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Insights on the origin and evolution of the Martian valley networks from erosion models : reconciling… Grau Galofre, Anna 2018

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Insights on the origin and evolution of the Martian valley networksfrom erosion models: Reconciling climate modeling andgeomorphological observations.byAnna Grau GalofreBSc, Physics, Universitat de Barcelona, 2012MSc, Geophysics, Universitat de Barcelona, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Geophysics)The University of British Columbia(Vancouver)August 2018c© Anna Grau Galofre, 2018Supervisory CommitteeThe following individuals certify that they have read and and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled: “Insights on theorigin and evolution of the Martian valley networks from erosion models: Reconciling cli-mate models and geomorphological observations.” Submitted by Anna Grau Galofre in partialfulfillment of the requirements for the degree of Doctor of Philosophy in The Faculty of Grad-uate and Postdoctoral Studies.Supervisory CommitteeProfessor A. Mark JellinekSupervisorProfessor Catherine L. JohnsonSupervisory Committee memberProfessor Christian SchoofSupervisory Committee memberProfessor Aaron BoleyUniversity ExaminerProfessor Sun KwokUniversity ExaminerAdditional Supervisory Committee MembersProfessor Marwan HassanSupervisory Committee memberAssociate Professor Sean A. CroweSupervisory Committee memberiiAbstractThe surface of Mars is incised with hundreds of ancient valley networks, the physical record offlowing liquid water during the early stages of Mars evolution (3.5-3.8 Byr ago). Their remark-able similarity to terrestrial rivers has historically motivated their interpretation in terms ofrainfall and surface runoff, indicating that Mars climate was significantly warmer than presentday. Protracted surface liquid water stability is, however, hard to reconcile with results fromstate of the art Global Climate Models, which predict that under a fainter young Sun and athicker primitive atmosphere, the Martian southern hemisphere would be largely under icecover. Distinguishing whether early Mars harbored surface water or was covered by an exten-sive cryosphere is key to understanding the nature of any habitable environments. The goalof this dissertation is to reconcile the climate and geomorphological characterizations of earlyMars by establishing quantitative constraints on the origin of the Martian valley networks. InChapter 2, I develop a methodology to quantitatively characterize valley networks in termsof their predominant erosional mechanism, including fluvial, glacial, sapping, and subglacialregimes. Chapter 3 uses constraints from a detailed field characterization of subglacial chan-nels to establish their reliable identification from remote sensing data. In Chapter 4, I presentthe main results: the identification of subglacial channels among the Martian valley networks.These results support climate model predictions and are consistent with morphological ob-servations. Chapter 5 builds on Chapter 4 to further understand the dynamics of landscapeevolution on early Mars. In particular, I demonstrate that only a small fraction of valley net-works are in a steady-state, and that erosion rates were likely very low on early Mars. I con-clude (1) that subglacial erosion is widespread on the Martian Highlands and best explains thepuzzling characteristics of valley networks, (2) that fluvial erosion was short-lived and onlyconcentrated in narrow topographic corridors, and (3) that glacial and sapping erosion wererare on early Mars. In marked contrast to the popular view that Mars was “warm and wet”,my results show that early Mars had a climate akin to Antarctica: extensive ice sheets withlocalized melting.iiiLay SummaryMars is, besides Earth, the only planet where systems of valleys and canyons carved by liquidwater dissect the surface, evidence that Mars harbored surface water and an active hydrolog-ical cycle in its ancient past. What was the character of Mars’ climate during the formationof the valleys? What mechanisms formed these features? Even after 40 years of increasinglydetailed observations, answers to these questions remain elusive. In this thesis, I investigatethe formation of the Martian valley networks in terms of the physics of erosion by rivers,glaciers, groundwater springs, and subglacial channels. Comparing theoretical predictionswith observations, I find that a majority of the valley networks on Mars were likely producedby subglacial channels, which drained meltwater from large ancient ice sheets. These find-ings show that the climate on early Mars was likely similar to current day Antarctica, and thatMars’ surface was covered by an extensive cryosphere.ivPrefaceThis thesis is original work completed by Anna Grau Galofre. Guidance, support, and inspi-ration was given by my PhD supervisor Mark Jellinek, as well as the supervisory committee,including professors C. Johnson, C. Schoof, M. Hassan, and S. Crowe, and through an exten-sive collaboration with professor G. Osinski.This thesis includes two published articles and two submitted manuscripts. The publishedmanuscripts are presented in Chapter 2 and Chapter 3. The submitted manuscripts are pre-sented in Chapter 4 and Chapter 5, in turn.A version of the work in Chapter 2 is published in Journal of Geophysical Research: Earth Surface,Grau Galofre and Jellinek (2017). Mark Jellinek is the co-author. I am responsible for developingthe modeling Principal Component Analysis-based technique, implementation, data compi-lation and analysis presented in Chapter 2. I took the lead in writing the paper with MarkJellinek.A version of the work in Chapter 3 is published in The Cryosphere, Grau Galofre et al. (2018).Mark Jellinek, Gordon Osinski, Mike Zanetti and Antero Kukko are co-authors. Mark Jellinekand I are responsible for designing and carrying out the field campaign. All co-authors con-tributed in field data compilation. Antero Kukko, Mike Zanetti and I processed and analyzedthe data presented in Chapter 3. I took the lead writing the paper with Mark Jellinek.A version of the work in Chapter 4 is submitted for publication. Mark Jellinek and GordonOsinski are the co-authors. I am responsible for data compilation and data analysis, with con-tribution from Mark Jellinek and Gordon Osinski in discussing the geological implications. Itook the lead writing the paper and Mark Jellinek provided extensive feedback on the dataanalysis and writing.A version of Chapter 5 is submitted for publication. The study is in collaboration with RoseGallo and Mark Jellinek. I am responsible for the idea, the numerical and theoretical model-ing, part of the data analysis, and interpretation. Rose Gallo performed the data compilationand part of the data analysis. Mark Jellinek provided extensive and detailed feedback on thevresults and writing of the Chapter.Grau Galofre, A. and Jellinek, A.M. (2017), The geometry and complexity of spatial patterns ofterrestrial channel networks: Distinctive fingerprints of erosional regimes, Journal of Geophysi-cal Research: Earth Surface, 122(4), 1037–1059.Grau Galofre, A., Jellinek, A. M., Osinski, G. R., Zanetti, M., and Kukko, A. (2018). Subglacialdrainage patterns of Devon Island, Canada: detailed comparison of rivers and subglacial melt-water channels. The Cryosphere, 12(4), 1461.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Supplementary Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi1 Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation and driving question . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Structure and organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Extended outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5.1 The Martian valley networks: age, distribution and morphological vari-ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5.2 Early Mars: Warm and wet or cold and icy? . . . . . . . . . . . . . . . . . 71.5.3 The geomorphological insights of using valley networks as paleoclimateindicators: terrestrial analogs . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.4 The problem of visual analog studies and the need for a quantitativecomparison of terrestrial and Martian drainage systems . . . . . . . . . 13vii2 The geometry and complexity of spatial patterns of terrestrial channel networks:Distinctive fingerprints of erosional regimes . . . . . . . . . . . . . . . . . . . . . . . 152.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Methods I: Landscape metrics and patterns in channel networks . . . . . . . . . 172.3.1 Five distinctive properties of channel networks . . . . . . . . . . . . . . 172.3.2 Data compilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.3 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Methods II: Physical basis for the metrics . . . . . . . . . . . . . . . . . . . . . . 232.4.1 Aspect Ratio, R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.2 Minimum channel width, λ . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.3 Tributary junction angle, γ . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.4 Stream Order, Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.5 Fractal Dimension, D f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.6 Characteristic length-width scaling relationships . . . . . . . . . . . . . . 302.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.1 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . 322.5.2 Aspect ratio R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5.3 Minimum channel width λ . . . . . . . . . . . . . . . . . . . . . . . . . . 362.5.4 Tributary junction angle γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.5.5 Stream Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5.6 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5.7 L ∼Wδ scaling exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.6.1 Comparative channel geometry and morphology of channel networks . 412.6.2 Limitations and challenges . . . . . . . . . . . . . . . . . . . . . . . . . . 452.6.3 Earth viewed from Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.6.4 Can we constrain Earth’s glacial-interglacial climatology from the struc-ture of the channel networks? . . . . . . . . . . . . . . . . . . . . . . . . . 472.6.5 Are these metrics applicable to valley networks in other planets such asMars or Titan? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Subglacial drainage patterns of Devon Island, Canada: Detailed comparison ofrivers and subglacial meltwater channels . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2.1 Field site: Devon Island . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.1 Preliminary remote sensing characterization . . . . . . . . . . . . . . . . 55viii3.3.2 Longitudinal profile data . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.3 Airborne imagery and photogrammetry . . . . . . . . . . . . . . . . . . . 563.3.4 Kinematic LiDAR Scan acquisition . . . . . . . . . . . . . . . . . . . . . . 583.4 Results: Quantitative characterization of river and subglacial channels . . . . . 593.4.1 River and subglacial channels’ longitudinal profiles . . . . . . . . . . . . 593.4.2 LiDAR observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.3 Photogrammetry observations . . . . . . . . . . . . . . . . . . . . . . . . 633.5 Identification of subglacial channel networks . . . . . . . . . . . . . . . . . . . . 653.5.1 Morphometric comparison of river and subglacial channels . . . . . . . 653.5.2 Comparison of lateral and subglacial meltwater channels . . . . . . . . . 663.6 Detailed morphology of subglacial channels in Devon Island . . . . . . . . . . . 683.6.1 Network characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.6.2 Channel characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.6.3 Other characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.7.1 Undulations, obliquity, shape factor, and the remote sensing characteri-zation of subglacial channels . . . . . . . . . . . . . . . . . . . . . . . . . 713.7.2 Identification of subglacial channels from remote sensing data . . . . . . 733.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 An Antarctic-style ice sheet on ancient Mars . . . . . . . . . . . . . . . . . . . . . . . 754.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.5 Geological and climate implications . . . . . . . . . . . . . . . . . . . . . . . . . 794.6 Extended methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.6.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.6.2 Measurement strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.6.3 Details of the PCA technique . . . . . . . . . . . . . . . . . . . . . . . . . 834.6.4 Metric predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835 Lessons from the shape of Martian valley networks’ longitudinal profiles: steady-state, and the link between climate and tectonics on early Mars. . . . . . . . . . . . 855.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3 Landscape evolution equations and steady-state solutions . . . . . . . . . . . . 875.3.1 Fluvial erosion: the stream-power formulation . . . . . . . . . . . . . . . 885.3.2 Glacial erosion: Bedrock abrasion . . . . . . . . . . . . . . . . . . . . . . 895.3.3 Sapping erosion: it’s all about the boundaries . . . . . . . . . . . . . . . 91ix5.4 Model results: the steady-state shape of Martian longitudinal profiles . . . . . . 925.5 Are valley networks in steady-state? . . . . . . . . . . . . . . . . . . . . . . . . . 945.5.1 Evaluating longitudinal profiles: Modeling and observation insights . . 945.5.2 Valley networks out of steady-state . . . . . . . . . . . . . . . . . . . . . 965.6 Valley networks in steady-state: implications . . . . . . . . . . . . . . . . . . . . 965.6.1 Steady-state valley network distribution . . . . . . . . . . . . . . . . . . 975.6.2 Spatial correlation with surface tectonics . . . . . . . . . . . . . . . . . . 985.6.3 Inferred time-integrated erosion rates from the analysis of steady stateprofiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.1.1 Chapter 2: Any journey to Mars starts on Earth. . . . . . . . . . . . . . . 1026.1.2 Chapter 3: The missing piece of the puzzle is always the most significant. 1036.1.3 Chapter 4: The origin of the Martian valley networks. . . . . . . . . . . . 1046.1.4 Chapter 5: Standing on the shoulders of terrestrial giants, one can seeMars a bit better. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2.1 Why are signs of ancient glaciation so rare on Mars’ surface? . . . . . . . 1066.2.2 Does the location of younger valley networks follow the time evolutionof the ice equilibrium line? . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2.3 How long does it take for fluvial erosion to become the predominanterosional fingerprint in a previously glaciated landscape? . . . . . . . . 1106.2.4 The landscape signature of glacial/interglacial cycles on Mars . . . . . . 111Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A Governing equations for flow and landscape evolution . . . . . . . . . . . . . . . . 129A.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.2 General driving equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.2.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.2.2 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.2.3 Constitutive law: Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.2.4 Landscape evolution equation . . . . . . . . . . . . . . . . . . . . . . . . 130A.3 The fluvial regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131A.4 Glacial flow regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A.5 The sapping valley regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133A.6 The subglacial regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134xB Synthetic valley networks: Models and metrics predictions . . . . . . . . . . . . . . 136B.1 Summary and content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136B.2 Physical models of morphometry: model predictions . . . . . . . . . . . . . . . 136B.2.1 Fluvial regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136B.2.2 Glacial regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138B.2.3 Sapping regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139B.2.4 Subglacial regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140B.3 Metric Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142C Details of the PCA technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.2 Details of the PCA technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.4 PCA end-members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147D Martian parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148E Parameters, notation, and units for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . 151E.1 Mars parameter space: Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 151E.2 Caption for supplementary table . . . . . . . . . . . . . . . . . . . . . . . . . . . 151E.2.1 Applicability of the stream power law to Mars . . . . . . . . . . . . . . . 152E.2.2 Empirical fit: basin hydrology factor . . . . . . . . . . . . . . . . . . . . . 153E.2.3 Table of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153E.3 Geography of Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154xiList of TablesTable 2.1 Description of parameters: Metrics . . . . . . . . . . . . . . . . . . . . . . . . 24Table 2.2 Notation: flow and erosion mechanics . . . . . . . . . . . . . . . . . . . . . . 31Table 2.3 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Table 3.1 Diagnostic criteria for the identification of subglacial channels . . . . . . . . 52Table 3.2 Morphometric characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Table 3.3 Summary of morphological characteristics . . . . . . . . . . . . . . . . . . . . 68Table 4.1 Morphological evidence for subglacial erosion . . . . . . . . . . . . . . . . . . 79Table 5.1 Subset of valley networks considered in this study, lat/long location (deci-mal degrees), incision mechanism (f for fluvial, g for glacial, s for sapping,Grau Galofre et al., (2018) (In review)), uplift-erosion number N, erosion expo-nent ne, rms misfit, and terrain age (N for Noachian, H for Hesperian, A forAmazonian, e for early, m for middle, l for late). . . . . . . . . . . . . . . . . . 95Table 6.1 Timing of incision for 30 valley networks. Ages are in Byr, MN correspondsto Middle Noachian, LN corresponds to late Noachian, EH corresponds toEarly Hesperian, LH corresponds to Late Hesperian, and EA corresponds toEarly Amazonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Table B.1 Metric predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Table C.1 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Table E.1 List of parameters: general notation . . . . . . . . . . . . . . . . . . . . . . . . 151Table E.2 List of parameters: fluvial erosion . . . . . . . . . . . . . . . . . . . . . . . . . 152Table E.3 List of parameters: glacial erosion . . . . . . . . . . . . . . . . . . . . . . . . . 152Table E.4 List of parameters: sapping erosion . . . . . . . . . . . . . . . . . . . . . . . . 153xiiList of FiguresFigure 1.1 Global topographic map of Mars derived from the Mars Orbiter Laser Al-timeter, overlapped with the location and extent of the Martian valley net-works, adapted from Hynek et al. (2010). Note how valley networks accumu-late in the southern elevated hemisphere (Highlands) and leave the northernplains unincised. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.2 Examples of the different types of Martian valley networks as presented inthe literature. 2a: single channel morphology, 2b: dendritic valley networkon a volcano flank, 2c: longitudinal valley network, 2d: well-integrated val-ley network, 2e: young dendritic valley network on a volcano flank, 2f: fila-mentous valley network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.3 Diagram showing the timing of processes relevant to the Martian surfaceevolution (adapted from Gulick (2001); Ehlmann et al. (2011); Wordsworth(2016)). Although absolute ages are widely unconstrained (crater countingtechniques permit error bars up to 100 Myr), the relative timing of theseprocesses is robust. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 1.4 The adiabatic cooling effect on early Mars. (a) Annual mean surface temper-ature from a three-dimensional general circulation model (GCM) simulationwith 0.125 bar surface pressure. (b) Annual mean surface temperature froma three-dimensional GCM simulation with 1 bar surface pressure. (c) Scat-ter plot of surface temperature versus altitude for simulations with a 0.125,1, and 2 bar surface pressure. The dry adiabat g/cp is also indicated (grayline, constant potential temperature line). Data for the plots were acquiredfrom the 41.8◦ obliquity, fixed relative humidity simulations described byWordsworth et al. (2015). Adapted from Wordsworth (2016). . . . . . . . . . . 9xiiiFigure 1.5 Examples of analogs used in the literature as a basis to build compara-tive understanding between Martian (right) and terrestrial (left) valley net-works. 1.5a shows a comparison of the Grand Canyon and a dendritic val-ley network in Alba Patera (Ramirez and Craddock, 2018), 1.5b shows NanediValles compared to a section of the Colorado River Canyon as a terrestrialsapping analog (Ramirez and Craddock, 2018), 1.5c shows a comparison ofdeposits of a glacier at the base of Olympus Mons and a glacier in MullinsValley, Antarctica (Head et al., 2005) and 1.5d shows the comparison of smalldendritic valley networks with a subglacial channel network in Devon Is-land (Lee, 1997). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.1 Examples of channels emplaced through the action of different processes.a) A glacial valley on NE Devon Island, Canada b) A river in Western Aus-tralia d) A sapping valley in the Florida Panhandle e) An exposed subglacialmeltwater channel in NW Devon Island, Canada. Imagery: Landsat 7 (USGS). 17Figure 2.2 Cartoon depicting the metrics enumerated above. Panel (a), junction angleγ, (b) length L, (c) bankfull width W, (d) stream order (following the con-vention by Strahler) Sn, all tributary-less channels receive an order 1. Uponmerging, two order ones become an order two, two order two tributaries be-come an order 3, and so on, (e) minimum width λ, and (f) fractal dimensionD f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 2.3 Definition and examples of the measurement process behind the 5 metrics.(a) shows the angle in one junction of the channel network γ. (b) showsthe length of the channel network L (see color bar for reference). (c) showsthe channel bankfull width W using the 1 arc-sec derived SRTM curvature,that complements L to calculate the aspect ratio R. (d) shows the stream(Strahler) order Sn (see color bar for reference). (e) shows the minimumwidth λ measured either with curvature maps (left panel) or high resolutionimagery (right panel). (f) shows the measurement of the fractal dimensionD f using a box counting technique (plot) on the stream lines (bottom right). 21xivFigure 2.4 Mass balance, cross section and longitudinal section of the four erosionmechanisms described. Coordinate systems are shown in red dashed lines.(1a) mass balance in a river: flow enters the tributaries and exits in the outlet.(2a) and (3a) show the flow dynamics in a river with width (W) and depth(h), with blue arrows for the velocity field and eddies indicating turbulence(bigger size indicates larger discharge). (b1) mass balance in sapping val-leys: water enters the valley following the water table gradient (dashedblack line). (b2) and (b3) describe the flow dynamics in a sapping valley. Insubglacial meltwater channels, (c1) shows the balance between melting andice creep. (c2) and (c3) show the velocity field inside a channel. In glaciers,(d1) shows the snow accumulation and ablation zones, and (d2) and (d3)show the velocity field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.5 1st and 2nd Principal Components (PC) displaying the data projection in thePC space. Rivers appear in red squares, glaciers in blue diamonds, sappingvalleys in green triangles and subglacial meltwater channels in magenta ro-tated triangles, with data clusters color shaded in the same color convention.Metrics increase in the direction of the 5 dashed arrows, scaled to the rela-tive weight of each metric. λ for minimum channel width, R aspect ratio, Snstream order, D f fractal dimension, and γ junction angle. . . . . . . . . . . . 33Figure 2.6 Aspect ratio model predictions for (a) rivers, (b) sapping valleys, (c) sub-glacial channels and (d) glaciers. Each panel shows the predicted R curveagainst a control parameter: the non dimensional shear stress θt in rivers,sapping valleys and subglacial meltwater channels and the ice thickness hin glaciers. The different curves correspond to the variation of the predictedR in terms of a secondary parameter: angle of repose of sediment in waterφr in rivers, hydraulic conductivity Kh in sapping valleys, fraction of en-ergy loss to melt fm in subglacial meltwater channels and sliding velocity usin glaciers. Vertical shaded areas represent the range of control parameterstypically observed in nature (θt and h), whereas the horizontal shaded areasare the span of our aspect ratio data. Color bars show the variation withineach free parameter described above. . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.7 Comparison of observations (crosses) and model (colored background) fortributary junction angles of glaciers and rivers in 18 locations in North Amer-ica, Western Australia, and northern Africa. Crosses are color coded for themeasured junction angle according to the color bar and compared with themodel predictions (background color) for a given pair of main stem and trib-utary slope. Thus, similar colors indicate a low relative error, and deviationsbetween model and data appear with noticeable color differences. . . . . . 38xvFigure 2.8 Observational L-W scaling relationships: a) fluvial, b) glacial, c) sapping val-leys and d) subglacial channels. Horizontal axis and vertical axis are ln(W)and ln(L) (units are m) for rivers, sapping valleys and subglacial channels.Axes for glaciers are inverted ln(L) vs. ln(W) to better represent the scal-ing by Bahr (1997a). The best least square linear fit to the data in log space(solid line) in each panel gives the observational scaling exponent. Verticaland horizontal error bars are a result of DEM error propagation. Correla-tion coefficients for the fits are r2 = 0.77 for the fluvial scaling, r2 = 0.778for glaciers, r2 = 0.04 for sapping valleys (note that H valleys are located inHawaii), and r2 = 0.58 for subglacial channel networks. . . . . . . . . . . . 40Figure 2.9 Map of Earth with location and type of channels classified according to thePCA results presented in Figure 2.5. We use the same color scheme thanFigure 2.5. Solid black triangles indicate the points that led to equivocalresults from the PCA classification. On the basis of the distribution of glacialand subglacial channel networks at high elevations and latitudes, comparedto fluvial and sapping valleys dominating the lower latitudes, we identify acurrent interglacial period for Earth. . . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.1 (a) satellite imagery of Devon Island within the Arctic Archipelago (whitebox). (b) satellite image of Devon Island, with a white box indicating theselected field site. The map also shows the Innuitian ice sheet termini linesdigitalized from Dyke (1999), with age reference in the legend (refer to radio-carbon years). (c) Field site (UTM zone 16), with boxes around each networkinvestigated. White boxes are for sublgacial networks (SG1, SG2, SG3, andSG4), whereas black boxes indicate fluvial networks (R1 and R2) . . . . . . 54Figure 3.2 Aerial and field imagery of subglacial channels and rivers. 2(a) correspondsto helicopter imagery of a group of subglacial channels (89.13◦ W,75.28◦ N),channel widths approx. 35 m. 2(b) corresponds to a groups of subglacialchannels located at 89.37◦ W, 75.18◦ N, network is approx. 300 m wide. 2(c)corresponds to a subglacial channel emerging underneath the Devon Islandice cap, notice the similar morphology to 2(a) and 2(b), each channel beingapprox. 30 m wide. 2(d) shows the cross-section of a deeply incised canyonemerging from under the ice cap. Canyon cross-section measures 200 mapprox. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57xviFigure 3.3 Longitudinal profiles of river and subglacial channels normalized to totaltopographic loss and length along the channel, with WorldView satelliteimagery for each channel network. In the longitudinal profiles, blue crossesrepresent the raw GPS data for each channel, blue dashed lines are the dataafter filtering, and orange solid lines represent the LiDAR sections that over-lap GPS data for comparison. The profiles obtianed with the Arctic DEM at5 m resolution are shown in green color. . . . . . . . . . . . . . . . . . . . . 61Figure 3.4 KLS LiDAR observations. Panels (a) and (b) show the color coded pointcloud files (dark is low return), see the scale for spatial reference. Panels (c)and (d) show the raster produced using the point clouds. . . . . . . . . . . 63Figure 3.5 Stereo-photogrammetry derived from helicopter borne photography. Thetop panels (a) and (b) show the digital elevation model (DEM) at a resolu-tion of 0.48 and 0.56 m/pixel respectively, with the colorbar indicating theelevation of the model surfaces. The images underlying the panels corre-spond to the textured orthoimages in both locations. . . . . . . . . . . . . . 64Figure 3.6 cross-sectional evolution of a fluvial (upper row) and subglacial (bottomrow) channel, with satellite imagery for context on the right column. In thesubglacial case, the initial width and the shape remain largely unchangedover length, whereas the river cross-section grows monotonically both inwidth and depth with distance. Notice the differences in depth and lengthin the section scale bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 3.7 Hillshade and contour map of the four subglacial channel networks inves-tigated. Contour lines are separated 15 m, and hillshade resolution is 2m/pixel. In the bottom right corner, the black arrows indicate the overalldirection of the channels in the networks, whereas red indicates the regionalslope direction. Red arrows indicate anastomosing sections. . . . . . . . . . 67Figure 3.8 cross-section field imagery and profiles. Upper row shows a satellite im-agery context on the location where the image and the cross-section are ob-tained, together with a scale reference. Middle row shows images of fourcross-sections, obtained by this expedition on July 24th and 25th, 2017. Themiddle panel corresponds to a main channel whereas the other three im-ages correspond to tributaries. cross-section profiles below show elevation(m) vs. distance (m) obtained from the Arctic DEM at 2 m/pixel. . . . . . . 70Figure 3.9 Field images of the shallow depressions and potholes observed. Satelliteimagery provide context for the photographies through the camera icons.In photos (a) and (b), notice the human figures for scale. Photos (c) and (d)contain a scale bar for reference. Image (c) is an example of an overhangingvalley (here covered in snow) followed by a pothole. . . . . . . . . . . . . . 71xviiFigure 3.10 Cartoon representing the definitions of the three remote sensing based met-rics proposed in this study. (1) Shows our definition for longitudinal profileundulations Ψ, where the grey line represents the longitudinal profile of achannel (elevation vs. distance). (2) Represents the deviation between thedirection of a set of channel networks (red arrow) and the topographic gra-dient (black arrow), together with the axis notation and the ice and topo-graphic surfaces zi and zb in equation 3.4. (3) Shows the definition of shapefactor with two cartoons representing a trapezoidal and a V-shaped cross-section, where top width WT and depth D are represented (adapted fromWilliams and Phillips (2001)). . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 4.1 PCA analysis, 1st and 2nd PC (a) and 1st and 3rd PC (b). Shaded regions cor-respond to Montecarlo model predictions: fluvial (red), glacial (blue), sub-glacial (yellow), sapping (green), magenta (undifferentiated). The 6 metricsappear with arrows scaled to the variance they capture. (c) and (d) showrepresentative examples of the fluvial (c) and subglacial (d) end-members(Appendix C, Figure C.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Figure 4.2 Global geological map of Mars showing the main time periods and subdi-visions, overlaid with the location and characterization results for the 66valley networks considered in this study. Red dots represent fluvial sys-tems, black push-pins represent subglacial channels, blue asteriscs repre-sent glaciers and green diamonds correspond to sapping-like valleys. Thetriangles in magenta are not resolvable using this technique. . . . . . . . . 80Figure 4.3 (a) angle at a junction between two streamlines. (b) length of the streamlines. (c) stream lines color coded by their stream order. (d) measurement ofvalley network width. (e) CTX image with resolved first order tributaries,from which we extract λ. (f) longitudinal profile indicating the total topo-graphic drop to the uphill section at an undulation, required to calculate Ωfollowing equation 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 5.1 Model results for the steady-state shape of fluvial (a), sapping (b), and glacial(c) longitudinal profiles for Mars. The family of curves plotted shows parameter-induced variability in the non-dimensional uplift-erosion parameters Ne, Ngand Ns, which increase in the direction indicated by the arrow. Γ is kept con-stant in panel (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 5.2 Observed valley network profiles (dashed lines) overlapped by steady-statebest fit model results for fluvial (Naktong Valles, (a)), sapping (Abus Val-lis, (b)), and glacial erosion (unnamed valley network, (c)). The horizontalaxis is the normalized distance along the valley, and the vertical axis is theelevation normalized by the total topographic drop. . . . . . . . . . . . . . 96xviiiFigure 5.3 Profiles of valley networks not consistent with steady-state models of ero-sion (examples). (a) shows an irregular profile (Loire Valles), (b) shows astepped profile (Himera Valles), and (c) shows a convex profile (Nirgal Valles). 97Figure 5.4 Global geological map of Mars including the surface age (blue for Noachian,green for Hesperian and beige for Amazonian) and surface tectonic struc-tures (grey lines), overlaid with the results of our longitudinal profile anal-ysis. Check marks indicate valley networks in steady-state, color codedaccording to their origin: red for fluvial, purple for sapping, and blue forglacier. Black crosses indicate a valley network out of steady-state. Griddivisions correspond to the USGS quadrangles (45◦ across) . . . . . . . . . . 98Figure 6.1 Drainage of an ice sheet through channelized (left, a,b,c) and distributeddrainage (right, d,e,f). From left to right, cross-sectional perspective of thedrainage, planar view, and resulting landscape signature, respectively. Theblue layer represents meltwater accumulation in the ice-bedrock interface. . 107Figure 6.2 Global topographic map of Mars (MOLA) overlapped with the location andtiming of incision of the valley networks presented in the previous table.Note how late Hesperian and Early Amazonian drainage systems concen-trate at the high elevation areas around volcanic centers, whereas incisionup to the Early Hesperian period is widespread. . . . . . . . . . . . . . . . . 109Figure 6.3 Hypothesized ”transient landscape” path between fluvial and glacial steadystate landscape configurations as captured by the PCA. The figure showsexpected data from fluvial valleys spanning from Southern Washington toAlaska sampled every 50 km moving northwards (transient valleys (?), inorange dots), together with metric predictions for steady state fluvial andglacial systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Figure 6.4 Obliquity oscillations on Mars (black solid line) and Earth (red dashed line).Horizontal axis shows time in millions of years, and vertical axis showsdegrees of inclination. Dashed grey areas indicate periods of ice advance.Adapted from Head et al. (2003). . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure C.1 Principal Component (PC) linear decomposition in terms of the metrics,scaled to the variance captured by each of the metrics. From left to right,the X axis displays junction angle γ, fractal dimension D f , stream order Sn,width of 1st order tributaries λ, aspect ratio R, and undulations Ω. . . . . . 145Figure C.2 Representative examples and longitudinal profiles of the PCA erosional end-members: (a) fluvial, (b) glacial, (c) sapping, (d) subglacial. . . . . . . . . . 147xixFigure E.1 Approximate constraints on the slope-area region where the stream powerlaw is adequate to describe bedrock incision by fluvial processes. Suggestedupper and lower slope bounds and width of process transition zones areapproximate. Adapted from Sklar and Dietrich (1998), fig.1 . . . . . . . . . . 154Figure E.2 The thirty cartographic quadrangles of Mars, as defined by the United StatesGeological Survey. North is at the top; 0◦N 180◦W is at the far left on theequator. The quadrangles are overlapping a black-and-white MOLA topo-graphic map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155xxList of Supplementary MaterialsName Typeubc-2018-september-graugalofre-anna-ch1-dataset-Earth.csvdatasetubc-2018-september-graugalofre-anna-ch3-parameters.xlsmodelparametersubc-2018-september-graugalofre-anna-ch3-sensitivity-analysis.xlsdatasetubc-2018-september-graugalofre-anna-ch3-sg-evidence.xlslong tableubc-2018-september-graugalofre-anna-ch3-vn-dataset.csvdatasetubc-2018-september-graugalofre-anna-ch4-parameters.xlsmodelparametersxxiList of SymbolsRoman SymbolsSymbol Definition UnitsA drainage area m2a stress-erosion exponent (-)Ai ice flow law parameter Pa3/sb hydraulic geometry exponent (-)b˙ glacier mass balance sources 1/sBT hardness parameter 1/Pa3sBs sliding law parameter Pa3/m2sc basin hydrology exponent (-)C Dimensionless ice sliding constant (-)C f dimensionless bed friction factor (-)cp glacial sliding bed frictionparameterm/sd valley network depth mD10 10th percentile grain size of thesedimentmxxiiD50 50th percentile grain size of thesedimentmD84 84th percentile grain size of thesedimentmD90 90th percentile grain size of thesedimentmD f fractal dimension (-)ds average sediment grain size mE erosion rate m/sEa glacial erosion rate by sliding m/sE f fluvial erosion rate m/sEp glacial erosion rate by plucking m/sF shape factor (-)g acceleration due to gravity m/s2H topographic drop mhi ice thickness mK erodibility coefficient m1−2m/sKa Hack’s law parameter m2−hKb bedrock erodibility m/sPa−aKg glacial erosion constant m1−l sl−1Kh substrate hydraulic conductivity m/sxxiiiKi ice creep closure constant (-)Kp substrate permeability m2Kq basin hydrology coefficient m3−2c s−1Ks substrate hydraulic conductivity m/sKw hydraulic geometry coefficient m1−3b sbL length along the main stem mLi latent heat of ice fusion J/kgm stream-power area exponent (-)n stream-power erosion exponent (-)N effective pressure PaNe fluvial uplift-erosion number (-)Ng glacial uplift-erosion number (-)Ns sapping uplift-erosion number (-)Pi ice cryostatic pressure PaPw subglacial water pressure PaQ discharge m3/sQo characteristic discharge m3/sR aspect ratio (-)Rb glacial bed roughness (-)xxivRh hydraulic radius mRρ dimensionless density (-)S slope of the flow surface (-)Sn stream Strahler order (-)So slope of the valley floor (-)SS steady-state (-)t time su flow velocity m/sU uplift rate m/sUo characteristic uplift rate m/svtcw valley network width to innerchannel width ratio(-)W valley network width mx position along valley or channel mx∗ dimensionless position alongstreammz elevation perpendicular to valleyfloormz∗ dimensionless elevation mzo spring elevation mzr bed roughness mxxvGreek Symbolsα slope of main stem (-)β slope of merging tributary in ajunction(-)Γ non-dimensional ice thickness (-)γ tributary junction angle (-)δ length-width scaling exponent (-)ε˙ rate of strain tensor (1/s)η viscosity Pasηe f f effective viscosity Pasθc dimensionless critical Shieldsstress parameter(-)θt dimensionless Shields stressparameter(-)λ order 1 channel width mρi ice density kg/m3ρs sediment density kg/m3ρw water density kg/m3σ geometric standard deviation (-)τ stress tensor N/m2τb basal shear stress Paxxviτo yield stress N/m2φ slope angle of channel walls (-)φr static angle of friction (-)Ω profile undulation (-)xxviiAcknowledgmentsAt some point in the last two years of my physics degree I realized two things. First, I lovedfundamental physics. Second, I did not want to spend my career investigating such an abstracttopic I would not see the results of my research validated. That is when one of my old passions- Earth science - came to the rescue, and I realized I wanted to work in applying physics toEarth science.The journey (which is not finished yet!) started with me emailing Mark Jellinek a weekafter my birthday in 2011. He made the mistake of saying that he “wanted to hear more aboutthe questions I was interested in”. Little he knew he would end up hearing question, afterquestion, after question, after question, and so on, for the next 5 years. He does know, though,that he made the answer to some of those questions possible. This is a thank you, Mark, forthis incredible scientific journey that started the moment you answered to that email. I willalways be grateful to you for teaching me what it means to do interesting and good science,and to bravely and shamelessly tackle seemingly impossible questions. You taught me howthink critically about all the possible different ways there are to solve a big problem, to thinkbig and start small. Your ability to guide and not to boss, to listen to the emerging scientificvoices of your students instead of imposing your own is short of admirable. Thank you aswell for all your edits to my writing, the whole pages of manuscript bleeding in red color,the paragraphs crossed with little sentences of the kind “ you don’t actually need this” nextto them, my remarkable skill to mix up “constrain” and “constraint”, etc. As heartbroken asI was the first times I saw those, I can now write and speak much better English. You arean amazing supervisor, but you are more, you are a great friend. Our meetings over a beerdiscussing anything from my research to yours, to the best places to see flamenco in Vancouver,or even personal life, are not over Mark. Cheers to them! May they last for a long long time.What else can I say, other than I wish one day I will be half as good of an advisor as you havebeen with me. Gra`cies Mark.I would like to thank my family for a lifelong unconditional love and support, my dadRamon, my mom Neus, and my dear brother Marc. Thank you for always being there, eventhousands of kilometers away, even in the middle of the night. This journey does not actuallybegin with this PhD. This journey began a long time ago, with a 7 year old girl looking forfossils, hunting lizards, stargazing, and exploring the world in the summer months of anothercontinent. You, my dear family, made this possible in the first place by feeding my hungryxxviiicuriosity with answers to some of those questions, and by encouraging me to ask many more.You knew, deep inside you, I would one day take off and leave home seeking for answers toquestions, and although you knew that you would miss me dearly, you always encouragedme to find my own path and to keep the flame of my curiosity and passion for science alive.My brother, Marc, is the best of us two. He is a better person, pushing himself to the extreme tomake everyone he loves happy. He is also more intelligent, coming up with wonderful ideasand simple solutions to ridiculously hard problems. He is also more sensible, full of goodjudgment and common sense. No matter where I go, Marc, you will always have a home withme, for you are one of the most important people in my life. The farewell in the airport, whenI left home with a one-way ticket to Canada, was for me one of the hardest moments of mylife. With my family, I would like to thank my boyfriend Francesco for his love, support, andpatience over this last difficult couple of months. This is over, but another adventure awaits,one that we will tackle together. I will be forever grateful of how you welcomed me into yourlife and into your home, and how you taught me back what it means to be Mediterranean.I admire very much how methodical, sensible, intelligent, and loving you are, and how youlisten to others and not just to yourself. I have a lot to learn from you.My gratitude also goes to my extended family - my friends and colleagues from back home(Miriam, Raquel, Clara, Albert, Lluı´s, Aitor, Joan, Oscar, Isma, David), from the MJCJ researchgroup, and my other more-than-awesome partners in crime (Mike, British, Gesa, Thibaut,Sam, Gavin, Jilmarie, etc!). Thanks Kathi for letting me stay in your place until I found ahome in Vancouver, and for welcoming me to the family. Catherine, I admire you a lot, thankyou for being an example of how important is the contribution of women to science. Georgiaand Tasha, Manar, thank you for a long and extended friendship with a pinch of enjoyablecompetitiveness that developed over the years into profound respect. Yoshi, I saw you growfrom a kid into a very good scientist. You still don’t know how much you are capable of, butyou will discover it soon! Megan, you are a wonderful person with a lot of scientific talent anda good dose of stubbornness. The combination will bring you to the path of success whateveryou choose to do. Big boy - yes, Thomas has a nickname - you are my academic brother whomI admire as a person and as a scientist. You are brilliant. How did you manage to get 5 papersout?! Colin, you are a wonderful person and a brilliant scientist, with a terrible taste for greenshirts. We have had some awesome adventures up in the far north that I will never forget. Iam honored to call you a friend. Gabriela and Deni, you are absolutely amazing people thatI wish I had met a long time ago. So much vitality, so much energy, so much happiness. Youturn a grey day into bright. The Mikes (Mike and British), Gesa, and Thibaut, thank you forsome of the most fun and adventurous episodes of my life. May your desire of explorationand your mountain hunger never fade (please don’t eat mountains though). I know one dayyou will check all the mountains in the west coast off your list! My dear other Martian, thankyou bonita for these years of great friendship. Look at us in that picture in Galiano Island, howmuch we have changed! You turned into a stellar climber and a wonderful scientist - you stillxxixdon’t realize, my friend, how brilliant you are: precise, thorough, clever, good planner, clearthinker. You are one of the best friends I have. Thank you.I would also like to thank my collaborators, Antero Kukko, Mike Zanetti, Rose Gallo, Eti-enne Godin, and in particular Gordon Osinski. Oz, without you and the opportunities yougranted me, this dissertation would have been half as interesting as it is. During the course of3 weeks in Devon Island with you and Mark I learned more about surface processes and ero-sion than I did reading papers and textbooks over the course of two years. I owe my learningexperience as a field geologist to you, and for this I am very thankful. Cheers to a long-lastingcollaboration!To the vastly unexplored surface of Mars, so full of promises and answers, and to the silentand cold paths of Devon Island. The hunger for discovery has led science your way. May therevelation of your secrets lead to more questions, always more. Keep the flame of curiosityalive.xxxDedicationThis journey is dedicated to my family: my very dear brother, mum, and dad. E´s per vosaltres,per totes les vegades que ens hem trobat i despedit a l’aeroport. Per tu Marc, per 25 anys depreciosa germanor i profunda amistat, per totes les aventures que hem corregut, les sortidesde sol a Cala Pedrosa o a Cadaque´s, i totes les aventures que correrem! Per tu papa, perdespertar-me la curiositat que m’ha portat per aquest camı´, per aquelles sortides a Tavertetbuscant fo`ssils i aquelles nits en que enganxaves refredats a Sant Hilari mentre jo jugava ambel telescopi. I per tu, mama, per ensenyar–me a ser millor persona, per tenir la pacie`ncia d’unasanta amb mi quan era un petit dimoni i ara que encara ho so´c, i per demostrar que a la vidales coses bones sempre prenen temps i cura. Us estimo molt.xxxiChapter 1Introduction and motivation1.1 OverviewThe 1970s Mariner 9 and Viking missions to Mars revealed a landscape dissected by hundredsof valley networks (Figure 1.1 and 1.2), the remnants of ancient drainage systems likely carvedby liquid water (e.g., Sharp and Malin, 1975; Clifford, 1993; Carr, 1995; Gulick, 2001; Howard et al.,2005; Hynek et al., 2010). Since the 70s, and in spite of the data bonanza resulting from 18successful missions (11 orbiters and 7 landers), the origin of these valley networks, their rolein the hydrologic cycle of ancient Mars, and the climate implications of their emplacementremain a subject of heated debate (e.g., Sharp and Malin, 1975; Laity and Malin, 1985; Bakeret al., 1992; Kargel and Strom, 1992; Carr, 1995; Craddock and Howard, 2002; Hynek et al., 2010;Wordsworth, 2016; Ramirez and Craddock, 2018). Indeed, the presence of valley networks andtheir remarkable similarity to terrestrial river systems has motivated the view that early Mars(∼ 3.8− 3Byr ago, see Figure 1.3) was warm enough to sustain prolonged flowing water atthe surface (e.g., Pollack et al., 1987; Baker et al., 1991; Parker et al., 1993; Clifford, 1993; Craddockand Howard, 2002; Mangold et al., 2004; Irwin et al., 2005; Howard et al., 2005; Hynek et al., 2010;Ramirez and Craddock, 2018). In striking contrast to this view, 3D state-of-the-art climate modelsfor early Mars under a fainter young Sun (e.g., Feulner, 2012) show that surface temperatureswere likely never significantly higher than the water melting point for an extended period oftime (Wordsworth et al., 2013; Forget et al., 2013; Wordsworth et al., 2015; Wordsworth, 2016). I referto the extended outline below for more elaborate descriptions of these competing “warm andwet” and “cold and icy” scenarios. In addition to climate considerations, there I also present areview of the morphology of valley networks and relevant literature, and a detailed overviewof the most commonly used terrestrial analogs.1.2 Motivation and driving questionEfforts to understand the origin and climate implications of valley network emplacement havetriggered a large body of literature aimed at meticulously describing their morphology, and11.2. Motivation and driving questionled to widely-ranging interpretations (e.g., Group, 1983; Laity and Malin, 1985; Gulick, 1993;Carr, 1995; Lee, 1997; Carr and Malin, 2000; Gulick, 2001; Craddock and Howard, 2002; Howardet al., 2005; Lamb et al., 2006; Hynek et al., 2010; Grotzinger et al., 2015; Fastook and Head, 2015;Ramirez and Craddock, 2018). The morphological resemblance of valley networks to terrestrialdrainage systems (a.k.a, analogs) has been compelling (Figure 1.5).The three leading hypothesis regarding valley network formation are (e.g., Group, 1983;Laity and Malin, 1985; Lee, 1997; Gulick, 2001; Craddock and Howard, 2002; Howard et al., 2005;Hynek et al., 2010):1. River incision, involving rainfall and fluvial erosion (Craddock and Howard, 2002; Howardet al., 2005; Irwin et al., 2005; Ramirez and Craddock, 2018).2. Erosion through groundwater sapping at a spring site, leading to headwall underminingand eventual collapse (Pieri, 1980; Laity and Malin, 1985; Kochel and Piper, 1986; Laity, 1990;Gulick, 2001).3. Meltwater erosion in an otherwise glaciated Mars (Carr and Head, 2003; Head and Marchant,2014; Fastook and Head, 2015) .4. Erosion by wet-based (Kargel and Strom, 1992) or cold-based glaciation (Fastook and Head,2015)However, the physical basis behind such interpretations is, at best, questionable: The sur-face gravity on Mars is one-third of that on Earth, which leads to a proportionally smallerfluvial erosional power (Gulick, 2001; Howard et al., 2005), and has even more profound effectson glacial (Pelletier et al., 2010), subglacial (Kargel and Strom, 1992), and groundwater spring-fed (sapping) (Lamb et al., 2006) incision. Indeed, resemblance with terrestrial drainage systemsmay only be accidental, which leads to the question that motivates this thesis:What is the origin of the Martian valley networks, and what constraints do they providefor understanding the early climate and tectonic histories for Mars?To address this question, the thesis builds on models of landscape evolution, fluid dynam-ics, and erosion mechanics that describe four distinct mechanisms of incision proposed forMars: riverine, glacial, sapping, and subglacial. These mechanisms capture four fundamen-tally distinct climate settings for early Mars:1. A warm and wet scenario involving rainfall and widespread fluvial erosion.2. A cold and wet scenario, represented by sliding glaciers in a warm-based glaciation set-ting.3. A cold and dry scenario, represented by extensive and mostly cold based ice sheets withlocalized melting triggering subglacial channelization.21.3. Structure and organization4. A dry surface scenario, in which most water accumulates in aquifers and emerges insprings, triggering sapping erosion.Section 5 in this chapter explores the details of each of these climate scenarios in the contextof state-or-the-art climate model results for early Mars.1.3 Structure and organizationTo answer to the question posed above, I organize this thesis into four chapters. Chapter 2develops a quantitative characterization and classification technique for valley and channelnetworks, which allows for a physical understanding of their origin in terms of the analyticformulation of five quantitative landscape metrics. The test of this technique in terrestrialchannel networks is widely successful at characterizing steady-state fluvial, glacial, and sap-ping landscapes, but fails to distinguish networks incised by subglacial erosion.Chapter 3 builds on this knowledge gap and uses the products of an extensive field cam-paign to Devon Island (high Canadian Arctic) together with models of subglacial drainage todevelop three additional metrics that distinguish subglacial from fluvial incision from remotesensing data.Chapter 4 addresses the main question motivating this thesis (cf. section 2). In this chapterI apply the technique developed in Chapter 2, complemented with results from Chapter 3, toclassify the Martian valley networks and build a quantitative understanding of their origin.The results of this chapter, in particular the identification of widespread subglacial erosion onMars, reconcile the outcome of state of the art climate models with the increasingly higherresolution morphological observations relative to early Mars. Contrary to the most popularunderstanding of early Mars morphology, the presence of valley networks is not proof thatMars was once warm and wet but the opposite: they are evidence of an extensive, locallywarm-based late Noachian ice sheet, with punctuated warmer episodes allowing for forma-tion of proglacial, fluvioglacial valleys.In the last chapter, Chapter 5, I derive and use the distribution of steady-state and outof steady-state valley network longitudinal profiles to explore the coupled early climate andtectonic histories of Mars. The results of this chapter represent the first quantitative evidencethat the majority of fluvial valley networks did not reach a steady state, inconsistent with theprevailing view of a long-lived surface liquid water hydrological system. In addition, insightsinto the curvature of the few identified steady state fluvial longitudinal profiles show thaterosion rates were likely small on Mars in the fluvial systems, and that warm-based glaciationexisted late into Mars evolution during the Hesperian period.I conclude this thesis in Chapter 6 reviewing the main results from each chapter and dis-cussing the main implications for Mars early climate. In summary, the extensive and quanti-tative analysis developed in this thesis shows that:1. The channel and valley network classification and characterization technique devel-31.4. Outcomesoped in Chapters 2 and 4 is successful, and can be readily adapted to Earth, Mars, andpotentially other planetary bodies to build understanding of their surface evolution.2. Most of the valley networks on Mars are best explained by erosion in subglacial chan-nels, arguing that the climate of early Mars was likely cold and icy, with extensivecold-based ice sheets that covered the highlands and underwent local basal melting.3. Fluvial systems existed on early Mars, confirming that this cold climate was punctu-ated by episodes of warming that allowed for transient surface water stability.4. Valley networks rarely reached steady-state on early Mars.5. Erosion rates were likely small throughout Mars early history.1.4 OutcomesEach substantive chapter in this thesis corresponds to a publication. Chapter 2 is publishedin the Journal of Geophysical Research (The geometry and complexity of spatial patterns ofterrestrial channel networks: Distinctive fingerprints of erosional regimes). Chapter 3 is pub-lished in The Cryosphere (Subglacial drainage patterns of Devon Island, Canada: Detailedcomparison of rivers and subglacial channels). Chapter 4 is submitted as a Letter to Nature(An Antarctic-style ice sheet on ancient Mars). Chapter 5 is submitted to Icarus (Lessons fromthe shape of Martian valley networks’ longitudinal profiles: steady-state, and the link betweenclimate and tectonics on early Mars.).1.5 Extended outlineThe following extended outline complements the overview given at the beginning of this In-troduction with detailed descriptions of the Martian valley network morphology, distribution,timing, and most commonly used analogs, as well as a more extensive description of the cli-mate scenarios proposed for early Mars.1.5.1 The Martian valley networks: age, distribution and morphologicalvariabilityValley networks are ancient (3.9-3.5 Ga) branching systems of tributaries that morphologicallyresemble terrestrial river systems (e.g., Carr, 1995; Gulick, 2001; Carr, 2007). As shown in Figure1.1, they predominately concentrate on the heavily cratered southern hemispheric highlandsspanning low to mid latitudes, whereas the younger lava plains of the northern hemisphere(i.e., lowlands) appear mostly undissected. Within the southern Highlands, valley incisionis widespread across nearly all geological settings, including crater walls, the sides of vol-canoes and massifs, and intercratered plains, and particularly concentrated along the crustal41.5. Extended outlinedichotomy boundary. In terms of age, approximately 91% of mapped valley segments lie en-tirely within Noachian terrains (>3.7 Ga ago), 6% cross into or are entirely contained withinHesperian-aged surfaces (3.7-3.0 Ga), and 3% occur on Amazonian terrain (<3.0 Ga) (Fassettand Head, 2008a; Hynek et al., 2010).It is worth noting a direct, albeit unfortunate, possible interpretation of the word ”valley”when referring to the Martian drainage systems. By definition, a valley is any low-lying landbounded by higher ground on either side and usually traversed by a stream or river, whereasthe word channel refers to the lowest portion of a stream, bay or strait (e.g., Gilbert, 1877; Bateset al., 1984; Craddock and Howard, 2002). Water may be contained in a channel or stream, and achannel or stream may be contained in a valley. It is unclear whether Martian valley networksare indeed valleys or channels, which makes the interpretation about their origin complicated.The literature is inconclusive and unclear in this matter, and seems to point in both directions(Group, 1983; Carr, 1995; Carr and Malin, 2000; Craddock and Howard, 2002). I will further discussthis issue in Chapters 2 and 4.Figure 1.1: Global topographic map of Mars derived from the Mars Orbiter Laser Altime-ter, overlapped with the location and extent of the Martian valley networks, adaptedfrom Hynek et al. (2010). Note how valley networks accumulate in the southern ele-vated hemisphere (Highlands) and leave the northern plains unincised.Individual valleys are typically hundreds of meters to 10 km wide and up to a few hun-dreds of meters deep (e.g., Carr, 1995; Williams and Phillips, 2001; Hynek et al., 2010), with51.5. Extended outlinemean valley widths (W) of 〈W〉 ± σ = 2040 ± 1548 m and depths (d) of 〈d〉 ± σ = 109 ±137 m (Williams and Phillips, 2001). Cross sectional profiles can be V-shaped, U-shaped, andtrapezoidal-shaped (Williams and Phillips, 2001; Gulick, 2001), and are rarely dissected by in-ner channels even in high-resolution image data (Irwin et al., 2005). Network topologies varywidely from single-valley to fully developed dendritic systems consisting of more than 8000tributaries (Hynek et al., 2010).Figure 1.2: Examples of the different types of Martian valley networks as presented in theliterature. 2a: single channel morphology, 2b: dendritic valley network on a vol-cano flank, 2c: longitudinal valley network, 2d: well-integrated valley network, 2e:young dendritic valley network on a volcano flank, 2f: filamentous valley network.In terms of morphology, valley networks have been broadly classified as large and smallvalley systems (e.g., Baker, 1982; Gulick, 2001). Large, or longitudinal, valley systems, whichinclude Nanedi Valles, Ma’adim Valles, Al Qahira Valles, etc., are hundreds of kilometers long,and several kilometers wide. The headwater regions of longitudinal valleys display km-scalewide tributaries with rounded heads (theater-shaped), whereas the lower reaches display U-shaped or trapezoidal cross sections and sinuous channels (see panel (c) in Figure 1.2 for anexample) (Baker, 1982). Small valleys exhibit a wide array of morphologies, including singlechannels (Figure 1.2 a), “filamentous” tributaries (Figure 1.2 f), and well-integrated (Gulick,2001), dendritic networks (Figure 1.2 d). The locations of these features also span differentgeologic units (i.e., Figure 1.2 a is Late Hesperian, b is Noachian, c is Middle Noachian, d isLate Noachian, e is Early Amazonian, and f is Late Noachian), a point to which I will comeback in Figure 1.3 below.Valleys located in the heavily cratered terrains of the southern highlands form laterallyextensive networks with a large number of tributaries (Hynek et al., 2010). At the headwaters,valleys with no tributaries (1st order valleys) are hundreds of meters wide (see Chapter 4), andare typically degraded beyond the resolution of even the sub-meter scale of the High Resolu-tion Imaging Science Experiment (HiRISE) (up to cm/pixel). When traceable, 1st order valleystend to originate in crater slopes or fractures (Gulick, 2001). Further downstream, tributariesare often as deep as the main trunk, and display steep walls, flat floors, and relatively constant61.5. Extended outlinedown valley width (Gulick, 2001). None of these observations are consistent with terrestrialriver systems.Valleys located on the slopes of volcanoes, on the intercrater plains, or incising the rimsof craters and canyons are in general younger (see Figure 1.3 (Hynek et al., 2010), and lessdegraded than systems incised in the Noachian highland plateaus. Some of these youngervalley networks are structurally controlled, following the patterns of pre-existing fracturesand faults (Gulick, 2001), whereas some others follow topographic gradients, which is the casefor systems emplaced on volcano flanks (Figure 1.2 b and e).Figure 1.3: Diagram showing the timing of processes relevant to the Martian surface evo-lution (adapted from Gulick (2001); Ehlmann et al. (2011); Wordsworth (2016)). Al-though absolute ages are widely unconstrained (crater counting techniques permiterror bars up to 100 Myr), the relative timing of these processes is robust.1.5.2 Early Mars: Warm and wet or cold and icy?The surface of Mars is currently devoid of liquid water, with surface pressures averaging 0.64kPa (0.6% of that on Earth) and surface temperatures averaging 220 K. Consequently, the pres-ence of valley networks on Mars’ surface has historically represented one line of geologicalevidence arguing for a milder climate during the early stages of Mars’ evolution (see Figure71.5. Extended outline1.3). Dendritic valley networks (i.e., Figure 1.2 b, d, e), in particular, are commonly interpretedas evidence for fluvial erosion involving precipitation (rain or snowfall), and runoff (e.g., Bakeret al., 1992; Craddock and Howard, 2002; Howard et al., 2005; Ramirez and Craddock, 2018), withestimated discharges of the order of 1000 m3/s (Howard et al., 2005; Irwin et al., 2005) and for-mation timescales of 105 − 107 yr (e.g., Barnhart et al., 2009; Hoke et al., 2011). Other evidencesuggesting long-lived surface liquid water includes observations of open-basin lakes (Fassettand Head, 2008b; Wordsworth, 2016), detection of phyllosilicates, sulfates, carbonates, and otherhydrated minerals (e.g., Mustard et al., 2008; Taylor et al., 2010; Ehlmann et al., 2013; Viviano-Becket al., 2014), as well as the presence of inferred shorelines as a boundary for a northern ocean(Parker et al., 1993; Head et al., 1999; Perron et al., 2007). These signs of surface liquid water sup-port the so-called “warm and wet” climate regime, which would have allowed for an activehydrological cycle and potentially a northern hemisphere ocean (e.g., Pollack et al., 1987; Bakeret al., 1991; Parker et al., 1993; Clifford, 1993; Craddock and Howard, 2002; Mangold et al., 2004;Irwin et al., 2005; Howard et al., 2005; Hynek et al., 2010; Ramirez and Craddock, 2018). For surfaceliquid water to occur, the atmosphere of early Mars had to be thicker (Wordsworth et al., 2013),with a lower bound of 60-100 kPa of CO2 constrained by geomorphological observations ofthe Dorsa Argentea ice sheet formation (Scanlon et al., 2018), and an upper bound of 300 kPa toavoid atmospheric collapse (Forget et al., 2013; Soto et al., 2015), also supported by the analysesof Martian meteorites (Hirschmann and Withers, 2008).This warm scenario is, however, contentious. State-of-the-art climate models for early Marswith a ∼100 kPa atmosphere (e.g., Wordsworth et al., 2013; Forget et al., 2013; Wordsworth, 2016;Ramirez and Craddock, 2018) have to reconcile two facts that conspire against achieving warmsurface temperatures for extended periods of time on early Mars: a fainter young Sun andchaotic orbital parameters. Over timescales longer than the 105 yr required for valley net-work incision (Howard et al., 2005), the Martian eccentricity and obliquity behave chaotically,ranging from 0 to 0.125 and 10◦ to 60◦, respectively (e.g., Laskar et al., 2004; Wordsworth, 2016).The consequences of the lower solar insolation are profound. In detail, if we assume thatin the late Noachian, Mars’ received solar flux was 75% of today’s (Gough, 1981), and con-sidering Mars’ semimajor axis to be 1.524 au, Mars’ solar flux in the Noachian would havebeen 0.75x1, 366/1.5242 = 441.1 W/m2, where the terrestrial flux corresponds to 1, 366 W/m2.Assuming now a planetary albedo of 0 (all incoming radiation is absorbed), the surface equi-librium temperature would be of Tc = (441.1/4σ)1/4 = 210 K at the time of valley networkemplacement (∼ 3.8 Ga), 63 K short of that of surface liquid water stability (Wordsworth, 2016).The combination of both factors make a stable, long lived and warm climate for early Mars diffi-cult.In addition to orbital and solar flux constraints, an early thicker atmosphere would alsohave consequences for the temperature distribution. On present-day Mars (0.6 kPa), temper-ature variation is dependent mainly on latitude, whereas on early Mars (∼ 100 kPa) surfacetemperatures decreased with altitude. This effect, known as adiabatic cooling (e.g., Forget81.5. Extended outlineet al., 2013; Wordsworth et al., 2013, 2015; Wordsworth, 2016), is critical on Mars given the largetopographic difference between north and south, and enhances cooling on the southern high-lands (see Figure 1.4), which become cold traps (Fastook et al., 2012; Fastook and Head, 2015;Wordsworth et al., 2015). The high stability of this climate scenario means that, even whenforced with transient large climate oscillations involving e.g., cratering and massive volcaniceruptions, climate recovers in a similar manner to a damped oscillator (Wordsworth, 2016).Figure 1.4: The adiabatic cooling effect on early Mars. (a) Annual mean surface temper-ature from a three-dimensional general circulation model (GCM) simulation with0.125 bar surface pressure. (b) Annual mean surface temperature from a three-dimensional GCM simulation with 1 bar surface pressure. (c) Scatter plot of sur-face temperature versus altitude for simulations with a 0.125, 1, and 2 bar surfacepressure. The dry adiabat g/cp is also indicated (gray line, constant potential tem-perature line). Data for the plots were acquired from the 41.8◦ obliquity, fixed rel-ative humidity simulations described by Wordsworth et al. (2015). Adapted fromWordsworth (2016).Indeed, for moderate values of obliquity and average surface pressure and given any ini-tial configuration of ice or water reservoirs at the surface of Mars, coupled models of climate(3D Mars GCM) and ice evolution (University of Maine Ice Sheet Model) predict the build-upof large ice sheets covering the highlands, and consequently the regions incised by valley net-works (Fastook et al., 2012; Fastook and Head, 2015; Wordsworth et al., 2013; Wordsworth, 2016).This hypothesis is the so-called ”Icy Highlands” climate scenario. Under these conditions, aLate Noachian Icy Highlands (LNIH) ice sheet would form covering evenly the Highlands,with ice thickness ranging between a few hundred meters to three kilometers depending onsurface climate conditions (Fastook and Head, 2015). The ice stability line, defined by the equilib-rium between ice accumulation and ablation rates, would lie at +1 km of elevation (Wordsworthet al., 2013).91.5. Extended outlineTransient episodes of warmer temperatures driven by e.g., punctuated volcanism, crater-ing, extreme orbital configurations (∼ 50◦) (e.g., Laskar et al., 2004; Toon et al., 2010; Halevyand Head III, 2014; Fastook and Head, 2015), would induce surface (top-down) melting of theLNIH into the ground (Head and Marchant, 2014; Fastook and Head, 2015), as has been noted byepisodic landscape evolution events in the Antarctica Dry Valleys (Head and Marchant, 2014).The amount of LNIH meltwater released during transient warm periods is consistent withlower bounds on the volume of water required to incise the valley networks and mobilize thesediment (Carr, 1995; Carr and Malin, 2000) as noted by Fastook and Head (2015). However, theavailability of meltwater would have been heavily concentrated along the ice margin, whichfails to explain the complex branching structure of the more developed valley networks thatare fed from large drainage basins. In short, early Mars climate was likely very cold as a con-sequence of a fainter Sun, but also highly variable over long timescales as a consequence of thevariability of its orbital parameters. Both facts argue against the long-time stability of surfacewater seemingly required to carve the valley networks.1.5.3 The geomorphological insights of using valley networks as paleoclimateindicators: terrestrial analogsThe potential role for the Martian valley networks to be paleoclimate indicators has motivateda large body of literature aimed at meticulously describing their morphology, establishinglinks between morphology and erosion processes, and discussing the climate implications re-quired by the different erosional mechanisms to operate (e.g., Group, 1983; Laity and Malin,1985; Gulick, 1993; Carr, 1995; Lee, 1997; Carr and Malin, 2000; Williams and Phillips, 2001; Gulick,2001; Craddock and Howard, 2002; Howard et al., 2005; Lamb et al., 2006; Hynek et al., 2010; Ramirezand Craddock, 2018). The connection between morphology and formation mechanisms leadingto valley network incision relies heavily on morphological and morphometrical comparisonsto similar drainage systems originated on Earth (a.k.a, analogs) (e.g., Group, 1983; Carr, 1995;Carr and Malin, 2000). In the words of Victor Baker (Baker, 2001): “The key element of this inquiryis the formulation of one or more working hypotheses, which are most often suggested (but not proved)by analogies of form and context among landscapes of known origin and those under scrutiny ”.On the basis of analog studies, the three leading hypothesis regarding valley network for-mation are:1. Fluvial erosion related to rainfall and surface runoff (e.g., Craddock and Howard, 2002;Howard et al., 2005; Irwin et al., 2005; Ramirez and Craddock, 2018).2. Fluvial erosion related to groundwater sapping, leading to bedrock erosion and head-wall collapse (e.g., Pieri, 1980; Laity and Malin, 1985; Kochel and Piper, 1986; Laity, 1990;Gulick, 2001). This would potentially have been driven by a higher widespread (Squyres,1989) or localized (Gulick, 2001) geothermal heat flux.3. Fluvioglacial erosion related to meltwater runoff, produced either by sporadic melting101.5. Extended outlineof the LNIH ice sheet as detailed above (Fastook and Head, 2015), by basal melting of thicksnow and ice deposits (Carr and Head, 2003), or by melting, erosion and undermining ofthe active layer overlapping permafrost (Head and Marchant, 2014).4. Erosion by wet-based (Kargel and Strom, 1992) or cold-based glaciation (Fastook and Head,2015)Analogy to river valleysThe analogy to terrestrial river systems fed by rainfall (exemplified in Figure 1.5a) is histor-ically the oldest interpretation of valley networks (Masursky, 1973; Baker, 1981; Group, 1983;Hynek and Phillips, 2003; Hynek et al., 2010). This interpretation is largely based on the vi-sual resemblance of valley networks to rivers on Earth, as well as on morphometric compar-isons of drainage density (average length of a valley compared to drainage area, (e.g., Ritteret al., 1995)), stream length, and stream order (generally following the Strahler criteria (Strahler,1958), see Chapter 2 for a detailed discussion) (e.g., Hynek and Phillips, 2003; Hynek et al., 2010)between Martian and terrestrial valleys. The formation of valley networks by precipitationand fluvial runoff was historically the first leading hypothesis (Masursky, 1973; Sharp and Ma-lin, 1975). More recently, high resolution observations drawn first from the Mars Orbital Cam-era (MOC) (Hynek and Phillips, 2003) and then from the THermal EMission Imaging System(THEMIS) (Hynek et al., 2010) added support for this formation mechanism with the detectionof high drainage densities and stream orders comparable to the lower bound for terrestrialvalues. Other pieces of evidence for this hypothesis involve the remarks on crater degradationby creep, potentially involving rain splash and surface runoff by Craddock and Howard (2002),estimations of valley network discharge similar to terrestrial values (Howard et al., 2005), andthe evidence against sapping bedrock erosion (Lamb et al., 2006) reinforced the hypothesis oferosion by rainfall (Ramirez and Craddock, 2018).111.5. Extended outlineFigure 1.5: Examples of analogs used in the literature as a basis to build comparativeunderstanding between Martian (right) and terrestrial (left) valley networks. 1.5ashows a comparison of the Grand Canyon and a dendritic valley network in AlbaPatera (Ramirez and Craddock, 2018), 1.5b shows Nanedi Valles compared to a sectionof the Colorado River Canyon as a terrestrial sapping analog (Ramirez and Craddock,2018), 1.5c shows a comparison of deposits of a glacier at the base of Olympus Monsand a glacier in Mullins Valley, Antarctica (Head et al., 2005) and 1.5d shows thecomparison of small dendritic valley networks with a subglacial channel network inDevon Island (Lee, 1997).Analogy to sapping valleysSapping valleys form where the head of the water table encounters the surface, producinga spring at the site of seepage (e.g., Laity and Malin, 1985; Lamb et al., 2006). If the stressesinvolved in groundwater seepage are sufficiently large, the spring-fed flow can incise intosediment and bedrock, undermining the site of seepage and eventually leading to collapse.The case for sapping on Mars (exemplified in Figure 1.5b) is predominantly based on mor-phological comparisons between valley networks and terrestrial sapping valleys on the Col-orado Plateau (Laity and Malin, 1985) and the Hawaiian box canyons (Gulick and Baker, 1990),although both interpretations are contentious (Lamb et al., 2006). Sapping valleys on Earthand a large portion of Martian valley networks share many visual characteristics, such as the121.5. Extended outlineabrupt headward termination of valleys (Pieri, 1976), absence of inter-valley dissection (Malinand Carr, 1999), constant downstream valley width (Gulick, 2001), short and stubby tributaries(e.g., Gulick, 2001), and low drainage densities (e.g., Carr and Chuang, 1997; Hynek et al., 2010).This interpretation remains to date a popular hypothesis for the formation of some valley net-works (e.g., Ramirez and Craddock, 2018) in spite of increasing evidence showing that springscannot cut into bedrock (Lamb et al., 2006) (see also Chapter 2).Ice and meltwater fingerprintsAncient glaciation on the Martian highlands, with landscape evolution resulting from ice andmeltwater erosion is an additional possibility for early Mars climate (Allen, 1979; Clifford,1993; Kargel and Strom, 1992; Lee, 1997; Head and Marchant, 2003). This hypothesis was ini-tially based on the interpretation of sinuous ridges (essentially, inverted individual channelsor channel networks) as eskers, with surrounding terrains interpreted to be the result of glacialand periglacial action. Morphological similarities between sinuous ridges and eskers, whichmotivate this interpretation, include the range of spatial scales, planimetric patterns (single,dendritic, and anastomosing networks), and tabular distribution of the sedimentary deposits(Kargel and Strom, 1992). Visual resemblances between valley networks and subglacial chan-nels was first noted by Lee (1997): short, stubby tributaries which feed into a broad and shallowvalley, similar planimetric shapes, and absence of inner channels (e.g., Figure 1.5c). We willcome back to this hypothesis in Chapter 4.More recently, a proglacial version has been proposed (Fastook and Head, 2015; Carr andHead, 2003) in which episodic warming and top-down melting of the LNIH provide enoughmeltwater to drive valley erosion. The analogy is here noted with shallow, flat bottomed, anddendritic valleys incised by meltwater provided by episodic warming of the West Antarcticaice sheet at the McMurdo Dry Valleys site (e.g., Head and Marchant, 2014). Analogies to wet-based glaciation generally involve findings of striations or moraine-like deposits within thelarge, longitudinal valley networks or the short, stubby valleys at the rims of Argyre (e.g., Pel-letier et al., 2010). Other properties, such as the spacing between 1st order tributaries (Pelletieret al., 2010), provide quantitative evidence for glaciation beyond purely visual comparisons.Figure 1.5d provides an example of this particular interpretation.1.5.4 The problem of visual analog studies and the need for a quantitativecomparison of terrestrial and Martian drainage systemsThe different hypotheses briefly reviewed above share the common assumption that the sameerosional processes would result in landforms with similar morphologies on Earth and Mars.The physical basis for this assumption is questionable. Indeed, characterizations based ondirect comparison of Martian and terrestrial valleys are insensitive to the distinct channelarchitectures that arise as a result of the difference between Martian and terrestrial surfacegravities (cf. Chapter 2 and Chapter 4). Crucially, Mars’ surface gravity is one third of that131.5. Extended outlineon Earth, which linearly reduces the erosional power available in fluvial systems (e.g., Whippleand Tucker, 1999; Howard et al., 2005), and has an even more profound effect on ice sliding veloc-ities (e.g., Nye, 1952; Paterson, 1994), effectively retarding glacial erosion. The characterizationof the incision of sapping valleys on Mars is studied in detail in Lamb et al. (2006), where theynote that the low stresses involved in groundwater sapping are unlikely to result in bedrockincision. The effects of a smaller gravity on the erosion by subglacial channelized drainage hasnever been explored before this work. An accurate and quantitative study between Martianand terrestrial drainage systems, therefore, must take into account how the different gravitymodifies the final morphology related to each style of erosion, an aspect that analog studiesalone cannot capture appropriately. To establish a reliable comparison, these effects must beintroduced in models of physical geomorphology that motivate and define a series of quan-titative metrics. Such morphometric parameterizations of valley network morphology canthen be assessed statistically (how many different types of valley networks exist on Mars?)and their values compared with suitable model predictions (what is the origin of the Martianvalley networks?).The next chapter (Chapter 2) explores this knowledge gap by considering terrestrial val-ley networks that are representative end-members of fluvial, glacial, sapping and subglacialerosion. Using a combination of erosion models and data analysis, I establish a quantitativescheme to characterize and classify valley networks on Earth, that can then be readily exportedto the Martian valley systems.14Chapter 2The geometry and complexity of spatialpatterns of terrestrial channelnetworks: Distinctive fingerprints oferosional regimes2.1 SummaryThe morphology of channel networks related to long term erosion reflects the mechanismsinvolved in their formation. This study aims to identify quantitative metrics, drawn fromtopographic data and satellite imagery, that are diagnostic of the distinctive styles of erosionby rivers, glaciers, subglacial meltwater and groundwater sapping. From digital elevationmodels, we identify three geometric metrics: The minimum channel width, channel aspectratio (longest length to channel width at the outlet), and tributary junction angle. We alsocharacterize channel network complexity in terms of its stream order and fractal dimension. Tovalidate our approach, we perform a Principal Component Analysis (PCA) on measurementsof these five metrics on 70 channel networks. We build understanding of these results, inturn, using scaling analyses of appropriate physical models. We show that rivers, glaciers andgroundwater sapping erode the landscape in rigorously distinguishable ways. Whereas riversare characterized by nearly constant minimum width, variable aspect ratio and high streamorders, glaciers have highly variable minimum widths and aspect ratios, and much smallerstream orders. Erosion by subglacial meltwater remains poorly understood, and we arguethat we require an additional metric to fully characterize these systems. Our methodologycan more generally be applied to identify the contributions of different processes involved incarving a channel network. In particular, we are able to identify transitions from fluvial toglaciated landscapes and vice-versa.152.2. Introduction2.2 IntroductionReconciling how the large-scale morphologies of channels and the valley networks in whichthey occur record the main processes from which they originate has been an outstanding ques-tion in quantitative geomorphology for more than 100 years (e.g., Gilbert, 1877; Horton, 1945;Hack, 1973; Montgomery, 2002; Whipple, 2004; Roe et al., 2008). Recently, satellite-based digitalelevation models (DEM), and the use of Geographic Information Systems (GIS) for their pro-cessing and interpretation have enabled the acquisition of remote sensing topographic data tocharacterize channel and valley networks at higher resolution [e.g. Jacek, 1997]. As our abil-ity to resolve the structure of channel networks improves, some enduring questions emergewith greater clarity: What are the main incision processes that give rise to a particular chan-nel network geometry? Was a particular landscape glaciated before being carved by fluvialprocesses? What processes underlie the formation of the Martian valley networks? Finally, ifwe were standing on Mars, viewing the Earth through a telescope with only remote sensingtopography data, could we reliably determine that Earth is in an interglacial period?Characterizing the network geometries and constituent channel morphologies, and relat-ing them to the mechanics governing the landscape response to the flow of water or ice, hasbeen addressed for fluvial (e.g., Engelund and Hansen, 1967; Kennedy et al., 1975; Parker, 1978a,b;Perron et al., 2008) and glacial channel networks (e.g., Paterson, 1994; Herman and Braun, 2008;Herman et al., 2018), and to a lesser extent for sapping valleys (e.g., Laity and Malin, 1985;Howard, 1988; Devauchelle et al., 2011), and subglacial meltwater channels (e.g., Ro¨thlisberger,1972a; Weertman, 1972; Nye, 1976; Walder and Hallet, 1979; Walder and Fowler, 1994). However,there exists no systematic approach for attributing the geometry of a channel network to ex-plicit contributions from these four erosional processes only from topographic data. This lackof methodical characterization is particularly important for locations where field studies ofchannel networks are difficult to perform, but topographic data are available (e.g. high Arctic,Antarctica, Mars, Titan, Pluto, etc.).To make progress, our paper has two objectives. The first is to build a geomorphologicfingerprint scheme that reliably identifies 5 distinctive (i.e., linearly independent) elements ofthe spatial pattern of each of the four channel network types exemplified in Figure 2.1, thatwe refer to as metrics. A second goal is to use forward modeling and Principal ComponentAnalysis (PCA) to uniquely relate this fingerprint to the predominant underlying processes offormation. To understand relationships that emerge in the PCA results, and to consequentlymake predictions, we find geometrical scaling relationships drawn from physical models offluvial, glacial, subglacial and sapping erosion regimes. The ultimate goal of this methodol-ogy is to provide an understanding of the relationships among the geometry and complexityof fluvial, glacial, sapping and subglacial channel networks and the underlying mechanicsgoverning their erosion.162.3. Methods I: Landscape metrics and patterns in channel networks2.3 Methods I: Landscape metrics and patterns in channel networks2.3.1 Five distinctive properties of channel networksFigure 2.1 shows examples of landscapes carved by glaciers (1a), rivers (1b), groundwatersapping (1c) and subglacial meltwater channels (1d). Apparent visual differences among thechannels in Figure 2.1 include length along the longest flow path, channel width, and mini-mum channel width. In addition, differing network complexities are indicated by propertiessuch as the maximum stream order and tributary junction angle (see Figure 2.2 for a definingcartoon).d) Unnamed (NW. Devon Island, Canada) 4Km3210Source: Esri, DigitalGlobe, GeoEye, i-cubed, Earthstar Geographics, CNES/Airbus DS,USDA, USGS, AEX, Getmapping, Aerogrid, IGN, IGP, swisstopo, and the GIS UserCommunity Apalachicola tributaries (Florida, US)c)Km0 421 30 0.5 1 1.50.25KmEnacheddong Creek (Western Australia)b)0 25 50 75 100KmUnnamed (E. Devon Island, Canada)a)d)Figure 2.1: Examples of channels emplaced through the action of different processes. a)A glacial valley on NE Devon Island, Canada b) A river in Western Australia d) Asapping valley in the Florida Panhandle e) An exposed subglacial meltwater channelin NW Devon Island, Canada. Imagery: Landsat 7 (USGS).Glacial channel networks (Figure 2.1a) show large tributary junction angles (> 80◦ on av-172.3. Methods I: Landscape metrics and patterns in channel networkserage) and stream orders (see Figure 2.3 for a definition) that are typically below 6. Individualchannels are wide (∼1-10 km), and short (∼ 100 km) relative to other systems in Figure 2.1.Fluvial channel networks (Figure 2.1b), by contrast, feature comparatively intricate networkswith large stream orders (typically 9–12), as well as relatively small tributary junction angleson average (∼ 60-70◦). The minimum width of incipient channels (i.e., order 1 channels) issmall (< 1 m) compared to the other types of channel networks.Groundwater sapping produces landforms with unique characteristics (Figure 2.1c). Sappeddrainage systems differ in channel geometry and valley morphology from their fluvial coun-terparts (Laity and Malin, 1985; Dunne et al., 1990), although there is discussion on whethervalley morphology is a reliable indicator of seepage erosion (e.g., Lamb et al., 2006). Streamorders are typically below 5, and tributary junction angles around 72◦ (Devauchelle et al., 2012).Individual channels feature short lengths (∼ 10 km) and are a few hundred meters wide,with characteristic cross sectional U-shapes and rounded, theater-shaped valley heads (Laityand Malin, 1985; Lamb et al., 2006). Uniquely relating the rounded valley head morphologyto sapping processes is, however, disputed in several studies. In particular, Lamb et al. (2006)questions the ability of springs to incise into resistant rocks, and discusses how mass wasting,waterfall-driven erosion and runoff processes can lead to similar morphologies. Therefore, inthis study we do not use the shape of the valley head to identify groundwater sapping incision.Subglacial meltwater channels (Figure 2.1d) feature intricate networks with braided chan-nels, variable stream orders up to 6, and also variable tributary junction angles. Channelswithin a network are approximately parallel and oriented in the direction of paleo-ice flowlines, with relatively uniform width (Walder and Hallet, 1979). Longitudinal profiles of individ-ual channels have concave and convex sections (undulations) and significant overdeepenings(Kehew et al., 2012). Networks may be non-dendritic (e.g., anastomosing or parallel). Herein,we will refer to these channel networks as subglacial channels.On the basis of the distinctions drawn from Figure 2.1, we hypothesize 5 metrics to char-acterize a channel network:1. The aspect ratio R of length along the longest flow path in the network to its channelwidth measured where the channel achieves the highest stream order (see Figure 2.2, (b)and (c));2. the minimum channel width λ or bankfull width of order 1 channels in the network (seeFigure 2.2, (e));3. tributary junction angles γ, measured at the junction of a tributary and the main stem(see Figure 2.2, (a));4. the maximum stream order of a channel network, measured using the Strahler orderingsystem Sn (see Figure 2.2, (d)).;182.3. Methods I: Landscape metrics and patterns in channel networks5. the fractal dimension D f as a measure complexity of the channel network branchingpattern (see Figure 2.2, (f)).Figure 2.2: Cartoon depicting the metrics enumerated above. Panel (a), junction angle γ,(b) length L, (c) bankfull width W, (d) stream order (following the convention byStrahler) Sn, all tributary-less channels receive an order 1. Upon merging, two orderones become an order two, two order two tributaries become an order 3, and so on,(e) minimum width λ, and (f) fractal dimension D f .To test the effectiveness of these 5 metrics at characterizing and distinguishing the differentchannel network types, we construct a database with measurements of these 5 parameters on70 channel networks including rivers, glaciers, sapping valleys and subglacial channels fromarid and semi-arid areas. Our dataset (see supplementary dataset “ch1datasetEarth.xls”) in-cludes 40 river networks mostly distributed in the tropics, around 30◦ N and 30◦ S, includingexamples from central America, north Africa, the middle East, central Asia, Patagonia andAustralia. Also included are data from 14 glacial networks, which are located mostly northof 60◦ N, with a few additional alpine glacial network examples from the Himalayas. The 10data points corresponding to sapping valleys are located in Florida (3), the Colorado Plateau(3), and Hawaii (4). We are aware of the contentious discussion regarding the formation ofsapping channel networks in bedrock in Colorado and Hawaii (e.g., Lamb et al., 2006). How-ever, we consider the addition of these systems to be of interest, as our objective is to give aquantitative comparison of channel networks of different origin. Finally, our dataset includes6 data points corresponding to exposed subglacial channels, all located above 75◦ N in thehigh Arctic. See supplementary dataset “ch1datasetEarth.xls” for the location coordinates ofeach channel network.192.3. Methods I: Landscape metrics and patterns in channel networksWe perform a Principal Component Analysis (PCA) with the aim of identifying statisti-cally significant data groups (clusters) in terms of our 5 metrics. Through this process, we alsodetermine the extent to which these groups relate to the original 4 types of channel networks.To analyze the relative relevance of each metric at characterizing a given type of channel net-work, we calculate the fraction of variance captured in each principal component (PC), andthe weight of each metric within each PC. Comparison among cluster spread, cluster centralposition and overlap of channel network of different types in one same cluster govern dis-cussions about similarities among erosion regimes and valleys incised by a combination ofprocesses. Although a careful assessment of the metrics is sufficient to characterize the chan-nel networks in our data, the PCA adds statistical rigor, which enables the pattern recognitionto be understood with the physical models we present in section 3.Figure 2.3 introduces the definition and an example measuring process of the 5 metrics inthe particular example of a river channel network, and Figure 2.2 presents a series of cartoonsto help visualize them. The junction angle (Figure 2.3a) is the measure of the acute angle be-tween tributary and main stem at different locations in the channel network. At each location,we evaluate the angle existent between the stream lines we digitize according to the detailsdescribed in the next section. The three crosses in Figure 2.3a exemplify the spots where wetake this measurement.Figure 2.3 (b) and (c) are the two components of the channel aspect ratio, length and width.We measure channel length using the Flow Length tool of the hydrology catalog in ArcGIS,with Figure 2.3b displaying an example of the results of this routine. We take width and lengthmeasurements at the same location, when the channel network achieves the highest stream or-der and before a depositional fan regime. We measure channel width as the bankfull widthof the channel (Figure 2.3c, spacing between black arrows) and take this measurement as thespacing between the two lines of maximum curvature between the minimum attributed tothe channel. Abrupt changes in slope between the center of the channel and the surroundingflatter floodplain are, in general, a good indicator of the bankfull width (Harrelson et al., 1994;Bjerklie, 2007, e.g.,). In rivers, this measure corresponds to the size of the channel requiredto convey the water, with discharges up to 1.5-year flood events (Harrelson et al., 1994). Weunderstand that bankfull width is dependent on a series of hydrologic criteria in addition togeomorphic. However, under the assumption this paper builds on (the landscape has reacheda steady state response to the flow regime of each system), curvature gives an estimate of bank-full width that both reflects a time averaged landscape response to the hydrologic conditionsand satisfies the geomorphic criteria. The channel aspect ratio is therefore a ratio of lengthalong the longest path and the channel width as defined in Figure 2.3.The stream order of a channel network corresponds to the highest stream order achievedin the network (maximum Strahler order (e.g., Horton, 1945)). The cartoon in Figure 2.2 definesthe metric in more detail. We measure this metric according to Figure 2.3d, which shows theresults of the Stream Order tool in the ArcGIS hydrology catalog. In this example, the stream202.3. Methods I: Landscape metrics and patterns in channel networksFigure 2.3: Definition and examples of the measurement process behind the 5 metrics. (a)shows the angle in one junction of the channel network γ. (b) shows the length of thechannel network L (see color bar for reference). (c) shows the channel bankfull widthW using the 1 arc-sec derived SRTM curvature, that complements L to calculate theaspect ratio R. (d) shows the stream (Strahler) order Sn (see color bar for reference).(e) shows the minimum width λ measured either with curvature maps (left panel)or high resolution imagery (right panel). (f) shows the measurement of the fractaldimension D f using a box counting technique (plot) on the stream lines (bottomright).order of this channel network would be 8.In Figure 2.3e, we show an example of definition and measurement of the minimum chan-nel width λ, corresponding to the bankfull width of order 1 streams (note, i.e., that a glacialstream is completely filled with ice, fading the distinction between valley and channel). Weuse the same methodology presented for the channel bankfull width (Figure 2.3c) to measureminimum channel width in glaciers, sapping valleys and subglacial channels. However, influvial channel networks high resolution topography (upper left panel) only enables the iden-tification of order 1 channels. In this particular case we use high resolution imagery to providea measurement for the metric (right panel) on order 1 channels, which corresponds to thebankfull width of tributary-less channels as seen from the image (yellow arrows).Finally, Figure 2.3f shows an example of a fractal dimension calculation. Parting fromthe stream line representation of the channel network, we derive a black-and-white profile(bottom right) that we use in a box counting algorithm. The fractal - or Hausdorff- dimensioncorresponds to the slope of the regression here exemplified. Figure 2.2f exemplifies the fractaldimension significance in the context of tree-like streams.We measure the 5 metrics in a similar manner for sapping valleys and subglacial channels212.3. Methods I: Landscape metrics and patterns in channel networkson the supplementary dataset “ch1datasetEarth.xls”. We obtain data for glacial networks fromthe Global Land Ice Measurements (GLIM) (Raup et al., 2007), the Randolph Glacier Inven-tory (Pfeffer et al., 2014) and the World Glacier Inventory (WGI) also using ArcGIS resources.The data catalogs in WGI and Pfeffer et al. (2014) already include information about the meanwidths and lengths of glacial networks.2.3.2 Data compilationDigitizing stream linesWe follow a well established approach to digitize stream lines (e.g., Moore et al., 1991). We useDigital Elevation Maps (DEM) at 1 and 3 arc-sec resolution, which corresponds to a latitude-dependent resolution of approximately 30 m/pixel and 90 m/pixel respectively (sources arelisted below). After void-filling the elevation data, we use the output from the Flow Accu-mulation tool in the ArcGIS hydrology tool set. Flow accumulation in its simplest form isthe number of up-slope cells that flow into each cell. By applying a threshold value to theresults of the Flow Accumulation tool using either the Con or Set Null tools, we delineate thestream network lines that we use to calculate tributary junction angles and network fractaldimensions (see digital stream lines in Figure 2.3 panels (a) and (f)).Data sources3 arc-sec DEMs are freely available at the WWF hydroSHEDS hydrographic database (Lehneret al., 2006), and global 1 arc-sec DEMs are available at the USGS earth explorer interface, bothderived from the Shuttle Radar Topography Mission (SRTM). We use the 0.75 arc-second reso-lution DEMs extracted from the Natural Resources Canada data repository for latitudes above75◦ in Canada. We use imagery data for the measurement of the minimum channel width (Fig-ure 2.3e) from LANDSAT8 and the World Imagery ArcGIS base map, available for ArcGIS, toresolve details smaller than the DEM resolution. The imagery resolution is 1 m/pixel at ar-eas mapped with GeoEye IKONOS, i-cubed Nationwide Prime, Getmapping, AeroGRID, anddown to 0.3 m/pixel at areas mapped with DigitalGlobe.2.3.3 Principal Component AnalysisWe introduce our 70 channel network x 5 components (γ, D f , Sn,λ,R) dataset matrix (supple-mentary dataset “ch1datasetEarth.xls”) into a Principal Component Analysis (PCA) algorithm(e.g., Cle´ment and Pie´gay, 2003; Morris et al., 2009). This technique is a multidimensional linearstatistical analysis that uses an eigenvalue decomposition of the data covariance matrix to de-fine a new, orthogonal basis set of linearly independent vectors called Principal Components(PC). By definition of the transformation, the first principal component captures the largestpossible variance, and each succeeding component captures a progressively smaller fraction.222.4. Methods II: Physical basis for the metricsIn general, the first two PCs capture nearly 90% of the total variance in the data, and are suffi-cient for our characterization.Using PCA consequently serves three objectives. First, it reduces the dimensionality ofour problem from five to two dimensions, which allows for a presentation of the data in a 2Dprincipal component space. In this 2D space the data are clustered where channel networkshave similar properties in terms of our 5 metrics, whereas large euclidean distances indicateproportional disparity among data points and clusters. Second, the PCA identifies the mostrelevant metrics for capturing the differences among the four types of channel networks inFigure 2.1. Third, the PCA allows the treatment and statistical analysis of large datasets asopposed to the individual analysis of the morphological characteristics of every single channel,facilitating the interpretation of the dataset “ch1datasetEarth.xls” and adding statistical rigorto the channel network classification.2.4 Methods II: Physical basis for the metricsWhereas performing a PCA analysis on the dataset “ch1datasetEarth.xls” is straightforward,understanding the mechanics underlying the resulting clusters, such that reliable predictionscan be made is challenging. In particular, building understanding of the positioning andspreading of clusters, as well as providing a methodology to distinguish channel networksincised by a single process, or a combination of processes, requires an individual analysis andjustification of each of the metrics.Here we analyze the underlying flow mechanics and erosional regimes for the four styles oferosion in Figure 2.1, together with new and published scaling arguments, to understand themetrics (γ, Sn, D f ,λ, R), justify their choice, and understand the results from the PCA analysisquantitatively.Our main assumption is to consider a steady state relationship between the flow of ice orwater and a fully developed stress coupling to the topography. This picture implies the me-chanical stability of the channel cross-section, hydrostatic conditions in all flows, and steady-state landscapes. Table 2.1 contains the notation and units of the parameters used in thissubsection. Appendix A provides more information on the basic equations governing each ofthe four channel networks discussed, together with the parameters required for their interpre-tation in table Aspect Ratio, ROne of the most profound differences among the four channel network types exemplified inFigure 2.1 is the aspect ratio R, which is the ratio of length of the longest flow path L to thebankfull width W. A major control on R is the stress regime in which a given flow operates(e.g., Bahr, 1997a; Savenije, 2003; Crowe et al., 2009; Devauchelle et al., 2011).232.4. Methods II: Physical basis for the metricsTable 2.1: Description of parameters: Metricsmetrics R Aspect ratioλ Minimum channel width (m)γ Tributary junction angleSn Stream orderD f Fractal dimensionδ Scaling exponentR Rd depth-width ratioht topographic elevation (m)θt non-dimensional stressφ slope of channel wallsds average grain size (m)Rρ non-dimensional densityφr angle of repose of sediment in waterL, W, h length, bankfull width and depth (m)C f friction factorS slope of the flow surfaceSo topographic slopeKh hydraulic conductivity (m/s)g gravity (m/s2)fm energy fraction lost to meltinghi ice thickness (m)Rb glacier bed roughnessρi ice density (Kg/m3)us sliding velocity (m/s)λ τb basal shear stress (N/m2)ηe f f effective viscosity (Pa s)γ α main channel slopeβ tributary channel slopeSn N1 number of order 1 streamsD f D1 Fractal dimension of order 1 streamsrb Horton’s bifurcation ratiorl Horton’s length ratioFluvial and sapping valley networks far from springsDefining the L-W aspect ratio requires setting mechanical constraints on length L and width W.The stress distribution acting on sediment grains of various sizes in a channel ultimately setsthe bankfull width W of this channel (e.g., Lane, 1955; Savenije, 2003; Devauchelle et al., 2011),Figure 2.4. Assuming the cross-sectional mechanical stability of the channel (Lane, 1955), wecapture the ratio of tangential to normal stresses acting on a channel bed with lateral wallsinclined an angle φ (Figure 2.4) in terms of a non dimensional stress θt. A typical descriptionof this non-dimensional Shields stress θt is given in equation A.1. We manipulate this equationusing a trigonometric identity: sec2(φ) = 1+ tan2(φ) and relate the slope of the channel wallsφ to the average depth-width aspect ratio Rd = h/W = 2 tan(φr)/pi2 (Savenije, 2003, eq.16),242.4. Methods II: Physical basis for the metricswith φr as the angle of repose of sediment in water. We use equation (16) in Savenije (2003) toobtain an expression for R, taking into account that the slope of the flow surface S corresponds,to first order, with the topographic slope So = ∆htL in the hydrostatic approximation, with∆ht = ht(L)− ht(0).R = L/W =Rρ∆htds2 tan(φr)pi2√1+ tan(φ)2θ2t − tan(φ)2, (2.1)where the effective density Rρ = ρw/(ρs − ρw) ∼ 1.61 compares the sediment and waterdensity difference and ds ∼ 0.45 mm is typically taken to be the average grain size (Devauchelleet al., 2011, e.g.,). As a final remark, in developing equation 2.1 we assume a fully turbulentopen channel flow (see Appendix A equation A.5), with θt > tan(φ), which is typically in therange 0.29− 0.6 (Whiting and Dietrich, 1990).Sapping valleys close to the springIn contrast to the overland flow of rivers and sapping valleys far from the spring, boundaryconditions of flux continuity at the seepage site substantially alter the aspect ratio given byequation 2.1 (see equation A.16 in Appendix A) (Devauchelle et al., 2011). We do not explore thederivation details here, but it is straightforward to relate algebraically equations 3.7 and 3.13in Devauchelle et al. (2011) to leading order to obtain a description of R close to the spring:R = L/W = 4ds/3Rρpi√2√2κ(1/2)g3 sin(2pi/3)√θ6tC f KhS2, (2.2)where κ is the complete elliptic integral of the first kind (Devauchelle et al., 2011), and Kh is thehydraulic conductivity.Subglacial channelsThe dynamics of flow and erosion in subglacial meltwater channels differ from fluvial channelnetworks in that energy is extracted from the velocity field to do the work of both incisingthe channel and melting the surrounding ice (see description of subglacial flow mechanicsin Appendix A, equations A.17 and A.18). In detail, this and other complications are poorlyunderstood and are beyond the scope of this study. To make progress, we will simplify thedescription of the subglacial aspect ratio by adding parametrically the fraction of energy lossto melting fm to the balance of momentum in a pressurized subglacial conduit:R = L/W =(1− fm)Rρ∆htds2 tan(φr)pi2√1+ tan(φ)2θ2t − tan(φ)2. (2.3)252.4. Methods II: Physical basis for the metricsb) sapping valleyLongitudinal sectionCross sectionMass balancec) subglacial channelzxWhyxz2d) glacierzxaccumulationablationWhyxz11 233a) river 1zxlhWyxz2 3φzxlWhyxz213Figure 2.4: Mass balance, cross section and longitudinal section of the four erosion mech-anisms described. Coordinate systems are shown in red dashed lines. (1a) massbalance in a river: flow enters the tributaries and exits in the outlet. (2a) and (3a)show the flow dynamics in a river with width (W) and depth (h), with blue arrowsfor the velocity field and eddies indicating turbulence (bigger size indicates largerdischarge). (b1) mass balance in sapping valleys: water enters the valley followingthe water table gradient (dashed black line). (b2) and (b3) describe the flow dynam-ics in a sapping valley. In subglacial meltwater channels, (c1) shows the balancebetween melting and ice creep. (c2) and (c3) show the velocity field inside a chan-nel. In glaciers, (d1) shows the snow accumulation and ablation zones, and (d2) and(d3) show the velocity field.Glacial networksIce and compacted snow tend to fill the valley containing the glacier, resulting in strong me-chanical coupling to, and erosion of, the valley side walls in addition to the bed. In this case,the aspect ratio R reflects the relative contributions of both processes, depending on the ratioof the basal shear stress to the shear stress acting on the lateral walls of the channel. In gen-262.4. Methods II: Physical basis for the metricseral, glacial basal shear stresses increase with basal sliding speed us and with bed roughnessRb [Paterson, 1994, equation 6, chapter 7]:τb =(R4busC)1/2, (2.4)where C = 10−20m2s/kg is a constant (Paterson, 1994). Because glacial ice flows relative tothe valley side walls (i.e., ice or the glacial margins yields) the stresses imparted to an iceelement on the wall, i.e., an element flowing downstream towards the center of the glacierand downstream towards the terminus, are approximately set by a yield stress that reflects theprojection into both slopes of the form:τ = ρighiSo cos(φ), (2.5)where φ is the slope of the side valley, and hi the ice thickness. The ratio of equation 2.4 and2.5 gives the average glacial aspect ratio for a uniform distribution of the sliding velocity us:R =√R4bus/CρighiSo cos(φ). (2.6)2.4.2 Minimum channel width, λFor glaciers, Pelletier et al. (2010) derives the length-scale of order 1 systems by means of linearstability analysis on an ice-covered slope. The width of order 1 valleys is set by a competitionbetween the thickening of ice in incipient valleys, which increases flow speed and erosion,and viscous stresses/ sidewall drag, which acts to impede glacier motion and erosion. Theminimum glacial width they derive is:λ =2τbS3√pi2ηe f f c/(ρig)2, (2.7)where c is an empirical bed-friction parameter such that ηe f f c ∼ 10−2 m Pa−1, and the basalshear stress is in the range τb = 0.5× 105− 2.4× 105 Pa. For these parameter ranges, λi ≈ 1− 3km.Descriptions of minimum channel width are challenging for rivers, sapping valleys andsubglacial channel networks. In the particular case of fluvial channel networks, although de-scriptions of minimum valley width are available (Perron et al., 2008, 2009), there is no quantita-tive relationship between channel and valley width that enables a straightforward descriptionof this metric.272.4. Methods II: Physical basis for the metrics2.4.3 Tributary junction angle, γThe range of junction angles γ between a main stream and its tributaries distinguishes glaciers(generally, γ = 80− 100◦) and river networks (typically, γ = 40− 90◦). A number of classicalstudies (e.g., Horton, 1932, 1945; Howard, 1971) have addressed the main controls on river net-work junction angles. Here we give a general approach that is consistent with this work, butgenerally applicable to any class of gravity current, including glaciers. We will also discussthe controls on junction angles within sapping valley and subglacial channels networks.Consider a surface flow (river or glacier) driven by gradients in hydrostatic pressure. Theslope of a main stem with respect to a flat, sea level equivalent surface is described by an angleα. We use the angle β to indicate the slope of the tributary with respect to the same flat surface.Assuming that the stream follows the path of maximum topographic gradient (Horton, 1932),a rotation of coordinates of angle α from the flat surface to the path of the main stem gives thegravitational acceleration acting on a fluid element:gmain = (g sin(α), 0, g cos(α)) . (2.8)For our axis convention, we take the x axis to be parallel to the main stem downstream direc-tion, the z axis to be perpendicular to the main stem and pointing upwards, and the y axis tocomplete the pair wise set. We require two rotations on the gravity vector starting at the sealevel plane to describe the downwards component of the gravity vector in the plane of the sidevalley. The first rotation has an angle α along the y axis and the second an angle β along x inthe following wayRβ(x)Rα(y)g = cos(α) 0 sin(α)0 1 0− sin(α) 0 cos(α)1 0 00 cos(β) − sin(β)0 sin(β) cos(β) 00−g (2.9)Applying this rotation to equation 2.8 to obtain the effective gravitational acceleration experi-enced by an element of fluid in the side valley, and using basic trigonometry to solve for thejunction angle γ we obtain:γ = arctan(− cot(α) sin(β)) (2.10)Equation 2.10 is similar, in form and significance, to those obtained by Horton (1932) andHoward (1971), but now generalized to any gravity current. The difference between our workand those in these two previous studies lies in the flow mechanics approach we present, incomparison with the geometrical argument presented in Howard (1971) or the empirical obser-vations by Horton (1932).Junction angles in sapping valleys are, by contrast to fluvial and glacial systems, a productof upstream bifurcation rather than merging (see Figure 2.4 b) 1). The processes of sappingvalley formation and evolution are a consequence of spring erosion and valley head under-282.4. Methods II: Physical basis for the metricsmining, leading to valley bifurcation angles distributed around a characteristic angle of 72◦(Devauchelle et al., 2012).In subglacial channels, flow is driven by lateral gradients in ice thickness, and therefore,equation 2.10 is not appropriate for their description. Indeed, we can expect the tributary junc-tion angles of subglacial channels to be largely independent of the local topography (Walderand Hallet, 1979; Kehew et al., 2012). We return to this issue in section Stream Order, SnStochastic patterns in precipitation, pre-existing topography, climate conditions, lithology andsoil permeability affect the character of a stream network. The number, distribution and com-plexity of branching in a dendritic channel network define its stream order Sn and fractaldimension D f (Meir et al., 1980; Kirchner, 1993).Following Meir et al. (1980) we characterize the probability Ω{Sn} of a network having astream order Sn in terms of the number of 1st order tributaries N1, for a Gaussian probabilityof merging between two channels. Their result is:Ω{Sn} = 1/2 log2 N1 +O(1), N1  1 (2.11)The stream order of a system increases logarithmically with the number of first order streams.2.4.5 Fractal Dimension, D fThe fractal dimension of a system is a statistical measure of its scale invariant complexity(Mandelbrot, 1983). Channel networks are too complex to be considered one dimensional sys-tems since they are closer to covering a surface than a line. However, they do not completelycover this surface and consequently their dimension is between one and two.Applying the same approach in Rodrı´guez-Iturbe and Rinaldo (2001), we link D f to both thechannel order and Horton’s laws (Horton, 1945):D f = D1rbrl(2.12)Where D1 is the fractal dimension of 1st order tributaries and rb and rl are the bifurcation andlength ratios from Horton’s laws.Kirchner (1993) argued that Horton’s laws are a consequence of the randomness of streamnetworks and consequently, their utility describing complexity is not limited to fluvial net-works. This conclusion is consistent with the statistical and probabilistic description that Meiret al. (1980) derived for the stream order. Following their argument, Horton’s laws derive fromany type of tree-like network with an outlet, a number of sources, and downstream mergingof the paths from sources to outlet. Therefore, they can be applied generally to each class ofdendritic network considered here.292.4. Methods II: Physical basis for the metrics2.4.6 Characteristic length-width scaling relationshipsThe L−W scaling relationships for the four types of channels in Figure 2.1 reflect their distincterosion mechanisms and flow dynamics. Generalizing the approaches of Bahr (1997b) and Bahr(1997a) applied to glaciers, we look for scaling relationships of the form:L ∼Wδ. (2.13)Other authors, such as Devauchelle et al. (2011) for sapping valleys and Walder and Fowler (1994)for subglacial channels have derived length-width relationships from which it is straightfor-ward to derive a scaling law of the form of equation (13). To augment this existing work, wederive L−W for fluvial systems, and list the model parameters in table 2.2.For rivers, we begin with conservation of volume to express the bankfull width W in termsof the length of the longest path L in an open channel, which is given by a downstream flowrate of:Q ∼Wh f ux, (2.14)where the downstream velocity ux is unknown. We next substitute equation 2.14 into a de-scription of the basal shear stress in fluvial systems (equation A.5 in Appendix A) to expressh f in terms of the other parameters:h3/2 ∼ QW(C fgS)1/2, (2.15)A typical description of the friction factor (see equation A.6 in Appendix A) gives a scalingof C f ∝ h−2/7, and for hydrostatic flows, S ∼ ∆ht/L (e.g., Crowe et al., 2009). We require anadditional closure for the flow rate Q to arrive at a L−W scaling.At the basin scale, the steady state river discharge must balance the rate of precipitation Ptimes the drainage area A if we neglect evapo-transpiration, transient changes in storage, andinfiltration rates. In turn, Montgomery and Dietrich (1992) showed empirically how drainagearea scales like A ∼ L2. Since channel length L is typically proportional to basin length, wefindQ ∼ L2. (2.16)The theory of stable channel cross sections (Lane, 1955; Savenije, 2003) , which recognizes thatthe turbulent stresses acting on the sides of a channel are ultimately derived from motionscausing erosion at the base of the channel, states that bankfull width and flow depth are relatedlinearly:W ∝ h. (2.17)Combining a description of the friction factor with equations 2.16 and 2.17 into equation 2.15,302.4. Methods II: Physical basis for the metricsTable 2.2: Notation: flow and erosion mechanicsGeneral mechanics x, y, z axis (m)u velocity field(m/s)ρ flow density (kg/m3)fdrive driving stress (N/m2)ηe f f effective viscosity (Pa/s)τo yield stress (N/m2)τ stress tensor (N/m2)ε˙ strain rate tensor (1/s)BT hardness parameter (s−1Pa−3)p flow-law exponentU uplift rate (m/s)E erosion rate (m/s)zo stream bed roughnessRivers b˙ sources of velocity field divergence (1/s)ux downstream velocity (m/s)E f fluvial incision rate (m/s)K f erosional efficiency (m1−2m/s)A drainage area (m2)m drainage area exponentn slope exponentD Soil diffusivity (m2/yr)h f water flow depth (m)Glaciers Ea glacial abrasion rate (m/s)Ep glacial plucking rate (m/s)Ka abrasion efficiency constant (yr/m)s f sediment entrainment constantsc sediment damping parameterPi hydrostatic ice pressure (Pa)Pw water pressure (Pa)Sapping hw water table elevation (m)Pr precipitation rate (m/s)Subglacial Rsg hydraulic radius (m)ms subglacial drainage area exponentns subglacial slope exponentthe scaling we derive for fluvial systems isL ∼W44/35, (2.18)which is consistent with the empirical scaling range δ = 1.71 to δ = 1.2 resulting from al-gebraic manipulation of Hack’s law, L = 1.4A0.6 (Hack, 1957), the relationship for hydraulicgeometry Q ∝ W2 and a relation for basin hydrology Q ∝ Ac, where 0.7 ≤ c ≤ 1 (e.g.,Howard and Kerby, 1983; Whipple and Tucker, 1999). The assumptions listed to derive this scal-ing relationship, and in particular the linearity between width and depth in equation 2.17,312.5. Resultsdo not hold over the many orders of magnitude characteristic of rivers systems. Consistentwith our approximations, this result assumes uniform, steady flow, stable bankfull width, andconservation of mass without contributions from major sinks into the groundwater system orevapo-transpiration.Several authors have derived similar scaling relationships for the other types of chan-nels exemplified in Figure 2.1, and here we give the power law exponents for complete-ness: (1) glacial valleys, δ = 1/0.6 (Bahr, 1997a); (2) subglacial channels, δsilt = −21/4 andδgravel = −29/15 (Walder and Fowler, 1994); and (3) sapping valleys δ = −0.88 after algebraicmanipulation of the work by Devauchelle et al. (2011).2.5 ResultsIn this section we describe the correlation between PCA clusters and the four styles of chan-nel networks and analyze the results in terms of the metrics. We use the information in theprincipal component variance to establish a hierarchy within the metrics in terms of their effi-ciency at capturing the different aspects of channel network geometry. We then compare themetrics observations in our database with the model predictions in equations 2.1, 2.2, 2.3, 2.6,2.7, 2.10, 2.11, and 2.12 using the parameters listed in table 2.3. We focus our attention on thenovel metrics introduced: aspect ratio R, tributary junction angle γ and L−W fluvial scalingrelationship, and limit our analysis of the metrics already introduced in other studies.2.5.1 Principal Component AnalysisFrom the variance in the supplementary dataset “ch1datasetEarth.xls”, 95% is explained bythe first three Principal Components (PC), with 88% explained by the first two. To simplifyour analysis and aid with the interpretation of the results, we restrict our attention to a 2D plotwith the first two PC in Figure 2.5. We also show the explicit contributions of the differentmetrics to the data variance with a set of dashed arrows, pointing at the direction of metricincrease and length-scaled to their relative weight in the PC (Figure 2.5).A linear decomposition of all principal components in terms of their dependence on themetrics quantifies the relevance of each of the metrics at capturing the total variance. Thisfeature is a unique advantage of using PCA for this characterization as opposed to using themetrics individually. The 1st PC (67% of the total variance) is largely dominated by the mini-mum channel width λ, the aspect ratio R and the stream order Sn. The second PC (21% of thetotal variance) is dominated by the junction angle γ, R, λ and Sn. In order of relevance, theminimum channel width λ captures most of the data variance (40.6%), followed by the aspectratio R (25%), the stream order Sn (up to 18.5%), the junction angle γ (14%), and finally thefractal dimension D f capturing only 1.5% of the variance. The first two metrics, λ and R, cap-ture up to 65% of the variance and therefore are the most relevant at characterizing a channelnetwork from the 5 metrics introduced in this study.The PCA results in Figure 2.5 show three distinct clusters that correspond to fluvial net-322.5. Results−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.6−0.5−0.4−0.3−0.2− Principal Component 2nd Principal Component λγRSnDfGlacier valleysRiver valleys Sapping valleysSubglacial channelsFigure 2.5: 1st and 2nd Principal Components (PC) displaying the data projection in thePC space. Rivers appear in red squares, glaciers in blue diamonds, sapping valleysin green triangles and subglacial meltwater channels in magenta rotated triangles,with data clusters color shaded in the same color convention. Metrics increase in thedirection of the 5 dashed arrows, scaled to the relative weight of each metric. λ forminimum channel width, R aspect ratio, Sn stream order, D f fractal dimension, andγ junction angle.works (red squares), glacial networks (blue diamonds), and a third cluster that is mixture ofsubglacial channel networks and sapping valley networks (pink and green triangles respec-tively). The fluvial network cluster distributes along a line of nearly constant λ, increasingin ratio R and stream order Sn. By contrast, the glacial cluster shows a comparatively largespread along the minimum channel width λ and the junction angle γ, with a relatively smallvariability in the aspect ratio R (see dataset “ch1datasetEarth.xls”).Sapping valley networks (green triangles) split between two data sub-groups in the PCA,which correlate with systems in Hawaii (PC1 < −0.4) and sapping valley networks in Col-orado and the Florida Panhandle (PC1 > −0.4). The metric controlling the separation betweenthe two groups is the minimum channel width λ. In fact, a major difference from fluvial net-332.5. Resultsworks is the strong variability in λ, ranging from values similar to those of fluvial networks(PC1 > −0.1) in systems in Florida and Colorado, to values similar to glacial networks inHawaiian systems (PC1 < −0.4). In the discussion section, we will discuss this difference andthe possibility for different formation mechanisms between Hawaii and the Colorado Plateauand Florida channel networks, as suggested by Lamb et al. (2006) .Subglacial channel networks lie within the −0.3 < PC1 < −0.1 range, overlapping theFlorida /Colorado sapping valley cluster. The data spread is similar to fluvial networksin approximately constant minimum channel width λ, and increasing aspect ratios R andstream orders, both values comparatively smaller than those displayed by fluvial networks.We note that although R contributes strongly to the variance in subglacial channels, herethis feature of the data ultimately reflects variations in the length L. In our data set (see“ch1datasetEarth.xls”), the variance of bankfull width for subglacial channels is 1− 2 ordersof magnitude smaller than for the rest of channel networks.2.5.2 Aspect ratio RIn Figure 2.6, we compare the predictions of equations 2.1, 2.2, 2.3 and 2.6 with parameterslisted in table 2.3, to observations in the supplementary dataset “ch1datasetEarth.xls”. Figure2.6a shows the aspect ratio variation in fluvial networks for different values of the Shield stressparameter, from θt = 0 to θt = 20. We also explore the variation of the curve R(θt) withdifferent values of the angle of repose of sediment in water φr(color bar in Figure 2.6a). Aspectratio data collected for fluvial networks is in the range of 452 < R < 2800, with a mean valueof R = 1301 and a standard deviation σ = 560. For values of the angle of repose of sediment inwater corresponding to φr = 30◦ (Carrigy, 1970), equation 2.1 predicts values of R ranging from0 to a peak 3200, with the maximum at a value of the Shields stress parameter θt = 1.3; and amean value of Rmodel = 532 and σ = 550. For typical values of the Shield stress (0.3 < θt < 1),the average predicted is Rmodel = 1670 and σ = 400.Figure 2.6b shows the aspect ratio for sapping valley networks R(θt) predicted with equa-tion 2.2. Here we explore the role of soil hydraulic conductivity Kh in the color-coded curves(see corresponding color bar). Typical aspect ratio R measurements of sapping valleys span3 < R < 73, with an average of R = 29 and σ = 0.21. We find equation 2.2 to predict thedata range more accurately than equation 2.1, with modeled values of 11 < R < 123 for val-ues of 0.3 < θt < 0.5 characteristic of sapping valleys close to the spring (Devauchelle et al.,2011). We find the soil hydraulic conductivity Kh to strongly influence the aspect ratio, with Rincreasing up to a factor 5 as Kh decreases from 10−4 to 10−6 m/s, which are typical hydraulicconductivity values of coarse to very fine sand (Bear, 2013). There is consistency among modelpredictions and the overlapping shaded area, that indicates the intersection between expectedvalues of θt and observed R values.Figure 2.6c shows the R(θt) curve for subglacial channels as described in equation 2.3. Inthe color coded curves we explore the effect of varying the fraction of energy lost to melting342.5. Results0 2 4 6 8 10 12 14 16 18 200500100015002000250030003500L / W aspect ratio RData rangerange of θt  Shields stress parameter θt3525φr0.4 0.5 0.6 0.7 0.8 0.9 101002003004005006007008009001000L / W aspect ratio R range of θt  Data rangeShields stress parameter θt10-610-4K0 2 4 6 8 10 12 14 16 18 2005001000150020002500L / W aspect ratio R? Unknown range of θt   Data rangeShields stress parameter θtf01L / W aspect ratio RIce thickness h (m)Data rangeTypical range of h0 100 200 300 400 500 600051015202530354045500.150ub(a) Rivers (b) Sapping valleys(c) Subglacial channels (d) GlaciersFigure 2.6: Aspect ratio model predictions for (a) rivers, (b) sapping valleys, (c) subglacialchannels and (d) glaciers. Each panel shows the predicted R curve against a controlparameter: the non dimensional shear stress θt in rivers, sapping valleys and sub-glacial meltwater channels and the ice thickness h in glaciers. The different curvescorrespond to the variation of the predicted R in terms of a secondary parameter:angle of repose of sediment in water φr in rivers, hydraulic conductivity Kh in sap-ping valleys, fraction of energy loss to melt fm in subglacial meltwater channels andsliding velocity us in glaciers. Vertical shaded areas represent the range of controlparameters typically observed in nature (θt and h), whereas the horizontal shadedareas are the span of our aspect ratio data. Color bars show the variation within eachfree parameter described above.the ice fm in the R(θt) curve according to the color bar. Subglacial channel R values span95 < R < 351, with an average of R = 248 and σ = 83. We assume a sediment angle ofrepose of φr = 30◦. Taking fm = 0.5, equation 2.3 predicts 85 < R < 1203. Predicted values ofthe aspect ratio decrease with increasing θt. However, typical values of the Shields parameterin the subglacial channel environment are as yet unknown (Beaud et al., 2016), although thehigh hydraulic pressures (> 5 MPa from Walder and Fowler (1994)) suggest larger values thanis typical for overland flows, leading to a decrease of R in comparison with fluvial systems.352.5. ResultsModel predictions are consistent with data for all values of fm, at higher θt values than thosefound typically in fluvial systems. We do not explore the possible dependence of fm with θtand treat them as linearly independent parameters.Figure 2.6d shows the R (h) curve predicted for glacial networks in equation 2.6. Herewe explore the role of the basal sliding speed ub at varying the values of R(h) according tothe corresponding color bar. We use typical values of ub spanning from 0.5 < ub < 50m/yr(Paterson, 1994), although higher and lower values also can be found in nature (Paterson, 1994).Observations for glacial channel networks indicate 5.5 < R < 31, with an average of R =17.36 and σ = 7. We compare these observations with predictions from equation 2.6, using aroughness factor Rb = 0.1, a basal velocity in the range 0.1 < ub < 50 m/yr, and ice thicknessin the range 10 < h < 2000 m (Paterson, 1994). Using these values we obtain a range of aspectratios 0.24 < R < 46, with a value of R = 14.7 for average values of basal ub speed and icethickness h, which is consistent with the average aspect ratio measured (Figure 2.6d, dashedarea).Comparisons of natural data with our theoretical predictions for the R metric reliably dis-tinguish the four groups: fluvial (0 < R < 1800), sapping valleys (11 < R < 1000), glacial(0.2 < R < 46). Although there is considerable uncertainty over θt and fm for subglacialchannels, an R in the lower range of the predictions (85 < R < 1200) is supported with theobservations. The agreement of observations and models indicates that the aspect ratio is oneof the most useful metrics at characterizing and distinguishing valley types and identifyingthe underlying formation process.2.5.3 Minimum channel width λWe compare minimum channel width observations, corresponding to the column named WidthO1in the supplementary dataset “ch1datasetEarth.xls”, with predictions drawn from equation 2.7(c.f., Pelletier et al. (2010)) for glaciers. Glacial minimum channel widths span 400 < λ < 1870m, with an average at λ = 1170 m and σ = 430. The analysis of λ for glaciers using equation2.7 gives a glacial spacing of λmodel = 1092 m with parameters listed in table 2.3, which is inagreement with our data.2.5.4 Tributary junction angle γIn Figure 2.7 we compare model predictions on the basis of equation 2.10 with junction angledata for fluvial and glacial networks. We use color coded crosses (see color scale for reference),with plotted size scaled to the DEM resolution, to show 18 observational junctions. We use thesame color scale to represent the model predictions in the colored background. The horizontalaxis corresponds to the slope of the main stem to a flat reference surface (α), and the verticalaxis to the slope of the tributary with reference to the same surface (β) .Observations fall within the shallow slope region at the top left of the plot, with junctionangles distributed between 10◦ and 90◦. It is in this region where the model best captures the362.5. ResultsTable 2.3: Model parametersλ model parametersBT = 5.7 ∗ 10−27 softness parameter (Pa−3 s−1)T = −10 (◦C) mean ice temperatureRb = 3× 10−17 bed friction parameter (m/Pa2s)h = 102 ice thickness (m)ηe f f = 1014 effective ice viscosity (Pa)τb = 0.75 ∗ 105 threshold shear stress for motion (Pa)R model parametersRiversRρ = 1.61 Non-dimensional densityδht = 1000 Topographic difference (m)ds = 4510−3 average grain size (m)25◦ < φr < 35◦ angle of repose of sediment in water0.3 < θt < 20 Shield stress parameter variationGlaciersC = 10−20 empirical constant (SI units for p=3)Rgs = 0.1 bed roughness factorub = 5 average basal velocity (m/yr)h = 100 ice thickness (m)BT = 2.4 ∗ 10−24 hardness parameterS = 7 ∗ 10−4 ice slope10 < h < 600 ice thickness (m)Subglacial channelsRρ = 1.61 Non-dimensional densityδht = 1000 Topographic difference (m)ds = 4510−3 average grain size (m)25◦ < φr < 35◦ angle of repose of sediment in water0.3 < θt < 20 Shield stress parameter variation0 < fm < 1 fractional energy loss to melting variationSapping valleysRρ = 1.61 Non-dimensional densitySo = 0.14 Average topographic slope (m)ds = 4510−3 average grain size (m)φr = 31◦ angle of repose of sediment in water0.1 < θt < 1 Shield stress parameter variation10−6 < Kh < 10−4 hydraulic conductivity (m/s)C f = 0.18 friction factordata. We can explain up to 80% of the γ measures within the error of the DEM. Qualitatively, asthe slope of the longitudinal profile approaches 0◦, α→ 0◦, the junction angle grows γ→ 90◦.By contrast, when main channel walls are very steep α → 90◦, γ → 0◦, indicating that steepermain valley slopes will be characterized by smaller junction angles. Thus, we expect the widestjunction angles in main valleys with α→ 0 and with steep walls. The model also predicts thatin the special case of β → 0◦ a tributary will run parallel to the main stream, maximizing the372.5. Resultsα  main stem slopeβ tributary slope0 2 4 6 8024681020304050607080γ angle variation786543021105 15 2005101520Figure 2.7: Comparison of observations (crosses) and model (colored background) fortributary junction angles of glaciers and rivers in 18 locations in North America,Western Australia, and northern Africa. Crosses are color coded for the measuredjunction angle according to the color bar and compared with the model predictions(background color) for a given pair of main stem and tributary slope. Thus, similarcolors indicate a low relative error, and deviations between model and data appearwith noticeable color differences.release of potential energy, and then γ → 0◦. These results agree with those of the classicaljunction angle theory (Horton, 1932; Howard, 1971).Our junction angle data (supplementary dataset “ch1datasetEarth.xls”) show different fre-quency distributions for each channel type. Fluvial networks are distributed around an aver-age of γ = 59◦, with σ = 7; glaciers show an average of γ = 80◦, with σ = 8; sapping valleysshow a distribution around γ = 64◦ with σ = 4, which is smaller than the value of γ = 72◦predicted by Devauchelle et al. (2012). Finally, subglacial channels show γ = 68◦ with σ = 8.382.5. Results2.5.5 Stream OrderData corresponding to the stream order Sn of the different channel networks in the supplemen-tary dataset “ch1datasetEarth.xls” highlight the relevance of this metric for characterizing thetype of channelized flow. For fluvial networks, we obtain an average Sn = 7.2 with a standarddeviation of σ = 1.5. Glacial networks show an average of Sn = 3.8 with σ = 1.4. These twosystems show the largest stream orders as well as the largest variances in this metric. Sappingvalleys distribute around an average of Sn = 2.8 with a standard deviation of σ = 0.8, mak-ing them the type of channels with the smallest average stream orders. Finally, for subglacialchannels we obtain an average of Sn = 3.7, with σ = 0.8. These results emphasize the dif-ference in pattern complexity between fluvial and any other type of channel networks, whichis already apparent in the PCA results in Figure 2.5 and in the high variance captured by thestream order in PC1.2.5.6 Fractal dimensionFractal dimension is the metric that overall captures the least variance in the data. Individually,fluvial channel networks have an average of D f = 1.73 with a standard deviation σ = 0.13, aresult remarkably similar to that for glacial networks D f = 1.70 with σ = 0.08. Sapping valleynetworks distribute around D f = 1.4 with a standard deviation of σ = 0.15. Finally, datafor subglacial channel networks shows D f = 1.63 and σ = 0.09. These results highlight thedifficulty of classifying the channel networks on the basis of the fractal dimension result. Inaddition, any conclusions regarding channel classification we draw from the fractal dimensiondata distribution are identical to those drawn from the stream order, which is a much moreefficient metric at capturing the data variance as we discussed in section L ∼Wδ scaling exponentsThe L −W scaling relationships enable us to identify a network as part of one of the fourgroups in Figure 2.5, or as an explicit outlier to that group. This approach is useful for achannel network located between two clusters in the PCA space, or when its characterizationfrom the values of the metrics is unclear.In Figure 2.8 we compare the power law scalings from equation (13) with the suite of obser-vations in the supplementary dataset “ch1datasetEarth.xls”. Observations of ln(W) vs. ln(L)for fluvial channel networks (Figure 2.8a) give an empirical scaling of δobs = 0.73± 0.10, whichis not consistent with our predictions of δpred = 44/35, although our prediction matches theempirical scaling derived from Hack’s law and the hydraulic relationship.Observations for glaciers (Figure 2.8b) give an empirical scaling coefficient of δobs = 0.83±0.13 from the best fit line, which deviates significantly from Bahr (1997a) scaling for valleyglaciers δpred = 1.67 .Sapping valley data (Figure 2.8c) show significant scattering and no linear scaling, with thebest fit at δobs = 0.1± 0.4. The L−W scaling derived from the equations by Devauchelle et al.392.5. Results1011121314153 5 7 9r² = 0.7736ln(L) = 0.7311 ln(W) + 8.6057 67891011129 10 11 12 13 14ln(W) = 0.8352ln(L) - 0.8039 r² = 0.778288.599.5104.5 5.5 6.5 7.5r² = 0.0412ln(L) = 0.1165ln(W) + 8.404610.410.811.211.6125.9 6 6.1 6.2ln(L) = -5.6738ln(W) + 45.865r² = 0.576a) L-W  river valleys scaling b) L-W  glacier valleys scalingd) L-W  subglacial channels scalingln(W)ln(W)ln(W)ln(W)ln(L)ln(L)ln(L) ln(L)c) L-W  sapping valleys scalingHHHHFigure 2.8: Observational L-W scaling relationships: a) fluvial, b) glacial, c) sapping val-leys and d) subglacial channels. Horizontal axis and vertical axis are ln(W) andln(L) (units are m) for rivers, sapping valleys and subglacial channels. Axes forglaciers are inverted ln(L) vs. ln(W) to better represent the scaling by Bahr (1997a).The best least square linear fit to the data in log space (solid line) in each panel givesthe observational scaling exponent. Vertical and horizontal error bars are a resultof DEM error propagation. Correlation coefficients for the fits are r2 = 0.77 for thefluvial scaling, r2 = 0.778 for glaciers, r2 = 0.04 for sapping valleys (note that Hvalleys are located in Hawaii), and r2 = 0.58 for subglacial channel networks.(2011) does not accurately capture the spread in the data, with a δpred = −0.88, which is notobserved. An interesting observation that arises from the L−W curve in this panel is the op-posing slopes for Hawaiian systems, marked with an H, (negative slope) and sapping valleysin Colorado /Florida (positive slope). Although we require more data from both locations tostatistically verify these trends, both the PCA and scaling point out a fundamental differenceamong these locations. We will return to this point in the discussion.Finally, subglacial channels fit to a power law with exponent δ = −5.6± 0.3 that is con-sistent with the scaling derived from the equations by Walder and Fowler (1994) for the case ofsand/silt channels δmodel = −5.3. More data is required, however, to establish a significant testfor this relationship, in particular given the size of the horizontal error bars.402.6. Discussion2.6 Discussion2.6.1 Comparative channel geometry and morphology of channel networksThe PCA results in Figure 2.5 and accompanying analyses provide a rigorous way to distin-guish 3 of the 4 channel network types in Figure 2.1 fluvial channel networks, sapping channelnetworks and glacial channel networks. In Figure 2.9 we show a world map with the locationand type of the channels studied in the database, classified entirely using the algorithm devel-oped in this paper. Here we discuss implications of our characterization technique, and alsodiscuss challenges and directions for future work.#*#*#*#*#*#*#*XWXWXWXWXWXW XWXWXW XWXWXW")") ")")")")")")")")")")")")")")")")") ")")")")")")")")")")")")") ")")")")")")")")Source: Esri, DigitalGlobe, GeoEye, i-cubed, Earthstar Geographics, CNES/Airbus DS,USDA, USGS, AEX, Getmapping, Aerogrid, IGN, IGP, swisstopo, and the GIS UserCommunity180°0'0"180°0'0"120°0'0"E120°0'0"E60°0'0"E60°0'0"E0°0'0"0°0'0"60°0'0"W60°0'0"W120°0'0"W120°0'0"W180°0'0"180°0'0"30°0'0"N 30°0'0"N0°0'0" 0°0'0"30°0'0"S 30°0'0"S60°0'0"N 60°0'0"N60°0'0"S 60°0'0"SLegend")XW#*#* Subglacial channelsSapping valleysGlaciersRiver networks*# Unclear from PCAFigure 2.9: Map of Earth with location and type of channels classified according to thePCA results presented in Figure 2.5. We use the same color scheme than Figure2.5. Solid black triangles indicate the points that led to equivocal results from thePCA classification. On the basis of the distribution of glacial and subglacial channelnetworks at high elevations and latitudes, compared to fluvial and sapping valleysdominating the lower latitudes, we identify a current interglacial period for Earth.Fluvial networksFluvial systems are the channel networks that our metrics capture most accurately in Figure2.5. They are typically characterized by small minimum channel widths λ ∼ 1 m, the largest412.6. Discussionaspect ratios R > 2000, junction angles typically between ∼ 60◦ and ∼ 90◦ (although smallervalues are also possible, particularly in arid climates), and the largest stream orders and fractaldimensions, a result that is consistent with the theoretical expectations in equations 2.1, 2.10,2.11 and 2.12.We find λ and R to be the most useful metrics at characterizing fluvial channel networks.Indeed, model predictions for R that explain the data are consistent with the expected turbu-lent stresses in rivers (see equation A.5 in Appendix A), and we show that large values of theaspect ratio (R > 2000) are a unique diagnostic of rivers. Whereas the characterization is interm of R and λ, the stream order Sn captures most of the variability within the fluvial datacluster (shaded red in Figure 2.5).The tributary junction angle γ for fluvial channel networks is generally small comparedto glacial, subglacial and sapping systems. We find equation 2.10 to be more accurate fortributaries merging into streams that differ by 3 or more stream orders in Figure 2.7. Thedata with the greatest departure from the model predictions correspond to areas with highsediment deposition rates that can lead to concave up longitudinal profiles, particularly withinengineered or highly perturbed urban areas.Fractal dimension D f and stream order Sn values are large in fluvial networks. The smallvariance captured by D f is particularly surprising. Although not linearly correlated with thestream order, these two metrics are functionally dependent through the logarithm of the ratioof Horton’s bifurcation and length ratios (see equation 2.12). Their similarity is captured in thePCA space, where both metrics capture variance in the same direction. The large differencesin the variance captured (18% vs. 1.5%) can be attributed to this logarithmic dependence in thefractal dimension definition. Furthermore, whereas the fractal dimension varies by less than afactor of 2 in the supplementary dataset “ch1datasetEarth.xls”, the stream order can be in therange 1 to 11. The insensitivity of the fractal dimension makes it particularly susceptible to theDEM resolution. Therefore its variance in our data set is more data-compromised than Sn.In addition to PCA results, the L −W scalings in equation 2.18 show discrepancy withthe observations in Figure 2.8. Equation 2.18 describes reasonably well the L−W logarithmiccurve for values of ln(W) < 7 m. Data points above this threshold, which include two sys-tems in Patagonia, show a significant deviation from the predictions. This discrepancy amongPatagonian systems could be a consequence of fluvial channel morphologies being influencedby pre-existing landscapes sculpted mostly by glaciers, therefore mixing the fluvial and glacialL−W signatures. More data are required for a comprehensive evaluation of the scaling rela-tionship, and a more complete understanding of the discrepancy of the Patagonian systems.As a final remark, on the basis of our PCA results, we find that channel networks that arereliably fluvial in Figure 2.5 distribute in arid regions in mid or low-mid latitude areas abovethe Tropic of Cancer (23◦27′N lat), and below the Tropic of Capricorn (23◦27′S lat) in Figure2.9. We did not sample channels in temperate zones where vegetation affects our predictionsfor the aspect ratio in complex ways, and also complicates our measurement of the minimum422.6. Discussionchannel width. We return to this issue below.Glacial networksGlacial networks show the largest minimum widths λ on the order of ∼ 1 km, junction anglesγ that are consistently around 90◦, and smallest aspect ratios in the range 5 < R < 30. Thephysical separation from other clusters in Figure 2.5, and the property that most glacial chan-nel networks have negative PC1 and PC2 makes identification of glacial networks from thePCA analysis in Figure 2.5 relatively straightforward.Equation 2.6 successfully predicts the R values observed in the supplementary dataset“ch1datasetEarth.xls” and highlights the importance of ice thickness and sliding velocity. Themodel predicts the highest aspect ratios at high sliding velocities and small ice thickness,which is in agreement with our limited dataset as well as the extensive length, width andice thickness data from the WGI.Although they are generally characterized by λ > 1 km, the variance in the data in thisdirection is very large, and is at least in part related to the well known range of processes thatcan enhance and limit sliding. For example, high levels of meltwater leading to faster slidingrates us, a change in the deposition or ablation rates that modifies ice thickness h, etc. (Paterson,1994).Our model for junction angles predicts values around 90◦ for glacial networks with a dom-inantly flat longitudinal profile. We attribute that to intensive erosional activity at the cirques,and comparatively downstream smaller slopes (e.g., Herman and Braun, 2008), for which equa-tion 2.10 predicts angles approaching 90◦. This is consistent with observations of the channelnetworks in the supplementary dataset “ch1datasetEarth.xls”. However, glacial systems differmore than fluvial systems from the junction angle model, particularly for angles larger than90◦ which our model fails to predict.Sapping valleysThe PCA results in Figure 2.5 show the sapping valley cluster in the 2nd quadrant, split into twosub-clusters. Sapping valleys are characterized by variably low aspect ratios (11 < R < 123),L −W data that does not collapse into a straight line, low stream order values (2 < Sn <4), and junction angles around 64◦. Figure 2.5 shows data to be divided into two groupscoinciding with well characterized sapping valley systems in Florida (e.g., Lamb et al., 2006;Devauchelle et al., 2011, 2012) and the Colorado Plateau (e.g., Laity and Malin, 1985; Lamb et al.,2006), and systems in Hawaii with a more contentious origin (e.g., Kochel and Piper, 1986; Lambet al., 2006).A similar classification arises from the slopes of the L −W log-log relationship in Figure2.8c: the positive slope corresponds to Colorado/Florida systems and the negative slope tosystems in Hawaii (marked with an H in the plot). Although this correlation and the agree-ment between PCA and scalings is provocative, the number of data points is not large enough432.6. Discussionto draw statistically significant conclusions. However, this qualitative relationship suggests afundamental difference in the formation mechanism of channel networks in Hawaii and Flori-da/Colorado that agrees with the conclusive remarks by Lamb et al. (2006). Our limited datais consistent with the sapping origin of the systems in Colorado and Florida, and suggestswith a formation other than sapping erosion for the Hawaiian channel networks. The mainmechanisms governing sapping valley evolution are incompletely understood but involve,e.g., headwall undermining and collapse as a consequence of erosion in the source springs(Laity and Malin, 1985; Devauchelle et al., 2012), which according to Lamb et al. (2006) is difficultto achieve in basaltic rocks.The aspect ratios for sapping valley networks are, on average, one order of magnitudesmaller than those typical of fluvial and glacial networks. Our model successfully captures thisobservation at values of the Shield stress close to the critical parameter of sediment removal(θt ∼ 0.3 in Figure 2.6), where the fluvial aspect ratio predicted in equation 2.1 diverges.Although we can characterize the aspect ratio of sapping systems (Figure 2.6), significantchallenges remain. For example, we find disagreement when comparing the junction angleswith the model by Devauchelle et al. (2012). Whereas this model performs well in the sappingvalleys of the Florida Panhandle, it does not match the observations taken from the ColoradoPlateau, suggesting the crucial role of soil characteristics, such as permeability and sedimentcohesion, governing erosion and groundwater emergence. Our results show that more workis required to fully understand how groundwater sapping affects the landscape, and that theHawaiian channel networks are most likely not formed by sapping erosion.Subglacial channel networksSubglacial channel networks form a cluster in Figure 2.5 that is distinct from glacial and fluvialnetworks, but overlaps the sapping valley cluster, in particular the systems from Colorado andFlorida. To distinguish subglacial channel networks from those related to rivers, we recognizethat subglacial channels are consistently marked by larger minimum channel width λ, and alsosmaller aspect ratios R and stream orders Sn. The small variability in minimum channel widthin Figure 2.5 might be influenced by the close proximity of our data, which are consistentlylocated 75◦N to 80◦N latitude on similar geological units. Future studies should considerchannels in the eastern side of Greenland, Siberia, and Antarctica to evaluate this aspect.The overlaps in Figure 2.5 between sapping valleys and subglacial channel networks, andtheir close proximity to fluvial channel networks highlight the need for an additional metricto successfully distinguish these channel networks. Obtaining predictions for the minimumchannel width and tributary junction angle is also challenging. In particular, deriving a suc-cessful model for the minimum channel width can prove problematic, given that the modelsof subglacial erosion and landscape evolution are mostly underdeveloped Beaud et al. (2016),and require a constraint on the overlying ice thickness. Although the model for the aspectratio in equation 2.3 is consistent with the data, the proportionality between R and ∆ht high-442.6. Discussionlights the sensitive dependence on the lateral variations of ice thickness. Therefore, furtherconsiderations on the dependence of R on ∆ht and on the non-dimensional stress regime θt inthe channels are required to improve the scope of the model.2.6.2 Limitations and challengesFluvial channel networksIn our characterization of fluvial processes, our steady state model is limited in several ways.The first is the lack of any explicit sediment-flow coupling stress which governs the abrasionprocess on the river bed (Sklar and Dietrich, 2001). The introduction of sediment mass balanceand corresponding additional momentum equations, including a self-consistent critical stressfor sediment removal, would improve this model by providing more accurate predictions ofthe shear stress at the base as a function of sediment abrasion. Quantitatively, the addition ofthese effects would contribute to more precise descriptions of R.A second limitation is our neglect of vegetation effects by limiting our measurements toarid regions with minimal vegetation cover. Vegetation can effectively decrease or increase thesediment erodibility, which affects the hillslope diffusion term D in the landscape evolutionequation (equation A.4 in Appendix A). Furthermore, vegetation can affect the cohesion ofsediment, thus altering the angle of repose of sediment in water (φr) in equation 2.1 (e.g.,Manga and Kirchner, 2000).The third and arguably most significant limitation of this study regarding fluvial channelsis the measurement of the minimum channel width λ, limited to the resolution of the satelliteimagery and topography data. The best resolution available (0.3 m/pixel) can be in somecases still larger than or comparable to the potentially centimeter-scale of order 1 channel rills,in which case our fluvial λ dataset can be biased towards larger values. This bias wouldtranslate into a greater separation between fluvial channels and sapping valleys /subglacialchannels in Figure 2.5 because the river networks would generally plot at lower λ values.Higher resolution imagery and LIDAR topography data will eventually help reduce this biasand obtain better measurements.Glacial channel networksThere is an increasing effort in the literature to understand how glaciers interact with theirboundaries (i.e., ice-bedrock, ice-water, ice-air, etc.). In developing the equations in section 3,we considerably simplified this description by characterizing the boundary interactions withan effective shear stress and ignoring complications related to cavities and other accumula-tions of subglacial water. For the sake of a tractable model, we also did not introduce a tem-perature dependence to the softness parameter, thus avoiding an additional energy balanceequation to describe temperature variations throughout the ice. However, a vertical temper-ature gradient in the glacier will introduce a relative large gradient in effective viscosity, and452.6. Discussiontherefore an increase in flow speed. This behavior affects the shear stress at the walls and theground and thus will alter the predicted aspect ratio: we can expect higher aspect ratios forglaciers with strong temperature gradients. Thus, our model probably under predicts the as-pect ratios of glaciers with steep temperature gradients. The scatter in the data for minimumchannel width can also be related to temperature-dependent viscosity changes, affecting di-rectly equation A10 in Appendix A. Furthermore, warm-bed glaciers will have larger slidingrates (us) than cold-bed glaciers, affecting the R predictions in equation 2.6. This potentiallyenhanced sliding is consistent with observations from the WGI of glacier widths in moun-tain glaciers in low and mid latitudes compared to sea level glaciers at latitudes higher than60◦N. Understanding the varied temperature dependences of ice flow can therefore provide away to understand the variability of Figure 2.5 in terms of meltwater availability and climateconditions.Sapping valley systemsA comprehensive model for sapping valleys evidently requires some characterization of theresponse of different types of sediment and different degrees of rock cohesion to the spring-driven erosional forcing. The separation of sapping valley data into a group of Hawaiian,Colorado and Florida channel networks in Figure 2.5, and the opposing slopes of the L−Wscaling in Figure 2.8 suggest that these channel networks formed by different processes. Theaspect ratio predictions from equation 2.2 show that the dynamics at the interface betweengroundwater and surface flows largely control the geometry of the sapping valley. Therefore,further studies should investigate in more detail the role of rock cohesion and sediment erodi-bility in the geometry of sapping valleys in particular their behavior close to the spring.Subglacial channel networksThe characterization of subglacial channels represents one of the biggest limitations in thisstudy. By definition, the subglacial environment makes it impossible to observe the activechannel geometry at a network scale, or obtain reliable information about their dynamics.High resolution digital elevation models that include exposed subglacial channel networksare mostly limited to the Canadian Arctic, and therefore our dataset is constrained to systemsin that region. A second challenge regarding subglacial channels is the lack of understandingof the erosional regime at the floor and walls of the channel, which is in part consequence of thelimitations described above. Modeling the aspect ratio in equation 2.3 requires a descriptionof the non dimensional stress distribution in the channel, which is not well understood. Onthe basis of our model results and the comparison with data in the supplementary dataset“ch1datasetEarth.xls”, we can argue that a turbulent regime with higher typical Shield stressparameters than observed in fluvial systems describes the data more accurately, but we cannottest the reliability of the model in detail until further studies better constrain this parameter.To our knowledge, there is no comprehensive study on the erosion by subglacial chan-462.6. Discussionnels that develops an appropriate landscape evolution equation (Beaud et al., 2016), or givespredictions for the minimum channel width. Understanding the processes that control thejunction angles require a model for the topography of the overlying ice-sheet and the pressuregradient distribution. Mechanically, at least two issues emerge. First, our model for junctionangles assumes a gravity driven flow, and therefore it is not applicable for flow in subglacialchannels driven by lateral variations in ice thickness. Indeed, our observations in the supple-mentary dataset “ch1datasetEarth.xls” indicate that angles in subglacial channels span 20◦ to90◦, which is consistent with field studies about non-dendritic morphology (Walder and Hal-let, 1979) and undulating longitudinal profiles (Lonergan et al., 2006). Second, the shear stressregime in subglacial channels is unclear, and as are the additional mechanical consequences ofenergy lost to melting, and both effects contribute to R through equation 2.3. A careful assess-ment of the coupling of these terms requires a full energy conservation model that includeswater temperature variations and latent heat (e.g., Walder and Fowler, 1994; Clarke, 2003), whichwe do not discuss here and which is objectively difficult to constrain. The main controls onthe width and height of an individual channel and how they evolve with distance to the outletdepend on ice thickness variations, and therefore are hard to predict. Nevertheless, our aspectratio model successfully reproduces the data range (see Figure 2.6) if we take values of thenon-dimensional shear stress to be much larger than fluvial systems θt > Earth viewed from MarsOne of the motivations of this paper was a simple thought experiment in climatology: If wewere standing on the surface of Mars and viewing channel networks on Earth, could we reli-ably conclude that the planet is in an interglacial period without the aid of geological context?To answer this question, we plotted the location of data points as classified from our PCA ina global map in Figure 2.9, and shaded in black the data points that were unclear from Figure2.5. The dominance of glacial and subglacial channels above 60◦N and fluvial and sappingvalleys below this line provides quantitative evidence of Earth’s current interglacial period.2.6.4 Can we constrain Earth’s glacial-interglacial climatology from the structureof the channel networks?Considering now the topographic signatures of overprinted processes such as fluvial systemsincised within glacial valleys, it is also possible to characterize aspects of Earth’s recent glacialcycles. Take as an example the transition between a glacial and a fluvial dominated landscapesince the end of the Younger-Dryas (YD) or Last Glacial Maximum (LGM). In this context,quantifying the aspect ratio in a succession of older to recently glaciated channel networkscould provide a calibration on the time scale of transient landscape evolution, because thetiming of the YD and LGM are well constrained. Other metrics, such as the tributary junctionangle variation along the same succession of landscapes, could also potentially identify theglacial to fluvial transition in the landscape.472.7. ConclusionsThe transient between a subglacial channel and a fluvial channel can become apparentin the same channel network, particularly in areas with a strong meltwater production orprecipitation input. In this aspect, further investigations should address this transition in moredetail.2.6.5 Are these metrics applicable to valley networks in other planets such asMars or Titan?The metrics developed in this study are readily applicable to other planetary surfaces and of-fer a quantitative understanding of the mechanisms leading to valley network incision. Water(ice or liquid) is believed to be the dominant erosional flow on Mars. Adapting the models forEarth (section 3) to understand the valley network formation mechanisms under the Martianconditions is consequently simple, and requires the adjustment of some parameters such asgravity, grain size, roughness factors, ice flow exponent, etc. to the Martian surface. However,processes such as ice CO2 flow or sublimation driven mass waste on Mars could lead to ero-sional signatures that are not observed on Earth, nor captured by these metrics, requiring theaddition of new metrics and the development of the respective models of erosion.Modeling the physics underlying these metrics in other planetary bodies such as Pluto orTitan is not as straightforward, given the uncertainty in the primary composition and rheo-logical properties of the respective flows causing erosion, and the unknown relative roles ofthermal to mechanical erosion. However, deriving qualitative differences between valley net-works and applying a PCA to find clusters of valley networks of different characteristics isstraightforward in any planetary surface. Future advances in the understanding of the flowregimes in the surface of these bodies will enable the full adaptation of these and other metricsto characterize and classify the valley networks in Titan and Pluto.2.7 ConclusionsWe developed a methodology using 5 metrics drawn from remote sensing topographic data todiscriminate between channel networks incised by fluvial, glacial, subglacial and groundwatersapping processes. We tested the relative success of these metrics for characterizing thesefour different types of channel networks by using a principal component analysis (PCA), theresults of which are understood with physical models, to the extend permitted by the data andexisting knowledge.Our main results (Figure 2.5) highlight the success of the metrics at characterizing statisti-cally significant fluvial, glacial, and sapping/subglacial data clusters, with a divided sappingvalley cluster that overlaps the glacial and subglacial data points. In the case of the division be-tween data from Florida/Colorado and Hawaii in both the PCA and the scaling relationships,we agree with Lamb et al. (2006) and propose a fundamentally different formation mechanismto sapping erosion for the Hawaiian box canyons.As shown in Figure 2.5, the most useful metrics at characterizing a channel network are the482.7. Conclusionsminimum channel width λ, aspect ratio R and stream order Sn. According to the physical de-scription of the aspect ratio in section 3, the work by Pelletier et al. (2010) and Meir et al. (1980)at respectively characterizing the significance of the minimum channel width and the streamorder, we can interpret the main differences between the 4 erosion styles in terms of these met-rics as: (1) the stable cross sectional profile in relation with the stress regime in the channel(R), (2) the erosional regime established by a landscape evolution equation (λ) and the point atwhich instabilities incise a channel, and (3) the number of first order channels (Sn) as the maindifferences between the 4 channel types. We find the tributary junction angle to reflect theslopes in the longitudinal profile of gravity driven currents (fluvial and glacial), which agreeswith the classical theory by Horton (1932) and Howard (1971). Our results indicate that fractaldimension D f and stream order Sn capture the same aspect of channel network complexity.Given that Sn captures a much more significant portion of the overall data variance, we con-clude that is a better description of the complexity in channel network spatial patterns thanD f . We also present the scaling relationships derived here for fluvial networks, and make useof other relationships derived in other studies (e.g., Bahr et al., 1997) as a way to complementthe PCA results and aid in their interpretation.The models presented in this study predict and constrain the possible geometries withineach channel network type in Figure 2.1, individually for each of the metrics and statisticallywith the PCA . Mixed signatures in the metrics in one individual channel network, such asthe aspect ratio geometry predicted for a sapping valley and the minimum channel widthpredicted for a river, might indicate that both processes were at play in carving the channel. Asa final remark, our methodology represents a tool to quantitatively compare different channelnetwork types in different parts of the world using remote sensing topographic data. Wecan reliably identify and characterize multiple erosion processes involved in the incision ofa channel, the usefulness of which can be applied to identify the landscape signatures of theglacial and interglacial recent climatic cycles.49Chapter 3Subglacial drainage patterns of DevonIsland, Canada: Detailed comparison ofrivers and subglacial meltwaterchannels3.1 SummarySubglacial meltwater channels (N-channels) are attributed to erosion by meltwater in sub-glacial conduits. They exert a major control on meltwater accumulation at the base of icesheets, serving as drainage pathways and modifying ice flow rates. The study of exposedrelict subglacial channels offers a unique opportunity to characterize the geomorphologic fin-gerprint of subglacial erosion as well as study the structure and characteristics of ice sheetdrainage systems. In this study we present detailed field and remote sensing observations ofexposed subglacial meltwater channels in excellent preservation state on Devon Island (Cana-dian Arctic Archipelago). We characterize channel cross-section, longitudinal profiles, andnetwork morphologies and establish the spatial extent and distinctive characteristics of sub-glacial drainage systems. We use field-based GPS measurements of subglacial channel longi-tudinal profiles, along with stereo imagery derived Digital Surface Models (DSM), and novelkinematic portable LiDAR data to establish a detailed characterization of subglacial channelsin our field study area, including their distinction from rivers and other meltwater drainagesystems. Subglacial channels typically cluster in groups of∼ 10 channels and are oriented per-pendicular to active or former ice margins. Although their overall direction generally followstopographic gradients, channels can be oblique to topographic gradients and have undulatinglongitudinal profiles. We also observe that the width of first order tributaries is one to twoorders of magnitude larger than in Devon Island river systems, and approximately constant.Furthermore, our findings are consistent with theoretical expectations drawn from analyses of503.2. Introductionflow driven by gradients in effective water pressure related to variations in ice thickness. Ourfield and remote sensing observations represent the first high resolution study of the subglacialgeomorphology of the high Arctic, and provide quantitative and qualitative descriptions ofsubglacial channels that revisit well-established field identification guidelines. Distinguishingsubglacial channels in topographic data is critical for understanding the emergence, geometryand extent of channelized meltwater systems and their role in ice sheet drainage. The final aimof this chapter is to facilitate the identification of subglacial channel networks across the globeby using remote sensing techniques, which will improve the detection of these systems andhelp to build understanding of the underlying mechanics of subglacial channelized drainage.3.2 IntroductionSubglacial meltwater channels, often referred to as N-channels after Nye (1976), are the ero-sional expression of turbulent flows in pressurized subglacial channels. Together with sub-glacial channels incised in overlying ice (R-channels after Ro¨thlisberger (1972b)), they modulatemeltwater accumulation at the base of ice sheets and serve as highly efficient drainage path-ways carrying meltwater to the ice terminus (e.g., Ro¨thlisberger, 1972b; Weertman, 1972; Nye,1976; Sugden et al., 1991; Greenwood et al., 2007; Kehew et al., 2012). In particular, the transitionfrom distributed to channelized drainage leads to a reduction in ice flow rates, modifying iceloss rates and suppressing surging (e.g., Schoof , 2010). Subglacial channelized drainage plays akey role in deglaciation, and so their spatial characteristics, density, and distribution can helpunderstand the patterns of glacial retreat (e.g., Sugden et al., 1991; Greenwood et al., 2007).In spite of their importance, some outstanding questions remain: What are the typicallength scales that characterize subglacial drainage systems (i.e., what is the drainage area,how many individual valleys form, how many tributaries do they have, etc.)? Can we reli-ably identify subglacial channels from rivers and other meltwater channels by using remotesensing techniques, including imagery and topographic data? And how do the characteristicsof remarkably well preserved channels compare with channels elsewhere? To answer thesequestions, here we perform a detailed geomorphological study of exposed subglacial chan-nels on Devon Island (Canadian Arctic Archipelago). This work represents the first field andhigh resolution remote sensing characterization of subglacial channels in the high Arctic, oneof the areas with the best exposures of such features worldwide.Well preserved exposed subglacial channels are rare. During glacial recession, meltwa-ter released from the ice sheet accumulates at the ice marginal area and erodes the channel,with post-glacial sediment accumulation causing burial or partial burial (Le Heron et al., 2009).Vegetation overprint and fluvial incision makes the detailed study of channel geometry andmorphology difficult (e.g., Walder and Hallet, 1979). Exceptions include areas with polar desertclimate in the Antarctic Dry Valleys (e.g., Sugden et al., 1991) and the Canadian High Arctic(e.g., Dyke, 1993, 1999). The reduced rainfall conditions of these sites, recent ice retreat, andnull or minimal vegetation cover are key for the preservation of these features. The morphol-513.2. IntroductionTable 3.1: Diagnostic criteria for the identification of subglacial channelsDistinctive characteristics referencesProfile undulations Greenwood et al. (2007); Kehew et al. (2012), this studyNetwork direction Sugden et al. (1991); Greenwood et al. (2007), this studyOther subglacial landforms Greenwood et al. (2007); Kehew et al. (2012)Cavity systems and potholes Sugden et al. (1991), this studyStepped confluences Sugden et al. (1991), this studyAbrupt beginning and end Sissons (1961); Glasser et al. (1999)Absence of alluvial fans Sissons (1961), this studyOther characteristics referencesAbandoned loops Clapperton (1968), this studyHigh sinuosity Clapperton (1968)Bifurcating patterns Sugden et al. (1991); Greenwood et al. (2007), this studyTributary variable scales Sissons (1961)Presence of steep chutes Sissons (1961), this studyogy of subglacial channels at our field study area on Devon Island (Figure 3.1), the second-largest of the Queen Elizabeth Islands in the Canadian Arctic Archipelago, is consequentlywell-preserved. The retreat of the Devon Island ice cap, in addition, offers a unique opportu-nity to compare recently exposed subglacial channels with systems incised during the YoungerDryas Innuitian de-glaciation.The systems identified by Dyke (1999) as subglacial meltwater channels on Devon Islandare ∼10-20 m wide and ∼3-6 m deep, which is consistent with subglacial channels observedelsewhere, and one to two orders of magnitude smaller than tunnel valleys (e.g., Cofaigh, 1996;Kehew et al., 2012; Livingstone and Clark, 2016). Although subglacial channels have been de-scribed in detail in the field in northern Europe (e.g., Kleman, 1992; Clark et al., 2004; Piotrowskiet al., 2006), the Antarctica Dry Valleys (e.g., Denton et al., 1984; Sugden et al., 1991), Canada (e.g.,Kor et al., 1991; Beaney and Shaw, 2000; Shaw, 2002), and the United States (e.g., Walder and Hal-let, 1979; Booth and Hallet, 1993), their rigorous distinction from other drainage systems fromremote sensing imagery and topographic data is limited (Greenwood et al., 2007). Field iden-tification of subglacial channels consists of (1) identification from fluvial runoff (proglacialchannels or river systems) and (2) distinction from other meltwater features such as lateralmeltwater channels, which originate and follow the ice margin (e.g., Beaney and Shaw, 2000;Greenwood et al., 2007; Syverson and Mickelson, 2009; Margold et al., 2013). A set of subglacialchannel identification criteria is presented by Greenwood et al. (2007) and summarized in table3.1. These guidelines are qualitative, and the characteristics listed here may or may not be allpresent in a set of subglacial channels (e.g., Sugden et al., 1991; Beaney and Shaw, 2000).To categorize and characterize the features presented as subglacial meltwater channels inDyke (1999), we conducted fieldwork on central Devon Island. In section 3.3 we provide a de-tailed description of our field data acquisition and processing. We acquired GPS based channellongitudinal profiles along with stereo imagery and derived photogrammetry digital elevation523.2. Introductionmodels. We also used a Kinematic LiDAR Scanning (KLS) portable system, a novel method ofmeasuring ultra-high resolution topography (< 2 cm/pixel Digital Elevation Models (DEMs),which is described in more detail in section 3.3.4. In section 3.3 we present a quantitativecharacterization of the observed channel networks, which we apply to distinguish subglacialchannels from lateral meltwater channels and rivers in section 3.5. In section 3.6 we present adetailed qualitative description of subglacial channel morphology and the shape of subglacialdrainage networks, which serves to further characterize and distinguish subglacial channelnetworks.3.2.1 Field site: Devon IslandDevon Island was covered by an extensive Innuitian Ice Sheet that reached its maximum ex-tent during the last glacial maximum (e.g., England, 1987; Dyke, 1999; England et al., 2006).Shortly after the Younger Dryas, around 10 radiocarbon years BP, the margin of this ice sheetbegan retreating towards the current coast line, and the final remnants in central Devon Islandvanished around 8.8 radiocarbon years BP (Dyke, 1999), leaving a landscape of plateaus, fiordsand deeply incised canyons. We refer to the work by Dyke (1999) and England et al. (2006)for a detailed discussion of the glacial history of the island during and since the Innuitian icesheet. Since deglaciation, the landscape evolution has been mainly the result of periglacialprocesses and erosion by ephemeral seasonal streams (e.g., McCann et al., 1972; Dyke, 1999).Fluvial incision represents only a small and highly localized contribution to the overall land-scape evolution as a consequence of the island’s polar desert climate conditions (e.g., French,2013). Aerial photography obtained from the National Air Photo Library (National ResourcesCanada) of Devon Island reveals highly directional channel networks, which Dyke (1999) de-scribed as meltwater channels. These systems are incised into the otherwise flat plateaus thatcomprise the majority of the topography (regional slopes are 1 to 6◦), and typically drain intodeeply incised canyon systems that are believed to pre-date Innuitian glaciation (Dyke, 1999).To categorize and study these meltwater channels in detail, we selected a study site thatcomprises an area of approximately 15 km2 to the E-SE of the Haughton impact structure incentral northern Devon Island (Figure 3.1). This is a well-preserved 23 km diameter (Osin-ski and Spray, 2005) 23 Myr. old (Young et al., 2013) meteorite impact structure, which is awell-established Mars analogue terrain. It has been the focus of numerous planetary analoguestudies including crater morphology and erosion, periglacial landscape evolution on Earthand Mars, and evolution of ancient lake beds (Lee and Osinski, 2005). The location providesaccess to both exposed subglacial channel and river networks, allowing for systematic com-parisons of their geometry and longitudinal profiles. Geologically, the study area lies entirelywithin carbonate strata of the Upper Ordovician Allen Bay Formation, specifically the LowerMember, which comprises a uniform succession of medium bedded to massive limestone withdolomitic labyrinthine mottling (e.g., Thorsteinsson and Mayr, 1987; Osinski and Spray, 2005),and is overlain by Quaternary glacial till (e.g., Dyke, 1999; Osinski and Spray, 2005). Other land-533.2. Introductionforms typical of glaciated landscapes, including eskers, glacier moraines, and striations arerare on the plateau surface of Devon Island (Roots et al., 1963) and only occur sporadicallywithin the Haughton impact structure (Osinski and Spray, 2005).Figure 3.1: (a) satellite imagery of Devon Island within the Arctic Archipelago (whitebox). (b) satellite image of Devon Island, with a white box indicating the selectedfield site. The map also shows the Innuitian ice sheet termini lines digitalized fromDyke (1999), with age reference in the legend (refer to radiocarbon years). (c) Fieldsite (UTM zone 16), with boxes around each network investigated. White boxesare for sublgacial networks (SG1, SG2, SG3, and SG4), whereas black boxes indicatefluvial networks (R1 and R2)543.3. Methodology3.3 Methodology3.3.1 Preliminary remote sensing characterizationFigure 3.1(a) shows a Digital Globe satellite image of the Canadian Arctic including DevonIsland and the rest of the Arctic Archipelago for context. Also shown are the Innuitian icesheet termini as digitized from the work by Dyke (1999). Figure 3.1(b) shows a high resolutionsatellite view of our selected field area. Our target locations consist on four distinct subglacialchannel networks and two fluvial networks. To identify the areas with potential subglacialincision, we looked for highly directional channel networks parallel to former ice flow linesin areas easily accessible from the Haughton structure, and also based on the locations whereDyke (1999) found evidence for meltwater channels. Using high resolution WorldView im-agery of the site (resolution of 2 m/ pixel), CDEM (0.15 arc-sec DEM corresponding to 20by 36 m/ pixel at a latitude of 75◦ N, obtained from the Natural Resources Canada website(http://geogratis.gc.ca/site/eng/extraction), and the recently released Arctic DEM (release5 at a resolution of 2 m/pixel available for free at https://www.pgc.umn.edu), we chose sixchannel networks, from which we inferred four to be subglacial and two to be fluvial.From these six channel networks, we selected and visited twenty individual channels fordetailed, in situ characterization (see Figure 3.1). For this study, we selected only first ordertributaries and characterized them from the origin until the first junction. In most cases melt-water accumulates into surface streams, and fluvial incision is apparent from field and remotesensing data in the profiles and cross-sections of the channels once the network develops astream order of 2 or more. In addition to in situ data, we acquired helicopter airborne imageryof sites located in central and eastern Devon Island, and identified subglacial channels as theyare exposed by the current retreat of the ice cap at its terminus.3.3.2 Longitudinal profile dataA distinctive characteristic of subglacial channels is the presence of vertical undulations intheir profiles (e.g., Sissons, 1961; Sugden et al., 1991; Greenwood et al., 2007). To detect these fea-tures we obtained longitudinal profile data (i.e., elevation vs. distance data) of the 20 targetchannels using a GARMIN gpsmap 64s with a 1-3 m horizontal resolution, depending on polarsatellite availability (3-7 satellites available at a given time), and a vertical barometric resolu-tion of 3-6 m. We acquired the data by walking or driving an All Terrain Vehicle (ATV) alongeach inferred river or subglacial channel from the head until the first junction, and averagingthe multiple profiles acquired in two to three runs.To minimize the effects of the variable GPS resolution in processing the data, we groupedthe channels into 5 groups that correspond to sites visited on the same day during the sea-son. We also recorded the speed of the traverse during the collection of the longitudinal pro-files, which varied between walking speed (vwalk ∼ 3-5 km/h) and ATV speed (vdrive ∼ 10-15km/h). We then use this information to filter the GPS data corresponding to each field day553.3. Methodology(Figure 3.3).We processed the GPS raw data by high band-pass filtering the signal with an upper fre-quency equivalent to twice the Nyquist sampling frequency, which corresponds to 1 GPS pointper 3 seconds in the hiking traverses and 1 point per second in the ATV traverses, and a lowerfrequency corresponding to the inverse of the time required to drive/ walk along the channellength, that is, L/vdrive in ATV traverses and L/vwalk in hiking traverses (where L is the chan-nel length). This filtering operation removes the data spikes related to avoiding obstacles ontraverses including stream paths, large boulders, and snow patches.3.3.3 Airborne imagery and photogrammetryWe complemented our in-situ channel characterizations with an extensive collection of highresolution aerial photography of over 50 subglacial channels throughout the island, fromwhich we derived Digital Surface Models (DSM). DSM generation through stereo-photogrammetryprocessing of image data involves the reconstruction of a three-dimensional body employingmeasurements in two or more overlapping images, acquired from different positions. Accu-rate reconstructions require an overlap of more than 50% between each image on a basis ofat least 10 images per location, common features identifiable in different images for reference,and detailed spatial coordinates for each site.For this purpose, we acquired over 1000 helicopter airborne images to capture the topogra-phy of multiple inferred subglacial channel networks. To build this image database, we useda GPS-referenced CANON EOS 6D with an image resolution of 72ppp (5472 by 3648 pixels)(e.g., Smith et al., 2009). The built-in GPS has a horizontal spatial resolution of ∼ 10 m and avertical resolution of∼ 5 m. Although the camera GPS resolution also depends on polar satel-lite availability, resolution variations are minimal given the very small time lapse of imageacquisition of all helicopter data.To construct a Digital Surface Model (DSM), we obtained geo-referenced helicopter borneimages for more than 50 channels including the center-east of the island and the margin of theDevon Island ice cap. Significantly, this survey includes imagery of subglacial channels cur-rently emerging under the active Devon Island ice cap margin (Figure 3.2(c)), which enabledus to ground truth our identification scheme. Figure 3.2 includes examples of these imagesacquired at different points in the island.We process the data in several steps. For each site, we first upload the images into AG-ISOFT software (e.g., Tonkin et al., 2014), together with the camera-generated EXchangeableImage Format (EXIF) files that include the geo-reference information. The software automat-ically aligns the imagery using the overlap existent between images. We improve the initialautomated alignment with manual alignment of the images by selecting and matching com-mon features (control points). Next, we produce a dense point cloud, a meshed surface model,and a surface model with ground texture. At this stage, we use the recently released Arc-tic DEM (available for free download at http://www.agic.umn.edu/arcticdem, release 5 at563.3. MethodologyFigure 3.2: Aerial and field imagery of subglacial channels and rivers. 2(a) correspondsto helicopter imagery of a group of subglacial channels (89.13◦ W,75.28◦ N), channelwidths approx. 35 m. 2(b) corresponds to a groups of subglacial channels locatedat 89.37◦ W, 75.18◦ N, network is approx. 300 m wide. 2(c) corresponds to a sub-glacial channel emerging underneath the Devon Island ice cap, notice the similarmorphology to 2(a) and 2(b), each channel being approx. 30 m wide. 2(d) shows thecross-section of a deeply incised canyon emerging from under the ice cap. Canyoncross-section measures 200 m approx.http://www.pgc.umn.edu) to manually introduce markers in the model with known coor-dinates and elevation. This step improves the resolution of the final product by an order ofmagnitude. With this improved 3D model, we produce an orthophoto and the Digital SurfaceModel (DSM). In turn, at the final stages of processing we manually crop the DSM to removenoisy areas.The DSM model reconstructions range in resolution between 0.4 m/pixel to 10 m/pixeldepending on helicopter elevation and speed, number of images captured at each site and theiroverlap, the number of manually introduced control points, and other factors. All productsare available upon request from the authors in point cloud (.LAS) and Geotiff (.tif) formats.573.3. Methodology3.3.4 Kinematic LiDAR Scan acquisitionWe used a novel kinematic backpack LiDAR scanning (KLS) system to capture the detail ofsubglacial channel topography from the ground. This study is the first time the KLS systemhas been deployed in the Arctic, and the first time it has been applied to detailed subglacialchannel morphometric measurements. The goal of the survey was to reproduce the surfacetopography at cm resolution, but also to make a proof of concept for the capabilities of kine-matic LiDAR. The system consists of a LiDAR scanner, a GNSS/GPS positioning system, andan inertial measurement unit (IMU) which are mounted within a backpack frame, allowingthe user to make ultra-high-resolution LiDAR point clouds (>5,000 points/m2) of features tra-versed by the user. KLS enables the reproduction of surface topography at cm resolution (2cm/pixel; Figure 3.4), which is a higher level of accuracy compared to e.g., airborne LiDARdata, which would also be expensive to obtain in remote areas such as the high Arctic. The KLSdataset and derived surface models improve cross-sectional and longitudinal profile analysisand comparison with river valleys, and are a ground-truth for hand-held GPS topographicdata acquisition.The LiDAR data was acquired using the AkhkaR3 kinematic backpack LiDAR developedat the Finnish Geospatial Research Institute, which is an updated version from those presentedin Kukko et al. (2012) and Liang et al. (2015). The system is based on GNSS-IMU (Global Navi-gation Satellite System-Inertial Measurement Unit) positioning system, consisting of NovatelSPAN: Flexpak6 receiver, UIMU-LCI inertia measurement unit and 702GG antenna, a 360 de-grees of field of view cross-track profiling laser scanner (Riegl VUX-1HA) synchronized to thepositioning and operated by a tablet computer (Panasonic Toughpad ZF-G1).River and subglacial channel topography data were collected by traversing the center lineof the channel by ATV, with the operator carrying the LiDAR system on his/her back. Con-tinuous scanning was done using 150 Hz profiling, and 500 kHz pulse repetition frequencies.With these settings the maximum range was about 200 m from the scanner (i.e. a 400 m-widechannel could be completely scanned), and the along-track line spacing about 1 cm with an-gular resolution of 1.8 mrad.For accurate trajectory computations we set up a GNSS base station (Trimble R10) at theHaughton river valley base (75◦ 22.42’ N, 89◦ 31.89’ W) constituting of about 5 km base linelength to the target channels. The raw GNSS observables were recorded at a 5 Hz frequency,as were those at the KLS mapping system. The altitude data for the KLS system were recordedat a 200 Hz data rate, and the positions and altitude trajectory were computed in a post-processing step for the point cloud generation. For post-processing we used the base stationposition using PPP method and the tightly coupled KLS trajectory Waypoint Inertial Explorer8.60 (NovAtel, Canada).To produce an elevation model the raw point cloud data was further processed: the pointsresulting from multiple reflections were removed as well as points with weak return signal(intensity less than 800 in the scanner scale). Some remaining points resulting from the laser583.4. Results: Quantitative characterization of river and subglacial channelsbeam hitting the rear of the ATV during the capture were manually cleaned out of the datausing CloudCompare software. Post-processed data was exported as .LAS files.The final DEMs (see example in Figure 3.4 (c) and (d)) from the georeferenced LiDAR pointclouds were created using .LAS Dataset tools in ArcGIS (LAS Dataset to Raster: Bin, Avg,Simple). The effective pixel resolution of 2 cm/pixel in the DEMs represent the average valueof the point cloud within a 2 cm bin (typically 5-10 LiDAR points, depending on proximity tothe scanner). Due to the very high spatial coverage of the LiDAR points, minimal interpolationwas needed, except in areas of LiDAR shadow. These areas were interpolated using the simpleinterpolation method outlined in the ArcGIS help section.3.4 Results: Quantitative characterization of river and subglacialchannels3.4.1 River and subglacial channels’ longitudinal profilesReliably identifying subglacial channels on the basis of the criteria listed in Greenwood et al.(2007) and summarized in table 3.1, and in particular recognizing sections where the subglacialflow eroded against topographic gradients (e.g., Sissons, 1961; Sugden et al., 1991; Glasser et al.,1999) requires measuring longitudinal profiles in the field. Figure 3.3 shows the longitudinalprofiles of different channels visited within the field area (see Figure 3.1 and Table 3.2 forlocation reference). In this figure, SG Network 1 consists of a group of inferred subglacialchannels. R Network 1 is an inferred river system with two investigated channels, and RNetwork 2 includes four investigated river channels. SG Network 2 - D group is a groupof inferred subglacial channels, with two channels investigated, and so is SG Network 3 - Pgroup. SG network 4 - CAMF group consists of inferred subglacial channels, of which wevisited five.From these data we identify the channels that display undulating profiles, as well as quan-tify the maximum undulation in each profile which we define as the elevation difference be-tween the local minima at the beginning of a section with positive topographic gradient hmin,and the local maxima that follows it hmax, to the total topographic loss ∆H, that is:ψ = (hmax − hmin)/∆H, (3.1)see Figure 3.10 for a cartoon representation. We refer to this magnitude as the magnitude ofundulation ψ and we use it herein to quantify differences between the fluvial and subglaciallongitudinal profiles. Table 3.2 shows the magnitude of undulation ψ of the different channelnetworks in Figure 3.3, together with their detailed coordinates.Channels in SG Network 1 (j201, j202, j204, j233, and j234) show magnitudes of undulationψ corresponding to 24%, 3%, 27%, ψ = 3%, and ψ = 0% respectively, taking the percentageof the total topographic loss. The magnitude of the topographic drop in undulations is of593.4. Results: Quantitative characterization of river and subglacial channelsTable 3.2: Morphometric characteristicsIndividual channel latitude longitude ψ deviation from topography FSG 1 27◦j201 75◦ 20’ 0” 89◦ 25’ 5” 0.24 4.5j202 75◦ 20’ 3” 89◦ 23’ 13” 0.03 8j204 75◦ 20’ 11” 89◦ 22’ 49” 0.27 7j233 75◦ 20’ 14” 89◦ 22’ 19” 0.03 10j234 75◦ 20’ 02” 89◦ 24’ 44” 0 10R 1 -j211 75◦ 21’ 35” 89◦ 27’ 40” 0 2j212 75◦ 21’ 28” 89◦ 28’ 42” 0 3j213 75◦ 21’ 12” 89◦ 25’ 22” 0 4j214 75◦ 21’ 25” 89◦ 27’ 10” 0 4j215 75◦ 21’ 04” 89◦ 26’ 24” 0.01 3R 2 -j231 75◦ 20’ 56” 89◦ 27’ 5” 0.04 4.5j232 75◦ 20’ 25” 89◦ 27’ 39” 0 3SG 2 - D 30◦j251 75◦ 17’ 27” 89◦ 11’ 41” 0.07 5.5j252 75◦ 17’ 39” 89◦ 9’ 42” 0.01 30.5SG 3 - P 4◦j253 75◦ 16’ 43” 89◦ 4’ 55” 0.07 6j254 75◦ 17’ 06” 89◦ 5’ 17” 0.03 5.5SG 4 - CAMF 6◦j261 75◦ 17’ 28” 89◦ 27’ 41” 0 17j262 75◦ 17’ 23” 89◦ 27’ 48” 0.13 17.5j263 75◦ 17’ 23” 89◦ 27’ 26” 0.04 31j264 75◦ 17’ 24” 89◦ 28’ 01” 0.04 18j265 75◦ 17’ 24” 89◦ 28’ 22” 0 176.6 m, 1.5 m, 6.5 m, 1.6 m, and 0 m respectively for j201, j202, j204, j233, and j234. Channelsj201 and j204 display the largest undulations that we analyzed. R Network 1 corresponds toinferred fluvial systems, which we investigated and analyzed for comparison. In this network,no undulations are detected above the GPS confidence level, and in fact the profiles show asteady decrease of elevation with distance that is much more consistent with profiles of fluvialchannels in the literature (e.g., Howard, 1994; Sklar and Dietrich, 1998; Whipple and Tucker, 1999;Whipple, 2004). R Network 2 shows two profiles corresponding to active rivers (j231 and j232),which show ψ = 4% and ψ = 0% respectively. Subglacial channels in SG Networks 2 and 3were identified on the basis of their similar morphology and orientation to subglacial channelsin other networks, but do not present significant undulations. Within SG Networks 2 - D and 3- P, channels j251 and j253 display undulations of ψ = 7%, j252 has ψ = 1%, and j254 displaysψ = 3%. Finally, SG Network 4 - CAMF displays 5 channels with different levels of undulation,respectively ψ = 0%,ψ = 13%,ψ = 4%,ψ = 4% and ψ = 0%. Based on similar morphology603.4. Results: Quantitative characterization of river and subglacial channelsFigure 3.3: Longitudinal profiles of river and subglacial channels normalized to total to-pographic loss and length along the channel, with WorldView satellite imagery foreach channel network. In the longitudinal profiles, blue crosses represent the rawGPS data for each channel, blue dashed lines are the data after filtering, and orangesolid lines represent the LiDAR sections that overlap GPS data for comparison. Theprofiles obtianed with the Arctic DEM at 5 m resolution are shown in green color.and proximity to other channels with large ψ within the same network, and recurrent N−NWto S− SE orientation, we conclude that all channels in SG Network 4 to have originated in thesubglacial regime.Additionally, Figure 3.3 provides a comparison of LiDAR and GPS-acquired longitudinalprofiles. This was a useful indicator of the reliability of the hand-held GPS profile datasetwithin the GPS resolution range. We discuss the LiDAR results in more detail below. We613.4. Results: Quantitative characterization of river and subglacial channelsalso performed an additional comparison of our data (LiDAR and GPS) with correspondinglongitudinal profiles extracted from the Arctic DEM at 5 m/pixel resolution. In most of theprofiles the agreement is excellent and well within the GPS precision margin. However, pro-files j201, j211, j213, j214, j263 and j264 show substantial deviations. Profiles j201, j202, j203,and j253 are also substantially noisier than the rest of Arctic DEM derived data. We attributethe discrepancy, in particular the difference in concavity in profiles j263 and j264 with our data(and potentially also in j265, both in our data and the DEM), to the presence of snow coveringthe rivers and subglacial channels during the acquisition of the photogrammetry data used toderive the Arctic DEM. We did not observe in the field any of the spikes present in the ArcticDEM profiles j201, j202, j253, and j254, and therefore we argue that they are DEM artifacts.However, this figure also proves that channel profile analyses based on Arctic DEM are reli-able within the DEM limitations, and therefore subglacial channels can be identified and theirundulations quantified using remote sensing high resolution topography.3.4.2 LiDAR observationsKinematic LiDAR Scanning (KLS) was acquired in 5 subglacial channels and one river. TheLiDAR dataset provides very high resolution topography data, which adds robustness to GPS-based undulation observations. Furthermore, KLS highlights a difference in cross-sectionalshape, scale, and downstream evolution that has not yet been considered as a distinctive char-acteristic of subglacial erosion, and is not appreciable from GPS profile data.Figure 3.4 shows the results of using the kinematic backpack LiDAR approach to imagingthe topography of a channel. The first panel 3.4(a) shows the point cloud files produced wheninvestigating the channel cross-section. The data are colored by back scattered intensity at alaser wavelength of 1550 nm used in the KLS LiDAR system, resulting in darker values for wetsnow and ice as seen in the image. Panel 3.4(c) shows the raster derived from the point cloudfiles for a river valley (corresponding to j231), and 3.4(d) shows the raster for the subglacialchannels analyzed in network 1, at a resolution of 9 cm/pixel. Point spacing in 3.4(a) and 3.4(b)corresponds to 6 cm, with a total point count of 117,147,558 points for the river valley and forthe subglacial channels. Raster resolution corresponds to 9 cm in the subglacial channels innetwork 1, and 10 cm in the river valley corresponding to j231.Panels (c) and (d) offer a clear comparison of cross-sectional scale, shape, and evolution inthe case of river valleys (c) and subglacial channels (d). In (d), subglacial meltwater 1st ordertributaries have widths of 5-7 m, and maintain a remarkably constant cross-sectional scale asthe channels evolve downstream. The cross-sectional shape is flat bottomed with steep-sidedwalls at the angle of repose (∼ 15− 20◦) as shown in panel (b) and described in more detailin next section. In comparison, the river valley cross-section starts narrower but increasessignificantly towards the end of the channel, as shown in panel (c). Other features such asthe absence of internal channels inside the subglacial channel flat bottoms are also evident inLiDAR observations.623.4. Results: Quantitative characterization of river and subglacial channelsFigure 3.4: KLS LiDAR observations. Panels (a) and (b) show the color coded point cloudfiles (dark is low return), see the scale for spatial reference. Panels (c) and (d) showthe raster produced using the point clouds.3.4.3 Photogrammetry observationsDSM rasters produced with the technique described in section 3.3.3 allow for a detailed topo-graphic study at higher resolution than the CDEM (30 m/pixel) or the Arctic DEM (2 m/pixel),but lower than the LiDAR observations. The advantage of this dataset over LiDAR and GPSobservations is the mobility of the aircraft, which allowed for topographic data acquisitionin different parts of the island. Stereo imagery and photogrammetry results include DigitalSurface Maps (DSM), point clouds, and textured orthomosaics for an additional 10 subglacialchannel networks. These datasets complement the LiDAR observations in different parts ofthe island at lower resolution, and are available at variable resolutions upon request as pointclouds and Geotiff rasters. Figure 3.5 shows two DSM models and textured orthoimages oftwo different subglacial channel networks.Surface models acquired through photogrammetry enable the differentiation of three regimesin a channel network (Figure 3.5 (a)). In the first regime (zone (1)), subglacial tributariesoriginate as smooth depressions in the plateaus, merging into the topography and withoutclearly distinguishable heads. During the second zone (2), channels evolve into well devel-oped systems ∼ 15 m across and ∼ 4 m deep in this network, keeping the width remarkablyconstant as they deepen downstream (see orthoimage below for better reference). Finally, inthe last zone (3), tributaries merge into a deeper channel where fluvial incision by seasonal633.4. Results: Quantitative characterization of river and subglacial channelsFigure 3.5: Stereo-photogrammetry derived from helicopter borne photography. The toppanels (a) and (b) show the digital elevation model (DEM) at a resolution of 0.48 and0.56 m/pixel respectively, with the colorbar indicating the elevation of the modelsurfaces. The images underlying the panels correspond to the textured orthoimagesin both locations.meltwater streams is apparent (notice the deepening in the DEM). In Figure 3.5(b), DSM andorthoimage highlight the evolution of the tributary channels mostly by deepening as opposedto cross-section widening (see particularly the scale of tributaries and main stem in the or-thoimage). This high resolution local DEM highlights the size of the first order tributaries,which are 10− 20 m from the origin with no small scale channels or tributaries visible, theirquasi-periodic spacing, and the smooth merging with local topography at the origin, whichis an example of use of the topography data produced. Snow and ice accumulations were acommon view in some channels, particularly closer to the Devon ice cap. This was an issue atprocessing the DSM and textured image, although in some channels the thickness of the snowpack could be estimated.643.5. Identification of subglacial channel networks3.5 Identification of subglacial channel networks3.5.1 Morphometric comparison of river and subglacial channelsMorphometric differences between rivers and subglacial meltwater channels are apparenteven before analyzing the longitudinal profiles in Figure 3.3, only from the correlation to for-mer ice margins and direction consistent with estimated ice flow lines of subglacial channels(Dyke, 1999). On remote sensing data of the field site and surrounding area, inferred subglacialmeltwater channels appear in groups of ∼ 10, parallel to each other consistently in the N-NWto S-SE direction. Moving to the east of the island, channel directions change on average fromW to E, remaining oriented radially towards the current day Devon Island ice cap.Characteristic inferred subglacial channel lengths are ∼ 1− 2 km throughout the distinctchannel networks. The typical cross-section is trapezoidal, with widths of ∼ 40 − 60 m atthe initiation stages (∼ 150 m downstream) that contain flat floors ∼ 20 m wide, and depthsof under 5 m. Downstream (> 1.3 km), cross-sections evolve to a better defined trapezoidalshape and deeper channels (> 10 m), preserving roughly the same width (Figure 3.6).In comparison, the geometry of inferred river valleys on the island displays major differ-ences. River widths vary continuously downstream by one to two orders of magnitude fromthe origin (∼ 5− 10 cm) until the first junction (60 m across,∼ 5 m deep)∼ 150 m downstreamfrom the headwaters (i.e., Figure 3.4(c) and (d); Figure 3.6 initiation stages). Downstream, rivervalleys deepen up to ∼ 60 m and grow in width up to 400 m, forming deeply incised canyonswith V-shaped cross-sections (see Figure 3.6, developed stage and Figure 3.4). The evolutionof bankfull channel width we observe is consistent with observations elsewhere (e.g., Parker,1978a,b), and with the well-established hydraulic relationship for flow in river channels (e.g.,Leopold and Maddock, 1953; Parker et al., 2007; Gleason, 2015), relating the channel bankfull widthto the discharge:W = KwQb (3.2)Where for a gravel bed such as the ones considered, Parker et al. (2007) showed that Kw =4.63g−7/10D−5/250 and b = 0.467, with D50 the medium value of the grain size distribution, anddischarge that increases as tributaries merge into the main channel.The morphometrical characterization of the cross-sectional differences between rivers andsubglacial channels can be captured with a shape factor:F = WT/D, (3.3)defined as the ratio between channel top width and the depth (Leopold, 1970; Williams andPhillips, 2001) (see Figure 3.10 for a reference cartoon). We measured channel top width fol-lowing Grau Galofre and Jellinek (2017) as the distance between two points of maximum curva-ture along a cross-sectional line for consistency, which corresponds to valley width for riversand the width of the entire cross-section in subglacial channels. We present shape factor re-653.5. Identification of subglacial channel networksFigure 3.6: cross-sectional evolution of a fluvial (upper row) and subglacial (bottom row)channel, with satellite imagery for context on the right column. In the subglacialcase, the initial width and the shape remain largely unchanged over length, whereasthe river cross-section grows monotonically both in width and depth with distance.Notice the differences in depth and length in the section scale bars.sults in table 3.2, column 6, where all F measurements correspond to cross-sections beforetributary junctions. These results highlight the fundamental differences between fluvial andsubglacial cross-sections: whereas subglacial shape factors are in the range 4.5-31, with an av-erage 〈FSG〉 ± σ = 13.5± 9, fluvial shape factors are much smaller, in the range 2-4.6 with anaverage 〈FSG〉 ± σ = 3.4± 1. Furthermore, according with our observations in figure 3.6, weexpect the variation of this shape factor to be considerable for subglacial channels as the topwidth remains constant and the channel deepens, and less important for river valleys, as bothcross-section and width increase downstream.Another geometrical distinction between both erosional regimes is the width of first ordertributaries (Grau Galofre and Jellinek, 2017). Even at the tip of the channel, subglacial channelwidths are up to tens of metres (consistent with arguments in Weertman (1972)), as opposedto widths of first order river channels which are typically sub-meter in scale (Grau Galofre andJellinek, 2017).3.5.2 Comparison of lateral and subglacial meltwater channelsAlong with the distinction from river systems, it is relevant to distinguish subglacial channelsfrom channels formed by meltwater accumulated and released at the ice margins, i.e., lateralmeltwater channels (i.e., Greenwood et al. (2007); Syverson and Mickelson (2009); Margold et al.(2013)), which have also been identified in the area (Dyke, 1999). We follow the criteria pre-sented in (Greenwood et al., 2007, Table 1), to this end. Focusing now only on the meltwaterchannel networks and ignoring rivers, the systems we investigate display a number of char-663.5. Identification of subglacial channel networksacteristics that exclude lateral meltwater drainage: (1) they do not follow contour lines, butrather run parallel or slightly oblique to topographic gradients (Figure 3.7) (e.g., Price, 1960;Greenwood et al., 2007); (2) their longitudinal profiles often contain stepped sections (e.g., Sug-den et al., 1991; Greenwood et al., 2007) (Figure 3.3), and may or may not display significantundulations (Figure 3.3); (3) they display anastomosing patterns, with channel sections thatsplit in two to join again further downstream (anabranching) (e.g., Sugden et al., 1991); and (4)potholes and shallow depressions are a common sight (e.g., Sugden et al., 1991; Greenwood et al.,2007).Figure 3.7: Hillshade and contour map of the four subglacial channel networks investi-gated. Contour lines are separated 15 m, and hillshade resolution is 2 m/pixel. Inthe bottom right corner, the black arrows indicate the overall direction of the chan-nels in the networks, whereas red indicates the regional slope direction. Red arrowsindicate anastomosing sections.Figure 3.7 shows the relationship between the subglacial channel networks direction inves-tigated here and the topographic contour lines. Comparing both directions, and taking intoaccount that the channel networks we study are perpendicular to, and feed into, large canyons(see networks 1, 2, and 4) instead of forming terraces at their rims, we conclude that a lateralmeltwater origin is unlikely. We discuss more details regarding the morphology and charac-teristics of these networks in the next subsection, which also suggest the emplacement of these673.6. Detailed morphology of subglacial channels in Devon IslandTable 3.3: Summary of morphological characteristicsCharacteristic SG 1 SG 2 - D SG - P SG 4- CAMFtributary n◦ 10 17 10 5regional slope (%) 1.8 2.0 1.3 5.5plungepools? yes yes yes yesanabranching sections? yes yes yes nohanging valleys? no yes yes nonetwork length (km) 1.5 2.1 2.5 0.9network width (km) 1.6 1.3 1.5 1stepped profiles? yes, j201 yes, j252 no nonetwork shape dendritic dendritic dendritic parallelpresence of other subglacial bed-forms no no no nofeatures in subglacial conditions.3.6 Detailed morphology of subglacial channels in Devon IslandBased on the field and remote sensing observations presented above, we build a detailed de-scription of the morphology of the four networks of subglacial meltwater channels we visitedwhile in Devon, as specified in Figure 3.1 and Figure Network characteristicsThe overall channel systems range from 2.5 to 0.9 km long and 1.6 to 1 km wide. They are allincised into dolomite bedrock and coarse gravel. Regional slopes in the plateaus where thenetworks are incised are very small (see table 3.3), and the number of tributaries varies fromfive in network 4 to seventeen in network 2, consistent with networks elsewhere in the island.Table 3.3 contains a summary of morphological field observations of subglacial channels.The general pattern of the four subglacial channel networks studied is dendritic (in thatchannels merge to produce larger channels) with a main channel that wraps around the ex-terior of the network and tributaries that flow parallel to it, merging at acute angles (see net-works 1 and 3 in Figure 3.7), which gives the system of channels a finger-like appearance. Ina few cases in networks 1, 2, and 3, tributaries bifurcate to give the network an anastomosingpattern (see red arrows in Figure 3.7). All subglacial channel networks observed terminate indeeply incised canyons that predate glaciation through hanging valleys or very steep chutes.We observed at different sites on the island how the melting of snow accumulations withinsubglacial channels leads to the formation of meltwater ephemeral streams, which merge at thechannel junctions into larger streams. This transition is often associated with a gradual changein cross-section, from the shallow flat-bottomed form characteristic of subglacial channels to adeeply incised V-shape. Figure 3.5 exemplifies this morphological transition from a networkof subglacial channels to a single meltwater fed river.683.6. Detailed morphology of subglacial channels in Devon Island3.6.2 Channel characteristicsMain channelMain stems are 1.5 − 2 km in length and follow a NE-SW direction for about 1 km beforebending around tributaries. The profiles of these channels are stepped, with steps consistingon 2-3 segments about 300− 500 m long separated by sections of steeper gradient (Figure 3.3,see channels j201, j252, j254). In some occasions, there is a short section of reverse gradientfollowing these steps. Main stem cross-sections are generally trapezoidal, with flat bottomedfloors and steep sided walls ( ∼ 20◦ degrees). Variations in width from the channel originuntil the junction are small, accounting for no more than a few meters of change in any of thenetworks (c.f., Figure 3.4 and Figure 3.5).TributariesAll tributaries in each network typically formed at the same elevation and incised the substrateparallel to, or oblique to, the topographic gradients to meet the main channel (Figure 3.7,table 3.2 column 5). Typically, the subglacial channel networks we observe consist of 5 to17 tributaries oriented in the NE-SW direction. Within the same network tributary depth canvary between< 1− 10 m. This differential incision is particularly evident in networks 1 and 2,although this property is noticeable in all networks within the resolution of this study (Figure3.8).In general, tributaries display the same trapezoidal flat bottomed, steep sided shapes thatare characteristic of the main channels, although cross-sectional asymmetries appear here withmore frequency than in the main stems. In particular, tributaries in network 3 are incised moredeeply in the eastern side than the western side, allowing for shallow depressions to formalong the steeper side (Figure 3.8, left panel). Developed tributaries (at a distance of ∼ 1kmfrom their origin) are around ∼ 10 m in depth, with a flat floor ∼ 10 m across and steep sidedwalls up into the plateaus, as shown in the examples of Figure 3.6.Longitudinal profiles of tributaries are complex and vary across the different networks(Figure 3.3). Tributaries in networks 1 and 4 grade into the main channel continuously, whereasin networks 2 and 3 some confluences present hanging valleys followed by shallow potholesin the junctions between smaller and larger tributaries (see an example in Figure 3.9 (c)). Pro-file curvature is variable even within the same network: channels j202, j232, j261, j262, j265display concave profiles, whereas channels j231, j234, j251, j263, and j264 display convex pro-files. Shallow potholes (∼ 1 m deep) filled with water or snow are a common view across allnetworks, as detailed later.693.6. Detailed morphology of subglacial channels in Devon IslandFigure 3.8: cross-section field imagery and profiles. Upper row shows a satellite imagerycontext on the location where the image and the cross-section are obtained, togetherwith a scale reference. Middle row shows images of four cross-sections, obtainedby this expedition on July 24th and 25th, 2017. The middle panel corresponds to amain channel whereas the other three images correspond to tributaries. cross-sectionprofiles below show elevation (m) vs. distance (m) obtained from the Arctic DEM at2 m/pixel.3.6.3 Other characteristicsAnastomosing patternsAlthough the shape of the networks is mostly dendritic (channels merge to produce largerchannels), anabranching (bifurcation followed by re-junction downstream) patterns occur fre-quently at the beginning of the networks, typically before 1 km. Examples of this anabranchingbehavior are shown in networks 1, 3, and 4, where channels split to rejoin anywhere between5− 250 m downstream. Figure 3.7 shows a hillshade map of the subglacial networks, wherethe anastomosing patterns are easily identified (see red arrows). Also of interest here, the leftpanel in Figure 3.8 shows a high resolution cross-section taken across an anabranching sec-tion in network 1. The section clearly shows how one of the channels (in this case the easternchannel) is more deeply incised than the western one, which may suggest a time-transgressiveemplacement of the system (e.g., Beaney and Shaw, 2000; Brennand, 1994).PotholesPotholes appear frequently in the subglacial channel networks explored in the field (Figure3.9), and they are also evident from the photogrammetry and LiDAR DEMs we produced,falling at resolution edge of the Arctic DEM. They are shallow depressions (50 cm to 2 mdeep) typically filled by water that grade into the channel floors, and that vary in dimensions703.7. Discussionbetween ∼ 5-50 m long by ∼ 2-30 m across, with a particular example in network 1 wheredimensions are up to ∼ 125 m and ∼ 40 m across (Figure 3.9 panel (b)). We observed them tooccur (1) at the junction of two tributaries, (2) in the middle of a tributary channel associatedwith a channel widening, or (3) at the channel headwaters area. Typically, the larger sizesappear in case (1), whereas the smaller depressions occur in (2). Figure 3.9 shows field imagesof potholes of several sizes, fitting into type (1) (panel c and b), class (2) (panel d), class (3)(panel a). The location of these features is indicated in the aerial images at the side with acamera icon.Figure 3.9: Field images of the shallow depressions and potholes observed. Satellite im-agery provide context for the photographies through the camera icons. In photos (a)and (b), notice the human figures for scale. Photos (c) and (d) contain a scale bar forreference. Image (c) is an example of an overhanging valley (here covered in snow)followed by a pothole.3.7 Discussion3.7.1 Undulations, obliquity, shape factor, and the remote sensingcharacterization of subglacial channelsIn section 3.4.1 we introduce the magnitude of undulation ψ and discuss its role in identify-ing subglacial channels. This is a useful metric, but it requires the acquisition of very highresolution topographic data. At lower resolution, in addition, differences among channel di-rection and local topographic gradients can also be indicative of subglacial erosion, as long asice erosion rate by sliding is lower than meltwater erosion rate (e.g., Weertman, 1972; Shreve,1972; Paterson, 1994). Observations of channels incised obliquely to topographic gradients arecommon in the literature (e.g., Sissons, 1961; Walder and Hallet, 1979; Sugden et al., 1991; Living-stone et al., 2017). Quantifying and measuring these deviations in a set of subglacial channelsto establish a quantitative base for channel categorization has, however, not been done.Considering the confined flow of pressurized water in a subglacial channel at the base of713.7. Discussionan ice sheet to follow the x direction, with y perpendicular to ice flow and z perpendicularto the ground surface, so that zb and zi are the bed and ice surface elevations respectively. Atsteady-state, water flow at the base of the ice is driven by the water pressure potential gradient∇φ:∇φ = −ρig∇zi − ∆ρg∇zb +∇N. (3.4)Here ρi is the ice density, ∆ρ = ρw − ρi is the density difference between water and ice, gis the gravity, and N = pi − pw is the effective pressure, where pi = ρig(zi − zb) is the localhydrostatic pressure related to ice thickness and pw is the water pressure. The topography ofDevon Island’s plateaus is mostly flat, which implies that the controls on pressure potentialgradient arise mostly from variations in ice surface slopes, ∇zi, and not from surface topo-graphic gradients ∇zb. This picture is true generally if ice surface slope is more importantthan bed topography, such that ρig∇zi/∆ρg∇zb  1. Although ice surface slope is correlatedwith topography at a regional scale it can depart from topography at the scale of individualchannels (Figure 3.10), driving both channelized and distributed meltwater accordingly. Thisexplains the slight deviations we observe between subglacial channel direction and local to-pographic angles in our field site, recorded in table 3.2.Figure 3.10: Cartoon representing the definitions of the three remote sensing based met-rics proposed in this study. (1) Shows our definition for longitudinal profile un-dulations Ψ, where the grey line represents the longitudinal profile of a channel(elevation vs. distance). (2) Represents the deviation between the direction of aset of channel networks (red arrow) and the topographic gradient (black arrow),together with the axis notation and the ice and topographic surfaces zi and zb inequation 3.4. (3) Shows the definition of shape factor with two cartoons represent-ing a trapezoidal and a V-shaped cross-section, where top width WT and depth Dare represented (adapted from Williams and Phillips (2001)).However, where ice topography is nearly flat or bed slopes are important, bed topography723.8. Conclusionsdominates incision ρig∇zi/∆ρg∇zb  1, and drives meltwater flow. Under these conditions,undulations or departures from topographic gradients cannot occur. In this case, neither met-ric will identify channels as subglacial, and their characterization will depend on other obser-vations, such as cross-sectional characteristics or morphology. Morphological criteria includethe presence of anabranching patterns, consistent direction with former ice flow lines, and cor-relation with other subglacial features (i.e., eskers, moraines, out-wash fans) (Greenwood et al.,2007; Kehew et al., 2012, e.g.,).3.7.2 Identification of subglacial channels from remote sensing dataWe distinguish subglacial channels in our field area on the basis of four properties, whichare measurable at the Arctic DEM resolution: (1) consistent N-NW to S-SE direction, radialto the paleo-ice margins (Dyke, 1999, Figure 8) near our field area, changing to W to E nearthe ice cap margin; (2) Topographic undulations in the longitudinal profiles (Figure 3.3 andTable 3.2), and channel incision with orientations not parallel to the local topographic gradient(Figure 3.7); (3) Cross-section size and shape, i.e., shallow trapezoidal for subglacial channelsand deeply incised V shape for river valleys (Figure 3.6 and Figure 3.8, quantified in table 3.2column 6 in the shape factor), which evolve in distinguishable ways (Figure 3.6; and (4) large1st order channel widths on the order of ∼ 10 m (Figure 3.4). Not all channels we identify assubglacial from their morphology and direction have undulations in their longitudinal pro-files. However, none of the river profiles show any detectable undulations. We conclude thatthe magnitude of the undulation index ψ > 0 unequivocally distinguishes subglacial erosion,but ψ = 0 does not necessarily preclude it. Similarly, channel orientation deviations from localtopographic gradients for subglacial channels are much larger (> 5◦) than for rivers.We add three remote sensing indicators of subglacial erosion to the criteria in Greenwoodet al. (2007). The first indicator is the large cross-section widths at the origin of order 1 chan-nels (i.e., Figure 3.4) which are orders of magnitude larger than in river systems. The secondcriterion is the minimal variation in width downstream, from the beginning of the channeluntil the first junction, as visible from Figure 3.4, comparison of panels (c) and (d), and Figure3.6. The third is the remarkable difference between shape factors (top width to depth ratios)between river valleys and subglacial channels.3.8 ConclusionsIn this study we describe a population of subglacial channels (N-channels) exposed on De-von Island, Canadian Arctic Archipelago. In particular, we implement, and discuss the use of,remote sensing techniques to distinguish between systems of rivers, lateral meltwater chan-nels, and subglacial channels, which serves as a complement to existing field-based methods.We provide detailed field descriptions of 20 individual channels, including their longitudinalprofile characteristics, cross-sectional geometry, channel directionality and drainage networkmorphology, to then revisit and expand the identification methods listed in Greenwood et al.733.8. Conclusions(2007). Our field observations include GPS mapping of subglacial and fluvial incised chan-nels longitudinal profiles, photogrammetry, kinematic LiDAR data (KLS), and aerial imagery,allowing for both a qualitative and quantitative description.Subglacial channels appear in clusters of 10-17 parallel individual systems that roughlyfollow the topographic gradients (Figure 3.7). Tributaries and main stems are wide, ∼ 40− 60m at the initiation stages, with defined trapezoidal shapes that are relatively shallow (5− 10 m)and preserve roughly the same width downstream. Other patterns characteristic of subglacialdrainage, such as anastomosing networks, potholes, and hanging valleys at the junctions, arealso common in the channels investigated.We find that a quantitative measure of the undulation index ψ, defined as the topographicloss (local minima to local maxima of the undulation) at an upstream section to the total topo-graphic loss according to equation 3.1, reliably distinguishes fluvial and subglacial longitudi-nal profile (Figure 3.4), although the lack of undulations does not rule out subglacial erosion.We also argue that the departure in channel direction from local topographic gradients alsoreflects subglacial erosion (table 3.2), as well as a large top width-to-depth ratio (shape factorF, compiled in table 3.2, defined in equation 3.3.) We then discuss the limitations of both thesemetrics in identifying subglacial channels. If both metrics fail, other morphological observa-tions such as channel direction, anastomosing networks, cross-sectional scale and downstreamevolution, serve as a categorization guide.With our observations, we present the first high resolution study of the subglacial drainagechannels of the high Arctic, as well as revisit well-established field identification guidelines(e.g., Greenwood et al., 2007). We conclude with the following target characteristics of interest:(1) undulations in the longitudinal profile and changes in channel direction with respect tolocal topographic gradients, (2) consistent channel direction following former ice flow lines,close to the ice margin, and with the possible presence of other subglacial features in the area,(3) order 1 channel widths on the 5-10 m scale with minimal variation downstream, with wideand trapezoidal cross-sections, and (4) presence of anabranching sections.74Chapter 4An Antarctic-style ice sheet on ancientMars4.1 SummaryThe widespread presence of ancient valley networks is evidence that water flowed on, andsculpted, the surface of Mars early in the planet’s history (Carr and Clow, 1981; Baker et al., 1992;Gulick, 2001; Hynek et al., 2010). Characterizing the mechanisms and climatic environments ofvalley incision is vital to guide the search for ancient life and understand the early hydrolog-ical cycle of Mars. The origin of these valley networks is, however, contentious (Gulick, 2001;Lamb et al., 2006; Wordsworth et al., 2015). Current interpretations of the geological record re-quire precipitation and surface water runoff (Craddock and Howard, 2002; Howard et al., 2005;Hynek et al., 2010; Ramirez and Craddock, 2018), and are in profound contradiction with re-sults of state-of-the-art climate models predicting a cold, icy Mars under a fainter young Sun(Wordsworth et al., 2013; Fastook and Head, 2015; Wordsworth, 2016). Here we provide qualitativeand quantitative support for a new hypothesis for valley network formation that reconciles,for the first time, geomorphological observations with climate model predictions. We use aPrincipal Component Analysis constructed on the basis of 6 landscape metrics that capturedistinctive morphometrical properties of 66 valley networks, together with physical modelsof fluvial, glacial, sapping, and subglacial erosion, to establish the first rigorous global clas-sification scheme for Mars’ valley networks.(Grau Galofre and Jellinek, 2017; Grau Galofre et al.,2018). Our results show how subglacial incision was a widespread formation mechanism forvalley networks, with minor contributions from fluvial and sapping erosion. Our conclusionspresent early Mars as a frozen world akin to present day Antarctica. Beneath the ice, a dy-namic hydrological cycle transported water from areas undergoing basal ice melting, throughnetworks of channels terminating in subglacial lakes or ice marginal rivers. Our observationschallenge the current view that valley network incision required surface runoff (Craddock andHoward, 2002; Hynek et al., 2010; Fastook and Head, 2015). Furthermore, our work presents the754.2. Introductionfirst robust observational support for state-of-the-art climate models (Wordsworth et al., 2015;Wordsworth, 2016) and provides a new framework for identifying astrobiologically interestingenvironments.4.2 IntroductionAs described extensively in the Introduction 1, the Martian valley networks are ancient (3.8-3.5 Ga) (Gulick, 2001; Hynek et al., 2010) drainage systems mostly located in the southern hemi-spheric highlands. On the basis of similarities with terrestrial valleys (analogs), many au-thors (Carr and Clow, 1981; Laity and Malin, 1985; Kargel and Strom, 1992; Lee, 1997; Craddockand Howard, 2002; Howard et al., 2005; Hynek et al., 2010) have interpreted their morphology interms of contributions from fluvial and fluvio-glacial (Carr and Clow, 1981; Baker et al., 1992;Kargel and Strom, 1992; Craddock and Howard, 2002; Hynek et al., 2010), glacial (Kargel and Strom,1992; Head et al., 2005; Pelletier et al., 2010), and sapping (Sharp and Malin, 1975; Laity and Malin,1985; Gulick, 2001; Lamb et al., 2006) erosion, with most of these processes requiring the pro-longed action of surface liquid water. Protracted liquid water stability, however, profoundlycontradicts results from state-of-the-art 3D climate models. Under a fainter young Sun anda thicker atmosphere, climate models (Wordsworth et al., 2013; Forget et al., 2013) predict thebuild-up of thick ice sheets in the highlands (Fastook et al., 2012; Fastook and Head, 2015), di-rectly over the valley network basin areas (Wordsworth et al., 2015). To date, no hypothesis hassucceeded in reconciling predictions from climate models and morphological observations,failing to provide a unified theory for early Mars climate and surface evolution.4.3 MethodsIn this study we provide quantitative and qualitative evidence for subglacial erosion as thepredominant mechanism for valley network incision. First, moving away from classic qualita-tive analog-based interpretations, we use a combined statistical and physical analysis of val-ley network morphometry (Grau Galofre and Jellinek, 2017). We then compliment this analysiswith observations of valley network morphology as compared to known subglacially erodeddrainage systems on Earth (Sissons, 1961; Sugden et al., 1991; Greenwood et al., 2007; Kehew et al.,2012; Grau Galofre et al., 2018). Our results show that a large fraction of valley networks areeroded remnants of drainage conduits of ancient ice sheets, with other mechanisms such assapping or fluvial incision being comparatively less common.In detail, we analyze a set of 66 valley networks using 6 parameters targeted to capturedistinctive aspects of fluvial, glacial, sapping, and subglacial erosion, in turn. Valley networksincluded in the dataset (see supplementary table “ch3-VN-dataset.xls”) span distinct time peri-ods of incision, provide a spatially meaningful coverage, and reflect the variety of morpholo-gies on Mars (Gulick, 2001; Hynek et al., 2010). The morphometrical analysis of each valleynetwork includes: length to width aspect ratio (R), magnitude of longitudinal profile undula-764.4. Resultstions (Ω), width of first order tributaries (λ), junction angle between a tributary and the mainstem (γ), maximum network stream order (Sn) and fractal dimension (D f ). In particular, thewidth of order 1 tributaries and aspect ratio capture distinct aspects of the momentum balancewithin channelized flows, steady state tributary junction angle distributions reflect whetherthe flow is driven by topographic gradients alone, and stream order and fractal dimensioncapture the network branching complexities (Grau Galofre and Jellinek, 2017). The presence ofundulations in the longitudinal profile is a direct consequence of subglacial erosion, as pres-sure gradients in conduits beneath ice sheets can counteract gravitational pull and drive waterupstream (Greenwood et al., 2007; Kehew et al., 2012; Grau Galofre et al., 2018) (see Appendices Aand B).We first explore the statistical variability within valley network morphometry with a Prin-cipal Component Analysis (PCA, see extended methods at the end of the Chapter) algorithm inFigure 4.1. Valley networks cluster according to statistically common morphometries, whereasthe relative contributions of each metric (arrows) provide a measure of the major controls onmorphological variability. To understand the erosional mechanics giving rise to each datacluster, we use physical models (Grau Galofre and Jellinek, 2017) with appropriate parameteri-zations for Mars (Methods). In detail, theoretical predictions for the six metrics under fluvial,glacial, sapping, or subglacial erosion define synthetic valley networks (SVN), so that 1 SVN =(γ, D f , Sn,λ, R, ). A collection of SVN predictions from the same erosional mechanism definea region in the PCA (shaded) that can be directly compared with data clusters (points). Valleynetworks overlapping each of the shaded areas, therefore, have morphometrical propertiesconsistent with the steady-state form for that erosional mechanism (Grau Galofre and Jellinek,2017). To be significant statistically, these shaded regions must also be distinct from the dataaverage variance in the PCA space (magenta). Assuming the Martian landscape is in steady-state, this area represents the noise spectrum. Physically, the magenta region may also reflectcontributions from additional unidentified surface process or a local departure of the Marssurface from landscape equilibrium. I discuss this issue extensively in Chapter 5.4.4 ResultsValley networks cluster according to statistically common morphometries in the PCA spaceof Figure 4.1. Taken together, the relative contributions of each metric provide a quantitativemeasure of the major controls on valley network morphological variability.For example, fluvial valley networks overlap the red shaded area: they distribute along atrend of constant minimum width λ ∼ 100− 300 m (Figure 4.1a), display negligible undula-tions Ω (Figure 4.1b), and are characterized by the highest stream orders Sn and aspect ratiosR in the dataset (Figure 4.1c, also Figure C.2a in Appendix C). In marked contrast, valley net-works in the blue shaded area have characteristics predicted for glacial erosion: The largestminimum widths (λ = 3− 5 km) and junction angles γ ∼ 90◦ (Figure C.2b in Appendix C).Valley networks overlapping the green shaded area agree with predictions for erosion by sap-774.4. ResultsFigure 4.1: PCA analysis, 1st and 2nd PC (a) and 1st and 3rd PC (b). Shaded regionscorrespond to Montecarlo model predictions: fluvial (red), glacial (blue), subglacial(yellow), sapping (green), magenta (undifferentiated). The 6 metrics appear witharrows scaled to the variance they capture. (c) and (d) show representative examplesof the fluvial (c) and subglacial (d) end-members (Appendix C, Figure C.2).ping: Networks have small stream orders (Sn = 1− 2), low aspect ratios (R < 400), and littlevariance in junction angles γ ∼ 60 (Figure C.2c in Appendix C).Figure 4.1 consequently presents quantitative evidence for the presence of extensive sub-glacial erosion of Mars. Formally, river and subglacial systems appear morphologically simi-lar (see panel (a)), as they share characteristics such as moderate to high stream orders or highlength-to-width ratios (Grau Galofre and Jellinek, 2017; Grau Galofre et al., 2018). Crucially, how-ever, surface runoff flows follow topographic gradients strictly and cannot erode uphill (Figure4.1c). In contrast, channelized pressurized subglacial meltwater follows cryostatic pressuregradients and can flow parallel or oblique to topographic gradients, producing profile undu-lations and deviations between channel and topographic gradient direction (Figure 4.1d, alsoFigure C.2d in Appendix C) (Sugden et al., 1991; Greenwood et al., 2007; Grau Galofre et al., 2018)(see also extended methods). The undulations recorded in valley networks that remain unex-784.5. Geological and climate implicationsplained by tectonic processes, crater ejecta/ infill, or eolian dune infill (Figure 4.1c, shaded),thus require subglacial erosion.Table 4.1 presents further geomorphological evidence for valley network formation throughsubglacial incision, and shows how common morphologies attributed to subglacial erosion ex-plain several puzzling valley network characteristics: bifurcating tributaries, absence of innerchannels, absence of meter or sub-meter scale tributaries, low number of deltas and depo-sitional fans, large cross-sectional scales, undissected terrain surrounding networks, and flatbottomed, steep sided cross sections. The table is divided in quantitative evidence (undula-tions and departures from local topographic gradients) and qualitative evidence, and presentsexamples on Earth and Mars. See complete table with references in the supplementary table“ch3-SG-evidence.xls”.Table 4.1: Morphological evidence for subglacial erosionQualitative observations Observed on Earth in Observed on Mars in (i.e.,)Rare inner channels high latitudes UbiquitouslyChannel constant width Devon Island Nanedi, VNs in 22.2S, 344.9WNo m-scale tributaries Devon Island UbiquitouslyAnabranching patterns high latitudes Loire, Evros, VNs in 2S, 243WRare alluvial fans NY, Devon Island UbiquitouslyAbrupt beginning/end NY, Svalbard Nirgal Valles, VNs in (3S, 249W)Inter-valley undissected area Devon Island UbiquitouslyDrainage into main stem Devon Island VNs in (22.2S, 344.9W)Non-uniform distribution Devon Island Parana Valles regionLarge x-section scale variability high lat. (N) Ubiquitously (100 m to 10 km)Stepped long. profiles high latitudes Huygens crater tributariesTrapezoidal cross sections Devon Island Ubiquitously (40S to 40N)Quantitative observationsLarge width to depth ratio Devon Island Ubiquitously (40S to 40N)Long. profile undulations high latitude Ubiquitously, this studyOblique to topo gradients high latitude Work in progress4.5 Geological and climate implicationsOverlying the results of Figure 4.1 on a geological map of Mars (Figure 4.2) motivates a dis-cussion about the distribution and relative timing of each erosional process, and constrainsthe environmental conditions required for their operation. We identify fluvial and subglacialincision within the ancient Noachian Highlands. Whereas fluvial valley networks are mostlylocated along a corridor that connects Noachis Terra with Meridiani Planum (see Figure E.2for the geographical quadrants of Mars), subglacial valley networks are ubiquitous across thehighlands and concentrated at the dichotomy boundary (Figure 4.2).The small number of identified sapping systems are generally associated with youngerEarly Hesperian terrains and appear along the rim of Valles Marineris, the northern rim of794.5. Geological and climate implicationsFigure 4.2: Global geological map of Mars showing the main time periods and subdivi-sions, overlaid with the location and characterization results for the 66 valley net-works considered in this study. Red dots represent fluvial systems, black push-pinsrepresent subglacial channels, blue asteriscs represent glaciers and green diamondscorrespond to sapping-like valleys. The triangles in magenta are not resolvable us-ing this technique.Terra Sirenum, and the NE boundary of Xanthe Terra. Significantly, erosion by wet-basedglaciation is similarly rare on Mars, with glaciers identified at the rim of Argyre, the edge ofKasei Valles, and Terra Cimmeria. This observation seems to contradict a subglacial origin forthe valley networks. However, work in progress hints at how subglacial drainage by cavita-tion leading to glacial sliding and by channelization leading to conduits may in fact excludeeach other. Given the overwhelming lack of evidence for wet-based glaciation, we agree withprevious studies (Fastook and Head, 2015) in proposing a mainly cold-based ice sheet with lo-calized seasonal basal melting.Subglacial erosion as potentially the most important mechanism for valley network inci-sion is consistent with both climate models and geomorphologic observations for early Mars.Critically, subglacial erosion requires the presence of extensive ice sheets on the NoachianHighlands with marginal basal meltwater generation, in agreement with climate model pre-dictions (Wordsworth et al., 2015; Wordsworth, 2016). Comparing the spatial distribution of sub-glacial networks and areas of basal melting as predicted for ice sheets in the icy Mars scenario,(Wordsworth et al., 2015; Fastook and Head, 2015), we find a spatial correlation for model pa-rameterizations with a 1 km elevation ice stability line and a water Global Equivalent Layer(GEL) of 10–15 times the current water inventory of Mars (1X ∼ 34m GEL,∼ 5x106 km3). Be-cause we identify subglacial erosion on the basis of profile undulations, which do not occurin all subglacial drainage systems (Greenwood et al., 2007; Grau Galofre et al., 2018), our results804.6. Extended methodsare a conservative lower bound on the contribution of subglacial erosion to the incision of theMartian valley networks. Finally, the close spatial proximity between fluvial and subglacialdrainage (Figure 4.2, red arrows) is an additional important result. This finding implies pro-glacial networks of fluvio-glacial origin, and requires episodic warmer climate that allowedfor ice sheet melting and retreat. Overall, our observations require early Mars climate to bemuch alike present day Antarctica: widespread cold-based glaciation with marginal meltwa-ter accumulation drained through high-pressure subglacial networks of conduits.Understanding the origins of the Martian valley networks provides critical constraints onthe climate and hydrosphere of early Mars. Here we present the first formation hypothesis thatis broadly consistent with both climate model predictions and observations drawn from thegeomorphological record. Our Principal Component Analysis, understood with predictionsdrawn from physical models of erosion (Figure 4.1), rigorously distinguishes contributionsfrom fluvial, glacial, sapping, and subglacial incision. Detailed morphological observationsand comparisons with terrestrial subglacial drainage systems (Table 4.1) provide further qual-itative evidence for extensive subglacial erosion on the Martian Highlands. The combined re-sults show that the majority of valley networks were incised by pressurized meltwater flowingin subglacial channels, and provide evidence that early Mars’ climate was much like presentday Antarctica. This picture alters substantially the current understanding of early Mars’ hy-drological cycle, provides a new framework for how to identify new environments that areastrobiologically interesting, and sets the ground for a unified theory of early Mars climateand surface evolution.4.6 Extended methods4.6.1 DatasetsMeasurements of the metrics require high resolution topography and imagery, and an inven-tory of valley network stream lines (Hynek et al., 2010). We use elevation maps from the HighResolution Stereo Camera (HRSC), with resolution ranging from 75-12 m/pixel where avail-able, and the Mars Orbiter Laser Altimeter (MOLA) global topography with an equatorial res-olution of 256 m/pixel. For measurements requiring higher resolution, we use CTX imagerywith a resolution of 12 m/pixel. A list of valley networks and morphometric measurements isenclosed in the supplementary table “ch3-VN-dataset.xls” .4.6.2 Measurement strategyThe junction angle is a statistical measure of the acute angle between stream lines (Hynek et al.,2010) representing tributaries, averaged over different locations in the valley network. Thethree crosses in Figure 4.3a are examples of locations where we take this measurement.The stream order Sn corresponds to the highest Strahler number in the network (Horton,814.6. Extended methodsFigure 4.3: (a) angle at a junction between two streamlines. (b) length of the stream lines.(c) stream lines color coded by their stream order. (d) measurement of valley net-work width. (e) CTX image with resolved first order tributaries, from which weextract λ. (f) longitudinal profile indicating the total topographic drop to the uphillsection at an undulation, required to calculate Ω following equation 3.11945). We obtain measures of Sn directly from the valley network stream lines (Hynek et al.,2010). The same dataset (Hynek et al., 2010) includes valley lengths along the longest path.This process is exemplified in Figure 4.3b (L) and 4.3c (Sn).We measure the valley network top width (Williams and Phillips, 2001) as the mean distancebetween the points of maximum convexity defining the valley top, and then linking the pointsof minimum and maximum curvature used to evaluate the top width to their respective crosssectional positions in the profile. We take an average of 10 measurements on highest streamorder valleys (Figure 4.3d). We then calculate the valley network aspect ratio R as the lengthL over this width W.We obtain the width of order 1 tributaries λ by similarly measuring the width over a sta-tistical number of first order tributaries (> 10 if possible). Where topographic resolution is notsufficient, we measure λ from CTX imagery as shown in Figure 4.3e.From longitudinal profile data (Figure 4.3f), we quantify Ω as the elevation difference be-tween the local minimum at the beginning of a positive topographic gradient section hmin, andthe following local maxima hmax, to the total topographic loss ∆H: ψ = (hmax − hmin)/∆H(cartoon in Figure 4.3f). Other processes can produce local depressions followed by uphillsections: i.e., craters in the valley network, infill with impact ejecta from nearby craters, andsuperficial tectonic expressions such as faults, subsidence, etc. We use a geological map (Tanakaet al., 2014) to locate these features on the longitudinal profiles and remove their contributionsto the topography by filtering the profile accordingly.824.6. Extended methods4.6.3 Details of the PCA techniqueMachine learning techniques, in particular Self Organizing Maps, have been used to classifyvalley networks (Stepinski and Stepinski, 2005; Bue and Stepinski, 2006). For our purpose, we usea linear PCA algorithm, which facilitates the interpretation of the results in terms of physicalmodels. This technique is a multidimensional linear statistical analysis that uses an eigenvaluedecomposition of the data covariance matrix to define a new, orthogonal basis set of linearlyindependent vectors called Principal Components (PC). By definition of the transformation,the first PC captures the largest variance, and each succeeding component captures a progres-sively smaller fraction. In our valley network database, the first three PCs capture ∼ 90% ofthe total variance and are sufficient for our characterization. Further details on the PCA tech-nique, including a sensitivity analysis of the dependence of the results on our choice of valleynetworks, is included in Appendix C4.6.4 Metric predictionsThe interpretation of the PCA results starts with comparing the data with physical models ofsynthetic valley networks (SVN) (Grau Galofre and Jellinek, 2017; Grau Galofre et al., 2018) (seeAppendix B), defined by a set of 6 theoretical predictions, each corresponding to one metric:1 SVN = (γ, D f , Sn,λ, R,Ω). Shaded areas in Figure 4.1 correspond to these model predic-tions. Each area encompasses 100,000 SVN representing, in turn, fluvial, glacial, subglacial andsapping erosion mechanisms, and each SVN within is calculated using different combinationsof parameters. In detail, upper and lower reasonable bounds for Mars (supplementary table“ch3-VN-dataset.xls”) define the parameter space from which we calculate the theoreticallyplausible values for a given metric in a given erosion process. Using a Montecarlo method(see Appendix B) over this parameter space, we randomly select 105 possible combinations ofparameters that we then use to calculate the value of each metric. Table B.1 in Appendix Bpresents the average metric predictions, together with their maximum and minimum valuesextracted from these parameter bounds.The resulting 105 SVN are then projected in the PC space and colored according to themechanism of erosion considered. Valley networks outside the boundaries of a shaded areaare not appropriately explained by that mechanism of erosion, which leaves a number of un-explained valley networks. To formally define a set of valley networks that the PCA techniquecannot resolve, we define an area centered in the origin (average) and spanning one standarddeviation in the direction of each metric, which we then project in the PCA space. Our analysisof the results consists of identifying where is each valley network with respect to the spaces ofSVN predicted for fluvial, subglacial, sapping and glacial processes; or identifying it as unre-solvable if it belongs to the average valley network group. Valley networks that do not belongto any Montecarlo produced morphometrical space, or that belong to more than one, are char-acterized according to the erosional process that best approximates its morphometry (i.e., theone that minimizes the difference between the predicted and measured metric values). The834.6. Extended methodsresult of the characterization is indicated in the supplementary dataset “ch3-parameters.xls”.Combined, these analyses provide a robust constraint on the origin of the valley networksunder study.84Chapter 5Lessons from the shape of Martianvalley networks’ longitudinal profiles:steady-state, and the link betweenclimate and tectonics on early Mars.5.1 SummaryEstablishing whether the valley networks reached landscape equilibrium provides a criticalconstrain for the rates, intensities, and duration of the episodes of erosion on early Mars. Ero-sional episodes, in turn, are governed by the rate of tectonic uplift and modulated by climateand geology. In this chapter we address two specific questions, motivated by the lack of val-ley network morphological differentiation as evident from Figure 4.1 in Chapter 4: Are thevalley networks in steady-state? And what erosion rates were required for incising the valleynetworks on early Mars? To proceed, we first build quantitative understanding of the steady-state form of Martian valley networks by using models of landscape evolution that capture thedynamic exchange between uplift and erosion rates in fluvial, glacial, and sapping settings.Solutions of the landscape evolution equation can then be readily compared to the curvaturecharacteristics of observational valley network profiles extracted from the Mars Orbiter LaserAltimeter (MOLA) and the High Resolution Stereo Camera (HRSC) topographic data. Thiscomparison then permits the identification and investigation of fluvial, sapping, and glacialvalley networks in steady-state. Our results show that the majority of valley networks are notin steady-state. In addition, on the basis of inferred low uplift rates for the Noachian High-lands, time-integrated erosion rates were likely very low on early Mars, suggesting that valleyincision was driven by intense episodic periods of erosion, and was likely not a continuousand steady process.855.2. Introduction5.2 IntroductionAs detailed in the Introduction and Chapter 4, hundreds of valley networks incise the southernhemispheric highlands of Mars, standing as evidence that water sculpted the ancient surfaceof the planet. Although there is consensus for the role of liquid water in incising the valleys,controversy remains over the duration and intensity of the erosion events (Gulick, 2001; Irwinet al., 2005; Howard et al., 2005). Whether valley network incision involved high intensity, lowduration events or was mostly controlled by low intensity, prolonged steady flows, requirestwo completely different climate scenarios.Episodic or low intensity erosion rates could lead to very immature valley networks, whichwould not display the morphometries characteristic of steady-state fluvial, glacial, sapping orsubglacial channel networks. This lack of maturity could, in turn, explain the undifferentiatednature of a large portion of valley networks as evident from the magenta data plotted in thePrincipal Component Analysis in Figure 4.1 in Chapter 4. Metric predictions developed inChapters 2 and 3 would only partially succeed at capturing the valley network morphometryobserved on Mars, since a main assumption behind the models is a steady-state relationshipbetween channel and landscape.A way to build understanding is to interrogate the extent to which valley networks are insteady-state, understood as a dynamic balance between the rates of tectonic uplift and erosion.The slope characteristics of channel and valley longitudinal profiles can be used to addresswhether, and where, the Martian valley networks reached steady-state (SS). Additionally, ananalysis of profile curvature provides quantitative insight to understanding the interactionsamong tectonics, climate, and erosion, and has been widely explored in terrestrial channelnetworks (e.g., Howard, 1994; Whipple and Tucker, 1999; Roe et al., 2008; Herman and Braun, 2008;Whittaker, 2012).The steady-state longitudinal profile of a channel is a result of the long term landscape re-sponse to erosion, uplift, and hillslope diffusive processes, and reflects interactions among tec-tonics, climate, and erodibility (e.g., Howard, 1994; Sklar and Dietrich, 1998; Whipple and Tucker,1999; Whipple, 2004; Chen et al., 2014; Herman and Braun, 2008). Solutions to landscape evo-lution equations, which relate changes in ground elevation to explicit parameterizations forfluvial (e.g., Howard, 1994; Sklar and Dietrich, 1998; Whipple and Tucker, 1999; Whipple, 2004; Fer-rier et al., 2013), glacial (e.g., Hallet, 1979; Braun et al., 1999; Herman and Braun, 2008; Egholmet al., 2012), and sapping (Howard, 1988; Devauchelle et al., 2011; Rothman et al., 2009; Petroffet al., 2011) erosion, uplift and hillslope diffusion rates, give the dynamic equilibrium form ofa channel. General formulations for mechanical erosion are in form of power laws relatingerosion rates to flow basal stress (e.g., Whipple, 2004; Herman and Braun, 2008). At steady-state,therefore, the slope and curvature of a valley network profile are related to the rates of erosion,which themselves depend on climate, tectonic, and geological forcing.Here we model steady-state longitudinal profiles of rivers, glaciers, and sapping valleys onMars by using known terrestrial landscape evolution equations (e.g., Howard, 1994; Whipple and865.3. Landscape evolution equations and steady-state solutionsTucker, 1999; Hallet, 1979; Herman and Braun, 2008; Devauchelle et al., 2011). We then investigatewhether the valley networks are in steady-state by comparing the model results with a datasetof valley network profiles at MOLA and HRSC resolution. Our first objective is to introducethe landscape evolution equations that are relevant to each erosive process identified on Mars(Grau Galofre et al., In review), followed by a discussion of their limitations and underlyingassumptions. To gain physical insight, we simplify and non-dimensionalize these equationsand find analytic expressions for steady-state profiles (e.g., Whipple and Tucker, 1999). In moredetail, we: (1) test the profile slope and curvature sensitivity response to non-dimensionalterms that capture uplift, substrate erodibility, erosion- stress exponents, rheology, etc., and (2)identify the valley networks in steady-state by comparing the longitudinal profiles’ curvatureto steady-state solutions to the appropriate landscape evolution equations. Finally, we discussthe distribution of steady-state fluvial, sapping, and glacial valleys on Mars, and use theseprofiles to derive constrains for time-integrated erosion rates.5.3 Landscape evolution equations and steady-state solutionsA widely used, one-dimensional (1D) formulation of the landscape evolution equation acrossdifferent erosion styles is of the form (Whipple and Tucker, 1999):dzdt= −E(x, t) +U(x, t) + D∇2z (5.1)Where the left hand side is the rate of change of elevation (z), and the right hand side describesthe rates of erosion (E), uplift (U), and hillslope diffusive processes, in turn. At steady-state,for a given uplift rate, the landscape responds depending on the leading order erosional mech-anisms, and the hillslope diffusion rate. To constrain the 1D dynamic equilibrium shape of theprofile, it is necessary to define the form and dependence of the erosion rate E on the spatialcoordinates (z, x), which vary depending on the erosion process. The near vertical morphol-ogy of Martian valley network side walls apparent after billions of years since the carving ofthe networks suggests that valley networks are incised in bedrock, and that hillslope diffusiveprocesses contribute negligibly to the channel network evolution. Under this assumption, weignore the hillslope diffusivity term to simplify analytic treatment. We acknowledge, however,that the characteristics of the Martian substrate remain largely unexplored and this term maybe important in areas with less cohesive surface (i.e., sand deposits, regolith, etc.).Finally, the modification and degradation of valley networks through geological timescalesinvolving billions of years is an additional issue. However, detailed observations of valleynetwork morphology and cross sections (Baker and Partridge, 1986; Williams and Phillips, 2001,e.g.,), as well as the rapid decline in erosion rates after the Noachian period, involving 10−2 to10−1 nm/yr at the Hesperian (1.8-3.5 Ga) period (Golombek and Bridges, 2000), are evidence forminimal post-emplacement degradation.875.3. Landscape evolution equations and steady-state solutions5.3.1 Fluvial erosion: the stream-power formulationFluvial erosion models, and in particular, the stream-power equation, have been discussed ex-tensively in the terrestrial literature (Howard, 1994; Sklar and Dietrich, 1998; Whipple and Tucker,1999; Perron et al., 2009). The applicability of the stream power law is restricted to the bedrockareas of channels, and more generality in areas of high slope and discharge. Prior to our useof this model, we tested the applicability of this formulation by locating the Martian valleynetworks within the regime diagram presented in Sklar and Dietrich (1998), scaled for Mars’lower surface gravity. The first order approximation inherent to the stream-power law enablesalgebraic treatment of the problem, but is strictly true only at the headwall areas of the valleynetworks (Whipple and Tucker, 1999).River profiles carved into bedrock are governed by the following equation (Howard, 1994;Whipple and Tucker, 1999):dzdt= U(x, t)− KKma xhmSn, xc < x < L. (5.2)Here, Ka is Hack’s law parameter and h is Hack’s law exponent (Whipple and Tucker, 1999),m is the area exponent, n is the slope exponent, and S = |dz/dx| is the local slope. K is anerodibility parameter (Whipple and Tucker, 1999) that depends on channel hydraulic geometryKw, basin hydrology Kq, bedrock erodibility Kb, friction factor C f , density ρw, gravity g, andthe hydraulic geometry exponent b = 0.467 (Parker et al., 2007):K = KbK−nw K2n(1−b)q Cn/2f ρawgn. (5.3)Choosing the scales H, L and Uo for topographic drop, length of the channel, and uniformuplift (average rates), we non-dimensionalize equation 5.2 with:z∗ =zHx∗ =xLt∗ =t UoH; with, Uo = 〈U(x, t)〉 . (5.4)Assuming a steady, uniform uplift (dU/dx = 0), and keeping lithology, erodibility, precipi-tation, and other controls constant, the non-dimensional solution to the steady-state form ofequation 5.2 is (Whipple and Tucker, 1999):z∗(x∗) = z∗(1) + N1/ne U1/n∗ (1− hm/n)−1(1− x1−hm/n∗)for hm/n 6= 1z∗(x∗) = z∗(1)− N1/ne U1/n∗ ln(z∗) for hm/n = 1(5.5)Where the non-dimensional uplift-erosion number Ne is (Whipple and Tucker, 1999)Ne =UoKK−ma Ln−hmH−n. (5.6)Physically, the uplift-erosion number expresses a balance between the uplift and erosion885.3. Landscape evolution equations and steady-state solutionstime scales and provides a metric for tectonic forcing through changes in Uo, or climatic/ litho-logic forcing through K. The effects of variability in any of these two forcings are dynamicallyequivalent (Whipple and Tucker, 1999). Published studies show that equation 5.5 captures rea-sonably well the slope and curvature of profiles in terrestrial rivers (Howard, 1994; Whippleand Tucker, 1999; Perron and Royden, 2013). Applied to Mars, such a simplified treatment isconsistent with the limitations inherent to constraining the parameter space for K and Uo.5.3.2 Glacial erosion: Bedrock abrasionGlacier landscape evolution modeling involves several interacting mechanisms that include,but are not limited to, abrasion by sliding (e.g., Hallet, 1979; Braun et al., 1999; Herman andBraun, 2008), plucking of underlying bedrock fragments (e.g., Iverson, 2012; Egholm et al., 2012),chemical weathering, and subglacial meltwater incision (Walder and Fowler, 1994; Beaud et al.,2016), the effects of which are not well understood (Beaud et al., 2016).For tractability, our definition of the glacial landscape evolution equation builds on threemain approximations: shallow ice (Hutter, 2017; Paterson, 1994, e.g.,), the description of abra-sion rates in terms of a sliding law, which depends in turn on the basal shear stress (Hallet,1979; Paterson, 1994; Braun et al., 1999; Herman and Braun, 2008; Tomkin, 2009), and the con-sideration of wide, thick glaciers where the wall boundary conditions provide a very weakconstrain on ice surface slope (Reeh, 1982).We herein follow the same approach as Herman and Braun (2008) and relate the bedrockerosion rate E to the ice sliding velocity us (Hallet, 1979). The landscape evolution equation isthen:dzdt= U(x, t)− Kguls. (5.7)Here Kg is an abrasion erodibility constant, and the sliding velocity is given by (Herman andBraun, 2008):us =Bs(Pi − Pw) (ρig)ph(x, z)p|∇(h(x, z) + z)|p, (5.8)Where Bs is the sliding law parameter, the term (Pi − Pw) is the effective pressure (ice over-burden Pi minus water pressure Pw), ρi is the ice density, p is Glen’s flow parameter, whichprescribes ice rheology (e.g., Glen, 1958; Paterson, 1994), and h(x, z) is the ice thickness. Substi-tuting 5.8 into 5.7:dzdt= U(x, t)− Kg[BsPi − Pw (ρig)ph(x, z)p|∇(h(x, z) + z)|p]l. (5.9)To obtain an analytic solution to equation 5.9, we assume a constant effective pressure (Pi−Pw) (Herman and Braun, 2008), and give a closure relationship for the steady-state longitudinalshape of the ice thickness in terms of the topographic height and distance along profile h(x, z).We choose scales for uplift, topographic drop, length, as well as a characteristic ice thickness895.3. Landscape evolution equations and steady-state solutionsHi following previous studies (Herman et al., 2018), and introduce:z∗ =zHx∗ =xLU∗ =UUoh∗ =hHi. (5.10)Considering a glacier that is sufficiently wide, such that the effects of the boundaries do notinfluence the steady-state longitudinal shape of the center-line (i.e., a thin ice sheet), we canapproximate the ice thickness far from boundaries as (Reeh, 1982):h∗ =√2x∗ − x2∗. (5.11)Non dimensionalizing equation 5.9 and balancing uplift and erosion we obtain after algebra:U∗ = Nghp−1∗ (Γ∇∗h∗ +∇∗z∗)p . (5.12)Two control parameters emerge: Γ = Hi/H, and Ng, which is a non dimensional uplift-erosionnumber much like Ne for fluvial erosion:Ng =(KgUo)(BsPi pPi − Pw)(HL)p. (5.13)The ice uplift-erosion number Ng provides a basis for a similar discussion to the fluvial uplift-erosion number Ne. The first factor on the right hand side captures the relationship betweenthe ground erodibility Kg, which depends on lithology, and tectonic uplift Uo, highlightingthe same dynamic dependence on erodibility and uplift that Ne. The second factor dependson the glacier internal dynamics. It includes the ice sliding factor Bs, which is a function ofbedrock roughness and ice temperature (Paterson, 1994) (p. 142) and the effective pressureof basal meltwater (Pi − Pw), which controls the subglacial hydraulic regime (Schoof , 2010)and therefore the extend of basal lubrication. The last factor on the right is a scale for thetopographic slope to the power p = 3.Substituting equation 5.11 into 5.12 and solving for the steady-state slope∇∗z∗ we obtain:∇∗z∗ =(U∗Ng)1/pl √2x∗ − x2∗1−pl/pl− Γ 1− x∗√2x∗ − x2∗1 > x∗ > 0. (5.14)From equation 5.14 we can infer how the slope and curvature characteristics of a landscapeeroded by glacial abrasion depend on ice rheology p, the ice uplift-erosion number Ng andthe sliding-erosion exponent l. Taking p = 3 for a typical ice rheology, and choosing l = 2,905.3. Landscape evolution equations and steady-state solutionsequation 5.14 can be integrated to obtain an analytic form for the steady-state profile:z∗ =−6 6√1/Ng 12√(x∗ − 2)6 Re [1F2 ( 512 , 512 ; 116 ; 22−x∗)] +Hi/H (x2∗ − 2x∗)√(2− x∗)x∗+6 6√1Ng12√(2− x∗)x∗ x∗ 1 > x∗ > 0.(5.15)Where Re [2F1] indicates the real part of the hypergeometric function.5.3.3 Sapping erosion: it’s all about the boundariesErosion by sapping processes occurs where groundwater seeping from a spring through porousground delivers a basal stress that is sufficient to mobilize the sediment (e.g., Laity and Malin,1985; Howard, 1988; Devauchelle et al., 2011), which leads to sediment removal from the springsite, undermining, and eventual collapse of the headwall. Theater-shaped valley networksand canyons on Mars have been historically explained through this process (Laity and Malin,1985), however not without controversy (Lamb et al., 2006). Given the low stresses involvedin groundwater flow from springs, and considering the lower surface gravity on Mars, thefollowing equations are restricted to describing the erosion of cohesion-less sediment, such asunconsolidated sand or regolith (Lamb et al., 2006).Any successful theory that predicts sapping longitudinal profiles must account for the con-tinuity of water flux at the spring between groundwater in the aquifer and the flow of wateroverland (Devauchelle et al., 2011). The importance of this essential boundary condition re-stricts the use of a landscape evolution equation - which generally applies over lengthscalesmuch longer than the valley length - in favor of a more local development. We herein adoptthe steady-state profile derived as a solution to the Poisson equation describing the evolutionof the square of the height of the water table at the site of the groundwater emergence De-vauchelle et al. (2011). According to this model, the slope and curvature of sapping and fluvialsystems should only differ considerably at the headwall where sapping profiles are governedby the distinct boundary conditions related to groundwater seepage. These results are applica-ble only to the 1st order valley networks. Higher stream orders would be expected to displayfluvial profiles captured by equation 5.5.Imposing continuity of water flux at the spring between groundwater and overland flow,and setting the elevation of the spring equal to the elevation of the water table, the longitudinalprofile is given by Devauchelle et al. (2011) :z = zo −√32(QoKszo)1/3x2/3. (5.16)Where zo is the elevation of the spring and Ks is the substrate hydraulic conductivity. Qo isa function of the critical shields stress Θc, average grain size ds, a non dimensional density915.4. Model results: the steady-state shape of Martian longitudinal profilesRρ = ρw/(ρs − ρw), and a friction factor relating bed roughness zr to flow depth D: C f =0.18(zr/D)1/7:Qo =1.75C f√Θ3c g(dsRρ)5. (5.17)To gain physical insight, we again non-dimensionalize equation 5.16 with a similar choiceof parameters as for glaciers and rivers, noting that zo naturally corresponds to the topographicdrop H:z∗ =zHx∗ =xL. (5.18)With these scales, equation 5.16 gives a power law dependence on the distance along thestream:z∗ = 1− Nsx2/3∗ (5.19)In this case, the non-dimensional sapping number Ns is:Ns =√32(QoKsH2)1/3 ( L2H2)1/3. (5.20)Physically, Ns provides insight into the dynamics of seepage. The first bracketed term relatesthe critical flux required for sediment removal Qo to the flux of water seeping through a cross-section perpendicular to the spring (KsH2). This ratio behaves as a switch: When KsH2 > Qo,sapping processes are possible, whereas the opposite inhibits sapping erosion. The secondbracketed term on the right is a ratio between the basin area (L2) and the aquifer area, whichis proportional to the square of the water table elevation (H2). Larger basin areas with smalleraquifers naturally limit the water available to erode, which can slow down or inhibit the sap-ping process if L2  H2. In summary, Ns  1 inhibits sapping erosion, and Ns  1 enhancesit, with a switch occurring when Ns =√32(L2H2)1/3.5.4 Model results: the steady-state shape of Martian longitudinalprofilesEquations 5.5, 5.15, and 5.19 give the steady-state longitudinal profile curves for fluvial, glacial,and sapping erosion regimes. Adapting these models to provide a plausible range of valleynetwork profiles for Mars requires careful characterization of the parameter space that entersinto the calculation of the different scaling factors and non-dimensional numbers Ne, Ng, Ns.The scaling factors that capture valley geometry, e.g., L, H are readily obtained from the anal-ysis of each individual valley network. We take glacial ice thickness Hi to scale like the depthof a valley network. It is, however, more challenging to constrain parameters such as substrateerodibility and grain size distribution, for which we use a range of values appropriate to theMartian surface drawn from the existing literature (parameters included in the supplementarytable “ch4 parameters.xls”, see also Appendix E). Figure 5.1 shows the predicted steady-state925.4. Model results: the steady-state shape of Martian longitudinal profileslongitudinal profiles of Martian rivers (panel a), sapping valleys (panel b), and glaciers (panelc) for a range of non-dimensional uplift-erosion numbers calculated from the parameters doc-umented in the supplementary table “ch4 parameters.xls”.Figure 5.1: Model results for the steady-state shape of fluvial (a), sapping (b), and glacial(c) longitudinal profiles for Mars. The family of curves plotted shows parameter-induced variability in the non-dimensional uplift-erosion parameters Ne, Ng and Ns,which increase in the direction indicated by the arrow. Γ is kept constant in panel(c).As with drainage systems on Earth, the modeled steady-state shape of Martian profilesis concave (albeit flatter than their terrestrial counterparts due to the smaller Martian grav-ity), with maximum curvature at the headwaters. Fluvial profiles (5.1a) are concave for anyrange of erosion-uplift numbers Ne and erosion exponents 1− hm/n permitted by models andparameterizations (Whipple and Tucker, 1999; Gulick, 2001; Irwin et al., 2005). Typical profileslose most of the elevation (40-60%) within the first quarter of the valley length (x∗ < 0.25)and are relatively flat afterwards. In marked contrast to sapping erosion (b), changes in theuplift-erosion number Ne have a dramatic effect on curvature, which increases as the rate oferosion (higher erosional power and higher erodibility) exceeds tectonic uplift. An increase inthe erosion exponent 1− hm/n also contributes in deepening the curvature at the headwall.Sapping profiles (5.1b) present the smallest curvature in our results. Varying the two freeparameters Ns and the sapping exponent 2/3 < α < 15/22 within the bounds allowed bymodels (Devauchelle et al., 2011) and parameterizations (supplement “ch4 parameters.xls”) hasonly a very small effect on profile shape. Variations in Ns, related to a change in the flux ofgroundwater emerging the spring compared to the critical flux for sediment removal, reflectan inversely proportional change in profile curvature.Modeled steady-state glacial profiles (5.1c), for a sliding exponent of l = 2, display thelargest curvature of the three processes in agreement with terrestrial glacial erosion (Hermanand Braun, 2008; Egholm et al., 2012). Their curvatures are also consistently smaller than interrestrial contexts for the same choice of l, as expected from the lower surface gravity of Mars.In glacial profiles, 20-75% of elevation is lost at the headwaters within the first 10% of distancefrom the origin, leaving a relatively flat profile with minimal slope downstream. In contrastwith fluvial and sapping cases, two dimensionless numbers control glacial erosion by sliding:935.5. Are valley networks in steady-state?the uplift-erosion number Ng and the ratio of topographic drop to ice thickness Γ. Consistentwith fluvial and sapping erosion, an increase in Ng for a given Γ leads to higher concavity atthe headwall and flatter profiles downstream.The interpretation of Ng, however, is complex given its dependence on ice physical prop-erties: an increase of Ng can indicate either erosion rates overcoming uplift rates because ofhigher erodibility, etc., or because of faster ice sliding driven by an increase in ice softness(higher temperature) or a decrease in effective pressure (water accumulated at the base). Moreconcave glacial profiles therefore indicate easily eroded ground or wetter-based glaciation(larger Ng), as well as thinner ice or higher topographic drop.Fixing Ng, allows for an analysis of profile curvature as a function of Γ. Thick ice, or verylow topographic drops (large Γ), contribute in flattening the profile and can even slightly invertthe curvature for unrealistically large values (Hi  H). A decrease in Γ, that is, Hi  H leadsto more pronounced concavity.5.5 Are valley networks in steady-state?A fundamental question regarding the state of early Mars’ hydrological cycle is: Did valleynetworks carry discharges sufficiently large and for a long enough period to reach a dynamicequilibrium with the landscape? Using the valley network characterization results in GrauGalofre et al., (2018) (In review), we identify a subset of fluvial, glacial, and sapping valleynetworks and analyze their profiles by comparing them with the steady-state model solutionsfor each erosional process, in turn (equations 5.5, 5.15, 5.19). Knowing the erosional processthat incised each valley network, we can target our analysis of each individual profile by per-forming a non-linear fit to the appropriate model (fluvial, sapping, or glacial erosion), andfinding the combination of non-dimensional uplift-erosion numbers and erosion exponentsthat minimize the RMS misfit to each profile.5.5.1 Evaluating longitudinal profiles: Modeling and observation insightsTable 5.1 presents the results of comparing steady-state landscape evolution models (equations5.5, 5.15, 5.19) to observational profiles. The resulting uplift erosion number N (Ne, Ng, Ns) anderosion power law exponent (ne) are given by the best fit profile. For glacial erosion, the non-dimensional number Γ is reported in the supplementary table “ch4 parameters.xls”.To elaborate this table, we identify steady-state valley networks on the basis of the physicalplausibility of the erosion exponents found in the fit and a higher rms misfit bound of 0.5.According to the models of erosion introduced in the previous section, and for the range ofparameters presented in the supplementary table “ch4 parameters.xls”, the fluvial ne must be0.6− 3.7, and the sapping ne between 2/3− 15/22. Owing to the complexity of the glacial fitand the non-existence of a single characteristic exponent in equation 5.15, we base the decisionon the misfit and whether profile slope is consistent with the model. Convex profiles, profileswith stepped sections, or with undulating sections are considered out of steady state. Abus945.5. Are valley networks in steady-state?Table 5.1: Subset of valley networks considered in this study, lat/long location (decimaldegrees), incision mechanism (f for fluvial, g for glacial, s for sapping, Grau Galofreet al., (2018) (In review)), uplift-erosion number N, erosion exponent ne, rms misfit,and terrain age (N for Noachian, H for Hesperian, A for Amazonian, e for early, mfor middle, l for late).Name lat long Type SS? N ne misfit ageNaktong Vallis 6.0 32.2 f yes 0.4 1.3 0.03 mNLoire Valles -20.7 -15.3 f no 1.2 0.18 0.8 mNMa’Adim Valles -18.7 177.3 s yes 0.9 0.6 0.02 mNLicus Vallis -4.6 125.9 f no 1.13 0.4 0.05 mNEvros Valles -12.2 9.4 f yes 0.5 1.17 0.04 mNIndus Vallis 19.7 36.9 f no 1.04 0.5 0.05 mNCusus Valles 15.1 50.4 f yes 0.3 1.5 0.05 mNUnnamed 10.0 46.7 f yes 3E-4 2.2 0.03 mNAuqakuh Vallis 32.0 61.4 f no 0.1 1 0.8 eN-mNTagus Valles -6.0 114.6 f no 0.15 1 0.7 mNHuo Hsing Valles 31.1 66.3 g yes 5E-10 200 0.01 eNHimera Valles -14.71 -22 f no 0.9 0.5 0.05 mNArda Valles -19.6 -31.4 f no 0.11 1 0.08 mN-lNParana Valles 22.7 -10.6 f no 0.9 0.7 0.02 eN-mNWarrego Valles -42.7 -92.2 f no 0.11 1 0.05 eN-mNTader Valles -48.8 -152.5 f no 0.11 1 1.3 eN-mNAbus Vallis -5.5 -147.3 s yes 1 0.6 0.01 H-ALouros Valles -8.4 -83 s no 0.93 1.6 1.4 eHVichada Valles -18.9 89.1 f no 0.08 1 1.2 mNUnnamed 22.1 -72.3 g yes 6.4E7 215 0.3 mN-eHDubis Vallis -5.2 -148.2 s no 630 0.7 0.4 mNVallis and MaAdim Vallis conform well to the predicted steady-state profile for a sappingvalley, with only minor misfit either at the very end of the channel length, or in several veryshallow undulations along the length of the profile (Figure 5.2b). By contrast, Dubis Vallis andLouros Valles are both sapping systems, with convex or partially convex profiles that are notin steady-state.Naktong Valles, Evros Valles, Cusus Valles, and a tributary to Locras Valles are fluvial sys-tems that fit the steady-state fluvial profile as shown in Figure 5.2a, with some small sizedgentle undulations in the headwaters and a smoother profile in the lower reaches of the chan-nel. Loire Valles, Licus Valles, Indus Valles, Auqakuh Valles, Tagus Valles, Himera Valles, ArdaValles, Parana Valles, and Warrego Valles are out of steady-state fluvial systems. These sys-tems are fully (i.e., Licus Valles, Arda Valles, Parana Valles), or partially convex, with convexsections at the origin and a change in curvature downstream (i.e., Himera Valles, Dubis Vallis),markedly stepped sections (i.e., Himera Valles), or display flat sections with sharp changes inslope (i.e., Warrego Valles, Indus Valles, Auqakuh Valles).To improve the performance of the non-linear fit to the glacial landscape evolution equa-955.6. Valley networks in steady-state: implicationsFigure 5.2: Observed valley network profiles (dashed lines) overlapped by steady-statebest fit model results for fluvial (Naktong Valles, (a)), sapping (Abus Vallis, (b)), andglacial erosion (unnamed valley network, (c)). The horizontal axis is the normalizeddistance along the valley, and the vertical axis is the elevation normalized by thetotal topographic drop.tion 5.15, we fixed Hi/H to the value obtained from valley network geometrical characteris-tics (see Supplementary table) and limited the free parameters to the erosional exponent andthe non-dimensional uplift-erosion number. Huo Hsing Vallis, and in particular an unnamedvalley network at the rim of Kasei Valles, have profiles consistent with steady-state erosionby glacial sliding (5.2c). Observations and models coincide in an abrupt change in elevationwithin the first 25% of the valley length, although the shape of both profiles depart from modelpredictions. We attribute this difference to the assumptions underlying the glacial landscapeevolution formulation: Glaciers confined within even very wide valleys are not ice sheets.Drag at valley walls reduces, e.g., the erosion rates at the early stages of the valley network,which justifies the model over-predicted topographic drop in comparison with the smoothervalley network longitudinal profile slope.5.5.2 Valley networks out of steady-stateThe majority of valley networks considered are out of steady-state. This conclusion derivesfrom a large misfit between observations and model predictions, or from profile convexity,stepped profile sections indicating sudden uplift or base changes, and flat profiles with sud-den gradient changes, all indicating the incapacity of the valley network to come back to equi-librium.Figure 5.3 below presents examples of valley networks out of steady-state. In all cases,the erosional exponents ne found fitting the model are unreasonable according to the Martianparameter space (see references in supplementary table “AGG RG MJ parameters.xls”).5.6 Valley networks in steady-state: implicationsIn summary, from a set of 21 valley networks analyzed, only 8 display consistently concaveprofiles that agree with steady-state curves of fluvial, glacial, or sapping erosion regimes.965.6. Valley networks in steady-state: implicationsFigure 5.3: Profiles of valley networks not consistent with steady-state models of erosion(examples). (a) shows an irregular profile (Loire Valles), (b) shows a stepped profile(Himera Valles), and (c) shows a convex profile (Nirgal Valles).These results represent a very small fraction of the valley networks under study, the implica-tions of which we discuss in this section. First, we consider the spatial distribution of steady-state and unsteady valley networks. We then analyze the curvature characteristics of steady-state longitudinal profiles in terms of the non-dimensional uplift erosion numbers. Buildingon both observations, we discuss time-integrated erosion rates in terms of the tectonic contextto each valley network.5.6.1 Steady-state valley network distributionFigure 5.4 displays the global geological map of Mars by Tanaka et al. (2014), with the locationof the valley networks in this study. See also Figure E.2 for geographical reference.All of the SS fluvial systems identified in our dataset occur on mid-Noachian terrains (Fig-ure 5.4) and are distributed along the narrow topographic belt linking the Noachian High-lands with Arabia Terra. The proximity between steady and unsteady fluvial systems withinthe same geological unit is evidence that that valleys in the same region can be in differentstages of landscape evolution. It is, however, challenging to discuss the origin of the differ-ences in evolutionary stage between spatially close valleys without detailed information aboutthe relative timing of emplacement and the geological characteristics of the area, particularlyin terms of substrate erodibility.The number of data available for glacial and sapping SS valley networks is insufficient tomake robust observations regarding their distribution. Nevertheless, it is of interest to discuss,with caution, the results in Figure 5.4. The two SS sapping valleys are located in late Noachianunits (plateau and plateau ridged units), whereas younger Hesperian systems (e.g., LourosValles) display convex profiles far from steady-state. As with fluvial systems, two drainagesystems close to each other, and in the same surface geologic unit as are Abus and DubisValles display widely different profiles in different stages of network evolution.Both glacial examples studied here have profiles that are consistent with steady-state mod-eling results. Geologically, they belong to distinct units and time periods: Huo Hsing is locatedin the Noachian plains, whereas the unnamed valley network in Figure 5.2 is incised into Hes-975.6. Valley networks in steady-state: implicationsFigure 5.4: Global geological map of Mars including the surface age (blue for Noachian,green for Hesperian and beige for Amazonian) and surface tectonic structures (greylines), overlaid with the results of our longitudinal profile analysis. Check marksindicate valley networks in steady-state, color coded according to their origin: redfor fluvial, purple for sapping, and blue for glacier. Black crosses indicate a valleynetwork out of steady-state. Grid divisions correspond to the USGS quadrangles(45◦ across)perian chaotic terrain.5.6.2 Spatial correlation with surface tectonicsSurface tectonic expression appears to exert a strong control over the extent to which valleynetworks are in steady state (Figure 5.4). Fluvial valley networks in regions with large reportedsuperficial faults (Tanaka et al., 2014), such as Warrego Valles on the Tharsis rim, are out ofequilibrium, displaying abrupt changes in profile slope or stepped sections in agreement withobservations of terrestrial fluvial systems in regions of rapid uplift or sudden base level fall(Whipple and Tucker, 1999). Following the terrestrial literature, our results make possible tocapture the timescale of response to a tectonic forcing, which is the subject of work in progress,and beyond the scope of this study. By contrast, all fluvial networks in steady-state are locatedin terrains with very little expression of major surface tectonics (see red check marks in Figure5.4).Such a spatial correlation with the surface expression of major structures does not seemto influence whether the sapping ans glacial profiles reach steady-state, as Abus Vallis is inclose proximity to the fault systems at the boundary of the Tharsis province. Furthermore,the unnamed glacier next to Kasei valleys in Lunae Palus is the only valley network reaching985.6. Valley networks in steady-state: implicationssteady-state in an area with extensive surface tectonic features (Figure 5.4), a point to whichwe return to in next subsection.Detailed timing of the tectonic history around individual SS valley networks is crucial tounderstanding whether faulting and/or subsidence were active during the time of the incisionof SS valley networks. Drainage systems with disrupted profiles, such as Warrego Valles,suggest that changes in uplift rate occurred during or after the incision of the valley network.If faulting and changes in uplift rate develop before the valley incision, steady-state profileswould not necessarily preserve a record of it.5.6.3 Inferred time-integrated erosion rates from the analysis of steady stateprofilesFigures 5.1, 5.2, and 5.4, as well as the results in table 5.1, constrain Ne, Ng, Ns, and the erosionexponent n, and enable an interpretation of steady-state profiles. Here we address how theseerosion numbers provide information related to the time-integrated erosion rates required toincise the valley networks in steady-state. To facilitate the discussion, we will assume thatsurface tectonic expressions as mapped by Tanaka et al. (2014) can be used as a qualitativeproxy for uplift rates in the following way: In areas where surface structures are absent forhundreds of kilometers, we take the uplift rate to be very small.Of the fluvial valley networks in steady-state, Naktong Vallis, Evros Valles, and CususValles display similar best-fit uplift-erosion numbers on the order Ne ∼ 0.1, whereas the trib-utary to Locras Valles displays Ne ∼ 0.0001. The large Ne values of the first three valleynetworks expresses small erosion rates in comparison to uplift. The lack of large surface tec-tonic structures for hundreds of kilometers around these valley networks, however, suggeststhat uplift rates were very low, so that reaching steady-state would require equally low time-integrated erosion rates. The same is not true, however, in the Locras Valles system. In this sys-tem, and taking an equally low uplift, erosion rates were at least 3 orders of magnitude higher.Part of the explanation of this difference lies in the large topographic drop associated withthis system, which leads to higher local slopes and thus contributes to higher time-integratederosion rates (see the contributions of L and H to Ne in equation 5.6). When these differencesin valley network geometry are taken into account, erosion rates associated with climatic andgeological forcing become comparable, within a factor 3, to the other three fluvial valleys insteady-state.The largest difference in terms of uplift-erosion numbers come from the glacial valleys insteady-state, and it is particularly interesting to discuss why Ng varies over many orders ofmagnitude between the two studied systems. The variability in topographic drop and lengthof each valley network enter in the calculation of Ng (see equation 5.13) and account for 8out of the 17 orders of magnitude they differ in. The remaining 9 orders of magnitude arerelated to variations in uplift, climate and the dynamics of ice flow and sliding. Estimationsof uplift rates during the emplacement of Tharsis are on the order of 0.01 m/yr (Jellinek et al.,995.7. Conclusions2008). Although most of the Tharsis bulge was likely emplaced in the Hesperian period (e.g.,Fassett and Head, 2011), it is likely that the uplift rates for the unnamed glacier in Lunae Paluswere higher than for Huo Hsing. Overall, we do not assume this difference in uplift rates toaccount for more than 3 orders of magnitude of Ng variability. The dynamics of ice sliding,including variations in temperature affecting the ice sliding parameter Bs together with theeffective pressure of meltwater at the base (Pi − Pw) can account for the significant remainingdifference in Ng. In particular, when large amounts of meltwater accumulate at the base ofthe glacier, the effective pressure (Pi − Pw) ∼ 0 enhancing ice lubrication and the efficiencyof sliding-driven time-integrated rates of erosion. In warmer climates or exposed to highergeothermal fluxes, higher ice temperatures facilitate ice flow and sliding through the slidingcoefficient Bs. Consequently, we hypothesize that the glacier at the rim of Kasei Valles waslikely warm-based and carried a significantly larger amount of meltwater than Huo Hsing.These two combined effects can explain the remaining difference in Ng, and provide insighton the cold climate conditions that led to the emplacement of Huo Hsing and the developmentof warm-based conditions during the incision of the unnamed glacier in Lunae Palus.Our estimated low time-integrated erosion rates can either represent steady very low inci-sion rates, or intermittent high intensity erosion events separated by periods of repose. On thebasis of climate model results, which highlight the difficulty in sustaining protracted warmand wet conditions (Wordsworth et al., 2013; Forget et al., 2013; Wordsworth et al., 2015) we arguefor the latter explanation. This argument is additionally justified by the need of achieving acritical stress for sediment removal, which is not considered in this study, but would likely dif-ficult incision through hundreds of meters of bedrock in the case of steady, very low erosionrates.5.7 ConclusionsIn this study we use a combined observational and modeling study to address two questions:Are the valley networks in steady-state? What were the time-integrated erosion rates requiredfor incising the valley networks like on early Mars?To proceed, we first model the shape of steady-state longitudinal profiles for fluvial, glacial,and sapping valleys on Mars by using non-dimensional analytic solutions of landscape evolu-tion equations, which capture the dynamic exchange between uplift and erosion rates in termsof uplift-erosion numbers. We then use observations of individual valley network longitudinalprofiles derived from MOLA and HRSC topographic maps, together with the results from thevalley network characterization given in Grau Galofre et al., (2018) (In review), to quantitativelycompare appropriate fluvial, glacial, or sapping model predictions to each valley network pro-file. Non-linear fits that minimize RMS error between modeled and observed profiles indicatethe extend to which a network in our study reached steady-state, and constrain appropriatenon-dimensional uplift erosion numbers (N) and an erosion exponents (ne). Valley networkprofiles that diverge considerably from model predictions (i.e., convex, stepped, or irregular1005.7. Conclusionsprofiles), stand as quantitative evidence that the valley network in question is not in landscapeequilibrium. Our results indicate that less than half of the valley networks under analysis arein steady-state. For those steady networks, constraints on the uplift-erosion numbers enable adiscussion of the climatic and geologic characteristics that lead to their emplacement.Overlaying the respective networks with geologic and structural maps, we then discussthe interactions between valley network incision, age of the surface, and surface tectonic ex-pressions (i.e., faults and folds). We find that, with the exception of glacial networks, allvalley networks that have reached dynamic equilibrium are in ancient (mid-Noachain) unitswith negligible surrounding surface tectonic expressions. Dynamic landscape equilibrium isreached when uplift and erosion rates balance. Assuming very small uplift rates on the basisof the dearth of surface tectonic features around fluvial valley networks, we argue that time-integrated erosion rates in all steady fluvial networks were likely very small, and related tointense episodes of incision as opposed to steady erosion by prolonged water activity.Better constraints on the uplift rates on the different regions studied, a larger data set en-compassing the majority of valley networks, together with the relative timing between thetectonic forcing and the incision of the valley network are required, however, for a better, quan-titative assessment of the time-integrated erosion rates and a more detailed discussion.101Chapter 6Concluding remarksValley networks on Mars, as terrestrial valleys do, reflect the interaction between climate, ge-ology and tectonics. The main objective of this thesis was to build quantitative understandingof the erosive processes that lead to the incision and development of the Martian valley net-works, and discuss the implications for early Mars climate and tectonic history. In particular, Iwas interested in identifying, among the Martian valleys, contributions from fluvial, sapping,and glacial erosion, which are the common interpretations for valley network origin, but alsofrom subglacial erosion, which was bewilderingly ignored before this work. In this chapter, Iprovide a summary of the main findings and contributions, as well as the main implicationsof my work. I finish with a discussion of a series of open questions that arise from this thesis.6.1 Summary6.1.1 Chapter 2: Any journey to Mars starts on Earth.Chapter 2 is set as a thought experiment: If we were standing on Mars, looking at Earth withonly remote sensing imagery and topographic data available, could we reliably characterizethe terrestrial drainage systems, and assess whether Earth is in an interglacial period? Toanswer this preliminary question, I developed a technique based on a Principal ComponentAnalysis (PCA) of a set of terrestrial drainage systems, including rivers, glaciers, sapping val-leys, and subglacial channels, that I then interpret on the basis of model results for steady statemorphometric representations of their formation mechanisms. In more detail, I first designeda series of five morphometric parameters tailored to identify distinctive aspects of valley mor-phology that arise as a consequence of differences in flow dynamics and erosional mechanics.To test the performance of these metrics at identifying the underlying erosional mechanisms,I compiled a dataset that evaluates them in 67 terrestrial channel networks including rivers,glaciers, sapping valleys, and subglacial channels in steady state or as close as possible to it.The analysis of these data with a PCA allowed for:1. A classification of valley systems according to common morphologies as captured by the1026.1. Summarymetrics,2. a visualization of the dataset in a 2D parameter space, where all metrics are expressed aslinear combinations of the directions of maximum variance of the data (Principal Com-ponents),3. the identification of which metrics capture the maximum variance in the data, and aretherefore more relevant at characterizing distinct types of erosion.The results of the PCA analysis understood with quantitative interpretations of the morpho-metric parameters that characterize the fluvial, sapping, subglacial, and glacial clusters al-lowed for the quantitative classification and characterization of all but one type of terrestrialchannel networks: subglacial channels.Main outcomesThe major outcome of this chapter is the development of the characterization and classificationtechnique, together with the physical formulation of 5 quantitative landscape metrics. Theresults of this thesis presented in Chapter 4 and Chapter 5 build on the results of applying thisanalysis to Mars. This chapter is published in Journal of Geophysical Research: Earth Surface.6.1.2 Chapter 3: The missing piece of the puzzle is always the most significant.The results of Chapter 2 highlighted the difficulties in capturing the differences between sub-glacial and fluvial drainage systems from remote sensing data. This was the motivation behind5 weeks of field work in Devon Island (High Arctic) that led to the development of three addi-tional metrics tailored to capture the main differences between fluvial overland and subglacialpressurized drainage. Chapter 3 presents a morphological and morphometrical characteriza-tion of subglacial drainage channels on Devon Island from extensive field and remote sensingobservations, as well as a detailed comparison with river systems. In particular, my collab-orators and me compiled three datasets in the field: hand-held GPS longitudinal profiles offluvial and subglacial channels, photogrammetry-derived digital surface maps of up to 50 sub-glacial channel networks across the island, and high resolution LiDAR data for a fluvial anda subglacial channel network. Detailed field observations of subglacial channel morphology,including the identification of trapezoidal cross sections, low stream network stream orders,absence of inner channels or bed-forms, anabranching sections, potholes, etc. complimentedthe remote sensing characterization of subglacial channels. From these datasets acquired, to-gether with insights drawn from subglacial flow and erosional dynamics (Greenwood et al.,2007; Beaud et al., 2016), I derived three metrics aimed at identifying subglacial drainage: Un-dulations in the longitudinal profile, oblique directions respective to topographic gradients,and large cross-sectional shape factors. Adding any of these additional metrics to the tech-nique of Chapter 2 finally enables the identification of subglacial drainage on Earth or Mars.1036.1. SummaryMain outcomesThe main outcomes of this chapter are the identification of 3 metrics that successfully identifydrainage systems originated by subglacial erosion, as well as a detailed characterization of themorphology of subglacial channels. Of these metrics, 2 are quantitative based on fluid dynam-ics models for subglacial flow: the presence of undulations and the departure of flow directionfrom topographic gradients. The third is observational and is related to the characterization ofcross sectional shape. This chapter is published in The Cryosphere.6.1.3 Chapter 4: The origin of the Martian valley networks.Chapter 4 presents the main results of this thesis, which build on the technique derived inChapter 2 with an additional metric (undulations) developed in Chapter 3 to capture sub-glacial erosion. Following the steps presented in Chapter 2, I built a dataset including 66 valleynetworks x 6 metrics, sampling randomly in all geographic quadrangles, geological units, andtiming of incision. Applying, again, a PCA to the dataset provided statistical classification ofthe different types of valley networks, although the clear clustering retrieved for Earth, wherethe drainage systems were grouped according to common morphometries defined by the met-rics (Chapter 2) was not recovered. To quantitatively understand the results of the PCA, werequired predictions for the steady state morphometry of Martian valley networks. The firststep, constraining the parameter space required me to adapt the models for the metrics toMars. To do this, I carried out an extensive literature review to find upper, lower bounds, andan average value for each of the parameters that enter into the physical models. I then usedcorresponding model results to define synthetic valley networks, each characterized by the sixpredicted parameters. I then used a Montecarlo method sampling from the parameter ranges,to generate 500,000 synthetic valley networks. The projection of the Montecarlo results into thePCA defines spaces of possible steady state morphometries, which drive the interpretation ofthe clusters of valley networks in terms of contributions from fluvial, subglacial, sapping andglacial erosion. In addition, I represented on the PCA the variance inherent in the valley net-work dataset, which physically defines a set of undifferentiated valley networks. My resultsindicate:1. That subglacial drainage is responsible for the formation of a significant portion of val-ley networks, requiring the presence of extensive late Noachian ice sheets that locallyunderwent basal melting2. That fluvial systems developed on early Mars, but were constrained to only two verynarrow corridors: one connecting Terra Sabaeus with Oxia Palus, crossing the Arabiaand Margatifier quadrangles, and the second in the Mare Tyrrhenum quadrangle, alsofollowing a narrow topographic corridor (see geographical quadrants in Appendix E,Figure E.2).3. That sapping valleys and wet-based glaciation are rare on Mars.1046.1. Summary4. The close proximity of fluvial and subglacial drainage structures hints at a fluvioglacialorigin for the fluvial valleys, and raises the possibility of a dynamic LNIH ice sheet withintermittent episodes of advance and retreat.Main outcomesThe results of this chapter, in particular the identification of subglacial erosion on Mars, rec-oncile the results of state of the art climate models with the increasingly higher resolutionmorphological observations relative to early Mars. According to my results and contrary tothe current understanding of early Mars morphology, the presence of valley networks is notproof that Mars was once warm and wet but may hint at the opposite: a portion of them are ev-idence of an extensive, locally warm-based late Noachian ice sheet, with punctuated warmerepisodes allowing for formation of proglacial, fluvioglacial valleys. This chapter is submittedas Letter to Nature with an extensive supplementary information.6.1.4 Chapter 5: Standing on the shoulders of terrestrial giants, one can see Marsa bit better.The incapacity to explain, or make predictions for the morphometry of valley networks lo-cated at the PCA coordinate origin suggests that a large fraction of valley networks did notreach steady-state. Among those in steady-state, the curvature analysis of longitudinal profilesof fluvial, glacial, and sapping valleys motivated by studies of terrestrial landscape evolutioncan help unravel the links among climate, geology, and tectonics on early Mars. This chapterapplies insights from the extensive literature related to landscape evolution models and theinterpretation of their steady state solutions on Earth, to the results of Chapter 4 to addresstwo questions: Are the Martian valley networks in steady state? What were the climate andtectonic conditions during their emplacement? First, I review the landscape evolution equa-tions that model fluvial, glacial and sapping erosion and find solutions for the steady stateform of resulting longitudinal profiles. To address the first question, I analyze the longitudinalprofiles of 21 valley networks, extracted using the Mars Orbiter Laser Altimeter and the HighResolution Stereo Camera topographic datatsets, by comparing them to steady state modelingsolutions. I then use the curvature of the profiles of the valley networks found to be in steadystate to discuss the climate and tectonic conditions during their emplacement. My findings, inparticular the remarkable convexity of a majority of profiles, highlight that most of the valleynetworks on Mars never reached dynamic equilibrium. Moreover, I show that erosion rateswere likely very small on Mars to achieve a dynamic balance with the small uplift rates ex-pected from the lack of surface tectonic expressions, such as faulting, reported from geologicalmaps of Mars.1056.2. Future directionsMain outcomesThe results of Chapter 5 present the first quantitative evidence that the majority of valleynetworks did not reach steady state on early Mars, arguing against a long lived surface liquidwater hydrological system. In addition, insights into the curvature of the few steady statefluvial longitudinal profiles identified show that erosion rates were likely small on Mars in thefluvial systems, and that warm-based glaciation existed late into Mars evolution during theHesperian period. This chapter is in preparation for Icarus.6.2 Future directions6.2.1 Why are signs of ancient glaciation so rare on Mars’ surface?With the identification of widespread subglacial erosion on Mars in Chapter 4, a long standingargument against ancient glaciation on Mars becomes pressing: Why are geomorphologicalsigns of glacial erosion so rare? (Ansan and Mangold, 2006; Grotzinger et al., 2015; Wordsworth,2016). Predictions from climate models, starting either with a ”warm and wet or a ”cold andicy scenario, require widespread glaciation and glaciers to form on Mars (Wordsworth et al.,2015; Wordsworth, 2016), which is not observed. It is unreasonable to consider that this lack ofrecord is related to problems with preservation or exposure because it would affect equally thevalley networks, and it is equally unreasonable to think that glaciers never formed on Mars.Instead, my hypothesis is that we may have been looking for the wrong record of glaciationon Mars. I motivate this hypothesis with an observation from the high Arctic, presented infigure 6.1 below, in which glaciation patterns are obvious from panel (f), but not in panel (c),even though panel (c) belongs to the high Arctic in Devon Island and (f) belongs to northernQuebec. Indeed, no signs of sliding, no moraines, nor other obvious common signs of glacia-tion were apparent in the plateaus of Devon Island, over a range of different lithologies, eventhough Devon Island was under an ice sheet for longer than northern continental Canada. Thisobservation leads to the following hypothesis: Is it possible that subglacial meltwater drainage inDevon Island occurred through channels, which limited glacial sliding and consequent erosion? Andcould we be observing the same phenomena on the Martian surface? To build understanding, Iwill consider the mechanisms that control subglacial meltwater drainage, which ultimatelydrive ice sliding through ice lubrication and bed decoupling (Flowers and Clarke, 2002; Schoof ,2010). Glaciers and ice sheets can drain basal meltwater through distributed or channelizeddrainage 6.1. Subglacial meltwater is drawn into subglacial channels in panels (a), (b), (c),effectively draining the bed and coupling ice and ground. This is not the case for (d),(e),(f),where meltwater accumulates (blue layer) and lubricates the ice, accelerating sliding. An effi-cient subglacial channelized drainage, as shown in the figure (c and f), would potentially leavevery little morphological record of glaciation.1066.2. Future directionsFigure 6.1: Drainage of an ice sheet through channelized (left, a,b,c) and distributeddrainage (right, d,e,f). From left to right, cross-sectional perspective of the drainage,planar view, and resulting landscape signature, respectively. The blue layer repre-sents meltwater accumulation in the ice-bedrock interface.In a simplified view, the dynamic switch that controls the change from subglacial dis-tributed to channelized drainage occurs at a critical meltwater discharge Qc, which increaseslinearly with ice sliding velocity us according to (Schoof , 2010):Qc =4ρiLushΨ(6.1)Where ρi is the ice density, L is the latent heat of ice, h is ice thickness, and Ψ is the hydraulicgradient along the conduit or cavity. Ice sliding velocity is a factor ∼ 30 smaller on Mars dueto the lower gravity and the non-linear rheology of ice (us ∼ g3). Following this argument,we expect the onset of subglacial channelization to occur at smaller subglacial meltwater dis-charges on Mars than on Earth. This change in drainage regime may alter profoundly the wayice sheets behave on Mars and how they erode the substrate with respect to Earth, and mayjustify why sliding is not predominant on Mars.6.2.2 Does the location of younger valley networks follow the time evolution ofthe ice equilibrium line?Buffered crater counting techniques, together with detailed relative dating of valley networks(e.g., Fassett and Head, 2008a; Bouley and Craddock, 2014) have provided evidence that valleynetwork incision migrated from the heavily cratered highlands in the Noachain, to the north-ernmost margins of Arabia Terra and the rims of Valles Marineris in the Early Hesperian, and1076.2. Future directionsTable 6.1: Timing of incision for 30 valley networks. Ages are in Byr, MN corresponds toMiddle Noachian, LN corresponds to late Noachian, EH corresponds to Early Hes-perian, LH corresponds to Late Hesperian, and EA corresponds to Early Amazonian.Name lat long Neukum age Neukum periodLicus Vallis -3.15 126.30 3.83 LNunnamed 0.00 123.00 3.93 MNunnamed 8.0 131.00 3.74 LNunnamed 12.50 155.00 3.72 EHAl Qahira Vallis -18.21 162.40 3.86 MNEvros Valles -12.18 9.38 3.76 LNMeridiani 5.0 0.00 3.71 EHunnamed 5.0 9.0 3.79 LNunnamed 17.0 5.00 3.85 LNNaro Valles -3.96 60.60 3.9 MNParana Valles 22.75 -10.64 3.73 EHunnamed 7.0 13.0 3.79 LNunnamed 16.0 7.0 3.85 MNVichada Valles -18.90 89.10 3.8 LNunnamed 18.00 77.00 3.83 LNNaktong Valles 5.98 32.24 3.69 EHunnamed 16.0 62.50 3.75 LNTagus Valles -5.98 114.65 3.74 EHunnamed 22.0 132.50 3.7 EHunnamed 11.0 96.00 3.74 EHBrazos Valles -6.23 18.67 3.71 EHunnamed 9.0 65.00 3.81 LNCusus Valles 15.14 50.40 3.81 LNunnamed 11.0 162.0 3.79 LNWarrego Valles -42.76 -92.24 3.77 LNNanedi Vallis 6.70 -47.96 3.78 LNHecates Tholus 32.85 150.00 3.5 EACeranius Tholus 23.63 -96.33 3.55 LHEchus/ VM -1.04 -81.15 3.27 EAAlba Patera 44.88 -113.81 3.52 EAto the highly elevated volcanic flanks in Olympus Mons, Hecathes Tholus and Alba Paterain the Early Amazonian (Gulick, 2001) as indicated in table 6.1, adapted from Fassett and Head(2008a).Climate models (Wordsworth et al., 2015), as well as personal communications from Wordsworth(PC, 2018), confirm that the ice equilibrium line evolved in a similar manner through time asMars lost its atmosphere to space and the surface pressure dropped. For a wide range ofobliquity values, the high elevation areas around the Tharsis Montes, Alba Patera, and Ely-sium volcanic centers remained efficient cold traps until the Amazonian period, while thelower Highlands plateaus became drier after the Hesperian period (Wordsworth et al., 2015). In1086.2. Future directionscollaboration with R. Wordsworth, we will test the subglacial hypothesis by plotting the iceequilibrium line through time and the timing of incision of the valley networks, as shown infigure 6.2.Figure 6.2: Global topographic map of Mars (MOLA) overlapped with the location andtiming of incision of the valley networks presented in the previous table. Note howlate Hesperian and Early Amazonian drainage systems concentrate at the high el-evation areas around volcanic centers, whereas incision up to the Early Hesperianperiod is widespread.Subglacial erosion accumulates at the margins of the ice sheet, so we expect a high correla-tion between the ice equilibrium line predicted and the position of inferred sugblacial channelsat any moment of Mars history. If this correlation is observed for reasonable values of prim-itive atmospheric pressure and atmospheric escape rates (e.g., Lillis et al., 2015; Jakosky et al.,2015), this study would:1. Confirm the plausibility of the subglacial hypothesis presented in this thesis.2. Establish an independent constrain on atmospheric pressure on early Mars3. Suggest that an ice cover and geothermal heat flow high enough to provide basal melt-water accumulation existed on the Martian volcanoes during the Early Amazonian pe-riod.1096.2. Future directions6.2.3 How long does it take for fluvial erosion to become the predominanterosional fingerprint in a previously glaciated landscape?The scope of the analysis of terrestrial channel networks in Chapter 2 is limited to landscapesthat are in, or very close to, steady state. Capturing the transient response of a landscape, inparticular characterizing the transition between previously glaciated valleys to a purely flu-vial landscape, is a state of the art question in the surface process community. The west coastof Canada and the United States is an ideal setting to explore this question. The precise tim-ing of the retreat of the Cordilleran ice sheet from its extend at the Last Glacial Maximum,which covered almost all of British Columbia and extended south into the northwestern con-terminous United States, offers the opportunity to study this transient stages. Theoretically,one could, in principle, consider the steady state solution to the glacial landscape evolutionequation (equation 15 in Chapter 5) and find the solution for a small perturbation of upliftrate (Uo), ice thickness (Hi), or erosion coefficient (K) by means of a linear stability analysis.Non-dimensionalizing the result, it is possible to study the effects on the characteristic timescale of landscape response to changes in i.e., ice thickness, as the ice is retreating. Once theice disappears, capturing a fluvial erosion timescale should be in principle possible by non-dimensionalizing the stream power law (equation 2 in Chapter 5) and using as initial condi-tion the steady-state shape of the glacial profile and a known uplift rate. Observationally, thesame technique introduced in Chapter 2 can provide constrains on the timescale of responseof a previously glaciated landscape. Examining the fluvial valleys present between southernWashington state and southern Alaska in terms of the PCA, I could evaluate:1. Which of the fluvial valleys belong to the cluster of steady state fluvial systems (redgroup in figure 2.5), and how far north are they located?2. What kind of path do fluvial valleys follow in the PCA from a recently glaciated to asteady state fluvial landscape 6.3?3. Do valleys out of steady state appear with mixed signatures in the PCA?1106.2. Future directionsFigure 6.3: Hypothesized ”transient landscape” path between fluvial and glacial steadystate landscape configurations as captured by the PCA. The figure shows expecteddata from fluvial valleys spanning from Southern Washington to Alaska sampledevery 50 km moving northwards (transient valleys (?), in orange dots), together withmetric predictions for steady state fluvial and glacial systems.As in Chapter 2, learning about the transient response of landscapes on Earth can helpto shed light into the out-of-equilibrium valley networks on Mars, and address the question:How long were the valley networks active for?6.2.4 The landscape signature of glacial/interglacial cycles on MarsThe surface of Mars records the time-integrated landscape evolution history since the Noachian(locally up to the pre-Noachian (4.5- 4.1 Byr ago)) period. Super-imposed surface processeshave, since, modified the morphology and characteristics of the valley networks, obscuringtheir interpretation (Jakosky and Carr, 1985; Williams and Phillips, 2001; Gulick, 2001). Some ofthese mechanisms include eolian deposition, cratering and crater infill, hillslope, and perhapsmost notably the effects of glacial/ interglacial periods (Mustard et al., 2001; Head et al., 2003).These ice age cycles are driven by large chaotic obliquity variations spanning 15◦ - 35◦ (seeFigure 6.4), with a dominant periodicity of ∼ 100,000 kyr, similarly to Milankovitch cycles on1116.2. Future directionsEarth.Figure 6.4: Obliquity oscillations on Mars (black solid line) and Earth (red dashed line).Horizontal axis shows time in millions of years, and vertical axis shows degrees ofinclination. Dashed grey areas indicate periods of ice advance. Adapted from Headet al. (2003).The morphological overprint of such ice ages has undoubtedly modified the Martian land-scape, and it is unclear to which extend. A series of questions that follow from this argumentare: What are the landscape reactions to cycles of glaciation/ ice retreat, and how have theyaffected valley network morphology? Is this the reason why valley networks appear to be outof steady-state, as argued in Chapter 5? I will try to tackle this question both with insightsfrom erosion mechanics and from observations.Landscape evolution related to glaciationThe effects of glaciation on the landscape play a major role on its evolution, as discussed exten-sively in Chapters 2 and 3. In glacial periods, erosion may occur in the form of sliding glaciers(abrasion, plucking, etc.), subglacial meltwater processes including channels and linked cavi-ties, proglacial streams, and frost weathering depending on the climate conditions and waterbudget at the moment of the obliquity-driven glaciation. The cold and dry conditions expectedfor much of the history of Mars would, however, damper most of these processes.The landscape response to cold-based glaciation is significantly less understood. signifi-cant snow and ice infill of the valley networks alters the stress distribution of the VN walls,which could lead to a weakening and collapse of the side wall upon ice retreat. Such processwould modify VN morphology by increasing the system width, affecting two of the metricsdefined above: Aspect ratio R and minimum width λ. Valley networks on Mars would appear1126.2. Future directionswider than terrestrial counterparts. Several issues arise here. If such process had been in place,then valley network width would vary considerably along a system, often irregularly depend-ing on the bedrock strength at a given area, and width variations would be correlated withdepth. This is not consistent with observations reporting a quasi-constant width downstreamin valley networks, although more detailed tests of this idea are required to make a robust case.The other issue is with removal of the material at the site of collapse. Without active fluvial orglacial processes able to transport large boulders and fragments of bedrock, one would expectthe deposits to be left in-situ. Large deposits of fragmented bedrock lying on the side of thevalley networks are, again, not consistent with observations. Finally, an age dependence onwidth would also be expected. Prior to more extensive and detailed tests of these factors, itappears that this mechanism did not modify valley network morphology considerably.Another ice-age related resurfacing mechanism (Mustard et al., 2001; Head et al., 2003, e.g.,)consists on the deposition of water ice-rich mantling deposits. These deposits are typicallymeters in thickness, and have a strong latitude dependence (from mid-latitudes to the polesin both hemispheres) (Head et al., 2003). It is possible that the presence of mantle depositsobscures the smaller tributaries at latitudes larger than 30◦S. Although most of the valley net-works distribute between 30◦S and ◦N, a possible way to address this point would be byinterrogating the latitude dependence of minimum width (addressing degradation of smallertributaries) and stream order (addressing the fragmentation or degradation of fragments ofthe network).Other mechanisms, such as thermokarst reworking of the surface caused when permafrostundergoes cycles of freezing and sublimation (Soare et al., 2017, e.g.,), would further debilitatethe ground and potentially accelerate hillslope diffusion. The extend to which these mecha-nisms would degrade the final morphology of valley networks is thus dependent on latitude,bedrock or regolith properties, and the initial geometry of valley walls. Further studies shouldaddress modeling and experimental aspects of these processes, and ultimately evaluate eachvalley network independently.113BibliographyAllen, C. C. (1979), Volcano-ice interactions on Mars, Journal of Geophysical Research: Solid Earth,84(B14), 8048–8059.Ansan, V., and N. Mangold (2006), New observations of Warrego Valles, Mars: Evidence forprecipitation and surface runoff, Planetary and Space Science, 54(3), 219–242.Bahr, D. B. (1997a), Width and length scaling of glaciers, Journal of Glaciology, 43(145), 557–562.Bahr, D. B. (1997b), Global distributions of glacier properties: a stochastic scaling paradigm,Water Resources Research, 33(7), 1669–1679.Bahr, D. B., M. F. Meier, and S. D. Peckham (1997), The physical basis of glacier volume-areascaling, Journal of Geophysical Research: Solid Earth, 102(B9), 20,355–20,362.Baker, V. (1981), The geomorphology of Mars., Progress in Physical Geography, 5(4), 473–513.Baker, V., R. Strom, V. Gulick, J. Kargel, G. Komatsu, and V. Kale (1991), Ancient oceans, icesheets and the hydrological cycle on Mars.Baker, V. R. (1982), The channels of Mars, Research supported by NASA, NSF, and Australian-American Educational Foundation Austin, TX, University of Texas Press, 1982. 204 p.Baker, V. R. (2001), Water and the Martian landscape, Nature, 412(6843), 228–236.Baker, V. R., and J. B. Partridge (1986), Small Martian valleys: Pristine and degraded morphol-ogy, Journal of Geophysical Research: Solid Earth, 91(B3), 3561–3572.Baker, V. R., M. H. Carr, V. C. Gulick, C. R. Williams, and M. S. Marley (1992), Channels andvalley networks, in Mars, University of Arizona Press, Space Science Series.Barnhart, C. J., A. D. Howard, and J. M. Moore (2009), Long-term precipitation and late-stagevalley network formation: Landform simulations of Parana Basin, Mars, Journal of Geophys-ical Research: Planets, 114(E1).Bates, R. L., J. A. Jackson, et al. (1984), Dictionary of geological terms, vol. 584, Anchor Books.Beaney, C. L., and J. Shaw (2000), The subglacial geomorphology of southeast Alberta: evi-dence for subglacial meltwater erosion, Canadian Journal of Earth Sciences, 37(1), 51–61.Bear, J. (2013), Dynamics of fluids in porous media, Courier Corporation.Beaud, F., G. E. Flowers, and J. G. Venditti (2016), Efficacy of bedrock erosion by subglacialwater flow, Earth Surface Dynamics, 4(1).114BIBLIOGRAPHYBeaumont, C., P. Fullsack, and J. Hamilton (1992), Erosional control of active compressionalorogens, in Thrust tectonics, pp. 1–18, Springer.Bjerklie, D. M. (2007), Estimating the bankfull velocity and discharge for rivers using remotelysensed river morphology information, Journal of Hydrology, 341(3-4), 144–155.Booth, D. B., and B. Hallet (1993), Channel networks carved by subglacial water: Observationsand reconstruction in the eastern Puget Lowland of Washington, Geological Society of AmericaBulletin, 105(5), 671–683.Bouley, S., and R. A. Craddock (2014), Age dates of valley network drainage basins and sub-basins within Sabae and Arabia Terrae, Mars, Journal of Geophysical Research: Planets, 119(6),1302–1310.Boulton, G., R. Lunn, P. Vidstrand, and S. Zatsepin (2007), Subglacial drainage bygroundwater–channel coupling, and the origin of esker systems: part iitheory and simu-lation of a modern system, Quaternary Science Reviews, 26(7), 1091–1105.Boulton, G. S. (1979), Processes of glacier erosion on different substrata, Journal of glaciology,23(89), 15–38.Braun, J., D. Zwartz, and J. H. Tomkin (1999), A new surface-processes model combiningglacial and fluvial erosion, Annals of Glaciology, 28, 282–290.Brennand, T. A. (1994), Macroforms, large bedforms and rhythmic sedimentary sequencesin subglacial eskers, south-central Ontario: implications for esker genesis and meltwaterregime, Sedimentary Geology, 91(1-4), 9–55.Bue, B. D., and T. F. Stepinski (2006), Automated classification of landforms on Mars, Computers& Geosciences, 32(5), 604–614.Carr, M. H. (1995), The Martian drainage system and the origin of valley networks and frettedchannels, Journal of Geophysical Research: Planets, 100(E4), 7479–7507.Carr, M. H. (2007), The surface of Mars, vol. 6, Cambridge University Press.Carr, M. H., and F. C. Chuang (1997), Martian drainage densities, Journal of Geophysical Research:Planets, 102(E4), 9145–9152.Carr, M. H., and G. D. Clow (1981), Martian channels and valleys: Their characteristics, distri-bution, and age, Icarus, 48(1), 91–117.Carr, M. H., and J. W. Head (2003), Basal melting of snow on early Mars: A possible origin ofsome valley networks, Geophysical Research Letters, 30(24).Carr, M. H., and M. C. Malin (2000), Meter-scale characteristics of martian channels and val-leys, Icarus, 146(2), 366–386.Carrigy, M. A. (1970), Experiments on the angles of repose of granular materials, Sedimentology,14(3-4), 147–158.Chanson, H. (2004), Hydraulics of open channel flow, Butterworth-Heinemann.115BIBLIOGRAPHYChen, A., J. Darbon, and J.-M. Morel (2014), Landscape evolution models: A review of theirfundamental equations, Geomorphology, 219, 68–86.Clapperton, C. M. (1968), Channels formed by the superimposition of glacial meltwaterstreams, with special reference to the East Cheviot Hills, North-East England, GeografiskaAnnaler: Series A, Physical Geography, 50(4), 207–220.Clark, C. D., D. J. Evans, A. Khatwa, T. Bradwell, C. J. Jordan, S. H. Marsh, W. A. Mitchell, andM. D. Bateman (2004), Map and GIS database of glacial landforms and features related tothe last British Ice Sheet, Boreas, 33(4), 359–375.Clarke, G. K. (2003), Hydraulics of subglacial outburst floods: new insights from the Spring-Hutter formulation, Journal of Glaciology, 49(165), 299–313.Cle´ment, P., and H. Pie´gay (2003), Statistics and fluvial geomorphology, Tools in fluvial geomor-phology, pp. 597–630.Clifford, S. M. (1993), A model for the hydrologic and climatic behavior of water on Mars,Journal of Geophysical Research: Planets, 98(E6), 10,973–11,016.Cofaigh, C. O´. (1996), Tunnel valley genesis, Progress in Physical Geography, 20(1), 1–19.Craddock, R. A., and A. D. Howard (2002), The case for rainfall on a warm, wet early Mars,Journal of Geophysical Research: Planets, 107(E11).Crowe, C. T., D. F. Elger, B. C. Williams, and J. A. Roberson (2009), Engineering Fluid Mechanics,John Wiley and Sons, Inc.Culling, W. (1960), Analytical theory of erosion, The Journal of Geology, 68(3), 336–344.Davis, W. (1892), The convex profile of bad-land divides, Science, (508), 245–245.Denton, G. H., M. L. Prentice, D. E. Kellogg, and T. B. Kellogg (1984), Late Tertiary history ofthe Antarctic ice sheet: Evidence from the Dry Valleys, Geology, 12(5), 263–267.Derrida, B., and V. Hakim (1992), Needle models of Laplacian growth, Physical Review A,45(12), 8759.Devauchelle, O., L. Malverti, E. Lajeunesse, P.-Y. Lagre´e, C. Josserand, and K.-D. N. Thu-Lam(2010), Stability of bedforms in laminar flows with free surface: from bars to ripples, Journalof Fluid Mechanics, 642, 329–348.Devauchelle, O., A. Petroff, A. Lobkovsky, and D. H. Rothman (2011), Longitudinal profile ofchannels cut by springs, Journal of Fluid Mechanics, 667, 38–47.Devauchelle, O., A. P. Petroff, H. F. Seybold, and D. H. Rothman (2012), Ramification of streamnetworks, Proceedings of the National Academy of Sciences, 109(51), 20,832–20,836.Dunne, T., et al. (1990), Hydrology, mechanics, and geomorphic implications of erosion bysubsurface flow., Special Paper-Geological Society of America, (252), 1–28.Dyke, A. (1999), Last glacial maximum and deglaciation of Devon Island, Arctic Canada: sup-port for an Innuitian ice sheet, Quaternary Science Reviews, 18(3), 393–420.116BIBLIOGRAPHYDyke, A. S. (1993), Landscapes of cold-centred Late Wisconsinan ice caps, Arctic Canada,Progress in Physical Geography, 17(2), 223–247.Egholm, D., V. K. Pedersen, M. F. Knudsen, and N. K. Larsen (2012), Coupling the flow of ice,water, and sediment in a glacial landscape evolution model, Geomorphology, 141, 47–66.Ehlmann, B. L., J. F. Mustard, S. L. Murchie, J.-P. Bibring, A. Meunier, A. A. Fraeman, andY. Langevin (2011), Subsurface water and clay mineral formation during the early history ofMars, Nature, 479(7371), 53.Ehlmann, B. L., G. Berger, N. Mangold, J. R. Michalski, D. C. Catling, S. W. Ruff, E. Chassefie`re,P. B. Niles, V. Chevrier, and F. Poulet (2013), Geochemical consequences of widespread claymineral formation in Mars ancient crust, Space Science Reviews, 174(1-4), 329–364.Engelund, F., and E. Hansen (1967), A monograph on sediment transport in alluvial streams,Tech. rep., TEKNISKFORLAG Skelbrekgade 4 Copenhagen V, Denmark.England, J. (1987), Glaciation and the evolution of the Canadian high Arctic landscape, Geol-ogy, 15(5), 419–424.England, J., N. Atkinson, J. Bednarski, A. Dyke, D. Hodgson, and C. O´. Cofaigh (2006), TheInnuitian Ice Sheet: configuration, dynamics and chronology, Quaternary Science Reviews,25(7), 689–703.Fassett, C. I., and J. W. Head (2008a), Valley network-fed, open-basin lakes on Mars: Distribu-tion and implications for Noachian surface and subsurface hydrology, Icarus, 198(1), 37–56.Fassett, C. I., and J. W. Head (2008b), The timing of martian valley network activity: Con-straints from buffered crater counting, Icarus, 195(1), 61–89.Fassett, C. I., and J. W. Head (2011), Sequence and timing of conditions on early Mars, Icarus,211(2), 1204–1214.Fastook, J. L., and J. W. Head (2015), Glaciation in the Late Noachian Icy Highlands: Ice accu-mulation, distribution, flow rates, basal melting, and top-down melting rates and patterns,Planetary and Space Science, 106, 82–98.Fastook, J. L., J. W. Head, D. R. Marchant, F. Forget, and J.-B. Madeleine (2012), Early Marsclimate near the Noachian–Hesperian boundary: Independent evidence for cold conditionsfrom basal melting of the south polar ice sheet (Dorsa Argentea Formation) and implicationsfor valley network formation, Icarus, 219(1), 25–40.Ferrier, K. L., K. L. Huppert, and J. T. Perron (2013), Climatic control of bedrock river incision,Nature, 496(7444), 206.Feulner, G. (2012), The faint young Sun problem, Reviews of Geophysics, 50(2).Flowers, G. E., and G. K. Clarke (2002), A multicomponent coupled model of glacier hydrology1. Theory and synthetic examples, Journal of Geophysical Research: Solid Earth, 107(B11).Forget, F., R. Wordsworth, E. Millour, J.-B. Madeleine, L. Kerber, J. Leconte, E. Marcq, and R. M.Haberle (2013), 3D modelling of the early Martian climate under a denser CO 2 atmosphere:Temperatures and CO 2 ice clouds, Icarus, 222(1), 81–99.117BIBLIOGRAPHYFowler, A. (1981), A theoretical treatment of the sliding of glaciers in the absence of cavitation,Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineer-ing Sciences, 298(1445), 637–681.French, H. M. (2013), The periglacial environment, John Wiley & Sons.Gilbert, G. K. (1877), Report on the Geology of the Henry Mountains, US Government PrintingOffice.Gilbert, G. K. (1909), The convexity of hilltops, The Journal of Geology, 17(4), 344–350.Glasser, N. F., M. R. Bennett, and D. Huddart (1999), Distribution of glaciofluvial sedimentwithin and on the surface of a high arctic valley glacier: Marthabreen, Svalbard, Earth SurfaceProcesses and Landforms, 24(4), 303–318.Gleason, C. J. (2015), Hydraulic geometry of natural rivers: A review and future directions,Progress in Physical Geography, 39(3), 337–360.Glen, J. (1958), The flow law of ice: A discussion of the assumptions made in glacier theory,their experimental foundations and consequences, IASH Publ, 47, 171–183.Golombek, M., and N. Bridges (2000), Erosion rates on Mars and implications for climatechange: Constraints from the Pathfinder landing site, Journal of Geophysical Research: Planets,105(E1), 1841–1853.Gough, D. (1981), Solar interior structure and luminosity variations, in Physics of Solar Varia-tions, pp. 21–34, Springer.Grau Galofre, A., and M. A. Jellinek (2017), The geometry and complexity of spatial patternsof terrestrial channel networks: Distinctive fingerprints of erosional regimes., Journal of Geo-physical Research: Earth Surface, 122(4), 1037–1059.Grau Galofre, A., A. M. Jellinek, G. Osinski, M. Zanetti, and A. Kukko (2018), Subglacialdrainage patterns of Devon Island, Canada: Detailed comparison of river and subglacialchannels, The Cryosphere, 12, 1461–1478.Greenwood, S. L., C. D. Clark, and A. L. Hughes (2007), Formalising an inversion methodol-ogy for reconstructing ice-sheet retreat patterns from meltwater channels: application to theBritish Ice Sheet, Journal of Quaternary Science, 22(6), 637–645.Grotzinger, J., S. Gupta, M. Malin, D. Rubin, J. Schieber, K. Siebach, D. Sumner, K. Stack,A. Vasavada, R. Arvidson, et al. (2015), Deposition, exhumation, and paleoclimate of anancient lake deposit, Gale crater, Mars, Science, 350(6257), aac7575.Group, M. C. W. (1983), Channels and valleys on Mars, Geological Society of America Bulletin,94(9), 1035–1054.Gulick, V. C. (1993), Magmatic intrusions and hydrothermal systems: Implications for theformation of Martian fluvial valleys.Gulick, V. C. (2001), Origin of the valley networks on Mars: A hydrological perspective, Geo-morphology, 37(3-4), 241–268.118BIBLIOGRAPHYGulick, V. C., and V. R. Baker (1990), Origin and evolution of valleys on Martian volcanoes,Journal of Geophysical Research: Solid Earth, 95(B9), 14,325–14,344.Hack, J. T. (1957), Studies of longitudinal stream profiles in Virginia and Maryland, vol. 294, USGovernment Printing Office.Hack, J. T. (1973), Stream-profile analysis and stream-gradient index, Journal of Research of theUS Geological Survey, 1(4), 421–429.Halevy, I., and J. W. Head III (2014), Episodic warming of early Mars by punctuated volcanism,Nature Geoscience, 7(12), 865.Hallet, B. (1979), A theoretical model of glacial abrasion, Journal of Glaciology, 23(89), 39–50.Hallet, B. (1981), Glacial abrasion and sliding: their dependence on the debris concentration inbasal ice, Annals of Glaciology, 2(1), 23–28.Harrelson, C. C., C. L. Rawlins, and J. P. Potyondy (1994), Stream channel reference sites: anillustrated guide to field technique, Gen. Tech. Rep. RM-245. Fort Collins, CO: US Departmentof Agriculture, Forest Service, Rocky Mountain Forest and Range Experiment Station. 61 p., 245.Head, J., G. Neukum, R. Jaumann, H. Hiesinger, E. Hauber, M. Carr, P. Masson, B. Foing,H. Hoffmann, M. Kreslavsky, et al. (2005), Tropical to mid-latitude snow and ice accumula-tion, flow and glaciation on Mars, Nature, 434(7031), 346–351.Head, J. W., and D. R. Marchant (2003), Cold-based mountain glaciers on Mars: western ArsiaMons, Geology, 31(7), 641–644.Head, J. W., and D. R. Marchant (2014), The climate history of early Mars: insights from theAntarctic McMurdo Dry Valleys hydrologic system, Antarctic Science, 26(6), 774–800.Head, J. W., H. Hiesinger, M. A. Ivanov, M. A. Kreslavsky, S. Pratt, and B. J. Thomson (1999),Possible ancient oceans on Mars: evidence from Mars Orbiter Laser Altimeter data, Science,286(5447), 2134–2137.Head, J. W., J. F. Mustard, M. A. Kreslavsky, R. E. Milliken, and D. R. Marchant (2003), Recentice ages on Mars, Nature, 426(6968), 797.Henderson, F. M. (1961), Stability of alluvial channels, Journal of the Hydraulics Division, 87(6),109–138.Herman, F., and J. Braun (2008), Evolution of the glacial landscape of the Southern Alps ofNew Zealand: Insights from a glacial erosion model, Journal of Geophysical Research: EarthSurface, 113(F2).Herman, F., F. Beaud, J.-D. Champagnac, J.-M. Lemieux, and P. Sternai (2011), Glacial hydrol-ogy and erosion patterns: a mechanism for carving glacial valleys, Earth and Planetary ScienceLetters, 310(3), 498–508.Herman, F., J. Braun, E. Deal, and G. Prasicek (2018), The response time of glacial erosion,Journal of Geophysical Research: Earth Surface, 123(4), 801–817.Hewitt, I. J. (2011), Modelling distributed and channelized subglacial drainage: the spacing ofchannels, Journal of Glaciology, 57(202), 302–314.119BIBLIOGRAPHYHirschmann, M. M., and A. C. Withers (2008), Ventilation of CO2 from a reduced mantle andconsequences for the early Martian greenhouse, Earth and Planetary Science Letters, 270(1-2),147–155.Hoke, M. R., B. M. Hynek, and G. E. Tucker (2011), Formation timescales of large Martianvalley networks, Earth and Planetary Science Letters, 312(1), 1–12.Horton, R. E. (1932), Drainage-basin characteristics, Eos, Transactions American GeophysicalUnion, 13(1), 350–361.Horton, R. E. (1945), Erosional development of streams and their drainage basins; hydrophysi-cal approach to quantitative morphology, Geological society of America bulletin, 56(3), 275–370.Howard, A. D. (1971), Optimal angles of stream junction: Geometric, stability to capture, andminimum power criteria, Water Resour. Res, 7(4), 863–873.Howard, A. D. (1988), Groundwater sapping experiments and modeling, Sapping Features ofthe Colorado Plateau: A Comparative Planetary Geology Field Guide, 491, 71–83.Howard, A. D. (1994), A detachment-limited model of drainage basin evolution, Water re-sources research, 30(7), 2261–2285.Howard, A. D. (2007), Simulating the development of Martian highland landscapes throughthe interaction of impact cratering, fluvial erosion, and variable hydrologic forcing, Geomor-phology, 91(3-4), 332–363.Howard, A. D., and G. Kerby (1983), Channel changes in badlands, Geological Society of AmericaBulletin, 94(6), 739–752.Howard, A. D., J. M. Moore, and R. P. Irwin (2005), An intense terminal epoch of widespreadfluvial activity on early Mars: 1. Valley network incision and associated deposits, Journal ofGeophysical Research: Planets, 110(E12).Hutter, K. (1982), Dynamics of glaciers and large ice masses, Annual Review of Fluid Mechanics,14(1), 87–130.Hutter, K. (2017), Theoretical glaciology: material science of ice and the mechanics of glaciers and icesheets, vol. 1, Springer.Hynek, B. M., and R. J. Phillips (2003), New data reveal mature, integrated drainage systemson Mars indicative of past precipitation, Geology, 31(9), 757–760.Hynek, B. M., M. Beach, and M. R. Hoke (2010), Updated global map of Martian valley net-works and implications for climate and hydrologic processes, Journal of Geophysical Research:Planets, 115(E9).Irwin, R. P., A. D. Howard, R. A. Craddock, and J. M. Moore (2005), An intense terminal epochof widespread fluvial activity on early Mars: 2. Increased runoff and paleolake develop-ment, Journal of Geophysical Research: Planets, 110(E12).Iverson, N. R. (2012), A theory of glacial quarrying for landscape evolution models, Geology,40(8), 679–682.120BIBLIOGRAPHYJakosky, B. M., and M. H. Carr (1985), Possible precipitation of ice at low latitudes of Marsduring periods of high obliquity, Nature, 315(6020), 559.Jakosky, B. M., J. M. Grebowsky, J. G. Luhmann, and D. A. Brain (2015), Initial results from theMAVEN mission to Mars, Geophysical Research Letters, 42(21), 8791–8802.Jellinek, A., C. Johnson, and G. Schubert (2008), Constraints on the elastic thickness, heat flow,and melt production at early Tharsis from topography and magnetic field observations, Jour-nal of Geophysical Research: Planets, 113(E9).Kamb, B. (1993), Glacier flow modeling, in Flow and Creep in the Solar System: Observations,Modeling and Theory, pp. 417–506, Springer.Kargel, J. S., and R. G. Strom (1992), Ancient glaciation on Mars, Geology, 20(1), 3–7.Katul, G., P. Wiberg, J. Albertson, and G. Hornberger (2002), A mixing layer theory for flowresistance in shallow streams, Water Resources Research, 38(11).Kehew, A. E., J. A. Piotrowski, and F. Jørgensen (2012), Tunnel valleys: Concepts and contro-versiesA review, Earth-Science Reviews, 113(1), 33–58.Kennedy, J., et al. (1975), Hydraulic relations for alluvial streams, Sedimentation engineering:American Society of Civil Engineers Manual, 54, 114–154.Kirchner, J. W. (1993), Statistical inevitability of Horton’s laws and the apparent randomnessof stream channel networks, Geology, 21(7), 591–594.Kirkby, M. (1971), Hillslope process-response models based on the continuity equation.Kleinhans, M., H. Markies, S. De Vet, F. Postema, et al. (2011), Static and dynamic angles ofrepose in loose granular materials under reduced gravity, Journal of Geophysical Research:Planets, 116(E11).Kleinhans, M. G. (2005), Flow discharge and sediment transport models for estimating a mini-mum timescale of hydrological activity and channel and delta formation on Mars, Journal ofGeophysical Research: Planets, 110(E12).Kleman, J. (1992), The palimpsest glacial landscape in northwestern Sweden. Late Weichseliandeglaciation landforms and traces of older west-centered ice sheets, Geografiska Annaler. Se-ries A. Physical Geography, pp. 305–325.Kochel, R. C., and J. F. Piper (1986), Morphology of large valleys on Hawaii: Evidence forgroundwater sapping and comparisons with Martian valleys, Journal of Geophysical Research:Solid Earth, 91(B13).Kor, P., J. Shaw, and D. Sharpe (1991), Erosion of bedrock by subglacial meltwater, GeorgianBay, Ontario: a regional view, Canadian Journal of Earth Sciences, 28(4), 623–642.Kreslavsky, M. A., and J. W. Head (2000), Kilometer-scale roughness of Mars: Results fromMOLA data analysis, Journal of Geophysical Research: Planets, 105(E11), 26,695–26,711.Kukko, A., H. Kaartinen, J. Hyyppa¨, and Y. Chen (2012), Multiplatform mobile laser scanning:Usability and performance, Sensors, 12(9), 11,712–11,733.121BIBLIOGRAPHYLaity, J. E. (1990), Spring sapping and valley network development, Groundwater Geomorphol-ogy: The role of subsurface water in earth-surface processes and landforms, 252, 235–258.Laity, J. E., and M. C. Malin (1985), Sapping processes and the development of theater-headedvalley networks on the Colorado Plateau, Geological Society of America Bulletin, 96(2), 203–217.Lamb, M. P., A. D. Howard, J. Johnson, K. X. Whipple, W. E. Dietrich, and J. T. Perron (2006),Can springs cut canyons into rock?, Journal of Geophysical Research: Planets, 111(E7).Lane, E. W. (1955), Design of stable channels, Transactions of the American Society of Civil Engi-neers, 120(1), 1234–1260.Laskar, J., A. Correia, M. Gastineau, F. Joutel, B. Levrard, and P. Robutel (2004), Long termevolution and chaotic diffusion of the insolation quantities of Mars, Icarus, 170(2), 343–364.Le Heron, D. P., J. Craig, and J. L. Etienne (2009), Ancient glaciations and hydrocarbon accu-mulations in North Africa and the Middle East, Earth-Science Reviews, 93(3), 47–76.Le Meur, E., O. Gagliardini, T. Zwinger, and J. Ruokolainen (2004), Glacier flow modelling: acomparison of the Shallow Ice Approximation and the full-Stokes solution, Comptes RendusPhysique, 5(7), 709–722.Lee, P. (1997), A Unique Mars/Early Mars Analog on Earth: The Haughton Impact StructureDevon Island, Canadian Arctic, in Early Mars: Geologic and Hydrologic Evolution, Physical andChemical Environments, and the Implications for Life, vol. 916, p. 50.Lee, P., and G. R. Osinski (2005), The Haughton-Mars Project: Overview of science investiga-tions at the Haughton impact structure and surrounding terrains, and relevance to planetarystudies, Meteoritics & Planetary Science, 40(12), 1755–1758.Lehner, B., K. Verdin, and A. Jarvis (2006), HydroSHEDS technical documentation, version 1.0,World Wildlife Fund US, Washington, DC, pp. 1–27.Leopold, L. B. (1970), Landscape aesthetics, Ekistics, 29(l73), 27l–277.Leopold, L. B., and T. Maddock (1953), The hydraulic geometry of stream channels and some phys-iographic implications, vol. 252, US Government Printing Office.Levy, J. (2012), Hydrological characteristics of recurrent slope lineae on Mars: Evidence for liq-uid flow through regolith and comparisons with Antarctic terrestrial analogs, Icarus, 219(1),1–4.Liang, X., Y. Wang, A. Jaakkola, A. Kukko, H. Kaartinen, J. Hyyppa¨, E. Honkavaara, and J. Liu(2015), Forest data collection using terrestrial image-based point clouds from a handheldcamera compared to terrestrial and personal laser scanning, IEEE transactions on geoscienceand remote sensing, 53(9), 5117–5132.Lillis, R. J., D. A. Brain, S. W. Bougher, F. Leblanc, J. G. Luhmann, B. M. Jakosky, R. Modolo,J. Fox, J. Deighan, X. Fang, et al. (2015), Characterizing atmospheric escape from Mars todayand through time, with MAVEN, Space Science Reviews, 195(1-4), 357–422.122BIBLIOGRAPHYLivingstone, S. J., and C. D. Clark (2016), Morphological properties of tunnel valleys of thesouthern sector of the Laurentide Ice Sheet and implications for their formation, Earth Sur-face Dynamics, 4(3), 567.Livingstone, S. J., W. Chu, J. C. Ely, and J. Kingslake (2017), Paleofluvial and subglacial channelnetworks beneath Humboldt Glacier, Greenland, Geology, 45(6), 551–554.Lliboutry, L. A. (1994), Monolithologic erosion of hard beds by temperate glaciers, Journal ofGlaciology, 40(136), 433–450.Lonergan, L., S. C. Maidment, and J. S. Collier (2006), Pleistocene subglacial tunnel valleys inthe central North Sea basin: 3-D morphology and evolution, Journal of Quaternary Science,21(8), 891–903.Malin, M. C., and M. H. Carr (1999), Groundwater formation of Martian valleys, Nature,397(6720), 589.Mandelbrot, B. B. (1983), The fractal geometry of nature, vol. 173, Macmillan.Manga, M., and J. W. Kirchner (2000), Stress partitioning in streams by large woody debris,Water Resources Research, 36(8), 2373–2379.Mangold, N., C. Quantin, V. Ansan, C. Delacourt, and P. Allemand (2004), Evidence for pre-cipitation on Mars from dendritic valleys in the Valles Marineris area, Science, 305(5680),78–81.Margold, M., K. N. Jansson, J. Kleman, A. P. Stroeven, and J. J. Clague (2013), Retreat pat-tern of the Cordilleran Ice Sheet in central British Columbia at the end of the last glaciationreconstructed from glacial meltwater landforms, Boreas, 42(4), 830–847.Masursky, H. (1973), An overview of geological results from Mariner 9, Journal of GeophysicalResearch, 78(20), 4009–4030.McCann, S., P. Howarth, and J. Cogley (1972), Fluvial processes in a periglacial environment:Queen Elizabeth Islands, NWT, Canada, Transactions of the Institute of British Geographers, pp.69–82.Meir, A., J. Moon, and J. Pounder (1980), On the order of random channel networks, SIAMJournal on Algebraic Discrete Methods, 1(1), 25–33.Montgomery, D. R. (2002), Valley formation by fluvial and glacial erosion, Geology, 30(11),1047–1050.Montgomery, D. R., and W. E. Dietrich (1992), Channel initiation and the problem of landscapescale, Science, 255(5046), 826.Moore, I. D., R. Grayson, and A. Ladson (1991), Digital terrain modelling: a review of hydro-logical, geomorphological, and biological applications, Hydrological processes, 5(1), 3–30.Morris, A. J., J. J. Donovan, and M. Strager (2009), Geospatial analysis of climatic and geomor-phic interactions influencing stream discharge, Appalachian Mountains, USA, Environmen-tal modeling & assessment, 14(1), 73–84.123BIBLIOGRAPHYMustard, J. F., C. D. Cooper, and M. K. Rifkin (2001), Evidence for recent climate change onMars from the identification of youthful near-surface ground ice, Nature, 412(6845), 411.Mustard, J. F., S. L. Murchie, S. Pelkey, B. Ehlmann, R. Milliken, J. A. Grant, J.-P. Bibring,F. Poulet, J. Bishop, E. N. Dobrea, et al. (2008), Hydrated silicate minerals on Mars observedby the Mars Reconnaissance Orbiter CRISM instrument, Nature, 454(7202), 305.Niemann, J. D., N. M. Gasparini, G. E. Tucker, and R. L. Bras (2001), A quantitative evalua-tion of Playfair’s law and its use in testing long-term stream erosion models, Earth SurfaceProcesses and Landforms, 26(12), 1317–1332.Nye, J. (1952), The mechanics of glacier flow, Journal of Glaciology, 2(12), 82–93.Nye, J. (1976), Water flow in glaciers: jo¨kulhlaups, tunnels and veins, Journal of Glaciology, 17,181–207.Osinski, G. R., and J. G. Spray (2005), Tectonics of complex crater formation as revealed by theHaughton impact structure, Devon Island, Canadian High Arctic, Meteoritics & PlanetaryScience, 40(12), 1813–1834.Parker, G. (1978a), Self-formed straight rivers with equilibrium banks and mobile bed. Part 1.The sand-silt river, Journal of Fluid Mechanics, 89(01), 109–125.Parker, G. (1978b), Self-formed straight rivers with equilibrium banks and mobile bed. Part 2.The gravel river, Journal of Fluid mechanics, 89(1), 127–146.Parker, G., S. Dhamotharan, and H. Stefan (1982), Model experiments on mobile, paved gravelbed streams, Water Resources Research, 18(5), 1395–1408.Parker, G., P. R. Wilcock, C. Paola, W. E. Dietrich, and J. Pitlick (2007), Physical basis forquasi-universal relations describing bankfull hydraulic geometry of single-thread gravelbed rivers, Journal of Geophysical Research: Earth Surface, 112(F4).Parker, T. J., D. S. Gorsline, R. S. Saunders, D. C. Pieri, and D. M. Schneeberger (1993), Coastalgeomorphology of the Martian northern plains, Journal of Geophysical Research: Planets,98(E6), 11,061–11,078.Paterson, W. (1994), The physics of glaciers, Butterworth-Heinemann.Pelletier, J. D., D. Comeau, and J. Kargel (2010), Controls of glacial valley spacing on Earth andMars, Geomorphology, 116(1), 189–201.Penido, J. C., C. I. Fassett, and S. M. Som (2013), Scaling relationships and concavity of smallvalley networks on Mars, Planetary and Space Science, 75, 105–116.Perron, J. T., and L. Royden (2013), An integral approach to bedrock river profile analysis,Earth Surface Processes and Landforms, 38(6), 570–576.Perron, J. T., J. X. Mitrovica, M. Manga, I. Matsuyama, and M. A. Richards (2007), Evidence foran ancient martian ocean in the topography of deformed shorelines, Nature, 447(7146), 840.Perron, J. T., W. E. Dietrich, and J. W. Kirchner (2008), Controls on the spacing of first-ordervalleys, Journal of Geophysical Research: Earth Surface, 113(F4).124BIBLIOGRAPHYPerron, J. T., J. W. Kirchner, and W. E. Dietrich (2009), Formation of evenly spaced ridges andvalleys, Nature, 460(7254), 502–505.Petroff, A. P., O. Devauchelle, D. M. Abrams, A. E. Lobkovsky, A. Kudrolli, and D. H. Rothman(2011), Geometry of valley growth, Journal of Fluid Mechanics, 673, 245–254.Petroff, A. P., O. Devauchelle, A. Kudrolli, and D. H. Rothman (2012), Four remarks on thegrowth of channel networks, Comptes Rendus Geoscience, 344(1), 33–40.Pfeffer, W. T., A. A. Arendt, A. Bliss, T. Bolch, J. G. Cogley, A. S. Gardner, J.-O. Hagen, R. Hock,G. Kaser, C. Kienholz, et al. (2014), The Randolph Glacier Inventory: a globally completeinventory of glaciers, Journal of Glaciology, 60(221), 537–552.Pieri, D. (1976), Distribution of small channels on the Martian surface, Icarus, 27(1), 25–50.Pieri, D. C. (1980), Martian valleys- Morphology, distribution, age, and origin, Science,210(4472), 895–897.Piotrowski, J. A., N. K. Larsen, J. Menzies, and W. Wysota (2006), Formation of subglacialtill under transient bed conditions: deposition, deformation, and basal decoupling under aWeichselian ice sheet lobe, central Poland, Sedimentology, 53(1), 83–106.Pollack, J. B., J. F. Kasting, S. M. Richardson, and K. Poliakoff (1987), The case for a wet, warmclimate on early Mars, Icarus, 71(2), 203–224.Price, R. (1960), Glacial meltwater channels in the upper Tweed drainage basin, The Geograph-ical Journal, 126(4), 483–489.Ramirez, R. M., and R. A. Craddock (2018), The geological and climatological case for awarmer and wetter early Mars, Nature Geoscience, 11(4), 230.Raup, B., A. Racoviteanu, S. J. S. Khalsa, C. Helm, R. Armstrong, and Y. Arnaud (2007), TheGLIMS geospatial glacier database: a new tool for studying glacier change, Global and Plan-etary Change, 56(1), 101–110.Reeh, N. (1982), A plasticity theory approach to the steady-state shape of a three-dimensionalice sheet, Journal of Glaciology, 28(100), 431–455.Ritter, J. B., J. R. Miller, Y. Enzel, and S. G. Wells (1995), Reconciling the roles of tectonism andclimate in Quaternary alluvial fan evolution, Geology, 23(3), 245–248.Rodrı´guez-Iturbe, I., and A. Rinaldo (2001), Fractal river basins: chance and self-organization,Cambridge University Press.Roe, G. H., K. X. Whipple, and J. K. Fletcher (2008), Feedbacks among climate, erosion, andtectonics in a critical wedge orogen, American Journal of Science, 308(7), 815–842.Roots, E., Y. Fortier, R. Blackadar, B. Glenister, H. Greiner, D. McLaren, N. McMillan, A. Norris,J. Souther, R. Thorsteinsson, and E. Tozer (1963), Geology of the North-Central Part of theArctic Archipelago, Northwest Territories (Operation Franklin), Geological Survey of Canada,Memoir 320, 164.Ro¨thlisberger, H. (1972a), Water pressure in intra-and subglacial channels, Journal of Glaciology,11(62), 177–203.125BIBLIOGRAPHYRo¨thlisberger, H. (1972b), Water Pressure in Intra-and Subglacial Channels: Pres. at the Sym-posium on the Hydrology of Glaciers, 7-13 September 1969, Cambridge, Ph.D. thesis.Rothman, D., D. Abrams, O. Devauchelle, A. Petroff, A. Lobkovsky, K. Straub, B. McElroy,D. Mohrig, and A. Kudrolli (2009), Growth of a dendritic channel network, in AGU FallMeeting Abstracts, vol. 1, p. 01.Savenije, H. H. (2003), The width of a bankfull channel; Lacey’s formula explained, Journal ofHydrology, 276(1), 176–183.Scanlon, K., J. Head, J. Fastook, and R. Wordsworth (2018), The Dorsa Argentea Formationand the Noachian-Hesperian climate transition, Icarus, 299, 339–363.Schoof, C. (2005), The effect of cavitation on glacier sliding, in Proceedings of the Royal Societyof London A: Mathematical, Physical and Engineering Sciences, vol. 461, pp. 609–627, The RoyalSociety.Schoof, C. (2010), Ice-sheet acceleration driven by melt supply variability, Nature, 468(7325),803–806.Sharp, R. P., and M. C. Malin (1975), Channels on Mars, Geological Society of America Bulletin,86(5), 593–609.Shaw, J. (2002), The meltwater hypothesis for subglacial bedforms, Quaternary International,90(1), 5–22.Shreve, R. (1972), Movement of water in glaciers, Journal of Glaciology, 11(62), 205–214.Sissons, J. (1961), A subglacial drainage system by the Tinto Hills, Lanarkshire, Transactions ofthe Edinburgh Geological Society, 18(2), 175–193.Sklar, L., and W. E. Dietrich (1998), River longitudinal profiles and bedrock incision models: Streampower and the influence of sediment supply, Wiley Online Library.Sklar, L. S., and W. E. Dietrich (2001), Sediment and rock strength controls on river incisioninto bedrock, Geology, 29(12), 1087–1090.Smith, M. J., J. Chandler, and J. Rose (2009), High spatial resolution data acquisition for thegeosciences: kite aerial photography, Earth Surface Processes and Landforms, 34(1), 155–161.Soare, R., S. Conway, C. Gallagher, and J. Dohm (2017), Ice-rich (periglacial) vs icy (glacial)depressions in the Argyre region, Mars: a proposed cold-climate dichotomy of landforms,Icarus, 282, 70–83.Somfai, E., and L. Sander (1997), Scaling and river networks: A Landau theory for erosion,Physical Review E, 56(1), R5.Soto, A., M. Mischna, T. Schneider, C. Lee, and M. Richardson (2015), Martian atmosphericcollapse: Idealized GCM studies, Icarus, 250, 553–569.Squyres, S. W. (1989), Urey prize lecture: Water on Mars, Icarus, 79(2), 229–288.Stepinski, T., and A. Stepinski (2005), Morphology of drainage basins as an indicator of climateon early Mars, Journal of Geophysical Research: Planets, 110(E12).126BIBLIOGRAPHYStepinski, T., M. Collier, P. McGovern, and S. Clifford (2004), Martian geomorphology fromfractal analysis of drainage networks, Journal of Geophysical Research: Planets, 109(E2).Strahler, A. N. (1958), Dimensional analysis applied to fluvially eroded landforms, GeologicalSociety of America Bulletin, 69(3), 279–300.Sugden, D. E., G. H. Denton, and D. R. Marchant (1991), Subglacial meltwater channel systemsand ice sheet overriding, Asgard Range, Antarctica, Geografiska Annaler. Series A. PhysicalGeography, pp. 109–121.Syverson, K. M., and D. M. Mickelson (2009), Origin and significance of lateral meltwaterchannels formed along a temperate glacier margin, Glacier Bay, Alaska, Boreas, 38(1), 132–145.Tanaka, K. L., J. A. Skinner, J. M. Dohm, R. P. Irwin III, E. J. Kolb, C. M. Fortezzo, T. Platz,G. G. Michael, and T. Hare (2014), Geologic map of Mars, US Department of the Interior, USGeological Survey.Taylor, G. J., L. M. Martel, S. Karunatillake, O. Gasnault, and W. V. Boynton (2010), MappingMars geochemically, Geology, 38(2), 183–186.Team, R. (1997), Characterization of the Martian surface deposits by the Mars Pathfinder rover,Sojourner, Science, 278(5344), 1765–1768.Thorsteinsson, R., and U. Mayr (1987), The sedimentary rocks of Devon Island, Canadian arcticarchipelago, Memoir/Canada. Geol. survey.Tomkin, J. H. (2009), Numerically simulating alpine landscapes: the geomorphologic conse-quences of incorporating glacial erosion in surface process models, Geomorphology, 103(2),180–188.Tonkin, T. N., N. G. Midgley, D. J. Graham, and J. Labadz (2014), The potential of smallunmanned aircraft systems and structure-from-motion for topographic surveys: A test ofemerging integrated approaches at Cwm Idwal, North Wales, Geomorphology, 226, 35–43.Toon, O. B., T. Segura, and K. Zahnle (2010), The formation of Martian river valleys by impacts,Annual Review of Earth and Planetary Sciences, 38, 303–322.Tucker, G. E., and R. L. Bras (1998), Hillslope processes, drainage density, and landscape mor-phology, Water Resources Research, 34(10), 2751–2764.Tulaczyk, S., W. B. Kamb, and H. F. Engelhardt (2000), Basal mechanics of ice stream B, WestAntarctica: 1. Till mechanics, Journal of Geophysical Research: Solid Earth, 105(B1), 463–481.Van der Veen, C. J. (2013), Fundamentals of glacier dynamics, CRC Press.Viviano-Beck, C. E., F. P. Seelos, S. L. Murchie, E. G. Kahn, K. D. Seelos, H. W. Taylor, K. Taylor,B. L. Ehlmann, S. M. Wiseman, J. F. Mustard, et al. (2014), Revised CRISM spectral param-eters and summary products based on the currently detected mineral diversity on Mars,Journal of Geophysical Research: Planets, 119(6), 1403–1431.Walder, J., and A. Fowler (1994), Channelized subglacial drainage over a deformable bed,Journal of Glaciology, 40(134).127BIBLIOGRAPHYWalder, J., and B. Hallet (1979), Geometry of former subglacial water channels and cavities,Journal of Glaciology, 23(89), 335–346.Weertman, J. (1972), General theory of water flow at the base of a glacier or ice sheet, Reviewsof Geophysics, 10(1), 287–333.Whipple, K. X. (2004), Bedrock rivers and the geomorphology of active orogens, Annu. Rev.Earth Planet. Sci., 32, 151–185.Whipple, K. X., and G. E. Tucker (1999), Dynamics of the stream-power river incision model:Implications for height limits of mountain ranges, landscape response timescales, and re-search needs, Journal of Geophysical Research: Solid Earth, 104(B8), 17,661–17,674.Whiting, P. J., and W. E. Dietrich (1990), Boundary shear stress and roughness over mobilealluvial beds, Journal of Hydraulic Engineering, 116(12), 1495–1511.Whittaker, A. C. (2012), How do landscapes record tectonics and climate?, Lithosphere, 4(2),160–164.Williams, R. M., and R. J. Phillips (2001), Morphometric measurements of martian valley net-works from Mars Orbiter Laser Altimeter (MOLA) data, Journal of Geophysical Research: Plan-ets, 106(E10), 23,737–23,751.Wilson, L., G. J. Ghatan, J. W. Head, and K. L. Mitchell (2004), Mars outflow channels: Areappraisal of the estimation of water flow velocities from water depths, regional slopes,and channel floor properties, Journal of Geophysical Research: Planets, 109(E9).Wordsworth, R., F. Forget, E. Millour, J. Head, J.-B. Madeleine, and B. Charnay (2013), Globalmodelling of the early martian climate under a denser CO 2 atmosphere: Water cycle andice evolution, Icarus, 222(1), 1–19.Wordsworth, R. D. (2016), The climate of early Mars, Annual Review of Earth and PlanetarySciences, 44, 381–408.Wordsworth, R. D., L. Kerber, R. T. Pierrehumbert, F. Forget, and J. W. Head (2015), Compar-ison of warm and wet and cold and icy scenarios for early Mars in a 3-D climate model,Journal of Geophysical Research: Planets, 120(6), 1201–1219.Young, K. E., M. C. Soest, K. V. Hodges, E. B. Watson, B. A. Adams, and P. Lee (2013), Impactthermochronology and the age of Haughton impact structure, Canada, Geophysical ResearchLetters, 40(15), 3836–3840.128Appendix AGoverning equations for flow andlandscape evolutionA.1 SummaryThis Appendix provides the basic equations governing flow dynamics and erosional regimesin the fluvial, glacial, sapping and subglacial regime. In detail, I first present the mass andmomentum conservation equations for each erosion mechanism, in turn. I then present thelandscape evolution equation and formulations of the erosion rates of each respective erosiontype.IntroductionThe network scale characteristics and geometry of the above mentioned channel networks,including glaciers, rivers, subglacial meltwater channels and sapping valleys, are genericallygoverned by 4 equations:1. Mass conservation2. Momentum conservation3. Constitutive law describing the rheology of ice and water flows4. Landscape evolution equations.All the notation introduced here is listed in table 2.1 and table 2.2 in Chapter 2.129A.2. General driving equationsA.2 General driving equationsA.2.1 Mass conservationThe continuity equation expresses mass conservation. Assuming that water and ice flowsare incompressible with constant density, with external sources, the resulting conservation ofvolume is then∇ · u = b˙, (A.1)where u represents the velocity field and b˙ compiles all source and sinks.A.2.2 Momentum conservationFor gravity currents much longer than they are deep it is straightforward to show that fullydeveloped water or ice flows are approximately hydrostatic. Under these conditions, momen-tum conservation is:ρ(u · ∇u) = fdrive +∇ ·([2ηe f f +τo|˙ε|]ε˙), (A.2)where fdrive are the driving forces acting in each individual type of flow, and ηe f f is the effectiveviscosity of the flow, and ε˙ is the rate of strain tensor. For sliding glaciers, a zero strain-rateyield stress τo represents the threshold basal stress (e.g., Tulaczyk et al., 2000; Schoof , 2005).We ignore the effects of drag induced by vegetation, banks and bed-forms, although they canhave significant effects retarding local stream flow, modifying the bed texture, and changingsediment transport properties (e.g., Manga and Kirchner, 2000).A.2.3 Constitutive law: RheologyA generic constitutive law relates the stress tensor τ to the rate of strain tensor ε˙ through aneffective viscosity:2ηe f f ε˙ = τ + τo, (A.3)where 2ηe f f = BpTτ−(p−1) is the effective viscosity (Glen, 1958; Kamb, 1993; Paterson, 1994). Theaverage power law exponent for water ice is p = 3 (Glen, 1958; Kamb, 1993), whereas for water,p = 1, τo = 0 and η = 1 Pa · s. For ice, BT is the temperature dependent hardness param-eter that we take to be constant, which is justified in the absence of very steep temperaturegradients that generally occur only near boundaries (Kamb, 1993).A.2.4 Landscape evolution equationIn general, the rate of change of topographic height ht depends on the rates of tectonic upliftU and mechanical erosion E, and is captured by a landscape evolution equation of the form130A.3. The fluvial regime(e.g., Chen et al., 2014):∂ht∂t= U − E(x, y). (A.4)Here the rates of tectonic uplift (U) and spatially-varying erosion (E(x, y)) compete to shapethe landscape.Figure 2.4 in Chapter 2 summarizes the underlying mechanics and mass balances of theflows through a sketch, and contains the definition of the coordinate systems. Table 2.1 inChapter 2 contains the notation and units of all the parameters in this subsection regardingflow mechanics.A.3 The fluvial regimeIn channelized fluvial systems (figure 2.4a), lateral gradients of hydrostatic pressure drivevigorous water flows that are retarded primarily as a result of the production of complexturbulent overturning motions, the dynamical effects of which are conventionally captured interms of a net turbulent drag (e.g., Parker, 1978a,b; Chanson, 2004). Assuming a steady and welldeveloped incompressible flow, momentum conservation can be expressed with the Darcy-Weisbach equation:ρgSh f = 1/2ρC f u2x. (A.5)Here S is the slope of the water surface, which for hydrostatic flows is equivalent to the topo-graphic slope such that S = So. ux is the downstream component of the velocity field, h f is thewater flow depth, and C f is the friction factor (Katul et al., 2002). For fully developed turbulentflows,C f = 0.18(zo/h f )2/7. (A.6)The formulation of the landscape metrics defined in Chapter 2 requires a relationship betweenS, L and W. To this end, we require one additional equation to equations A.5 and A.6 toclose the problem. We use a condition for the cross sectional mechanical stability of a channelwith bankfull width W. A conventional equilibrium hypothesis formulated by Lane (1955) andHenderson (1961) defines this stability in terms of a non dimensional stress θt that captures theratio of tangential to normal stresses acting on a channel bed with lateral walls inclined anangle φ. Accordingly, here we follow Devauchelle et al. (2010) and Devauchelle et al. (2011) todefine a modified generalized Shields stress parameter:θt =√(RρSh f /ds)2+ sin2(φ)cos2(φ), (A.7)where the normalized density Rρ = ρw/(ρs − ρw) ∼ 1.61 compares the density of water andsediment and ds ∼ 0.45 mm is the average grain size. Other and more complete forms ofthe dimensionless shear stress exist in the literature [e.g., Kirchner, 1990]. However, for ourpurpose, equation A.7 is a reasonable description of the stresses imparted at the channel bed,131A.4. Glacial flow regimewith the additional practical benefit of being analytically tractable. Shields parameters arecommonly used in the literature to describe the onset of granular flow, although in this studywe will limit its use to a description of the stress regime on the channel bed.Parametrizations for the long-time averaged fluvial erosion rate E f usually involve contri-butions from both local stream channel incision processes (e.g., Kirkby, 1971; Somfai and Sander,1997; Whipple and Tucker, 1999; Niemann et al., 2001; Roe et al., 2008), and diffusive hillslope ero-sion (e.g., Gilbert, 1909; Beaumont et al., 1992; Davis, 1892). Semi-empirical steady state erosionrules, defined as a balance between erosion and uplift, are typically of the form (e.g., Howard,1994; Tucker and Bras, 1998; Perron et al., 2008; Chen et al., 2014):E f = K f Am|∇ht|n − D∇2ht. (A.8)Here, the first term on the right hand side is the stream incision rate, modulated by an erosionalefficiency coefficient K f and the drainage area A. Channel incision is typically related to theerosional power delivered by turbulent shear stresses through a stream power law, where mand n are positive constants (e.g., Howard and Kerby, 1983; Howard, 1994; Whipple and Tucker,1999). The second term is a topographic diffusion driven by gradients in local elevation andmodulated by an effective diffusivity D, which depends on mass wasting processes includinglandslides and soil creep (e.g., Culling, 1960; Beaumont et al., 1992; Tucker and Bras, 1998; Hermanand Braun, 2008; Chen et al., 2014).A.4 Glacial flow regimeThe relatively sluggish flow of ice (Figure 2.4d) reflects the balance between a driving forcerelated to lateral differences in ice thickness and to viscous stresses arising in response to theproduction of internal velocity gradients (e.g., Kamb, 1993; Paterson, 1994). For glacial flowswhere the depth of the ice hi is much smaller than its length, we can use the depth averagedshallow ice equation to simplify equation A.1, and express volume conservation as:∂hi/∂t = b˙− ∂/∂xhi∫0uxdz− ∂/∂yhi∫0uydz. (A.9)With A.3 and A.9, the stress balance driving glacier flow is given by (e.g., Fowler, 1981; Hutter,1982; Bahr, 1997b; Le Meur et al., 2004; Van der Veen, 2013):ρigShi = 2ηe f f ε˙. (A.10)We will not discuss possible time-dependent or stick-slip dynamics in this paper.Glacial erosion includes a combination of complex mechanisms such as abrasion, quar-rying and chemical weathering (e.g., Hallet, 1979; Boulton, 1979; Hallet, 1981; Lliboutry, 1994).Neglecting chemical weathering effects, the rate of mechanical erosion can be written as a132A.5. The sapping valley regimecombination of abrasion Ea and plucking Ep:Eg = Ea + Ep. (A.11)Here we follow Egholm (Egholm et al. (2012), equation 46) and define glacial abrasion in termsof the contributions from sliding rate us at the ice-rock interface as:Ea = Kas f sc|us|2. (A.12)Here Ka is an abrasion efficiency, s f is the entrained sediment concentration and 0 ≤ sc ≤ 1accounts for the reduced erosion power in a sediment-covered bedrock. Applying equationA.3 to equation A.10, we obtain an order of magnitude for the sliding velocity:us ∝BsPi − Pw (ρig)phi pSp, (A.13)where Pi corresponds to the hydrostatic ice pressure at the ice-bed interface and Pw the pore-water pressure at the same location. Hillslope diffusive mechanisms become relevant in ice-free regions (Herman and Braun, 2008).In contrast to abrasion, quarrying involves bedrock failure and depends on water pres-sure fluctuations. Following Egholm (equations (47), (48) and (49) in Egholm et al. (2012)), thiscontribution to erosion is:Ep = Kpsc|us| f (us), (A.14)where the function f (us) = 1/2[erf(−us∇ht0.4|us|)+ 1]vanishes on steep reverse slopes and ap-proaches 1 on surfaces dipping in the downstream direction, indicating the mechanical effi-ciency of quarrying, and Kp = 10−4 is an empirical parameter. Assuming shallow slopes, weTaylor expand to first order in ∇ht to obtain f (us) ' 1/2(erf(1)− 2/e+ 1).Finally, with equations A.12 and A.14, the full steady-state landscape evolution equationfor glacial erosion becomes:E = Kas f sc|us|2 + Kpsc|us| f (us) (A.15)with the additional hillslope diffusion term from equation A.11, Eg = −D∇2ht, active in ice-free regions (Egholm et al., 2012).A.5 The sapping valley regimeSapping valleys (Figure 2.4b) form when the height of the water table locally intersects theground level. The emergence of groundwater in springs can both incise channels, that growforward in the direction in which groundwater enters the tip, and contribute to the overlandflow (e.g., Laity and Malin, 1985; Devauchelle et al., 2012). In contrast to fluvial and glacialflows, mass conservation in this regime requires the squared water table elevation hw to satisfy133A.6. The subglacial regimePoisson’s equation [Devauchelle et al. 2010, equation 2.2]:∇2h2w = −2PrKh. (A.16)Here, the square of the water table height hw depends on the potential 2Pr/Kh, where Pr isthe precipitation rate and Kh is the soil hydraulic conductivity. Within a few meters from aspring, the flow is fully turbulent and equation A.5 is a suitable description of momentumconservation (Devauchelle et al., 2011).From observational data, groundwater sapping valleys show three different erosional regimesdepending on distance to the spring. Near the source, weathering and erosion concentrate atthe spring (e.g., Derrida and Hakim, 1992; Devauchelle et al., 2011; Petroff et al., 2011), which leadsto the undermining and collapse of the valley head and side walls, originating the typicalrounded shape of the heads (e.g., Laity and Malin, 1985; Devauchelle et al., 2011). In this regime,a description for the landscape evolution requires the solution of equation A.8 near the tipof the nascent valley, with suitable boundary conditions for continuity of flow velocity at thespring applied at the channel bed (Devauchelle et al. (2011), equation 3.6). Within a few me-ters from the tip, and once the overland flow is well developed and steady, the stream powerlaw (equation A.8) can be applied to describe incision. Far away from the stream erosion isgoverned by hillslope diffusion (Petroff et al., 2012).A.6 The subglacial regimeSubglacial channels (Figure 2.4c) originate where meltwater flows in pipe-like channels underlarge masses of ice. The turbulent water flow regime characteristic of incompressible sub-glacial meltwater leads to a formulation of momentum conservation similar in form to equa-tion A.5. However, an important difference is that here the flows are driven by variations in icethickness and are thus expected to be pressurized (e.g., Ro¨thlisberger, 1972a; Walder and Fowler,1994; Schoof , 2010), and retarded by turbulent stresses at both the roof and floor of the channel.Consequently, we modify equation A.5 to account for the gradients in pressurized water, withpressure Pw, and the hydrostatic ice overburden pressure Pi = ρigS. Under these conditions,and the assumption of a circular pipe with hydraulic radius Rsg, equation A.5 becomes:ρC f u2x = Rsg(dPwdx+ ρigS− ρwgSo). (A.17)The mechanisms governing subglacial bedrock incision are poorly understood and complex.Little work has, for example, been carried out on how erosion by confined meltwater differsfrom fluvial erosion, and how this process is modulated by sediment transport, which is be-lieved to be significant (Beaud et al., 2016). To move forward, we nevertheless assume that sub-glacial streams erode their bedrock substrate through processes similar to the stream-powerincision characteristic of runoff streams, with overlying ice acting to inhibit hillslope erosion134A.6. The subglacial regimeprocesses (Walder and Fowler, 1994). Consequently, a crude expression for erosion that modi-fies the stream power law by changing the power exponents to ms and ns, to be empiricallydetermined, is:E = K f fsqmsw |∇[ht]ns |. (A.18)135Appendix BSynthetic valley networks: Models andmetrics predictionsB.1 Summary and contentThis Appendix contains the physical models behind the definition of the Synthetic Valley Net-works (SVN) presented in Chapter 4. Here I describe in detail the models for the different met-rics and each specific type of erosion (fluvial, glacial, sapping and subglacial) as introduced inChapters 2 and 4. I then give four sets of average synthetic valley networks, as defined by theaverage parameter values in each model (see Appendix D).B.2 Physical models of morphometry: model predictionsThe generation of the synthetics used in interpreting the PCA starts with the physical descrip-tions of each metric (Grau Galofre and Jellinek, 2017; Grau Galofre et al., 2018). Below we presentthe models used to generate the metric predictions for each erosional mechanism, in turn, flu-vial, glacial, sapping, and subglacial. The models for each metric build on the following list ofassumptions (Grau Galofre and Jellinek, 2017):1. Steady and well-developed incompressible and hydrostatic flows.2. Shallow slopes.3. Steady state landscape response to the flow regime of each system.4. Cross-sectional mechanical stability.B.2.1 Fluvial regimeOpen channel flow, linear rheology, and a mechanical equilibrium between lateral gradientsof hydrostatic pressure and complex turbulent overturning motions, typically captured in136B.2. Physical models of morphometry: model predictionsterms of a net turbulent drag, are the main distinctive characteristics of fluvial systems. Withthe assumptions listed above, we derived in previous work (Grau Galofre and Jellinek, 2017;Grau Galofre et al., 2018) the following models to physically describe each of the metrics. Ap-pendix D contains the list of parameters, units and definitions as they appear on the followingequations.Aspect ratioR = L/W =Rρ∆htds2 tan(φr)pi2√1+ tan(φ)2θ2t − tan(φ)2. (B.1)Where L is length along the longest flow path, W is channel width, scaled to valley networkwidth using a linear conversion factor (Penido et al., 2013), ht is the channel depth, θt is a non-dimensional Shields parameter, φ is the inclination angle of the channel walls, φr is the angleof repose of sediment in water, the effective density Rρ = ρw/(ρs − ρw) ∼ 1.61 compares thesediment and water density difference and ds ∼ 0.45 mm is taken to be the average grain size.Width of order 1 tributariesOrder 1 fluvial valley spacing is a characteristic length scale of the landscape that emerges fromthe equilibrium between incision and mass wasting in the fluvial landscape evolution equation(Perron et al., 2008, 2009). If the ridges corresponding to first order tributaries are periodic,then valley width (distance from ridge to ridge) is equivalent to valley spacing (distance fromminimum to minimum). In these conditions, the width of order 1 tributaries scales like:λ ∼ DK12m+1(B.2)Where D is the soil diffusivity, K is the erodibility of the substrate, and m is the area exponentin the stream power equation (Whipple and Tucker, 1999; Perron et al., 2008).junction angleThe junction angle in a steady-state valley network carved by a gravity driven flow on anopen channel depends on the slopes of the main stem and tributary at the site of the junction(Howard, 1971; Grau Galofre and Jellinek, 2017), which can be described by:γ = arctan(− cot(α) sin(β)) (B.3)Where α is the slope of the main stem and β the slope of the tributary before the junction.137B.2. Physical models of morphometry: model predictionsStream OrderFluvial networks (precipitation or snow-fed), and glacial networks, behave mathematicallylike tree-like systems (Meir et al., 1980; Grau Galofre and Jellinek, 2017), with a Gaussian prob-ability distribution that describes when two tributaries of a given order merge. The streamorder of such systems increase logarithmically with the number of first order streams accord-ing to:Ω{Sn} = 1/2 log2 N1 +O(1), N1  1 (B.4)Where N1 is the number of order 1 tributaries in the network.Fractal DimensionThe fractal dimension of a system is a statistical measure of its scale invariant complexity(Mandelbrot, 1983; Rodrı´guez-Iturbe and Rinaldo, 2001). For a fluvial network, the followingequation links D f to both the channel order and Horton’s laws (Horton, 1945):D f = D1rbrl(B.5)Where D1 is the fractal dimension of 1st order tributaries and, again, rb and rl are thebifurcation and length ratios from Horton’s laws.UndulationsUndulations in the longitudinal profile arise from the presence of a pressure gradient thatopposes gravity and drives water uphill opposing topography. In the absence of such force,such as in fluvial systems, undulations are not significant Ω ∼ 0.B.2.2 Glacial regimeOpen channel flow, rheology following a power-law relationship (Glen, 1958; Paterson, 1994),and the balance between a driving force related to lateral differences in ice thickness and vis-cous stresses arising in response to the production of internal velocity gradients (Paterson,1994) are the main distinctive characteristics of glacial systems. With the assumptions listedabove, we derived in previous work (Grau Galofre and Jellinek, 2017) the following models tophysically describe each of the metrics. Appendix D contains the list of parameters, units anddefinitions as they appear on the following equations.Stream order, fractal dimension and junction angle derive from the assumption of a gravitydriven flow in an open channel, with a Gaussian probability of tributary merging, and thusdescribe both fluvial and glacial drainage systems (Grau Galofre and Jellinek, 2017). Below, wegive the equations for the remaining metrics.138B.2. Physical models of morphometry: model predictionsAspect ratioR =√R4bus/CρighiSo cos(φ). (B.6)Where us is basal sliding speed, Rb is the bed roughness, C = 10−20m2s/kg is a constant(Paterson, 1994), ρi is the ice density, hi is ice thickness, So is the topographic slope, and φ is theslope of the valley side (Grau Galofre and Jellinek, 2017)Width of order 1 tributariesThe width of order 1 valleys is set by a competition between the thickening of ice in incipientvalleys, which increases flow speed and erosion, and viscous stresses/ sidewall drag, whichacts to impede glacier motion and erosion (Pelletier et al., 2010).λ =2τbS3√pi2ηe f f c/(ρig)2, (B.7)where c is a bed-friction parameter such that ηe f f c ∼ 10−2 m Pa−1, and the basal shear stressis in the range τb = 0.5× 105 − 2.4× 105 Pa.UndulationsOverdeepenings and undulations exist in glacial valley longitudinal profiles, particularly inthose systems where subglacial hydrology strongly controls ice sliding speed (Herman et al.,2011). We expect glacial valley undulations to exist, and in occasions be significantly largerthan 0 (Ω > 0).B.2.3 Sapping regimeSapping valleys form when the height of the groundwater table locally intersects the groundlevel. The emergence of groundwater in springs can both incise channels, that grow forwardin the direction in which groundwater enters the tip, and contribute to the overland flow (Laityand Malin, 1985; Devauchelle et al., 2012).This has an important effect on network development:as erosion advances, tributaries bifurcate in the upstream direction (Devauchelle et al., 2011).Mass wasting represents an additional strong contribution in the areas of water seepage (Lambet al., 2006).Aspect ratioThe sapping valley aspect ratio is given by the following equation (Grau Galofre and Jellinek,2017):R = L/W = 4ds/3Rρpi√2√2κ(1/2)g3 sin(2pi/3)√θ6tC f KhS2, (B.8)139B.2. Physical models of morphometry: model predictionsWhere ds is the average grain size, C f is the friction factor, κ is the complete elliptic integralof the first kind (Devauchelle et al., 2011), S is the topographic slope, and Kh is the hydraulicconductivity.Width of order 1 tributariesTo our knowledge, there are no models or scaling relationships that predict the width of firstorder sapping valleys. In our predictions, to compromise the identification of sapping val-leys as little as possible, we adopted a value corresponding to the average width of order 1tributaries in our dataset.Junction angleObservations for junction angles in terrestrial sapping valleys distribute around 72◦ (Devauchelleet al., 2011), which has been explained analytically by solving the Poisson equation for thesquare of the water table height (Devauchelle et al., 2011; Grau Galofre and Jellinek, 2017) for neg-ligible precipitation rates and imposing the symmetrical bifurcation of two tributary streamsafter a junction. The solution does not depend on the characteristics of the terrain other thanassuming a basal impermeable layer and small precipitation rates, and therefore Martian sap-ping valleys should also distribute around 72◦ .Stream orderTributary sapping valleys do not merge into main stems as tree-like systems, which includefluvial or glacial valleys, but rather bifurcate in the upstream direction as the network de-velops, parting from a main stem. The mechanisms that lead to this bifurcation are poorlyunderstood, and therefore we cannot apply the model of Meir et al. (1980) to predict streamorder in a sapping network. The same is true for the fractal dimension metric, as the model-ing (Grau Galofre and Jellinek, 2017) depends on the Horton laws (Rodrı´guez-Iturbe and Rinaldo,2001), which are a consequence of the statistical merging pattern of tree-like networks (Kirch-ner, 1993)UndulationsWe do not expect undulations to appear in the steady state longitudinal profiles of sappingvalleys.B.2.4 Subglacial regimeSubglacial channels and the larger tunnel valleys originate where meltwater flows in pipe-like channels under large masses of ice, incised partly on the ice and partly on the bedrock.The turbulent flow regime characteristic of subglacial meltwater is similar to that of fluvialchannels. However, subglacial flows are driven by variations in ice thickness in addition to to-140B.2. Physical models of morphometry: model predictionspographic gradients, and are thus expected to be pressurized (e.g., Ro¨thlisberger, 1972a; Walderand Fowler, 1994; Schoof , 2010). Consequently, subglacial channels can either follow or divertfrom topographic gradients, and present some remarkable morphologies (e.g., Sugden et al.,1991; Greenwood et al., 2007; Kehew et al., 2012; Grau Galofre et al., 2018).Aspect ratioWe use the model presented in Grau Galofre and Jellinek (2017) to describe the aspect ratio ofsubglacial channels:R = L/W =(1− fm)Rρ∆htds2 tan(φr)pi2√1+ tan(φ)2θ2t − tan(φ)2. (B.9)Which is similar to the fluvial aspect ratio, but with the introduction of a parameter fm equiv-alent to the fraction of power lost to melt the icy roof. We give upper and lower bounds forthis parameter in Appendix D.Width of order 1 tributariesThe spacing between order 1 subglacial tributaries has been discussed in the literature, both forpurely ice-incised channels (R-channels) and depositional features such as eskers (e.g., Weert-man, 1972; Schoof , 2010; Hewitt, 2011). However, order 1 subglacial channel spacing has not yetbeen successfully modeled even though it is observed in the field (Grau Galofre et al., 2018). Toquantify this metric, we take the same approach than sapping 1st order spacing and adopt theaverage minimum width value.Junction angleJunction angles observations or modeling descriptions in the subglacial literature are nonexis-tent to our knowledge. To quantify this metric, we take the same approach than in the previousmetric and adopt the average junction angle value in our data.Stream order and fractal dimensionField observations and modeling results show that subglacial channels can develop into net-works (Sugden et al., 1991; Booth and Hallet, 1993; Boulton et al., 2007; Schoof , 2010). There isno quantification, however, of their maximum stream order other than our own observations(Grau Galofre and Jellinek, 2017; Grau Galofre et al., 2018). It is also unclear whether they followHorton’s laws or not (Horton, 1932), although their structure unlike tree-like systems suggestthey do not (Meir et al., 1980; Grau Galofre and Jellinek, 2017; Grau Galofre et al., 2018), whichmakes the calculation of fractal dimension values difficult. We adopted the empirical rangeobserved on Earth for stream orders, and the average value of this metric derived from thissame dataset for both the mean stream order and the fractal dimension for the same reasons141B.3. Metric PredictionsTable B.1: Metric predictionsMetric Predictions γ D f Sn λ R Ωfluvial 60 1.6 6 150 2430 1glacial 90 1.5 2 2000 6 5sapping 72 1.3 2 1025 38 0.1subglacial 60 1.5 3 1025 1215 15exposed above.UndulationsArguably the most distinctive characteristic of subglacial channels and tunnel valleys is thepossible presence of undulations in their profiles and of deviations from topographic gradi-ents, both characteristics noted extensively in the literature (e.g., Sugden et al., 1991; Greenwoodet al., 2007; Kehew et al., 2012; Grau Galofre et al., 2018). Reliable predictions for the magnitudeof undulations in the profiles of subglacial channels would require good constraints on boththe surface and the ice surface topographic gradients, which represents a significant challenge.Although we cannot quantify a number, we argue that undulations in the range between 0 anda significant fraction of the elevation drop are feasible for Martian subglacial channels.B.3 Metric PredictionsUtilizing the models presented above to make predictions about the morphometry of Mar-tian channels requires reasonable constraints on the Martian parameter space. Although theextensive exploration of the Martian surface has provided some reliable constraints on theseparameters, particularly the rover missions Sojourner and Opportunity (e.g., Team, 1997; Klein-hans, 2005), we acknowledge the fact that most of the Martian surface is unexplored and theseparameters may not accurately represent all areas on Mars.Appendix D provides a list of parameters required for the models. The upper and lowerbounds in this table give us the uncertainty space associated with each prediction, and with itthe space within the PCA where the morphometry is best explained by rivers, glaciers, sappingvalleys, or subglacial channels. Table B.1 presents the average (〈〉), upper (UB) and lowerbound (LB) predicted for each metric.The space of possible morphometries was produced by running 100,000 Montecarlo sim-ulations selecting random configurations of parameters between their respective upper andlower bounds, to produce 100,000 synthetic possible fluvial, glacial, sapping, and subglacialsystems. We then plotted these systems in the PCA space, and used them to characterize theerosional style inherent to each valley network.We chose an equal probability (white) distribution for the parameter space. When pa-rameters range over 2 or more orders of magnitude, we instead chose logarithmic distribu-142B.3. Metric Predictionstions to best characterize the span. The underlying assumption that all parameters distributerandomly is a first order approximation owing to the lack of information of the Martian pa-rameter space. In reality, parameters such as D10, D50, D84, S, d in the supplementary dataset“ch3-parameters.xls” distribute according to a power law. However, establishing parametervariations according to power-law probability distributions requires a much more extensiveempirical characterization of the Martian surface that we do not yet have (i.e., grain size dis-tribution has been measured at three sites in the planet).143Appendix CDetails of the PCA techniqueC.1 SummaryThis Appendix complements Chapter 4 with details on the PCA technique. In particular, hereI describe the mathematical base behind the technique and provide the results for a sensitivityanalysis to show that the results of Chapter 4 do not depend on our choice of valley networks.C.2 Details of the PCA techniqueEach Principal Component (PC) is by definition of the transformation a known linear combina-tion of the set of 6 metrics defined in the text (R, λ, γ, Sn, D f , andΩ), with coefficients scaled tothe variance captured by each metric. Figure C.1 shows the dependence on each of the metricsof the first, second and third PC (red dots), which capture up to 85% of the variance.The first PC captures 45% of the data variance, and it is dominated by the stream order,and followed by the width of 1st order tributaries and aspect ratio. Width of order 1 tribu-taries, stream order and undulations dominate the 2nd PC, in turn, which captures 24 % ofthe total dataset variance. The third PC is largely dominated by the undulations and captures16% of the total variance. Individually, the metric that explains a larger portion of valley net-work variability is the stream order Sn (31%), followed by the width of order 1 tributaries λ(29%), the magnitude of undulations Ω (17.2%), length to valley network width aspect ratio R(11.4%), junction angle γ (8.5%), and fractal dimension D f (2.9%), in turn.C.3 Sensitivity analysisWe assess the robustness of PCA based analysis of Mars VN, and therefore the dependenceof the results in Figures 4.1 and C.1 on our choice of VN, by performing sensitivity analysison different subsets of the Martian valley networks. From the suite of 60 networks presentedin this study, we first analyzed the variance captured by each metrics in subsets of all fluvial,all glacial, all subglacial, and all sapping networks as inferred from the results of the valley144C.3. Sensitivity analysisFigure C.1: Principal Component (PC) linear decomposition in terms of the metrics,scaled to the variance captured by each of the metrics. From left to right, the Xaxis displays junction angle γ, fractal dimension D f , stream order Sn, width of 1storder tributaries λ, aspect ratio R, and undulations Ω.network classification. Then we performed similar evaluations of the metric relative contribu-tion to the variance in 210 different runs with random data subsets. The supplementary table“ch3-sensitivity-analysis.xls” contains further detail on these results, together with all the runswe produced, sensitivity dependence on sample size, etc.Table C.1 shows the results of the sensitivity analysis in terms of the relative differencebetween the variance captured by a particular metric in a given run and the variance capturedby that metric considering the entire dataset. Columns show the different metrics, and rowsshow the sample size of the evaluation. The relative difference between the variance capturedby each metric in all the data and random subset diminishes in all metrics for as subset selec-tions grow larger, and becomes larger for smaller subsets, as expected.All metrics but the aspect ratio show average relative differences below 30%. Aspect ratio,however, shows a high relative difference up to 82%. We relate this higher difference to the145C.3. Sensitivity analysisTable C.1: Sensitivity analysisSample size (VNs) γ D f Sn λ R Ω41 0.08 0.01 0.11 0.08 0.85 0.1835 0.24 0.19 0.09 0.20 0.81 0.1420 0.30 0.21 0.11 0.10 0.71 0.1145 0.33 0.44 0.31 0.19 1.19 0.3731 0.13 0.13 0.03 0.20 0.70 0.0353 0.15 0.07 0.09 0.09 0.63 0.0051 0.02 0.01 0.07 0.09 0.45 0.0310 0.35 0.54 0.34 0.37 1.12 0.8359 0.13 0.08 0.10 0.16 0.86 0.0346 0.24 0.28 0.15 0.07 0.84 0.0253 0.05 0.09 0.07 0.08 0.60 0.0853 0.14 0.03 0.08 0.17 0.83 0.0259 0.02 0.16 0.13 0.03 0.19 0.2111 0.27 0.46 0.29 0.29 1.42 0.3225 0.36 0.38 0.21 0.16 0.95 0.3154 0.15 0.22 0.15 0.14 0.33 0.4315 0.35 0.41 0.35 0.29 1.18 0.646 0.39 0.43 0.21 0.48 0.96 0.8848 0.18 0.11 0.11 0.08 0.87 0.1453 0.21 0.19 0.05 0.18 0.68 0.0959 0.02 0.16 0.13 0.03 0.19 0.2111 0.27 0.46 0.29 0.29 1.42 0.3225 0.36 0.38 0.21 0.16 0.95 0.3154 0.15 0.22 0.15 0.14 0.33 0.4317 0.35 0.41 0.35 0.29 1.18 0.649 0.39 0.43 0.21 0.48 0.96 0.8844 0.25 0.15 0.18 0.10 0.83 0.1333 0.19 0.23 0.12 0.20 0.55 0.3437 0.29 0.31 0.17 0.14 1.05 0.0751 0.02 0.16 0.13 0.03 0.19 0.2128 0.26 0.40 0.18 0.19 0.98 0.2342 0.22 0.23 0.13 0.22 0.52 0.4426 0.35 0.40 0.31 0.21 1.09 0.4823 0.25 0.25 0.25 0.33 1.42 0.2813 0.34 0.43 0.38 0.19 0.96 0.68average 0.22 0.26 0.18 0.19 0.82 0.30std 0.1 0.2 0.1 0.1 0.3 0.3presence of two valley networks with substantially larger R than the rest, which makes therelative difference a large function of whether these two data points are part of the subset orno. This is however not a data bias, since these are the longest valley networks on Mars (Hyneket al., 2010), and also relatively narrow (Supplementary table “ch3-VN-dataset.xls”).146C.4. PCA end-membersC.4 PCA end-membersFigure C.2 shows four observational end-members corresponding to fluvial, sapping, glacial,and subglacial erosion. This figure complements Figure 4.1 in Chapter 4.Figure C.2: Representative examples and longitudinal profiles of the PCA erosional end-members: (a) fluvial, (b) glacial, (c) sapping, (d) subglacial.The fluvial end-member (a) is Loire Valles (γ = 71, D f = 1.65, Sn = 7, λ = 300 m, R = 800,and Ω = 14 %). The glacial end-member (b) is an unnamed glacier in Lunae Palus , at the rimof Kasei Valles (see Appendix E for geographical reference) (γ = 75, D f = 1.5, Sn = 4, λ = 1100,R = 100, and Ω = 9 %). The third, sapping example in (c) corresponds to Dubis Vallis (γ = 79,D f = 1.2, Sn = 1, λ = 820, R = 200, and Ω = 2 %). The final panel, (d), shows Mamers Vallis asan example of subglacial drainage (γ = 64, D f = 1.5, Sn = 2, λ = 950, R = 365, and Ω = 23 %)147Appendix DMartian parameter spaceSym Definition Units LB Average UB Referenceg S gravity m/s2 - 3.71 - -ρw water density kg/m3 - 1000 - -ρs sediment density kg/m3 2650 3025 3400 (Kleinhans, 2005)Rρ non dimensionaldensity- 0.42 0.49 0.61 calculatedds average grain size m - 0.045 - (Team, 1997)D10 10% percentilesediment grain sizem - 0.0071 - (Kleinhans, 2005)D50 50% percentilesediment grain sizem - 0.24 - (Kleinhans, 2005)D84 84% percentilesediment grain sizem - 0.63 - (Kleinhans, 2005)φr static angle of friction ◦ 30 35 40 (Kleinhans et al.,2011)φ inclination of channelwalls◦ 4 6 10 (Williams andPhillips, 2001)H topographic drop m 581 1640 4976 This studyL valley network length m 1.339E5 4.1E5 1.088E6 This studyS valley floor slope - 3.9E-5 4E-4 4E-2 This studyd valley network depth m 10 109 246 (Williams andPhillips, 2001)W valley network width m 84 770 6816 This studyC f friction factor - 6.4E-3 6.7E-3 7.4E-3 (Wilson et al.,2004; Kleinhans,2005)148vtcw valley width to topwidth ratio- - 0.12 - (Penido et al.,2013)Rh hydraulic radius m 30 70 105 (Wilson et al.,2004)θt Shield stress parameter - 0.107 0.11 0.125 J. Vendetti (PC2018)ρi ice density kg/m3 - 910 - -C Dimensionless icesliding constant- - 1E-010 - (Paterson, 1994)hi glacier ice thickness m 10 109 246 ∼ valley depthRb bed roughness - 0.04 0.1 6.5 (Paterson, 1994;Kreslavsky andHead, 2000)cp bed friction parameter m/sPa23E-17 3E-16 3E-15 (Pelletier et al.,2010)Kp permeability m2 3.5E-0131E-12 2.75E-12(Levy, 2012)µw dynamic viscosity ofwaterPa s 1.7E-4 1.8E-4 1.8E-4 -Kh hydraulic conductivity m/s 7.2E-7 2.1E-6 5.7E-6 calculatedµc empirical constant - - 0.05 - (Walder andFowler, 1994)θc critical shields stress Pa 13.8 16.9 20.0 Calculated(Parker et al.,1982)Ai Ice flow parameter (T =0◦)Pa3/s - 7E-24 - (Paterson, 1994)Li Latent heat of icefusionJ/kg - 3.33E5 - -Ki ice closure constant - 1 1.06 1.12 Calculated(Walder andFowler, 1994)fm fraction of poweravailable for erosion- 0.01 0.5 0.8 This studya erosion-stress exponent - 1 1.25 1.5 (Howard, 1994;Whipple andTucker, 1999)149b hydraulic geometryexponent- 0.4 0.467 0.6 (Whipple andTucker, 1999;Parker et al., 2007)c basin hydrologyexponent- 0.328 1 0.7 (Whipple andTucker, 1999;Irwin et al., 2005)(data fit)h Hack’s law exponent - 1.67 1.67 1.92 (Hack, 1957)Ka Hack’s law parameter m2−h 6.69 - - (Hack, 1957)D50 sediment size m 0.2 0.24 0.3 (Kleinhans, 2005)Uo uplift rate mm/yr 1 1.5 2 (Jellinek et al.,2008)Kq basin hydrologyparameterm3−2cs−1 1.39 - - (Irwin et al., 2005)(data fit)Kb bedrock erodibility m/sPa−a4.7E-12 6.27E-123.13E-12(Howard, 2007)Kw hydraulic geometryparameterm1−3bsb 65.5 - - (calculated)(Parker et al.,2007)K erodibility coefficient m1−2m/s 3.2E-10 - - (calculated)(Whipple andTucker, 1999)150Appendix EParameters, notation, and units forChapter 5E.1 Mars parameter space: NotationThe following tables (Tables E.1, E.2, E.3, and E.4) present the notation and units for the pa-rameters used in this study. Table E.1 provides the list of parameters corresponding to thegeneral notation followed in Chapter 5, whereas notation related to fluvial, glacial, and sap-ping erosion is compiled respectively in tables E.2, E.3, and E.4. Non-dimensional variablesare listed with a “−” symbol.Table E.1: List of parameters: general notationList of parameters definition unitsz topographic elevation mx length along channel mt time sg gravity m/s2ne erosion exponent -N generalized uplift-erosion num -E erosion rate m/sU uplift rate m/sD substrate diffusivity m2/sE.2 Caption for supplementary tableSupplementary table “ch4 parameters.xls” includes three excel sheets with information rele-vant to Chapter 5.151E.2. Caption for supplementary tableTable E.2: List of parameters: fluvial erosionList of parameters definition unitsK erodibility parameter m1−2m/sKa Hack’s law parameter m2−hh Hack’s law exponent -m Area exponent -S topographic slope -n slope exponent -Kw hydraulic geometry factor m1−3b sbKq basin hydrology factor m3−2c s−1Kb bedrock erodibility factor m/s Pa−aC f friction factor -ρw water density kg/m3b hydraulic geometry exponent -L channel length mH Topographic drop mUo characteristic uplift rate m/sx∗, z∗, t∗ non-dimensional variables -U∗ non-dimensional uplift rate -Ne fluvial uplift-erosion number -Table E.3: List of parameters: glacial erosionList of parameters definition unitsKg glacial erosion constant m1−l s l−1us ice sliding velocity m/sBs sliding law parameter Pa3/(m2 s)ρi ice destiny kg/m3l abrasion exponent -Pi ice cryostatic pressure PaPw subglacial water pressure PaPi − Pw effective pressure Pap Glen’s flow parameter -h ice thickness mHi characteristic ice thickness mh∗ non-dimensional ice thickness -Γ normalized ice thickness -Ng glacial uplift-erosion number -E.2.1 Applicability of the stream power law to MarsThe first sheet included in the supplementary table provides support for the applicability ofthe streampower law to the Martian valley networks. This table displays a list of valley net-works with their respective topographic slopes (averaged along the valley network length)and basin areas as calculated using Hack’s law. Hack’s law, which relates length and basin152E.2. Caption for supplementary tableTable E.4: List of parameters: sapping erosionList of parameters definition unitszo spring elevation mQo characteristic discharge m3/sKs substrate hydraulic conductivity m/sΘc critical Shields stress -ds average sediment grain size mRρ non-dimensional density -zr bed roughness mD flow depth mNs sapping uplift-erosion number -area in a valley network and can be derived from the fractal nature of the system, has beenpreviously applied to the Martian valley networks Stepinski et al. (2004) and found to hold.Basin area and slope, scaled to terrestrial values for consistency, are then compared with thezones defined in Figure E.1 Sklar and Dietrich (1998) (fig.1) to test the applicability of the streampower law.The slope-area constrains from supplementary sheet 1: Applicability stream power law(Area ∼ 13100 km2, slope ∼ 0.008) fall within the stream power applicability area.E.2.2 Empirical fit: basin hydrology factorWhereas most of the parameters necessary to apply the stream power law to the Martian pa-rameter space have already been constrained elsewhere (see references in sheet 3: List of pa-rameters), the basin hydrology factor Kq has not been calculated before for Mars. To constrainthis parameter, we use the dataset in Irwin et al. (2005) (see table in sheet 2) to find an empiricalrelationship between area and inferred discharge. Using a linear regression between the loga-rithms of normalized basin area and normalized discharge, the basin hydrology factor can bedirectly calculated using the following equation:ln(Q/Qmax) = ln(Kq) ln(A/Amax) + B (E.1)Where the factor r2 = 0.75.E.2.3 Table of parametersThe third sheet in the supplementary table “ch4 parameters.xls” compiles the parameters usedin the modeling efforts in Chapter 5. The table is organized in the following way. Columns1 through 7 present the notation for each parameter, definition, units, average value, lowerbound, upper bound, and references, respectively. The blocks separated by horizontal linesrefer to general, fluvial, glacial, and sapping erosion, in turn. For parameters that are measuredindividually for each valley network in Chapter 5, we use the notation “CD” (Case Dependent)153E.3. Geography of MarsFigure E.1: Approximate constraints on the slope-area region where the stream powerlaw is adequate to describe bedrock incision by fluvial processes. Suggested upperand lower slope bounds and width of process transition zones are approximate.Adapted from Sklar and Dietrich (1998), fig.1in the table.E.3 Geography of MarsThe United States Geological Survey (USGS) defines thirty cartographic quadrangles for thesurface of Mars. These can be seen below in E.2.154E.3. Geography of MarsFigure E.2: The thirty cartographic quadrangles of Mars, as defined by the United StatesGeological Survey. North is at the top; 0◦N 180◦W is at the far left on the equator.The quadrangles are overlapping a black-and-white MOLA topographic map.155


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