direction.Finally, and perhaps most importantly, future work should focus on learning hydrological lawsthat are applicable in undistributed natural environments and at scales that are more relevant tolarger regional studies (e.g., 1 km or 100 m). Based on our concurrent work on the identifiability ofadvection-diffusion parameters based on observations of plume snapshots [Guan et al., 2024, 2025], itmay be possible to learn the complex hydrological dynamics in undisturbed systems from geophysicalinversion data [Mitchell et al., 2011]. 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Extra figuresFigure A.2: The spatial maps of streamflow signatures. All signatures were normalized and trans-formed in this figure to simplify the spatial comparison.186A.1. Extra figuresFigure A.3: Tukeyâs honest significant difference test to determine whether values of streamflowsignatures between pairs of clusters, devolved by each framework, are statically different. The darkblue shade (light blue shade) suggest a statistical difference under 1% (5%) significance level. Eachpanel shows Tukeyâs p-values for one streamflow signature estimated by the three frameworks. Theframework with a larger number of dark or light blue squares is a stronger framework in identifyingcatchments (dis)similarities in streamflow signatures. a) shape-based streamflow signatures, b)magnitude-based streamflow signatures. This figure shows the results for clustering with six clusters.Figure 2.4 (in the main text) shows the results for clustering with five clusters.187A.1. Extra figuresFigure A.4: Maps of hydrologic clustering developed based on long-term streamflow observations(top panel) and the representative regime curve (mean regime curve) of catchments belonging toeach cluster (bottom panel). Representative regime curve of each cluster is colored according tothe clusterâs color. This figure was reproduced from clustering results released by Brunner et al.[2020]. These results are referred to as âindependent streamflow-based clusteringâ in the mainmanuscript. Brunner et al. [2020] labeled the representative regime curves belonging to the fiveclusters as: intermittent regimes (red), weak winter regimes (orange), strong winter regimes (grey),melt regimes (green) and New Yearâs regimes (blue).188Appendix BAppendix for âUltra-marginal FeatureImportance: Learning from Data withCausal GuaranteesâB.1 Mutual informationB.1.1 Properties of mutual informationTheorem B.1.1 (Symmetry of conditional mutual information [Yeung, 2002]).I(Y ;X|Z) = I(X;Y |Z)Theorem B.1.2 (Chain rule for mutual information [Yeung, 2002]).I(Y ;X,Z) = I(Y ;Z) + I(Y ;X|Z) = I(Y ;X) + I(Y ;Z|X)Theorem B.1.3 (Supermodularity under independence). Let S,X1, X2 be random variables suchthat X1 â¥â¥ (S,X2). Then, I(Y ;S,X1, X2) â I(Y ;S,X2) â¥ I(Y ;S,X1) â I(Y ;S) [Lau et al., 2022,Steudel and Ay, 2015].Proof.I(Y ;S,X1, X2)â I(Y ;S,X2)= I(Y ;S,X2) + I(Y ;X1|S,X2)â I(Y ;S,X2) (by chain rule)= I(Y ;X1|S,X2) = I(X1;Y |S,X2) (by symmetry)= I(X1;Y, S,X2)â I(X1;S,X2) (by chain rule)= I(X1;Y, S,X2) = I(Y, S,X2;X1) (by X1 â¥â¥ (S,X2) and symmetry)â¥ I(Y, S;X1) = I(X1;Y, S) (by monotonicity of mutual information and symmetry)= I(X1;Y |S) = I(Y ;X1|S) (by chain rule, X1 â¥â¥ S, and symmetry)= I(Y ;S,X1)â I(Y ;S) (by chain rule)Theorem B.1.4 (Data processing inequality). Let X,Y, Z be three random variables forming aMarkov chain X â Y â Z, i.e. X â¥â¥ Z|Y . Then, I(X;Y ) â¥ I(X;Z).189B.1. Mutual informationProof. The proof can be found in Cover and Thomas [2006, p. 32].Theorem B.1.5. Let F be a set of features used to predict the response Y . Then I(Y ;F ) â¥I(Y ; g(F )) for any deterministic function g. If g is injective, then I(Y ;F ) = I(Y ; g(F )).Proof. The first claim I(Y ;F ) â¥ I(Y ; g(F )) follows from the data processing inequality (TheoremB.1.4) since Y â F â g(F ) forms a Markov chain.If g is injective, then we may write F = h(g(F )) where h : Im(g)â F is the inverse of g restrictedto the image of g. By the data processing inequality and the fact that Y â¥â¥ h(g(F ))|g(F ) we knowthat I(Y ; g(F )) â¥ I(Y ;h(g(F ))). Then by the definition of h and g, we know that I(Y ;h(g(F ))) =I(Y ;F ), thus I(Y ; g(F )) â¥ I(Y ;F ). Combining with I(Y ;F ) â¥ I(Y ; g(F )) yields the desired claim,I(Y ; g(F )) = I(Y ;F ) when g is injective.B.1.2 Mutual information and feature importanceLet F = {x1, ..., xp} be a set of features used to predict Y . As shown in Griffith and Koch [2014],the mutual information I(Y ;F ) = I(Y ;X1, ..., Xp) can be visualized using a partial information(PI) diagram [Williams and Beer, 2010]. We may interpret the mutual information shared betweenY and F as a collection of non-negative pieces of information, whose sum forms I(Y ;F ). Eachof these pieces of information can be classified as unique, redundant, or synergistic (Figure B.1).Unique information is the information about Y that comes from only one feature and nowhere else.Redundant information is information about Y that comes from a single feature, but which can alsobe found elsewhere in F . Synergistic information is information about Y that cannot be extractedfrom a single feature, but is available when multiple features are considered. See Dutta et al. [2020,2021] for interesting connections between PID and fairness.We note that the distinction between feature importance methods that seek to explain data versusmethods that seek to explain or optimize a model comes from their treatment of redundant infor-mation [Wollstadt et al., 2021]. Methods for explaining data, such as MCI or UMFI, aim to countall of the redundant information pertaining to Xi in I(Y ;F ) towards the feature importance of xi.Indeed, even though this information can be found elsewhere by a model, redundant information stillconstitutes part of the information that Xi shares about Y in the data. Conversely, a method mainlymade for feature selection, such as conditional permutation importance (CPI), aims to count none ofthe redundant information towards the evaluation of a featureâs importance, since this informationis already found in another feature.Mutual information itself is a common choice in the context of feature selection [Battiti, 1994, Al-Aniet al., 2003, Yang and Ong, 2012, Bennasar et al., 2015]. However, due to the computational costand the limited number of observations available for the calculation of the high-dimensional jointprobability density function, it is not practical to compute I(Y ;S). For feature selection, users areonly interested in the importance given to the top k features. Therefore, mutual information-basedfeature selection methods typically bypass the computation of I(Y ;S) by instead studying the mutualinformation between the candidate feature and the response along with the mutual informationbetween the candidate and the previously selected features [Bennasar et al., 2015, Battiti, 1994].These methods are much less suitable for feature importance when the goal is to explain the datasince interactions cannot be considered, which is why the most common approach is to train machinelearning models to determine an approximation of the universal predictive power [Catav et al., 2021a,Chen et al., 2020a, Covert et al., 2020, Williamson and Feng, 2020].190B.1. Mutual informationFigure B.1: PI-diagrams taken from Griffith and Koch [2014] for I(Y ;F ) when |F | = 2 (left) and|F | = 3 (right). Magenta represents unique information, redundant information is colored withyellow, and synergistic information is in cyan. The starred regions represent a single region.Another connection between feature importance and mutual information comes from Louppe et al.[2013], who showed that when extremely randomized treesâ mean decrease in impurity (MDI) is usedas a feature importance score, the MDI of a single feature converges to a weighted sum of conditionalmutual information as the number of trees and the number of observations goes to infinity [Louppeet al., 2013]. Also, the sum of the MDI scores across the feature set F converges to I(Y ;F ).B.1.3 Mutual information and machine learning evaluation functionsThe evaluation function for a machine learning model νf (S) measures how well the response Y canbe predicted using the model f given the information subset S â I(F ). Intuitively, the predictabilityor best possible accuracy νf (S) should ideally mirror or at least covary with the mutual informationI(Y ;S) [DelSole and Tippett, 2007, Gong et al., 2013]. While entropy and mutual information areusually examined in discrete settings, the continuous entropy and mutual information are linearlyrelated to the analogous discretized version [Gong et al., 2013]. In the case of regression, one canalso closely relate mutual information to the explained variance of a model. Indeed, with someassumptions, mutual information and R2 accuracy are related. If we assume the response andpredictions are jointly Gaussian and the predictions are unbiased [Cover and Thomas, 2006], we canapproximate the mutual information between Y and F as:I(Y ;F ) â¥ I(Y ; g(F )) = I(Y ; YË ) = â12log[1â Ï2(Y, YË )] = â12log[1âR2].Machine learning evaluation functions and mutual information have been equated many times inthe feature importance literature. Covert et al. [2020] demonstrated equivalence when the Bayesclassifier is known and cross entropy loss is used. In a simple example, Catav et al. [2020] usedmutual information directly as the evaluation function. The connection between machine learning191B.2. Additional information about marginal contribution feature importance (MCI)evaluation functions and mutual information was further used by Sutera et al. [2021] to relate randomforest feature importance with Shapely values.B.2 Additional information about marginal contribution featureimportance (MCI)Two of the methods that are compared with MCI in Catav et al. [2021a] include ablation andbivariate association. Ablation methods determine feature importance based on the difference inaccuracy between the full model and the full model without the feature of interest, i.e. Aν(xi) =ν(F ) â ν(F {xi}). Bivariate methods are among the most popular methods for genome-wideassociation studies [Consortium et al., 2007, Easton et al., 2007, Sun et al., 2021c]. In this method,the feature importance is given by the difference in the evaluation function of the model with justthe feature of interest and the null model, i.e. Bν(xi) = Î½(xi)â Î½(â ). The three feature importanceaxioms proposed by Catav et al. [2021a] were partially motivated by the shortcomings of these twomethods.1. Marginal contribution: Ablation methods may underestimate the importance of featureswhen the correlation between features is high. In these scenarios, ν(F ) may be approximatelyequal to ν(F {xi}) even in cases where xi is highly related to the response. Because ofthis, the importance of a feature Iν(xi) should be at least as large as the importance given byablation methods: Iν(xi) â¥ Aν(xi) = ν(F )â ν(F {xi}) âxi â F .2. Elimination: Bivariate methods may underestimate the importance of features in cases whereinteractions exist between features. Many high-order interactions may be present in the data,so eliminating features from the feature set could prevent the detection of an important in-teraction. Thus, eliminating features from F should only be able to decrease the featureimportance of xi.3. Minimalism: Catav et al. [2021a] decided to impose the minimalism axiom so that MCI canbe unique. If Iν(xi) satisfies the first two axioms, then multiplying Iν(xi) by any constantλ > 1 would not change this. The minimalism axiom helps disambiguate MCI from thesetrivial variations.We intentionally excluded some of the MCI axioms and properties included by Catav et al. [2021a]when proposing axioms for explaining data in Section 2.3.2. Most importantly, the marginal con-tribution axiom is not included because it conflicts directly with the blood relation axiom. Indeed,ablation methods could give too much importance from a scientific inference perspective. For ex-ample, in the collider example presented by Harel et al. [2022], they present the causal graphY â S â G â E, where S is unmeasured. Let F = {E,G} be used to predict Y . Then, themarginal contribution axiom requires that feature E is given importance. Indeed, if we know G,then feature E can help predict the response by denoising G to recover information from the unob-served cause S. Thus, Iν(E) â¥ Aν(E) = Î½({E,G})â ν({G}) > 0. However, as stated in Harel et al.[2022], feature E has no relation to the response Y , since it can be thought of as a noise variable, soit would be more reasonable to give E zero importance. We note that E is given zero importanceunder the blood relation axiom, so the blood relation axiom is more reasonable and justified com-pared to the marginal contribution axiom. In contrast, G inherently contains information about Yvia S, but this information is noised up by E. Therefore, although E can be used to denoise G andpredict Y better, only G should be given importance when explaining the data when F = {E,G},192B.3. Additional information about ultra-marginal feature importance (UMFI)and indeed, G is blood related to Y . We note that UMFI obeys the blood relation axiom under someassumptions, and hence does not obey the marginal contribution axiom. We additionally excludethe minimalism axiom since we do not prioritize uniqueness.B.3 Additional information about ultra-marginal featureimportance (UMFI)The proof of the main theorem in our paper is as follows:Proof. Since ν(·) = aI(Y ; Â·)+ b for some a â¥ 0 and b â R, UF,yν = ν(SFxi âª{xi})âÎ½(SFxi), and scalingby a constant factor does not alter any axiom, without loss of generality, it suffices to prove theaxioms for the case where ν(·) = I(Y ; Â·).(i) Elimination: UF{xj},yν (xi) â¤ UF,yν (xi).Let xj be the eliminated feature, and let xi â F {xj} be the feature of interest. Theorem B.3.1in the Supplement shows that pairwise linear regression attains an optimal preprocessing SËFxi since(F, y) is multivariate Gaussian. After mean centering, pairwise linear regression transforms eachfeature xk â F {xj} into xËk = xk â βkxi, where βk is the regression coefficient of xk on xi. Hence,SËFxi = (SËF{xj}xi , xËj) andUF,yν (xi) = I(Y ; SËFxi , Xi)â I(Y ; SËFxi)= I(Y ; SËF{xj}xi , XËj , Xi)â I(Y ; SËF{xj}xi , XËj)â¥ I(Y ; SËF{xj}xi , Xi)â I(Y ; SËF{xj}xi )= UF{xj},yν (xi),where the inequality is given by the supermodularity of mutual information under independence(Theorem B.1.3), since Xi â¥â¥ (SËF{xj}xi , XËj) follows from the optimality of SËFxi .(ii) IRI: Let xË â I(F ), then UF,yν (xi) = UFâª{xË},yÎ½ (xi).By the optimality of the preprocessings and the invariance of mutual information under redundantinformation,UF,yν (xi) = I(Y ; SËFxi , Xi)â I(Y ; SËFxi)= I(Y ;F )â I(Y ; SËFxi)UFâª{xË},YÎ½ (xi) = I(Y ; SËFâª{xË}xi , Xi)â I(Y ; SËFâª{xË}xi )= I(Y ;F )â I(Y ; SËFâª{xË}xi )From Lemma B.3.3, [SËFxi ] â¡ [SËFâª{xË}xi ]. In other words, every optimal preprocessing SËFxi â [SËFxi ] alsoexists in [SËFâª{xË}xi ] and vice versa. Thus, we may select the same optimal preprocessing from eachequivalence class to ensure that I(Y ; SËFxi) = I(Y ; SËFâª{xË}xi ), which concludes the proof.SD: Let xË = h(xj) for some bijective function h. Then, UFâª{xË},yÎ½ (xË) = UFâª{xË},yÎ½ (xj).193B.3. Additional information about ultra-marginal feature importance (UMFI)We have UFâª{xË},yÎ½ (xË) = I(Y ;F )âI(Y ; SËFâª{xË}xË ) and UFâª{xË},yÎ½ (xj) = I(Y ;F )âI(Y ; SËFâª{xË}xj ). Similarlyto the proof of IRI, it suffices to show that [SËFâª{xË}xj ] â¡ [SËFâª{xË}xË ], which is proven in Lemma B.3.3 inthe Supplement.(iii) Blood relation: UF,yν (xi) > 0 ââ xi â BRG(Y ).We equivalently prove UF,yν (xi) = 0 ââ Xi Ì¸â BRG(Y ). First, we write UF,yν (xi) = I(Y ;SFxi , Xi)âI(Y ;SFxi) = I(Y ;xi|SFxi). Also, from the definition of conditional mutual information,UF,yν (xi) = 0 ââ I(Y ;Xi|SFxi) = 0 ââ Xi â¥â¥ Y |SFxi .Next, we claim that UF,yν (xi) = 0 ââ xi â¥â¥ Y . A conditional independence axiom that holds forthe variables in a structural causal model is the contraction axiom [Dawid, 1979]: X â¥â¥ Y and X â¥â¥W |Y ââ X â¥â¥ (W,Y ). From this, we obtain the statementXi â¥â¥ SFxi and Xi â¥â¥ Y |SFxi ââ Xi â¥â¥ (Y, SFxi)Since Xi â¥â¥ SFxi is true from the optimality of the preprocessing, this property can be reduced toXi â¥â¥ Y |SFxi ââ Xi â¥â¥ (Y, SFxi) ââ Xi â¥â¥ Y (B.1)where the last equivalence is due to Xi â¥â¥ SFxi and the fact that (SËFxi , Xi, Y ) is multivariate Gaussian[Steudel and Ay, 2015, Lauritzen and Sadeghi, 2018].All that is left to prove is Xi â¥â¥ Y ââ Xi Ì¸â BRG(Y ). The claim Xi Ì¸â BRG(Y ) =â Xi â¥â¥ Yfollows from the global Markov property. Indeed, it is easy to check that Xi and Y are d-separatedby the empty set in this case. The converse holds because we have assumed that the graphical modelobeys faithfulness. Indeed, Xi and Y would be d-connected by the empty set if Xi â BRG(Y ).Theorem B.3.1 (Existence of optimal preprocessing SËFxi when all features are jointly Gaussian). Letxi â F and suppose that all random variables in the random vector F are joint normally distributed,then there exists a preprocessing SFxi that is optimal.Proof. A preprocessing SFxi can be obtained via multiple linear regression (after mean centering)with the model:F {xi} = βxi + ϵ,where ϵ = SFxi , xi is a feature in F , and β is the column vector of size p â 1 containing regressioncoefficients β1, β2, ..., Î²pâ1 that minimize the sum of squared errors between xi and a linear functionof each other variable in F {xi}.To show that SFxi is an optimal preprocessing (Definition 1), it suffices to show that SFxi â¥â¥ Xi andthat I(Y ;F ) = I(Y ;SFxi , Xi), since SFxi is a function of F by construction.From the normal equations and the definition of covariance, we know that Cov(SFxi , Xi) = 0, asshown in the proof of Theorem B.5.3. Since SFxi = F {xi} â βxi, and all features in F are jointnormally distributed, it follows that (SFxi , xi) is joint normally distributed as well, since (SFxi , xi) canbe obtained via the linear transformation AF = (SFxi , xi), where the main diagonal entries of A are1, the other |F | â 1 entries of the column corresponding to xi are given by the entries of âÎ², and194B.3. Additional information about ultra-marginal feature importance (UMFI)all other entries are 0. Without loss of generality, we may reorder the columns of the matrix suchthat the last column is attributed to feature xi, and writeA =ï£®ï£¯ï£¯ï£¯ï£°1 0 . . . . . . âÎ²10 1 0 . . . âÎ²2.... . .0 0 . . . . . . 1ï£¹ï£ºï£ºï£ºï£» Aâ1 =ï£®ï£¯ï£¯ï£¯ï£°1 0 . . . . . . β10 1 0 . . . β2.... . .0 0 . . . . . . 1ï£¹ï£ºï£ºï£ºï£» .Hence, Cov(Xi, SFxi) = 0 =â SFxi â¥â¥ Xi from the properties of multivariate Gaussians.To prove the second claim I(Y ;F ) = I(Y ;SFxi , Xi), by Theorem B.1.5, it suffices to show that themap h(F ) = (SFxi , xi) = AF is injective. This is immediate from the fact that the matrix A, definedabove, is invertible and thus bijective.Theorem B.3.2 (Elimination axiom assuming optimal transport with chaining). Let xi â F ,xp+1 Ì¸â F , and ν(S) is positively linearly related to I(Y ;S). When preprocessing is performedusing optimal transport with chaining, UF,Yν (xi) â¤ UFâª{xp+1},YÎ½ (xi).Proof. Let SFâª{xp+1}xi be the preprocessed version of F âª {xp+1} relative to xi and let SFxi be thepreprocessed version of F relative to xi. By optimal transport with chaining [Johndrow and Lum,2019], we may assume that SFâª{xp+1}xi obeys the form SFâª{xp+1}xi = SFxiâªxË and that SFxi , Xi, XË are mu-tually independent. It follows from the supermodularity of mutual information under independence(Theorem B.1.3) thatUFâª{xp+1},YÎ½ (xi) = aI(Y ;SFâª{xp+1}xi , Xi) + câ (aI(Y ;SFâª{xp+1}xi ) + c)= aI(Y ;SFxi , XË,Xi) + câ (aI(Y ;SFxi , XË) + c)â¥ aI(Y ;SFxi , Xi) + câ (aI(Y ;SFxi) + c) = UF,Yν (xi).Lemma B.3.3 (Equivalence of optimal preprocessing equivalence classes under redundant informa-tion and duplicate features of interest). If xË â I(F ), then [SËFxi ] â¡ [SËFâª{xË}xi ]. If xË = h(xj) and h isbijective, then [SËFâª{xË}xj ] â¡ [SËFâª{xË}xË ].Proof. Recall that an optimal preprocessing given a feature set F and a feature of interest xi isdefined in Definition 1. To prove [SËFxi ] â¡ [SËFâª{xË}xi ], we prove that for any xi â F all optimalpreprocessings SËFxi are also optimal preprocessings in [SËFâª{xË}xi ], and that all optimal preprocessingsSËFâª{xË}xi are also optimal preprocessings in [SËFxi ].We first note that properties 1 and 2 in Definition 1 are equivalent for SËFxi and SËFâª{xË}xi . For property1, a function with repeated arguments can be defined to be equal to the same function withoutrepeated arguments. For property 2, the feature of interest xi is consistent across both optimalpreprocessings, so both preprocessings are independent of Xi. Lastly, since mutual information is195B.3. Additional information about ultra-marginal feature importance (UMFI)invariant under duplicate information and since SËFxi and SËFâª{xË}xi each satisfy their respective 3rdproperty,I(Y ;F, XË) = I(Y ; SËFâª{xË}xi , Xi) = I(Y ;F ) = I(Y ; SËFxi , Xi). (B.2)Hence, the final property is also equivalent across both equivalence classes of optimal preprocessings,and we conclude [SËFxi ] â¡ [SËFâª{xË}xi ].Similarly, to prove that [SËFâª{xË}xj ] â¡ [SËFâª{xË}xË ] if xË = h(xj) for some bijective function h, we note thatthe first property is equivalent for both optimal preprocessings, since they take place over the samefeature set F âª {xË}. Then, we note that SËFâª{xË}xj â¥â¥ Xj =â SËFâª{xË}xj â¥â¥ h(Xj) = XË, and similarly,SËFâª{xË}xË â¥â¥ XË =â SËFâª{xË}xË â¥â¥ hâ1(XË) = Xj . Finally, since mutual information is invariant underhomeomorphic reparametrization of marginal variables [Kraskov et al., 2004],I(Y ;F ) = I(Y ; SËFâª{xË}xË , XË) = I(Y ; SËFâª{xË}xË , hâ1(XË)) = I(Y ; SËFâª{xË}xË , Xj)I(Y ;F ) = I(Y ; SËFâª{xË}xj , Xj) = I(Y ; SËFâª{xË}xj , h(Xj)) = I(Y ; SËFâª{xË}xj , XË)Since all 3 properties are equivalent for both optimal preprocessings, we conclude that [SËFxi ] â¡[SËFâª{xË}xi ].We note that preprocessings SFxi and SFâª{xË}xi may be interchangeable without being optimal, andthat the interchangeability of these preprocessings is a sufficient condition for UMFI satisfying theredundant information invariance axiom, as long as ν(·) = I(Y ; Â·). For example, interchangeabilityof preprocessings also holds when the removal of dependencies on a feature xi is done in a pairwisefashion (see Algorithm 4), as well as when preprocessings is performed via optimal transport withchaining [Johndrow and Lum, 2019].Theorem B.3.4 (Blood relation axiom assuming faithfulness). Let xi â F , ν(S) is positivelylinearly related to I(Y ;S), and suppose that the data is generated from a structural causal model Cwith corresponding directed causal graph G so that the entailed distribution is faithful to G. Assumealso that there exists an structural causal model with graph Gâ² that contains the variables Xi, SFxi,and Y , where the distribution of all its variables is faithful to the graph Gâ². If the preprocessingSFxi â¥â¥ Xi, then UF,Yν (xi) > 0 if and only if Xi â BRG(Y ).Proof. As shown in the proof of Theorem 2.3.1 in the main text, we may apply the definition ofUMFI and ν, the assumption SFxi â¥â¥ xi, and properties of mutual information and conditionalindependence to obtainUF,Yν (xi) = 0 ââ aI(Y ;Xi|SFxi) + câ c = 0 ââ Xi â¥â¥ Y |SFxi ââ Xi â¥â¥ (Y, SFxi) =â Xi â¥â¥ YWe note that the last implication comes from the contraction axiom [Dawid, 1979], and it can bestrengthened to an equivalence as long as we can prove Xi â¥â¥ (Y, SFxi) using the assumptions Xi â¥â¥ Yand Xi â¥â¥ SFxi . Since Xi, Y , and SFxi belong to an SCM with graph Gâ², and the entailed distributionof this SCM is faithful to Gâ², then Xi â¥â¥ Y and Xi â¥â¥ SFxi imply that Xi is d-separated from bothY and SFxi by â . Therefore, Xi â¥â¥ (Y, SFxi), and UF,Yν (xi) = 0 ââ Xi â¥â¥ Y .Also, if the data is faithful to the causal graph G, then Xi â¥â¥ Y is equivalent to Xi Ì¸â BRG(Y ),which would conclude the proof of the blood relation axiom. We explicitly provide the details.196B.4. Additional information about other feature importance methodsIf Xi Ì¸â BRG(Y ), then Xi â¥â¥ Y follows from the global Markov property and the fact that Xi and Yare d-separated by the empty set. Indeed, every path from Xi to Y must have at least one collider.We consider two cases. (1) The edge coming out of Y is outgoing. Then since Xi is not a descendentof Y , the path must reverse its orientation at some vertex before meeting Xi. That vertex is acollider. (2) The edge connecting to Y points towards Y . Then the path must reverse its orientationat some point since Xi is not an ancestor of Y . The path must then reverse another time becauseotherwise, Xi would share a common ancestor with Y (the vertex of the first reversal). The vertexwith the second reversal is a collider.Conversely, let Xi â BRG(Y ). By the faithfulness assumption, it suffices to show that Xi and Y ared-connected by the empty set. Since Xi â BRG(Y ), there are two possible cases: either there is adirected path between Xi and Y , or Xi and Y share a common ancestor. In the first case, we simplychoose the directed path between Xi and Y and observe that there cannot be a collider. Similarly,in the second case, we may pick the path beginning at Y and trace it up to the common ancestorand then travel to Xi. There can be no colliders along the path since every vertex has at least oneoutgoing edge by construction. Also, the empty set cannot contain any non-colliders.Theorem B.3.5 (Blood relation axiom in the absence of interactions). Suppose that there is nosynergistic information Isyn(Y ;SFxi , Xi) about Y between Xi and SFxi for all xi â F , and that SFxi â¥â¥ Xi. Then, if the graphical model obeys the global Markov property and faithfulness and ν(S) ispositively linearly related to I(Y ;S), then UF,Yν > 0 if and only if Xi â BRG(Y ).Proof. As in the proof of Theorem B.3.4, it suffices to show that I(Y ;Xi|SFxi) = 0 if and only ifXi Ì¸â BRG(Y ). We may rewrite I(Y ;Xi|SFxi) = 0 as I(Y ;SFxi , Xi) = I(Y ;SFxi).Though it is fairly controversial [Williams and Beer, 2010, Griffith and Koch, 2014], some definitionsof partial information decomposition imply that independent predictors cannot contain redundantinformation between them [Kolchinsky, 2022, Harder et al., 2013]. Using partial information de-composition [Williams and Beer, 2010], and since SFxi â¥â¥ Xi =â Ired(Y ;SFxi , Xi) = 0, we maydecompose I(Y ;SFxi , Xi) asIunq(Y ;Xi) + Iunq(Y ;SFxi) + Isyn(Y ;SFxi , Xi).where we note that, because of the lack of redundancy, I(Y ;Xi) = Iuniq(Y ;Xi) and that I(Y ;SFxi)captures the unique information that SFxi shares with Y as well as synergistic information withinthe random vector SFxi that is shared with Y . As proven in Theorem B.3.4, I(Y ;Xi) = 0 if Xi Ì¸âBRG(Y ) and I(Y ;Xi) > 0 if Xi â BRG(Y ) by the global Markov property and faithfulness. SinceIsyn(Y ;SFxi , Xi) = 0 by assumption, this gives us the desired statement I(Y ;SFxi , Xi) = I(Y ;SFxi) ifand only if Xi Ì¸â BRG(Y ).B.4 Additional information about other feature importancemethodsHistorically, feature importance methods were developed in the pursuit of scientific questions, butcurrent research in this area typically focuses on model explainability or model optimization. Earlyforms of feature importance assessed the strength of the relationships between variables within197B.5. Preprocessing methods for removing dependenciesanimal biology or human psychology using methods such as the correlation coefficient [Galton, 1889],Spearmanâs rank correlation coefficient [Spearman, 1961], multiple linear regression [Darlington,1968], and partial correlation [Wright, 1921]. Although these methods are perfectly interpretable,they are inadequate for modelling and therefore explaining complex data, since they cannot quantifythe unknown interactions between multiple features. To counteract this severe limitation, Breimanwas instrumental with his introduction of variable importance within classification and regressiontrees [Breiman et al., 2017]. At that time, Breiman seemed more concerned about the true strengthof the relationships between the explanatory variables and the response, as he posited that a featurethat is related to the response should be given some importance even if it does not appear inthe final model [Breiman et al., 2017]. However, starting with Breimanâs random forests, featureimportance began to prioritize machine learning model explanation rather than data exploration.A good overview of the properties of some popular feature importance metrics is shown in Covertet al. [2020].B.5 Preprocessing methods for removing dependenciesFinding information preserving independent representations of our data is the central step of UMFI.These representations were first considered for AI fairness and privacy algorithms in order to giveunbiased predictions in the face of sensitive attributes. For example, if one wants to remove theinfluence of race on recidivism likelihood predictions, preprocessing methods can be used to alter theoriginal dataset such that the set of predictors are independent of race. In the following subsections,we discuss how optimal transport and linear regression can be used for finding these representations.B.5.1 Optimal transportMost of the results and methods explained in this section can be found in Johndrow and Lum[2019]. In this section, we denote features in the feature set F by Xj or Xi to emphasize thatthey are random variables, rather than the previously used xj and xi, where the former is used todenote observations xj sampled from Xj instead. To obtain a preprocessing SFXi, we may removethe dependencies of xi from each Xj â F {Xi} with minimal information loss with respect to Xj .To do so using optimal transport, we consider the Monge problem:gc(Xj , XËj) = infg:g(Xj)â¼XËjE[c(Xj , g(Xj))] = infg:g(Xj)â¼XËjâ«Rc(xj , g(xj))dµ(xj). (2.1.1)The quantity gc(Xj , XËj) represents the transportation cost of moving Xj to XËj with respect tosome cost function c, and in our case, we desire XËj â¥â¥ Xi. It is natural to use c(xj , xËj) = dq(xj , xËj),where d is the Euclidean norm. The transportation cost is also given by the Wasserstein-q distance,gc(Xj , XËj) =Wqq (Xj , XËj), defined below for one-dimensional distributions.Wq(Xj , XËj)q =â« 10|Fâ(p)â FËâ(p)|qdp,where Fj and FËj are the CDFs of Xj and XËj , and Fâj (p) = supxjâR Fj(xj) â¤ p. It can be shownthat given any continuous one dimensional distributions Xj and XËj , the optimal transport mapg : Xj â XËj is given by g = FËâj â¦ Fj .198B.5. Preprocessing methods for removing dependenciesTheorem B.5.1. Let X be a r.v. with density f and CDF F . Let XË have CDF FË . Then g = FËââ¦Fis the minimizer to (2.1.1). Hence, g optimally transports X to XË = FËâ(F (X)).Proof. We show E[|X â g(X)|q] = â« 10 |Fâ(p)â FËâ(p)|qdp for g = FËâ â¦ FE[|X â g(X)|q] =â« âââ|xâ FËâ(F (x))|qf(x)dx=â« âââ|Fâ(F (x))â FËâ(F (x))|qf(x)dx =â« 10|Fâ(p)â FËâ(p)|qdpTheorem B.5.2. Let Fj|xi(x) = P (Xj â¤ xj |Xi = xi) denote the CDF of Xj |{Xi = xi} . Theng = FËâ â¦ Fj|xi optimally transports Xj |{Xi = xi} to XËj â¥â¥ Xi for any CDF FËProof. We apply Theorem B.5.1 on the random variable Xj |{Xi = xi} and note that Xj |{Xi = xi}is independent of Xi. In particular, g(Xj |Xi = xi) â¥â¥ Xi for any choice of FË .Theorem B.5.2 suggests an algorithm for transporting data (xj1, ..., xjn) sampled from Xj , to(xËj1, ..., xËjn) â¥â¥ (xi1, ..., xin). Since xjk is taken jointly with xik, as they are attributes comingfrom the kth sample in the dataset, then xjk is a realization of the distribution Xj |{Xi = xik}.Consequently, for each k = 1, ..., n, we should transport xjk to xËjk = FËâ(Fj|xik(xjk)), where wemay pick any CDF FË . This procedure can also adapted for features sampled from discrete r.vâs, asshown in Johndrow and Lum [2019].Algorithm 3: Algorithm for removing dependencies of Xi from XjRequire: Xj = [xj1, ..., xjn], Xi = [xi1, ..., xin], Xj |(Xi = xik) â¼ Fj|xik , FË is a CDFfor k = 1, ..., n doxËjk = FËâ(Fj|xik(xjk))end forreturn XËj = [xËj1, ..., xËjn]We denote the result of the algorithm by XËj = FËâ(Fj|Xi(Xj)) and would ideally pick FË such that itminimizes the transportation cost gc(Xj , XËj) = gc(Xj , FËâ(Fj|Xi(Xj))) across all CDFs FË in orderto minimize information loss. However, in practice, the choice of FË does not matter much. In fact,as long as the support of FË is at least a large as the support of Fj , the cdf of Xj , then any rank-basedprediction rule, e.g. random forest, will be invariant to the choice of FËj [Johndrow and Lum, 2019].A standard choice for FËj is Fj so that we can recover the original quantiles of Xj .Furthermore, Fj|xik is not usually known and must be estimated from the data. For example,this can be done by splitting Xi into N quantiles and using the empirical CDF P (Xj â¤ xj |Xi âxikâs quantile). The ability of this method to remove dependencies on Xi from Xj relies significantlyon the accuracy of this estimate.We may iterate Algorithm 3 over each feature in F {Xi} to obtain pairwise independence betweenthe transported variables XËj and Xi. It is also possible to iterate Algorithm 3 via chaining toachieve mutual independence between the transformed variables XËj and Xi [Johndrow and Lum,199B.5. Preprocessing methods for removing dependencies2019, Section 3.2]. However, this is computationally expensive, and pairwise independence shouldsuffice for an accurate UMFI score, as will be explored further in Section B.6. Step 2 of Algorithm1 in the main paper can therefore be implemented with Algorithm 4.Algorithm 4: Algorithm for estimating SFXi via pairwise optimal transportRequire: Xi = [xi1, ..., xin], Xj = [xj1, ..., xjn] for Xj in F XiSFXi = â for Xj in F {Xi} doXËj = output of Algorithm 3 with Xj and Xiadd XËj to SFZend forreturn SFXiIn other words, we may estimate SFXi as:SFXi = {Fâj (Fj|Xi(Xj)) : Xj â F {Xi}}.B.5.2 Linear regressionThe most basic method for removing dependencies is linear regression. Even though it is quitesimple, it can be shown to be optimal with a few assumptions (Theorem B.3.1). This preprocessingtechnique is implemented in the popular Python package fairlearn [Bird et al., 2020, Matthijs et al.,2019].To reiterate, removing dependencies requires methods to make a feature or set of features S inde-pendent of a protected attribute xi, while keeping as much of the original information as possible.The overarching idea of linear regression as a preprocessing technique is that under the assumptionthat the residuals and the protected attribute are jointly Gaussian, the residuals can be utilized asa representation of S, which is independent of xi.Theorem B.5.3. The residuals ϵ of a simple linear regression model have zero covariance with thepredictor X.Proof. (1) From the normal equations, the definition of covariance, and the fact that E[ϵ] = 0, itfollows thatCov(X, ϵ) = E[XT Ïµ]â E[ϵ]E[X] = E[XT ϵ] = E[XT (Y âXÎ²)]= E[XT (Y âX(XTX)â1XTY ))] = E[XTY âXTX(XTX)â1XTY ] = E[XTY âXTY ] = 0Thus, in step 2 of the algorithm for UMFI (Algorithm 1), we can estimateSFXi = {ϵj = Xj â β0,j â β1,jXi : Xj â F {Xi}}where β0,j is the intercept term and β1,j is the slope term of the linear regression model Xj =β0,j + β1,jXi + ϵj .200B.6. Experiments comparing linear regression and optimal transportB.6 Experiments comparing linear regression and optimaltransportIn the following subsections, we compare the ability of linear regression and pairwise optimal trans-port to remove the information of a feature from data while distorting the original data as little aspossible. It can be concluded that while linear regression works optimally when the data is jointlyGaussian, on real data, such as the BRCA dataset, pairwise optimal transport can find independentrepresentations of the data, while linear regression fails (Section B.6.1).To implement UMFI paired with linear regression, we only remove dependencies when the regressionslope coefficient is statistically significant (p-value < 0.01). To implement UMFI paired with pairwiseoptimal transport, when removing dependencies on the feature xi from the dataset, we estimate Fj|xikby breaking up xi into quantiles of size 150 and running linear regression on each quantile. The newindependent predictors are then given by the values of the inverse empirical CDF of the residualsfrom the mentioned linear regression models.B.6.1 Removing dependenciesIt is crucial for our linear regression and optimal transport preprocessing methods to remove theinformation associated with the feature of interest, xi, from the rest of the dataset F{xi}. Therefore,we would like the preprocessed dataset SFxi to share zero mutual information with xi. The mutualinformation I(Xi;SFxi) is difficult to calculate, but it is closely related to the optimal predictor of xigiven SFxi [Song et al., 2019]. For example, if I(Xi;SFxi) = 0, as is desired, then the optimal predictorof xi will have zero accuracy when given SFxi as input. If the opposite is true and SFxi contains all ofthe information from xi, then an optimal predictor of xi should be able to perfectly predict xi fromthe given information in SFxi . In the following experiments, we assume that random forests can formthe optimal predictor of xi given SFxi . We use the OOB-R2 coming from the random forest modelas a approximate measure of the mutual information between xi and the transformed dataset SFxi .We used the BRCA dataset with 50 features to test the ability of optimal transport and linearregression to remove dependencies [Covert et al., 2020, Catav et al., 2020]. All 50 features arecontinuous and the response is categorical. For each individual feature, we first use random forestOOB-R2 to give a approximate measure of the mutual information I(Xi;F {xi}) between thefeature of interest xi and the other 49 features. We then consider the case where the 49 remainingfeatures are preprocessed to have dependencies on xi removed via linear regression or pairwiseoptimal transport. Similarly, random forestâs OOB-R2 is used to give a approximate measure ofI(Xi;SFxi).The results are plotted in Figure B.2. It is clear that the raw data (black line) shares considerableinformation across features. Most features can be predicted from the other untransformed featureswith an accuracy ofR2 > 0.2 and many can even be predicted with accuracies over 0.4. Since the datahas extremely nonlinear dependencies between features, simple linear regression is unable to removeall the mutual information between the protected attributes and the rest of the features. Indeed, thedata certainly cannot be approximated with multivariate Gaussians. Conversely, pairwise optimaltransport can successfully remove most of the mutual information present in the data. For all 50features in the dataset, xi cannot be predicted successfully by random forest (OOB-R2 = 0) fromthe other features after F xi is transformed with pairwise optimal transport.201B.6. Experiments comparing linear regression and optimal transportFigure B.2: The coefficient of determination (random forest OOB-R2) between the ith feature inthe BRCA dataset and all other features is plotted (black) for each i â {1, 2, ...50}. The R2 valuebetween the ith feature and all other features after preprocessing with linear regression (red) andoptimal transport (blue) is also plotted.B.6.2 DistortionNot only do we require that the transformed features are independent of the feature of interest,but we also require that as much of the information present in the original data is preserved in thetransformed data. To measure the amount of distortion imposed on the original data, we measurethe dependence between the original and perturbed data using the maximal information coefficient[Kinney and Atwal, 2014]. For each feature in the BRCA dataset with 50 features [Covert et al.,2020, Catav et al., 2020], the information from the current feature is removed from all other featureswith either linear regression or pairwise optimal transport (Figure B.3).Linear regression does not distort the transformed features in most cases. The dependence betweenthe original and perturbed features usually remains near 1, though the dependence does go as lowas 0.42 in one case (Figure B.3). While linear regression transformed these features with minimaldistortion, these results are moot since linear regression failed to remove the original dependenciesin a significant way, which was the main goal of the method (Figure B.2).Compared to linear regression, pairwise optimal transport has a much more sizable effect on thedistorted features, though this may have been necessary to completely remove dependence. Thedependence between original and perturbed features mostly ranges from 0.6-0.9, though some areas low as 0.37 (Figure B.3). While only the first 15 features are shown, the results are similar forthe other 35 features.202B.7. Further feature importance experimentsFigure B.3: Cell (i, j) indicates how similar the jth variable in the BRCA dataset is compared to itstransformation via pairwise optimal transport or linear regression with respect to feature i. This ismeasured with the maximal information coefficient, which is comparable to R2. To make the plotsmore clear and accessible, only the first 15 features are shown.B.7 Further feature importance experimentsThis section is comprised of additional experiments performed on the simulated data introducedin Section 2.3.4, the BRCA dataset with permuted random genes, the original BRCA dataset withunpermuted random genes [Tomczak et al., 2015, Covert et al., 2020, Catav et al., 2021a], andthe CAMELS hydrology dataset [Addor et al., 2017]. MCI and UMFI used either random forestsor extremely randomized trees [Breiman, 2001a, Geurts et al., 2006]. Both of these, as well asablation and permutation importance were implemented using the ranger R package [Wright andZiegler, 2017], while conditional permutation importance was implemented with the randomForestand permimp packages [Debeer et al., 2021, Liaw et al., 2002]. All experiments were run in MicrosoftR Open Version 4.0.2.B.7.1 Extra experiments on simulated dataWe repeat our previous experiments on simulated data from Section 2.3.4 to test how ablation,permutation importance (PI), and conditional permutation importance (CPI) behave in the pres-ence of nonlinear interactions (Section B.7.1), correlated interactions (Section B.7.1), correlation(Section B.7.1), and blood and non-blood related features (Section B.7.1). Further, we test howusing extremely randomized trees instead of random forests for MCI and UMFI changes the resultsof the same simulation experiments. Although other methods such as XGBoost [Chen et al., 2015]could have been implemented for these experiments, XGBoost requires greater care when optimiz-ing hyperparameters, so we chose to use extremely randomized trees instead, which is faster thanrandom forests and provides similarly good predictions [Geurts et al., 2006]. Both random forests203B.7. Further feature importance experimentsand extremely randomized trees are not very sensitive to hyperparameter selection [Probst et al.,2019]. For these simulation studies, we also perturb the size of the quantiles used by UMFI OT. Wenow use quantiles of size 30 instead of size 150. Quantiles of size 30 worked better on the hydrologydata used in later experiments, so we test to see if the simulation results are sensitive to this choicein quantile size for dependency removal via optimal transport.Nonlinear interactionsThe first experiment on simulated data handles the case where two variables, x1 and x2, interactin a nonlinear way in the response Y . As explained in Section 2.3.4, we should expect x1 and x2to contribute more than half of the total importance, while x3 and x4 should be important, butless important compared to x1 and x2. Figure B.4a shows that ablation, PI, and CPI all provideaccurate scores.When tested with extremely randomized trees, the nonlinear interactions simulation experimentresults for MCI and UMFI, shown in Figure B.4e, remain mostly unchanged compared to the resultsfrom the experiment with random forests given in Figure 2.8a.Correlated interactionsThe second experiment considers the case where two correlated variables, x1 and x2, interact togetherin the response Y . Thus, as explained in Section 2.3.4, we should expect x1 and x2 to have moreimportance compared to x3 and x4. Figure B.4b shows that ablation, PI, and CPI all correctlyweigh the importance of x1 and x2 as high relative to x3 and x4. The only notable difference is thatthe ablation method attributes an additional â¼ 3% importance to each of x1 and x2 compared toPI, CPI, MCI, and UMFI (Figure B.4b).When tested with extremely randomized trees instead of random forests, the correlated interactionsimulation experiment results (Figure B.4f) for MCI and UMFI are similar to the earlier resultsshown in Figure 2.8b. MCI gave slightly more importance to x1 and x2 compared to x3 and x4,though the differences are seemingly insignificant. On the other hand, both UMFI methods gavesignificantly more importance to x1 and x2 compared to x3 and x4, as expected.CorrelationThe third experiment tests how the metrics allocate importance to correlated features. As explainedin Section 2.3.4, x1 and x2 should remain around the same relative importance, and x3 = x1 + ϵ,should have just slightly less importance compared to x1 and x2. Figure B.4c indicates that CPIand ablation give near zero importance to the two heavily correlated features x1 and x3. Thisaligns with the discussion in Section B.1.2 about methods motivated by feature selection since thesemethods base their scores on the importance of a feature conditioned on all other variables presentin the model. Ablation performs similarly to CPI in this test, albeit with slightly less drastic results.Finally, we see that PI splits the importance detected from x1 and x3 proportionally across bothfeatures. This shows that PI can be viewed as a method for model explanation which in between thescientific inference and feature selection approaches. The scientific inference approaches (MCI andUMFI) allocate all of the redundant information to the feature. The feature selection approaches(CPI and ablation) allocate none of the redundant information to the feature. PI evenly splits theredundant information across the relevant correlated features.204B.7. Further feature importance experimentsWhen tested with extremely randomized trees, the correlation simulation experiment results (FigureB.4g) for MCI and UMFI change slightly compared to the experiment with random forests in Figure2.8c. MCI works well, though it still gives some non-zero importance to x4. With random forests, therelative importance of x4 was usually above 5%, but with extremely randomized trees, the relativeimportance dropped below 5%. The performance of UMFI with linear regression got slightly worseas now the importance of x1 is slightly greater than that of x2 on average. The performance ofUMFI with optimal transport changed for the better and now the importance of x1 and x2 arealmost identical which was not true before. In this experiment, UMFI OT performed the best.Blood relationFor the last simulation experiment, we revisit the blood relation experiment performed in Section2.3.4 using data generated from the causal graph in Figure 2.9. The feature S is unobserved, sothe only blood related features to Y in F are x3 and x4. x3 and x4 should therefore be given highimportance while x1 and x2 should receive zero importance. When tested on ablation, CPI, andPI, we notice that all three metrics fail to capture the desired importance, since they each givesignificant importance to x2, which is not blood related to Y . We also note that this experimentprovides an explicit example of UMFI not satisfying the marginal contribution axiom, which statesthat feature importance metrics should allocate at least as much importance as attributed by theablation metric. Indeed, as shown in Figure 2.8d, UMFI gives around zero importance to non-bloodrelated features x1 and x2, whereas ablation gives a significant portion of the importance to x2.When MCI and both implementations of UMFI were re-tested using extremely randomized treesinstead of random forest, we observe that UMFI LR and UMFI OT both continue to give positiveimportance to the blood related features x3 and x4, while giving near-zero importance to the tworemaining observed features (Figure B.4h). However, we note that x3 is given much more importancerelative to x4 when implemented with extremely randomized trees compared to random forests(Figure 2.8d). On the other hand, MCI gives positive importance to x2 in this experiment. Wenote that MCI correctly gave x1 almost zero importance while giving x3 and x4 significantly moreimportance compared to the random forest implementation. Across most simulation studies, itappears MCI performs slightly better using extremely randomized trees compared to random forests.BRCA experimentsWe use the same breast cancer (BRCA) classification dataset [Tomczak et al., 2015] used in previousfeature importance studies including Catav et al. [2021a] and Covert et al. [2020] to test the qualityand robustness of UMFI on real data. The original data contains over 17, 000 genes and 571 anony-mous patients with one of four types of breast cancer. We consider the same subset of 50 genes asin Catav et al. [2021a] and Covert et al. [2020] for easier computation and result visualization. Ofthe 50 selected genes, 10 are known to be associated with breast cancer, while the other 40 genesare randomly sampled. This data was downloaded from the MCI GitHub page. In Catav et al.[2021a] and Covert et al. [2020], these 40 randomly sampled genes are assumed to be unassociatedwith breast cancer. However, to ensure a more definitive ground truth, we also randomly permutethe values of these 40 genes across their respective 571 observations to further reduce the chancethat these genes have any association with breast cancer.Quality is then measured with the true positive and true negative rates: the 10 BRCA associatedgenes should have some non-zero importance (positive), and the other 40 genes should have exactly205B.7. Further feature importance experimentszero importance (negative). These experiments were run 200 times on different seeds and witha different random sample of 500 patients for each iteration. Robustness is measured using thestandardized interquartile range (SIQR) from the repeated experiments, which is calculated bydividing the average IQR across the 50 features by the average median. This experiment is toocomputationally intensive for MCI to be calculated exactly, so we implement MCI assuming soft2-size submodularity (see the supplement of Catav et al. [2021a] for details).We found that MCI and UMFI (UMFI LR and UMFI OT) correctly gave significant importance tothe 10 genes that are known to be associated with breast cancer (Figure B.5). Interestingly, theorder of important features was similar across methods, with BCL11A and SLC22A5 always rankingfirst, and TEX14 always being the least important of the 10 BRCA-associated genes. However, MCIconsistently gives non-zero median importance to all features, while UMFI correctly gives zeromedian importance to the majority of the randomized genes. Furthermore, UMFIâs performancein this experiment improves with increased iterations. After running the experiment 5000 times,both UMFI methods achieve perfect accuracy when distinguishing between important and permutedfeatures (Supplement B.7.2). Although UMFI scores have higher variability than MCI (Table B.1),Figure B.5 shows that UMFI provides a more accurate and interpretable scoring, since it betterseparates the 10 associated genes from the 40 unassociated genes.Table B.1: The standardized interquartile range (SIQR), true positive rate (TPR), true negativerate (TNR), overall accuracy (OA), and the number of features for which feature importance canbe calculated within 60 minutes are displayed after running the methods on the BRCA data.Method SIQR TPR TNR OA @1hrMCI (k=2) 6.6 % 1 0 0.20 130UMFI (LR) 41.9% 1 0.975 0.98 4010UMFI (OT) 28.5% 1 0.775 0.82 3000B.7.2 Extra BRCA experiments with known ground-truth feature importanceThe following experiments are performed on the BRCA dataset with 571 patients, each with oneof four breast cancer subtypes, and 50 continuous predictor genes. The experiments use the samesetting as in Section B.7.1, where the 40 randomly chosen genes are also permuted so that theground-truth feature importances are known. We observed that the overall classification accuracyof random forests for this dataset was 0.76.Running 5000 iterations of UMFIThe original BRCA experiment conducted in Section B.7.1 showed that UMFI LR and UMFI OTperformed impressively on real data, providing significantly more accurate feature importance scoresthan MCI after 200 iterations of the experiment. Both UMFI LR and UMFI OT correctly gave highimportance to the ten BRCA-associated genes, while giving zero median importance to about 80% ofthe unassociated genes. Additionally, in an overnight study spanning less than ten hours, UMFI LRand UMFI OT displayed ideal results after running 5000 iterations of the BRCA experiment. Asshown in Figure B.6, both implementations of UMFI achieve 100% overall accuracy by giving high206B.7. Further feature importance experimentsimportance to the ten BRCA-associated genes and zero median feature importance to all 40 unasso-ciated genes. These results indicate that UMFIâs relatively low computational cost can be leveragedvia aggregation to achieve superior performance on complex data within a reasonable time budget.Ablation, PI, and CPIWe also test the quality and robustness of other feature importance metrics including ablation, PI,and CPI, by running 200 iterations of the BRCA experiment from Section B.7.1 for each method.Results are shown in Figure B.7. Ablation importance scores are small and have large uncertaintiescompared to its median importance scores, which makes the scores impractical to interpret. Eightof the ten important genes are identified by ablation, but all other genes are given exactly zeromedian importance. All ten important genes are given non-zero importance by CPI, however, somerandomly permuted genes are given more importance than some genes known to be important, suchas CDK6. PI gave more reliable and stable results compared to ablation and CPI in this experiment,exhibiting similar performance to UMFI LR and UMFI OT from the analogous experiment shownin Figure B.5. We note that PI assigned zero importance to 29 of the 40 unassociated genes, makingits TNR of 0.725 slightly lower than UMFI in the analogous experiment from Section B.7.1.B.7.3 Experiments on unpermuted BRCA dataAdditional BRCA experiments were performed on the original unpermuted genes, as done in Covertet al. [2020] and Catav et al. [2021a]. The observed overall classification accuracy of random forestsfor this dataset was 0.79.Feature importance scores on this dataset were first computed with MCI, UMFI LR, and UMFI OTover 100 iterations, as shown in Figure B.8. The ordering of the BRCA associated genes is fairlysimilar across MCI and both UMFI methods. BCL11A and SLC22A5 are always the top two featuresand TEX14 is always the least important BRCA associated gene. While there are clear similaritiesin the results of all methods, the glaring difference is the number of features given zero importance.While MCI gives non-zero median importance to all 50 features, 14 features are given zero medianimportance by UMFI with linear regression, and 10 features are given zero median importance byUMFI with pairwise optimal transport. It is unlikely that all 40 randomly selected genes, which havenot shown any association with breast cancer in previous studies, share information about breastcancer, so in this respect, we conclude that UMFI performs better than MCI.Feature importance scores on the unpermuted BRCA dataset were also computed with ablation,CPI, and PI over 100 iterations, as shown in Figure B.9. When also considering these results,we observe that MCI, UMFI, and PI give similar importance scores, while ablation and CPI per-formed significantly worse. Once again, ablationâs high relative variance hampers its interpretability.Meanwhile, CPI gave by far the highest importance to SLC25A1, which is not known to have anyassociation with breast cancer. In the results of MCI, UMFI, and PI, BCL11A is the most importantwhile CST9L is always among the most important non-BRCA associated genes. Contrary to this,ablation and CPI give high importance to BRCA1, BRCA2, TEX14, EZH2, and IGF1R for BRCAassociated genes, and SLC25A1 for non-BRCA associated genes.207B.7. Further feature importance experimentsComputational complexityWe compare the computational complexity of UMFI and MCI against the other feature importancemethods that were explored in this section: ablation, PI, and CPI. To do so, we ran 10 iterationsof the BRCA experiment, which has 50 features, each with 571 observations. We recorded theaverage time for each method to compute feature importance for 5, 10, 15, 20, 25, 30, 35, 40, 45, and50 features. Figure B.10 shows that PI is the fastest method, processing 50 features in 50 millisecondson average, followed by ablation (50 features in 1.8 seconds), UMFI (50 features in 3 seconds whenparallelized), CPI (50 features in 30 seconds), and finally MCI with soft 2-size submodularity (50features in 205 seconds).B.7.4 Experiments on hydrology dataThe final experiments for this study were conducted on a large-sample hydrology dataset calledCAMELS [Addor et al., 2017]. This dataset records catchment averaged climate, soil, geology,topography, and land cover characteristics for 643 catchments across the contiguous United States.With these, there are 29 continuous explanatory variables. The response variable is averaged yearlystreamflow, which is also continuous. Extremely randomized trees were used in this experiment withan overall OOB-R2 of 0.91.Figure B.11, which is analogous to Figure B.2 in Appendix B.6, shows that both preprocessingmethods fail to completely remove dependencies from the CAMELS dataset. This can likely beattributed to the fact that each feature is extremely dependent on the other explanatory features(R2 â¥ 0.65).The feature importance scores indicated in Figure B.12 show that mean precipitation and aridityindex are the features with the strongest relationships with mean annual streamflow. Geology andsoil attributes such as bedrock permeability and soil porosity are always among the least importantfeatures. These conclusions are in line with previous studies [Addor et al., 2018, Jehn et al., 2020],thus, even when dependencies can not be completely removed, UMFI can still provide reasonablemeasurements of feature importance.208B.7. Further feature importance experiments(a) RF: Nonlinear interactions (b) RF: Correlated interactions(c) RF: Correlation (d) RF: Blood relation(e) ET: Nonlinear interactions (f) ET: Correlated interactions(g) ET: Correlation (h) ET: Blood relationFigure B.4: Results for the experiments on simulated data from Subsection B.7.1. The resultsfor ablation, conditional permutation importance (CPI), and permutation importance (PI) wereimplemented with random forest (RF), and are shown in Figures B.4a, B.4b, B.4c, and B.4d .The results for MCI, UMFI LR, and UMFI OT were implemented with extremely randomized trees(ET), and are shown in Figures B.4e, B.4f, B.4g, and B.4h. Feature importance scores are shown asa percentage of the total for each of x1 to x4 from 100 replications.209B.7. Further feature importance experimentsFigure B.5: Median feature importance scores provided by (a) MCI, (b) UMFI with linear regres-sion, and (c) UMFI with pairwise optimal transport, for each gene in the BRCA dataset after 200iterations. Genes colored in blue are known to be associated with breast cancer while genes coloredin grey are random permutations of randomly selected genes, which we assume to be unassociatedwith breast cancer. The first and third quantiles of the scores are visualized as error bars for eachgene.210B.7. Further feature importance experimentsFigure B.6: Median feature importance scores provided by (a) UMFI with linear regression, and(b) UMFI with pairwise optimal transport, for each gene in the permuted BRCA dataset after 5000iterations. Genes colored in blue are known to be associated with breast cancer while genes coloredin grey are random permutations of randomly selected genes, which we assume to be unassociatedwith breast cancer subtype. The first and third quantiles of the scores are visualized for each gene.211B.7. Further feature importance experimentsFigure B.7: Median feature importance scores provided by (a) ablation, (b) permutation importance,and (c) conditional permutation importance, for each gene in the permuted BRCA dataset after 200iterations. Genes colored in blue are known to be associated with breast cancer while genes coloredin grey are random permutations of randomly selected genes, which we assume to be unassociatedwith breast cancer subtype. The first and third quantiles of the scores are visualized for each gene.212B.7. Further feature importance experimentsFigure B.8: Median feature importance scores provided by (a) MCI, (b) UMFI with linear regression,and (c) UMFI with pairwise optimal transport, for each gene in the unpermuted BRCA dataset after100 iterations. Genes colored in blue are associated with breast cancer while genes colored in greyare randomly selected genes. The first and third quantiles of the scores are visualized for each gene.213B.7. Further feature importance experimentsFigure B.9: Median feature importance scores provided by (a) ablation, (b) permutation importance,and (c) conditional permutation importance, for each gene in the unpermuted BRCA dataset after100 iterations. Genes colored in blue are associated with breast cancer while genes colored in greyare randomly selected genes. The first and third quantiles of the scores are visualized for each gene.214B.7. Further feature importance experimentsFigure B.10: The average computation time for each method to process p features over 10 iterationsof the original BRCA data is plotted for each p â {5, 10, 15, 20, 25, 30, 35, 40, 45, 50}. We assume soft2-size submodularity to run the MCI results.Figure B.11: The coefficient of determination (random forest OOB-R2) between the ith feature inthe CAMELS dataset and all other features is plotted (black) for each i â {1, 2, ...30}. The R2 valuebetween the ith feature and all other features after preprocessing with linear regression (red) andoptimal transport (blue) is also plotted.215B.7. Further feature importance experimentsFigure B.12: Median feature importance scores provided by (a) MCI, (b) UMFI with linear regres-sion, and (c) UMFI with pairwise optimal transport, for each explanatory variable in the CAMELSdataset, aggregated after 100 iterations. The first and third quantiles of the scores are visualizedfor each feature.216Appendix CAppendix for âDynamically sparsehistorical functional linear modelswith applications to Earth scienceâC.1 Additional Real Data ResultsFigure C.1: The estimated function Î²Ë(s, t) for each lag s for the Koksilah River located in BritishColumbia (left) and Withlachoochee River, located in Florida (right). The lines within each plotshow a single representative day within the âWetâ, âDryingâ, âDryâ, and âWettingâ periods thatwe delineated. For the Koksilah River, February 19th is âWetâ, June 9th is âDryingâ, August 28this âDryâ, and October 17th is âWettingâ. For the Withlachoochee River November 16th is âWetâ,February 19th is âDryingâ, May 30th is âDryâ, and September 7th is âWettingâ.C.2 Additional Simulation ResultsC.2.1 Selecting the Sparsity ParameterAn example displaying how we choose the sparsity parameter is shown in Figure C.2 and explainedin Satopaa et al. [2011]. We note that in this example coming from our simulation study, the idealknee-point, where the fitted R2 begins to decrease nonlinearly, is quite prominent.C.2.2 Learned Dynamic Sparsity from Simulation Study217C.2. Additional Simulation ResultsFigure C.2: Each black point represents the fitted R2 after setting all coefficients within β(s, t) tozero which have a group norm less than the q parameter (standardized to be a quantile between 0and 1). The red dot displays the chosen sparsity level.Figure C.3: The estimated maximum time-varying lag Î´Ë(t) from 10 out of the 100 experiments inour simulation study from Section 4 (blue) versus the true δ(t) are visualized for two cases from oursimulation study. For the Koksilah River located in British Columbia (left), the maximum R2 was0.4 and the level of autocorrelation in the error was medium. For the Withlachoochee River, locatedin Florida (right), the maximum R2 was 0.8 and the level of autocorrelation in the noise was setas high. These two cases represent the second worst (left) and second best (right) results from theeight cases we tested in our simulation study.218Appendix DAppendix for âSelf-Explaining LSTMsfor hydrological inference andpredictionâFigure D.1: Performance comparison maps.219Appendix EAppendix for âTackling water tabledepth modeling via machine learning:From proxy observations toverifiabilityâE.1 IntroductionThe main article manuscript is considered a standalone document with sufficient details on thescientific approach, critical investigation of the results, as well as a thorough discussion on shallowgroundwater modeling. It also explores the intersection of physically-based modeling and MachineLearning in simulating mechanisms governing subsurface water flow. This document provides somefurther information, context and/or discussion, which enhances readerâs comprehension and fosterstransparency.E.2 Canadian well reportsE.2.1 QuebecData was downloaded from the Donne´es Quebec database. This database is titled as the hydro-geological database. It includes all documented well drillings across the province. It included noattributed or information classifying the producing aquifer type (confined/unconfined). Similarto many other datasets, Quebecâs hydrogeological dataset represents the static level and does notinclude any indication on current water use of wells.E.2.2 OntarioGroundwater and water table depth data was downloaded from WWIS2 database. The StaticWater Level attribute in WWIS2 is chosen to signify the depth of the water table. The databaseencompasses various features of the well of record, including casing depths, water found depth, staticlevel, and some details regarding pumping tests.E.2.3 Nova ScotiaNova Scotiaâs Well Logs were acquired from the Government of Nova Scotia Government website.Most of the info was accessed through the Department of Environment and Climate Change and theDepartment of Natural Resources and Renewables web pages. These databases include Well depths,220E.2. Canadian well reportscasing, bedrock depth, statics water level, yield, water use and information on location and drillercompany. To focus on unconfined aquifer well readings, wells with static water level deeper thanbedrock were excluded.E.2.4 AlbertaGroundwater database for the Alberta Province was acquired from the Government of Alberta webpage and Interactive GIS Platform. A large number of wells with static water level of zero weremasked out. These consist almost 170,000 of wells in the Alberta dataset. Most of them have noinformation on drilled depth or perforation depth. Among these wells (WTD=0) Only those forwhich the drilled depth and perforation depths were also documented as zero. It consists around30,000 wells out of 280,000 wells in the provincial database.E.2.5 British ColumbiaWater table depth across the British Columbian was extracted from the BC Groundwater Database.The database is in Excel format and it includes information on water use, total drilled depths, casingdepths and diameter, bedrock depths, static water level, artesian flows (where available), and manymore. Static water levels were filtered for depths below 100 meter, and static water levels deeperthan bedrock depths. Also, wells with artesian flow were also excluded.E.2.6 SaskatchewanGroundwater database was acquired through direct correspondences with Water Security Agency.Depths were converted from ft to m. Similar to Nova Scotia, flowing wells as well as those withstatic water levels shallower than screening depths were assumed to be located in confined aquifersand therefore masked out .E.2.7 New BrunswickThe New Brunswick Groundwater Level Dataset was accessed to extract WTD observations. Thisdataset was in Excel format and includes different attributes such as water use, pumping test data,drilled depth as well as the Initial Water Depth (IWD), bedrock depth, and several more. The IWDwas assumed to represent the static water level. Wells with IWDs deeper than bedrock depth wereexcluded as they were assumed to be in confined aquifers..E.2.8 NewFoundland and LabradorBased on correspondence with Government of The Newfoundland and Labrador (NL), the NL doesnot keep and update a database of the groundwater levels. Data on static water levels across theProvince were downloaded from the GIN databaseE.2.9 ManitobaManitobaâs Department of Environment was contacted to inquire about documented Provincialdatasets. The groundwater dataset, however, was not accessible for public use. As a result, allavailable GIN database in Manitoba was downloaded and used for the study.221E.2. Canadian well reportsE.2.10 YukonSimilar to a few other provinces and territories, groundwater level data across the Yukon territorywas accessed and downloaded from the GIN database.222Appendix FAppendix for âA Causal InferenceFramework for Catchment BoundaryDelineationâF.1 Granger causality, predictive power, and mutual informationTheorem F.1.1. Transfer entropy and Granger causality are equivalent, when Granger causality isdefined via the reduction of cross-entropy loss from an optimal Bayes classifier.Proof. In Grangerâs original paper, time series (Xt)tâ¥0 is said to âcauseâ time series (Yt)tâ¥0, if theoptimal least squares predictor for Yt, given previous values Ytâk:tâ1, achieves strictly lower errorwhen Xtâ1 is included in the prediction model [Granger, 1969][Definition 1]. Following advances inmachine learning in the ensuing decades, Granger causality has since been generalized beyond thelinear framework that was originally used (see Tank et al. [2021]).To formalize Granger causality in this context, first note that given a machine learning model f ,used to predict response Y , a set of predictor variables S, and a loss function â, it is standard toconsider [Covert et al., 2020, Janssen et al., 2023, Catav et al., 2021b]:νf (S) = E[â(f(â ), Y )]â E[â(f(S), Y )],which defines the predictive power of features S (according to the model f) as the reduction in errorcompared to the untrained null model. With this in mind, (Xt)tâ¥0 would Granger cause (Yt)tâ¥0 ifand only if the optimal model fâ achieves higher prediction loss when predicting Yt with the previoushistory Ytâk:tâ1 and Xtâ1 compared on only using the history of Y to train the model:E[â(f(Ytâk:tâ1), Yt)]â E[â(f(Ytâk:tâ1, Xtâ1), Yt)] > 0.Equivalently, (Xt)tâ¥0 Granger causes (Yt)tâ¥0 if the optimal model achieves strictly greater predictivepower for predicting Yt when it is additionally trained on Xtâ1:Î½fâ(Ytâk:tâ1, Xtâ1)â Î½fâ(Ytâk:tâ1) > 0. (F.1)Then, recall that given an optimal Bayes predictor fâ, i.e. fâ(S) = p(Yt|S), with cross entropy lossâ, the evaluation function is equivalent to the mutual information Covert et al. [2020]:Î½fâ(S) = I(Yt;S).By (F.1), we see that (Xt)tâ¥0 Granger causes (Yt)tâ¥0 if and only ifÎ½fâ(Ytâk:tâ1, Xtâ1)â Î½fâ(Ytâk:tâ1) = I(Yt;Ytâk:tâ1, Xtâ1)â I(Yt;Ytâk:tâ1) > 0, (F.2)223F.2. CausalityHowever, by the chain rule for mutual information, we obtainI(Yt;Ytâk:tâ1, Xtâ1)â I(Yt;Ytâk:tâ1) = I(Yt;Xtâ1|Ytâk:tâ1), (F.3)which is precisely the transfer entropy TXâY . We conclude that Granger causality is equivalent topositive transfer entropy.F.2 CausalityDefinition 6 (Adjustment Criterion Shpitser et al. [2010], Runge [2021], Perkovi et al. [2018]). Aset of variables Z satisfies the adjustment criterion for a graph G with mediators M between X andY if (1) Z â© {X,M, Y, des(M), des(Y )} = â and (2) Z blocks all non-causal (open but not directed)paths from X to Y .Then, within the set of valid adjustment sets, we may also define the optimal adjustment set.Definition 7 (Optimal adjustment set Runge [2021]). Given a input variable X and a responsevariable Y , a valid adjustment set Z is said to be optimal if it minimizes the asymptotic variance ofthe total causal effect estimate. To obtain the optimal adjustment set, one must maximize:I(Z;Y |X)â I(X;Z) (F.4)F.3 Extra Figures224F.3. Extra FiguresFigure F.1: Example classifications of the true catchment along with each of the three methods forthe DEA11040 catchment (1396 km2 catchment area) with 20 years of data and a maximum NSE(or signal to noise ratio) of 0.85. Black line shows the catchmentâs ground truth boundary polygon.Blue pixels indicate true (within the true catchment) for the first panel, and indicate true (within thepredicted catchment) for the remaining three panels/methods. Salmon pixels indicate false (outsidethe true catchment) for the first panel, and indicate false (outside the predicted catchment) for theremaining three panels/methods. Missing pixels (white) may appear due to the pixels not beingfully contained within Germany.225F.3. Extra FiguresFigure F.2: Plots showing the temporal autocorrelation of precipitation (top) and temporal partialautocorrelation of precipitation (bottom) of the first (left, DE110000) and last (right, DEG10620)catchment in the CAMELS-DE dataset. The dashed lines show significance at p=0.05.226
Thesis/Dissertation
eng
Geological Sciences
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Ameli, Ali
Robeva, Elina
Diagnosing spatiotemporal catchment hydrologic behavior via causal inference, information theory, and machine learning
Text