Arts, Faculty of
Geography, Department of
DSpace
UBCV
Callahan, Lilianne
Master of Science - MSc
University of British Columbia
As air temperature and precipitation patterns change globally, so too will stream thermal regimes which affect water quality and habitat suitability for aquatic life. Models are used to understand past and future stream temperatures and the primary processes that control them. Since water temperature is highly correlated with air temperature, this research investigates the use of reanalysis air temperature data from ECMWF Reanalysis v5 (ERA5) and North American Regional Reanalysis (NARR) as input in three different types of statistical models of stream temperature. ERA5 overall had the best performance, so it was then used as input in the hybrid air2stream model to test how well the model simulates stream temperatures for 23 streams in British Columbia, Canada, during the extreme conditions of a heat wave and a drought. The air2stream model had an average RMSE of 0.96 °C during the calibration period and 1.06 °C during the validation period. During the heat wave from June 25 to July 2, 2021, the average RMSE was 1.39 °C, and during the autumn drought from September 1 to October 31, 2022, the average RMSE was 1.08 °C. The model results indicate general accuracy with higher errors during anomalous conditions.
Predicting daily stream temperature in British Columbia,Canada, using reanalysis air temperature in statisticalmodels and the hybrid air2stream modelbyLilianne CallahanB.Sc., University of California, Santa Cruz, 2021A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Geography)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2023© Lilianne Callahan, 2023The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:Predicting daily stream temperature in British Columbia, Canada usingreanalysis air temperature in statistical models and the hybrid air2stream modelsubmitted by Lilianne Callahan in partial fulfillment of the requirements forthe degree of Master of Sciencein GeographyExamining Committee:Robert Daniel Moore, Professor, Geography, UBCSupervisorIan McKendry, Professor, Geography, UBCSupervisory Committee MemberSteven Weijs, Associate Professor, Civil Engineering, UBCAdditional ExamineriiAbstractAs air temperature and precipitation patterns change globally, so too will stream thermalregimes which affect water quality and habitat suitability for aquatic life. Models areused to understand past and future stream temperatures and the primary processes thatcontrol them. Since water temperature is highly correlated with air temperature, thisresearch investigates the use of reanalysis air temperature data from ECMWF Reanalysisv5 (ERA5) and North American Regional Reanalysis (NARR) as input in three differenttypes of statistical models of stream temperature. ERA5 overall had the best performance,so it was then used as input in the hybrid air2stream model to test how well the modelsimulates stream temperatures for 23 streams in British Columbia, Canada, during theextreme conditions of a heat wave and a drought. The air2stream model had an averageRMSE of 0.96 oC during the calibration period and 1.06 oC during the validation period.During the heat wave from June 25 to July 2, 2021, the average RMSE was 1.39 oC, andduring the autumn drought from September 1 to October 31, 2022, the average RMSE was1.08 oC. The model results indicate general accuracy with higher errors during anomalousconditions.iiiLay SummaryMathematical models can be used to understand how stream water temperatures will changedue to global warming. Since air temperature has a strong effect on water temperature, thisresearch looked at different air temperature data products as input into stream temperaturemodels, and found that the ECMWF Reanalysis v5 (ERA5) had the best performance. Thisdata was then used as input into a model called air2stream to recreate stream temperaturesduring a heat wave and a drought. Overall, the model was able to accurately recreate streamtemperatures over the entire data period, but it had higher errors during the extremeweather events.ivPrefaceThis thesis is original work completed by the author. Guidance was given by the supervisorycommittee (Dan Moore and Ian McKendry).A version of this work has been presented in May 2023 as a poster (Callahan, L. and Moore,R.D. Evaluation of reanalysis air temperature products for stream temperature modelingusing statistical models and the hybrid airstream model) on which the author acted aslead investigator, composing and presenting the poster at the 2023 Annual Meeting of theCanadian Geophysical Union (CGU) in Banff, Alberta.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Stream temperature models and their applications . . . . . . . . . . . . . . 21.3 Research objectives and thesis structure . . . . . . . . . . . . . . . . . . . . 62 Study site and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Study site: British Columbia, Canada . . . . . . . . . . . . . . . . . . . . . 82.2 Stream temperature data sources and processing . . . . . . . . . . . . . . . 82.3 Air temperature data sources and processing . . . . . . . . . . . . . . . . . 142.3.1 Station-based data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Reanalysis products . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Streamflow data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5.1 Spatial model for MWAT . . . . . . . . . . . . . . . . . . . . . . . . 192.5.2 Temporal models for daily stream temperature . . . . . . . . . . . . 202.5.3 Air2stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Evaluation of ERA5 and NARR air temperature data . . . . . . . . . . . . 223.1.1 Comparison to climate station observations . . . . . . . . . . . . . . 223.1.2 Comparison of reanalysis products and station-based interpolation asair temperature predictors in the regression model for MWAT . . . . 223.1.3 Performance of ERA5, NARR, and SBI data in linear regressions ofdaily stream temperature . . . . . . . . . . . . . . . . . . . . . . . . 253.1.4 Performance of ERA5, NARR, and SBI data in logistic curve relationsof daily stream temperature . . . . . . . . . . . . . . . . . . . . . . . 253.2 Evaluation of air2stream for predicting stream temperature . . . . . . . . . 323.2.1 Calibration and validation periods . . . . . . . . . . . . . . . . . . . 323.2.2 Heat wave and autumn drought periods . . . . . . . . . . . . . . . . 32vi4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.1 Assessment of ERA5 and NARR air temperature products for use in empir-ical stream temperature models . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 Air2stream performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1 Summary of key findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2 Avenues for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . 52References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61viiList of TablesTable 2.1 Station ID, name, and location information of each Water Survey ofCanada hydrometric station used in the air2stream model. . . . . . 12Table 2.2 Catchment characteristics of each Water Survey of Canada hydro-metric station used in the air2stream model. . . . . . . . . . . . . . 13Table 2.3 Information about the stream temperature time series record for eachWater Survey of Canada hydrometric station used in the air2streammodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Table 3.1 Seasonal errors between climate station air temperatures and NARRand ERA5 air temperatures. . . . . . . . . . . . . . . . . . . . . . . 22Table 3.2 Comparison of MWAT models using the temporal air temperaturecovariates computed from the ERA5, NARR, and SBI air tempera-tures. Error statistics are based on a leave-one-out cross-validation.𝑏9 is the model coefficient for the temporal air temperature covariate,𝛿𝑇𝐴, and df is the residual degrees of freedom, both based on fittingthe model to the full data set. . . . . . . . . . . . . . . . . . . . . . 24Table 3.3 Fitted parameters for each station’s air2stream model. . . . . . . . 33Table 3.4 RMSE, MBE, and MAE for each station during the calibration andvalidation periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Table 3.5 RMSE, MBE, and MAE for each station during the calibration andheat wave periods. Also included are the air temperature anomaliescompared to the 1979-2020 reference period and water temperaturechanges compared to the preceding five days during the heat wave. 42Table 3.6 RMSE, MBE, and MAE for each station during the calibration andautumn drought periods. Also included are the discharge anoma-lies compared to the 1997-2021 reference period during the autumndrought. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Table 3.7 RMSE, MBE, and MAE for each station during the heat wave andautumn drought for the regression model. . . . . . . . . . . . . . . 48viiiList of FiguresFigure 2.1 Map of each station used in the Moore et al. (2013) data set, coloredby the stream’s associated MWAT value. The three example stationsare highlighted with symbols. . . . . . . . . . . . . . . . . . . . . . . 9Figure 2.2 Map of Water Survey of Canada hydrometric stations used in theair2stream model, colored by regulation. The three example stationsare highlighted with symbols. . . . . . . . . . . . . . . . . . . . . . . 11Figure 2.3 Map of the ECCC climate stations used for comparison to reanalysisair temperatures, colored by the length of air temperature record inyears. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.4 Monthly lapse rates from Stahl et al. (2006) for daily maximum,minimum, and mean air temperatures. . . . . . . . . . . . . . . . . . 18Figure 3.1 Spatial pattern of RMSE between observed air temperatures and in-terpolated ERA5 (top row) and NARR (bottom row) air tempera-tures for each climate station during the summer. . . . . . . . . . . . 23Figure 3.2 Predicted versus observed MWAT for all stations for models fit us-ing the temporal air temperature covariate, 𝛿𝑇𝐴, computed fromERA5 (left column), NARR (middle column), and SBI (right col-umn) air temperatures. Data points are based on a leave-one-outcross-validation. Symbol colours indicate the July-August air tem-perature deviation for the monitoring year from the July-August av-erage over the reference period (1990-2003). The black line representsperfect agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 3.3 Scatterplots of daily stream temperature versus daily air tempera-ture by month using each air temperature source for three examplestations. The lines represent linear regressions by month. . . . . . . 26Figure 3.4 Boxplots of the 𝑟2 for each monthly linear regression using each airtemperature source. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Figure 3.5 Variation of the fitted y-intercept by month for the linear regressionsof daily stream temperature using each air temperature source. . . . 28Figure 3.6 Variation of the fitted slope by month for the linear regressions ofdaily stream temperature using each air temperature source. . . . . 29Figure 3.7 Scatterplots of daily stream temperature versus daily air temperaturewith fitted logistic curves for the cooling (September-February) andwarming seasons (March-August). . . . . . . . . . . . . . . . . . . . 30Figure 3.8 Boxplots of the RSE of the cooling (left column) and warming (rightcolumn) fitted logistic curves for each air temperature source. . . . . 31ixFigure 3.9 Time series of discharge (top row), air temperature (middle row),and stream temperature (bottom row) for the air2stream model forIngenika River above Swannel River (Station ID 07EA004). Observedstream temperature is in black, and simulated stream temperature isin red. The vertical dashed line separates the calibration and valida-tion periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 3.10 Time series of discharge, air temperature, and stream temperature forthe air2stream model for Renfrew Creek near Port Renfrew (StationID 08HA069). See caption for Figure 3.9 for details. . . . . . . . . . 35Figure 3.11 Time series of discharge, air temperature, and stream temperaturefor the air2stream model for Fraser River at Shelley (Station ID08MF040). See caption for Figure 3.9 for details. . . . . . . . . . . . 36Figure 3.12 Simulated versus observed stream temperaturea for the calibration(left column) and validation (right column) periods for each examplestation. Symbol colours indicate month. The black line representsperfect agreement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.13 Boxplots of error indices across all stations for the calibration andvalidation periods, separated by regulation status. . . . . . . . . . . 39Figure 3.14 Stream temperatures during the 2021 heat wave (left column) and2022 autumn drought (right column) from each example station. Sim-ulated stream temperatures from the air2stream model are shown inred, and observed are shown in black. . . . . . . . . . . . . . . . . . 41Figure 3.15 Boxplots of error statistics for the heat wave date range across allstations for the calibration period and the 2021 heat wave, separatedby regulation status. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 3.16 Spatial pattern of air2stream RMSE for the calibration period overthe heat wave dates (top row) and the heat wave period (bottomrow). Shapes represent the hydrologic regime. . . . . . . . . . . . . . 44Figure 3.17 Scatterplot of RMSE during the calibration period for the heat wavedate range vs RMSE during the 2021 heat wave for unregulated sta-tions. Symbol colours represent the air temperature anomaly com-pared to the 1979-2020 reference period and shapes represent thehydrologic regime. The black line represents perfect agreement. . . . 45Figure 3.18 Boxplots of error statistics for the autumn drought date range acrossall stations for the calibration period and the 2022 autumn drought,separated by regulation status. . . . . . . . . . . . . . . . . . . . . . 47Figure 3.19 Scatterplots of error statistics generated by the regression modelsversus mean absolute error generated by air2stream for the 2021 heatwave (left column) and the 2022 autumn drought (right column). Theblack line represents perfect agreement. Axis scales vary among plots. 49Figure .1 Schematic of the river reach considered in Equation .6 from Toffolonand Piccolroaz (2015). . . . . . . . . . . . . . . . . . . . . . . . . . . 62xAcknowledgementsThe support from my mentors, friends, and family are what has made my time at UBC sofulfilling.I would firstly like to thank Dan Moore for his dedication to furthuring my knowledgeduring every step of this project. Not only was he eager to provide feedback and answerquestions, but he also equipped me with invaluable experience in the field and set me upwith many tools for conducting good research. I feel extremely grateful that he took meinto his lab group after the first year of my degree.I would also like to thank the countless wonderful friends that I have made here in Van-couver, especially my lab group and my housemates from the last two years, who havewitnessed and experienced the highs and lows of graduate school alongside me.To my parents and grandmothers who always knew I could do it, thank you for your loveand unwavering encouragement.Funding for this project was provided by the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC) and the UBC Department of Geography.xi1 Introduction1.1 MotivationThe temperature of river water is a key environmental and ecophysiological variable becauseit regulates the chemical composition of the water and the productivity of aquatic life,including fish and insects (McQueen and Lean, 1987; Vannote and Sweeney, 2015; Parkinsonet al., 2016). Elevated river water temperature (𝑇𝑤) has been shown to support blue-greenalgal blooms and eventually eutrophication in lakes and rivers, and is considered one of themajor causes of streams not meeting certain water quality standards (McQueen and Lean,1987; Sridhar et al., 2004; Beretta-Blanco and Carrasco-Letelier, 2021).The thermal regime of streams also dictates habitat suitability for ectotherms, or cold-blooded animals, because temperature controls the rates of their biological processes. Ofparticular concern are salmonids, which are cold-water fish whose spawning, growth, anddistribution can be affected if waters warm above threshold temperatures (Eliason et al.,2011; Parkinson et al., 2016). Water above 23-25 oC can be lethal for salmonids, andtheir metabolism, habitat size, and coexistence among other species are sensitive to evenslight changes in temperature (Bjornn and Reiser, 1991; Hein et al., 2014; Watz et al.,2019; O’Sullivan et al., 2021). Hodgson and Quinn (2002) found that in watersheds where𝑇𝑤 exceeded 19 oC, sockeye salmon delayed their migration from the ocean to rivers formonths compared to the same species in colder rivers in order to avoid the hottest summertemperatures downstream when finding a place to spawn.Fish migration is in part determined by 𝑇𝑤 patterns, which cycle from warm to cold diur-nally and seasonally. These cycles can be altered by climatic changes, such as an increasein heat waves that raise regional air temperatures for extended periods of time, or land-usechanges, including forest harvesting that increases the amount of solar radiation reachingthe water surface (Sridhar et al., 2004; Piccolroaz et al., 2018). Air temperature (𝑇𝑎) andradiation, along with discharge from snowmelt, rain, groundwater, and dam releases, arethe primary influences on 𝑇𝑤 (Evans et al., 1998; Sridhar et al., 2004; Ruesch et al., 2012).The presence of salmon in the rivers along the Pacific Ocean has had ecological importanceand cultural significance to indigenous people for thousands of years (Carothers et al., 2021).They provide a source of food for bears and people alike, and in turn their carcasses aredispersed which adds nutrients to the riparian vegetation (Watkinson, 2000). As a changingclimate adds increased stress to the life cycle of Pacific Salmon, the traditional relationshipsbetween salmon, people, and the landscape may be interrupted (Jacob et al., 2010).1In British Columbia (BC), consistent stream temperature monitoring began in 2011, whichmeans there is a lack of historical data for the many streams which serve as habitats forsalmon and other important species (Northcote and Atagi, 1997; Hildebrand et al., 1999).This results in a need to develop stream temperature models to assess fish habitat suitabilityunder historic and current conditions. Additionally, modelling future climate projections ofstream temperature can advise management for anomalous conditions, such as hazardousweather events with precipitation and temperature extremes that are expected to occurmore frequently given changes in the global climate system (Stott, 2016).Since 𝑇𝑎 and 𝑇𝑤 are correlated, there is considerable concern over how global warming,especially in the form of heat waves, will affect aquatic ecosystems. Piccolroaz et al. (2018)analyzed the response of twenty-four rivers during major heat waves in Europe and foundthat low-land rivers without considerable snowmelt experience a reduction in streamflowas well as a notable spike in 𝑇𝑤 during extreme heat events.In BC from June 25 to July 2, 2021, there was a significant heat wave, which is definedas a stationary, warm air mass which creates temperatures that are higher than normalfor consecutive days (Luber and McGeehin, 2008; White et al., 2023). Many all-timetemperature records were broken across the Pacific Northwest with one town reaching49.6 oC, the highest recorded temperature in Canadian history (Bratu et al., 2022). Thesurrounding high air temperatures also raised sea water temperatures, and dead sea lifewas reported to have been found around the Salish Sea in southwestern BC (Raymond etal., 2022). In autumn 2022, there was an extended drought that persisted from Augustthrough October, resulting in lower-than-average streamflow and mass die-offs of salmon inthe province (Brend, 2022). Given the implications of elevated river temperatures duringthese events for the health of fish populations, it is important to have tools to predictstream temperature response under these extreme conditions.The next section reviews previous studies focused on stream temperature models and theirapplication. The goal is to identify specific knowledge gaps that will provide the focus forthis study’s objectives, which are provided in the last section of this chapter.1.2 Stream temperature models and their applicationsModels of 𝑇𝑤 cover a spectrum from deterministic models to data-based empirical models.Deterministic models are based on mathematical expressions of the governing physicalprocesses. Empirical models can be based on statistical models, stochastic models, andmachine learning approaches. For fish habitat assessment, empirical models are typicallybased on statistical relations with air temperature and other variables (Benyahya et al.,2007; Moore et al., 2013; Isaak et al., 2017).Deterministic models of 𝑇𝑤 simulate the energy and water fluxes between a stream andits environment. These processes include incident radiation from the sun and conductionwithin the streambed, longitudinal advection and dispersion from upstream, and lateraladvection from groundwater and tributary discharge (Sridhar et al., 2004; Bray et al., 2017).2Deterministic models can be applied at point, reach (Norton and Bradford, 2009; Wrightet al., 2009), or catchment scales (Noble and Jackman, 1979; Allen et al., 2007). Modelsat all scales typically resolve the energy exchanges across the water surface and the streambed. However, only reach- or catchment-scale models can explicitly account for longitudinaladvection of heat and water along the stream channel; these processes are either ignored orare parameterized in point-scale deterministic models (Bogan et al., 2004).Because deterministic models explicitly account for the meteorological, hydrological andland-cover factors that control stream temperature, they can be used to simulate howchanges in the controlling factors, such as climate change or riparian forest harvesting,will influence stream thermal regimes (Sinokrot and Stefan, 1993; Benyahya et al., 2007).However, a major limitation is that these models require extensive input data, includingphysical characteristics of the stream and watershed (e.g., shading by vegetation and streamgeometry), as well as hydrological and meteorological information (Benyahya et al., 2007).Elaborate field surveys are often necessary to collect data with a high enough level of detail,which may make these models more appropriate for a single catchment, though the use ofgeographical information system (GIS) and the increasingly detailed spatial data sets nowavailable expand their utility (Sridhar et al., 2004; Moore et al., 2013)Deterministic models have largely been used to inform land management practices by ex-amining the effects of shading from vegetation on radiation reaching the stream, which isincluded in the energy balance term (Rutherford et al., 1997; Chen et al., 1998; St-Hilaireet al., 2000; Sridhar et al., 2004; Allen et al., 2007; Norton and Bradford, 2009). Sridhar etal. (2004) developed a relatively simple model using GIS information that was evaluatedfor multiple rivers in basins across Washington state, but the simplicity of the vegetationparameters and stream geometry makes the model less applicable to larger stream systems.On the other hand, an even more general model for shading like BasinTemp, developed byAllen et al. (2007), had poor performance when modelling certain complex processes, likelow-flow discharge that varied with vegetation structure across a basin. Snow melt andbed conduction are also significant processes that require detailed measurements or furtherresearch to correctly model, as highlighted by MacDonald et al. (2014) and Bray et al.(2017).Statistical models have the advantage that they have less stringent data requirements. Sta-tistical models can be temporal (i.e., predict time series of stream temperature at a givenlocation), spatial (i.e., predict the spatial variation of a static temperature index), or spatio-temporal (i.e., predict time series of stream temperature at multiple locations). Temporalmodels are typically based on linear or nonlinear regression using air temperature timeseries as a predictor. They can also include variables, such as snowpack depth and timeof year (Siegel et al., 2022). Spatial models typically use linear regression or mixed-effectsmodels to fit models that treat 𝑇𝑤 as a function of predictor variables such as catchmentarea, land cover, and climatic characteristics (Benyahya et al., 2007; Isaak et al., 2010;Moore et al., 2013). Many of these variables can be readily accessed from spatial databases(e.g. drainage area determined from a digital elevation model) or collected in more detailfrom field surveys (e.g. canopy closure) (Isaak et al., 2010).Statistical models are usually simpler to implement than deterministic models (Benyahya3et al., 2007). Spatial models can be useful for predicting 𝑇𝑤 across large regions (Mohseni etal., 2003; Jackson et al., 2018), but because the coefficients are based on empirical relationsderived under existing conditions rather than physics, they typically are unable to simulatestream temperature response to changes to physical forcings (Noble and Jackman, 1979;St-Hilaire et al., 2000). However, some spatial and spatio-temporal statistical models haveincluded spatially explicit vegetation characteristics as a predictor, which allows the effectof vegetation change to be simulated by changing the value of the vegetation variable (Isaaket al., 2010; Jackson et al., 2018).A comparison of different spatial statistical models by Wehrly et al. (2009) showed that allapproaches had comparable prediction accuracies with root-mean-square errors (RMSEs)of 2.0-2.3 oC for Michigan and Wisconsin river temperatures. Similarly, Moore et al. (2013)saw prediction errors with a standard deviation of 2.1 oC for their regression model of BCrivers. These high metrics of error demonstrate that 𝑇𝑤 predictions for statistical modelsmay be coarse on regional scales, likely due to missing fine details of hydrological processesand riparian shading.Though many predictor variables in statistical models are static in time, climate informa-tion, particularly 𝑇𝑎, experiences interannual variability. Studies approach this in differentways. Both Wehrly et al. (1998) and Nelitz et al. (2007) collected 𝑇𝑤 data for several years,but chose to use a single year of data for each stream with 𝑇𝑎 data from the same year todevelop the model to avoid problems with lack of independence. Moore et al. (2013) rep-resented interannual variability of air temperature by interpolating data from surroundingweather stations and averaging the daily time series for a chosen multi-year period over thewarmest months, July and August. They included a temporal air temperature covariatecomputed as the deviation of the air temperature for the years of data for a given stationfrom the average over a historical reference period.Spatial statistical models predict a specific stream temperature metric. For example, Mooreet al. (2013) developed a model to predict the maximum weekly average temperature(MWAT). This is the annual maximum temperature of a seven-day running average foreach year studied. This metric works well for assessing summer thermal regime and hasbeen correlated with trout bioenergetics (Nelitz et al., 2007) and fish species assemblages(Parkinson et al., 2016), but does not capture temperature variability outside of summer,which can be relevant for studying fish that emerge in the winter like coho salmon fry(Holtby, 1988). Temporal and spatio-temporal models that operate on a daily time scaleare, in principle, more flexible, because the output could be used to compute any desiredthermal index, such as a measure of temperatures over winter.Ultimately, the drawbacks of each type of model possibly make them insufficient for large-scale studies of river water temperature under a changing climate. A deterministic modelof 𝑇𝑤 across a region like BC would require too much unavailable data and complex im-plementation, and the lack of underlying physics in statistical models could bring aboutunreliable results when extrapolated to different scenarios, especially those as extreme as aheat wave.4Toffolon and Piccolroaz (2015) developed a hybrid model, air2stream, which is based ona differential equation that represents a simplified point-scale heat budget, with surfaceenergy exchanges parameterized using air temperature and advective fluxes from upstreamusing stream discharge. In contrast to fully deterministic models, air2stream requires onlytime series of 𝑇𝑎 and streamflow (Q) to predict 𝑇𝑤 as a time series with the same frequencyas the input data, which can be subdaily, daily, or longer. The model is discussed in moredetail in Section 2.5.3.In its applications to date, air2stream has generally performed favorably so long as anappropriate calibration method is used (Piotrowski and Napiorkowski, 2018). One potentialdrawback is that it requires a time series of Q and 𝑇𝑤 to calibrate, so it can only be applied atlocations with stream gauges that have operated for a sufficient number of years to supportcalibration. Once calibrated, the model could be used to hindcast historic temperatures.For example, Islam et al. (2019) used air2stream to hindcast stream temperature in theFraser River basin over the period 1950 to 2015 (Islam et al., 2019). The model could alsobe used for climate change impact assessment if applied in conjunction with output from ahydrological model forced by data representing a future climate scenario.Both statistical models and air2stream rely heavily on reliable air temperature data. Whilestreamflow is measured for many rivers in North America and elsewhere, stream temper-ature records tend to be shorter and/or more sparse. Even where stream temperatureand streamflow data are available, many if not most do not include air temperature mea-surements. In many previous applications, modellers used nearby weather station data toestimate it when direct measurements were not possible (Evans et al., 1998; Sridhar et al.,2004; Ruesch et al., 2012). However, nearby weather stations may not be located at thesame elevation as the hydrometric station. For example, Wright et al. (2009) developeda simplified empirical model for the Colorado River using data from the nearest weatherstation which was at a higher elevation than the reach. Moore et al. (2013) addressed thischallenge by interpolating the data from surrounding weather stations to the locations ofeach stream temperature monitoring location using the approach of Stahl et al. (2006),which incorporates adjustments for elevation.As an alternative to working with weather station data, advanced climate models can pro-vide high-resolution reanalysis products that use numerical techniques to take into accountboth weather observations and orographic effects. These data products are gridded; thatis, temperature is represented over an evenly-spaced matrix across a region, and often hasfiner resolution than the distribution of weather stations. Arismendi et al. (2014) used theNational Climatic Data Center (NCDC) air temperature data from the nearest grid cell tothe study locations in order to develop a regression model for predicting 𝑇𝑤 in the US. Theair temperature at a specific site can be obtained even more precisely by interpolating fromthe surrounding grid cells (Trubilowicz et al., 2016; Szeitz and Moore, 2023).Two reanalysis products are available that cover the entirety of North America, includingBritish Columbia: the ECMWF Reanalysis v5 (ERA5) and the North American RegionalReanalysis (NARR). Both provide near-real time gridded climate data from 1979 to presentwith high spatial and temporal resolutions (Mesinger et al., 2006; Hersbach et al., 2020).5ERA5 is often used in major climate studies, and has been shown to be an accurate es-timation of surface temperature, especially in temperate regions (Hersbach et al., 2020;McNicholl et al., 2022). In previous hydrological models, NARR was nearly as effective asobserved data and produced reasonable results, and it has shown considerable accuracy forair temperature at high elevations (Choi et al., 2009; Trubilowicz et al., 2016). Becausethese databases are frequently updated, easily accessible, and high-resolution, both havepotential to be a dependable source for air temperature data to use in empirical models ofwater temperature and a hybrid model like air2stream.Islam et al. (2019) used gridded daily 𝑇𝑎 data provided by Natural Resources Canada fortheir application of the air2stream model in the Fraser River Basin. That product is basedon application of thin-plate spline interpolation to data collected at weather stations. Adownside of this product is that it exhibits latency due to the lag between acquisition ofupdated air temperature data and updating of the gridded product. A potential advantageof reanalysis products is that their latency is typically two or three months, allowing fortimely analysis.ERA5 and ERA5-Land have been successfully used as meteorological inputs in process-based stream temperature models for remote river reaches with sparse observations in theUS and Canada (Mihalevich et al., 2022; Gatien et al., 2023). However, no studies appearto have explored the use of reanalysis air temperature products as input to air2stream, orto have compared their utility as predictors in statistical models in comparison to station-based data.There has been an ongoing debate in the literature about the advantages and disadvantagesof empirical versus deterministic models for predicting stream temperature response toproject future climate scenarios, particularly in light of the fact that deterministic modelstend to produce greater responses than empirical models (Isaak et al., 2016; Leach andMoore, 2019). Given the extreme heat events of recent years, a major related questionconcerns the ability of stream temperature models to make predictions under conditions ofheat waves and extended drought periods.Since stream temperature models are used to predict important measures such as habitatsuitability (Schultz et al., 2017) and stream sensitivity (Trimmel et al., 2016; Seo et al.,2022; Tassone et al., 2023) under drought and heat wave conditions, it is important tounderstand how they function during these extremes before relying on their results.1.3 Research objectives and thesis structureThe literature review in Section 1.2 highlighted that an evaluation of gridded air tempera-ture products from reanalysis projects could be useful for guiding future stream temperaturemodelling efforts. A further question relates to the reliability of stream temperature modelswhen extrapolated to events that fall outside the limits of the data used to calibrate them.Drawing upon these knowledge gaps, this study focused on two key objectives:61. To evaluate the performance of two reanalysis-based air temperature products, ERA5and NARR, as predictors in both spatial and temporal statistical models. Air temper-atures derived from a station-based interpolation (SBI) were used as a benchmark.2. To compare the ability of a temporal statistical model and the air2stream model topredict stream temperature in BC during two extreme events: the 2021 heat waveand the 2022 autumn drought.The remainder of the thesis is structured as follows. Chapter 2 provides a description ofthe study area, data sources, and model methods. Chapter 3 presents the results of eachassessment and Chapter 4 discusses these results in detail. Chapter 5 summarizes the mainfindings and their relevancy for future research.72 Study site and methods2.1 Study site: British Columbia, CanadaThis study focused on the province of British Columbia, Canada. Located east of thePacific Ocean and southeast of the Aleutian Islands, the climate of BC is dominated bylow-pressure and high-pressure systems that flow west-to-east. Winters are characterized bygreater amounts of moisture and colder temperatures as low-pressure systems occur morefrequently. In the summer, extended periods of high-pressure bring dry, warm weather.As these systems make their way across the province, their effect is determined by theinteraction with the topography of the province, dividing it into five main physiographicregions (Pike et al., 2010).In the west, the island mountains and the Coast Mountains produce an orographic effectthat wrings moisture out of the air. While this is the wettest part of the province, thesouthwest is also the most temperate year-round. On the lee side of the Coast Mountainsare the north and south plateaus, which remain dryer due to the rain shadow effect andhave the greatest seasonality. In the northeast are the great plains, and in the southeast arethe Columbia and Rocky Mountains which create another region of moisture and preventcontinental arctic air masses from reaching the southern plateau (Pike et al., 2010).These unique topographic effects contribute to the variety of hydroclimatic regimes seenthroughout the province. Rain-dominated catchments, where peak flows generally occur inautumn and winter, are found in low-elevation areas near the coast. Snowmelt-dominatedand glacierized catchments are found at high elevations in the mountains on either side ofthe province and in the plateau; these generate peak flows during May. Hybrid-regimeshave mixed characteristics of the two and can be found near the coast and in the northeastof the province. Since streamflow and air temperature mainly control the temperature ofwater bodies, the weather patterns in the province shape each stream’s unique thermalregime (Pike et al., 2010).2.2 Stream temperature data sources and processingTwo sets of stream temperature data were used in this study. One is the data set compiledfrom multiple data donors and used in the analysis of Moore et al. (2013); the other includesdata collected by Water Survey of Canada at gauging stations.Moore et al. (2013) gathered raw stream temperature data from multiple donors forstreams around BC. After applying quality assurance/quality control procedures, data for8418 streams were retained. Time series of daily summaries (minimum, mean, maximum)were computed for each station (Figure 2.1). Moore et al. (2013) computed the annual max-imum weekly average temperature (MWAT) for each station-year of data for each stream.MWAT is determined by computing a seven-day running average of the daily mean streamtemperatures and then selecting the highest value in each year. For stations with multi-ple years of data, the MWAT values were averaged. Both the time series of daily meantemperatures and the MWAT values averaged by station were used in analyses to evalu-ate the utility of gridded reanalysis air temperature products in statistical models. Thesemonitoring sites will be referred to as MWAT stations.Figure 2.1: Map of each station used in the Moore et al. (2013) data set, colored by thestream’s associated MWAT value. The three example stations are highlightedwith symbols.For applying the air2stream model, hourly stream temperature data for all Water Survey ofCanada (WSC) hydrometric stations in BC with temperature measurements were obtainedfrom David Hutchinson, Regional Chief, Pacific and North Hydrometric Operations forEnvironment and Climate Change Canada. Data from days with less than 20 hourly mea-9surements were filtered out, and the daily average stream temperature was computed foreach stream. A total of 23 streams, with 15 unregulated and 8 regulated, were identified assuitable based on having at least 6 years of data with less than 40% missing. The locationsof these stations are shown in Figure 2.2, and information about catchment characteristicsand periods of record are provided in Tables 2.1, 2.2, and Table 2.3. These locations willbe referred to as WSC stations.Spot temperature measurements made by WSC technologists during visits to the gaugingstations were obtained from Environment and Climate Change Canada (ECCC). Thesespot measurements were used to help detect dewatering events, i.e., periods when thetemperature sensor was out of the water, and errors in the data were additionally filtered out.Dewatering events and errors were identified by generating interactive graphs of the spotmeasurements and hourly stream temperature data in plotly (Inc., 2015), and pinpointingwhere the hourly data either did not line up with the spot measurements or where theyreached unrealistic values, such as below 0 oC or above 30 oC.In order to find the stream temperature change at each WSC station during the heat wave,the stream temperatures for the 5 days before the heat wave (June 20, 2021 to June 24,2021) were averaged and then subtracted from the average stream temperature during theheat wave (June 25, 2021 to July 2, 2021).10Figure 2.2: Map of Water Survey of Canada hydrometric stations used in the air2streammodel, colored by regulation. The three example stations are highlighted withsymbols.11Table 2.1: Station ID, name, and location information of each Water Survey of Canadahydrometric station used in the air2stream model.Station ID Station name Latitude Longitude Elevation(m)07EA004 Ingenika River above Swannell River 56.73 -125.11 68507EB002 Ospika River above Aley Creek 56.52 -123.94 73908CG001 Iskut River below Johnson River 56.73 -131.67 2108FF001 Kitimat River below Hirsch Creek 54.05 -128.69 1708FF003 Little Wedeene River below Bowbyes Creek 54.14 -128.69 6908GA077 Seymour River below Orchid Creek 49.52 -123.00 24908HA002 Cowichan River at Lake Cowichan 48.83 -124.05 16408HA011 Cowichan River near Duncan 48.77 -123.71 2008HA069 Renfrew Creek near Port Renfrew 48.64 -124.29 30808HA070 Harris Creek near Lake Cowichan 48.72 -124.23 26508HB023 Ash River below Moran Creek 49.37 -124.98 7608JA017 Nechako River below Cheslatta Falls 53.69 -124.84 71708JB008 Nadina River at outlet of Nadina Lake 53.90 -126.95 90108KB001 Fraser River at Shelley 54.00 -122.62 57508KH006 Quesnel River near Quesnel 52.84 -122.23 54008LB078 Lemieux Creek near the Mouth 51.43 -120.20 39608LE031 South Thompson River at Chase 50.76 -119.74 34908LF051 Thompson River near Spences Bridge 50.35 -121.39 20808MF040 Fraser River above Texas Creek 50.61 -121.85 19408NE074 Salmo River near Salmo 49.05 -117.29 59408NG065 Kootenay River at Fort Steele 49.61 -115.64 77108NH118 Duncan River below Lardeau River 50.23 -116.96 55208NK002 Elk River at Fernie 49.50 -115.07 99712Table 2.2: Catchment characteristics of each Water Survey of Canada hydrometric station used in the air2stream model.Station ID Gross drainagearea (km2)Glacier cover in2013 (% area)Lake cover in2013 (% area)Regulation Hydrologic regime07EA004 4.14e+03 0.00 1.06 Unregulated Snow07EB002 2.19e+03 0.00 0.79 Unregulated Snow08CG001 9.50e+03 0.94 2.24 Unregulated Snow08FF001 1.99e+03 2.10 1.25 Unregulated Glacial/Hybrid08FF003 1.80e+02 0.32 0.48 Unregulated Hybrid08GA077 6.30e+01 0.00 1.31 Unregulated Hybrid08HA002 5.94e+02 0.00 10.95 Regulated Rain08HA011 8.26e+02 0.00 8.17 Regulated Rain08HA069 8.12e+00 0.00 0.00 Unregulated Rain08HA070 2.80e+01 0.00 0.00 Unregulated Rain08HB023 3.87e+02 0.24 3.89 Regulated Rain08JA017 1.55e+04 0.47 10.65 Regulated Snow08JB008 3.69e+02 0.25 5.02 Regulated Snow08KB001 3.24e+04 1.66 1.89 Unregulated Snow08KH006 1.15e+04 1.31 5.59 Unregulated Snow08LB078 4.43e+02 0.00 5.64 Regulated Snow08LE031 1.58e+04 0.40 4.41 Unregulated Snow08LF051 5.54e+04 0.73 3.54 Unregulated Snow08MF040 1.54e+05 0.83 0.00 Regulated Snow08NE074 1.24e+03 0.00 0.33 Unregulated Snow08NG065 1.15e+04 0.17 0.94 Unregulated Snow08NH118 4.08e+03 3.43 2.91 Regulated Glacial/Hybrid08NK002 3.09e+03 0.28 0.66 Unregulated Snow132.3 Air temperature data sources and processing2.3.1 Station-based dataFor comparing the ERA5 and NARR reanalysis air temperature data to observations, his-torical daily mean air temperature data from 654 weather stations across BC were obtainedfrom Environment and Climate Change Canada. The stations cover a broad spatial extentacross the province and were chosen to have a minimum of 5 years of data during thereanalysis period (1979-2022). The daily mean air temperature is defined as the averagebetween the daily maximum and minimum air temperatures. The location of these stationsis shown in Figure 2.3.Figure 2.3: Map of the ECCC climate stations used for comparison to reanalysis air tem-peratures, colored by the length of air temperature record in years.For fitting temporal and spatial statistical models using the stream temperature data fromMoore et al. (2013), air temperatures were generated by interpolating data from nearbyweather stations using the constrained lapse rate method applied by Stahl et al. (2006). The14Table 2.3: Information about the stream temperature time series record for each WaterSurvey of Canada hydrometric station used in the air2stream model.Station ID Start date End date Length of record(years)Data missing (%)07EA004 2011-01-01 2022-10-31 11.83 1.1607EB002 2011-01-01 2022-10-31 11.83 3.6508CG001 2016-01-22 2022-10-31 6.77 24.0008FF001 2014-01-01 2022-10-31 8.83 12.3808FF003 2014-01-01 2022-10-31 8.83 4.0508GA077 2014-03-11 2022-10-31 8.64 8.3408HA002 2012-10-04 2022-10-31 10.07 1.6908HA011 2015-03-16 2022-10-31 7.63 22.8008HA069 2014-01-01 2022-10-31 8.83 7.0908HA070 2016-01-01 2022-10-31 6.83 6.7608HB023 2016-01-01 2022-10-31 6.83 3.9408JA017 2011-01-01 2022-10-31 11.83 5.3908JB008 2016-01-07 2022-10-31 6.81 10.5208KB001 2011-01-01 2022-10-13 11.78 10.7208KH006 2013-08-14 2022-10-31 9.21 1.8708LB078 2016-01-01 2022-10-31 6.83 0.2508LE031 2011-01-01 2022-10-31 11.83 17.6208LF051 2011-01-01 2022-10-31 11.83 4.7108MF040 2011-01-01 2022-10-31 11.83 6.9608NE074 2014-02-04 2022-10-31 8.74 1.9908NG065 2011-01-01 2022-10-31 11.83 8.9608NH118 2012-06-08 2022-10-31 10.40 1.2308NK002 2011-01-01 2022-10-31 11.83 19.5215station data were drawn from a variety of sources, including ECCC stations, fire weatherstations, and stations operated by the Ministry of Transportation and Highway (Stahl etal., 2006). For fitting temporal models, the time series of daily mean air temperatures wereused.For fitting a spatial model to predict MWAT, the annual July-August mean air temperaturewas calculated and then subtracted from the July-August mean air temperature over thereference period of 1990-2003. This deviation from the average climate captures the climaticvariability over each year experienced at each stream location and can be expressed as𝛿𝑇𝑎(𝑖, 𝑡) = 𝑇𝑎(𝑖, 𝑡) − 𝑇𝑟𝑒𝑓(𝑖) (2.1)where 𝑇𝑎(𝑖, 𝑡) is the July-August mean air temperature for station 𝑖 and year 𝑡 (oC), 𝑇𝑟𝑒𝑓(𝑖, 𝑡)is the July-August mean air temperature averaged over the reference period of 1990-2003(oC), and 𝛿𝑇𝑎(𝑖, 𝑡) is the air temperature deviation from the reference period average forstation 𝑖 (oC). For each station, the temperature deviations were averaged over all station-years for use as a predictor.2.3.2 Reanalysis products2.3.2.1 ECMWF Reanalysis v5The European Centre for Medium-Range Weather Forecasts (ECMWF) provides theECMWF Reanalysis v5 (ERA5) through the Copernicus Climate Change Service (C3S).This product includes hourly atmospheric reanalysis data on a 0.25 degree global gridfrom 1950 to present, an improvement from ERA-Interim, which spanned 1979-2019 ata 3-hourly interval with an 80 km grid resolution. This product was chosen becauseit is publicly available and regularly updated; preliminary data are updated daily witha 5 day lag, and final data have a 2-3 month lag. Additionally, ERA5 is used by theIntergovernmental Panel on Climate Change (IPCC) and the World MeteorologicalOrganization (WMO) for climate assessments (Hersbach et al., 2020).Using the CDS API client in Python, ERA5 daily mean 2-m air temperature was down-loaded for the region bounded by the latitudes -145o to -105o and longitudes 46o to 62ofrom 1979-2022 (Cucchi). The original data were in the WGS 84 coordinate system, andwere reprojected to the BC Albers Equal Area Conic projection using the ‘projectRaster’function within the raster package in R so the coordinates could be used in distance cal-culations (Hijmans and Etten, 2012). The corresponding surface geopotential data for thesame region were converted to elevation as follows:𝑧𝑒 = 𝑧/𝑔 (2.2)where 𝑧𝑒 is the elevation at the center of a grid point (𝑚), 𝑧 is the surface geopotentialat the center of a grid point (𝑚2𝑠−2), and 𝑔 = 9.80665 𝑚𝑠−2 is Earth’s gravitationalacceleration.162.3.2.2 North American Regional ReanalysisNorth American Regional Reanalysis (NARR) is a long-term atmospheric reanalysis prod-uct provided by the National Center for Environmental Prediction (NCEP). It coversCanada, the US, and Mexico for the period 1979 to present with a ~32 km horizontalresolution and a 3-hourly temporal resolution. This product was chosen due to its regionalfocus on North America and near real-time availability (Mesinger et al., 2006).NARR daily mean 2-m air temperature data from 1979-2022 were downloaded using thedownload URL http://www.esrl.noaa.gov/psd/thredds/fileServer/Datasets/NARR/Dailies/monolevel/. The original data are in the Lambert conformal conic projection, and were re-projected to the BC Albers Equal Area Conic projection using the ‘projectRaster’ functionand then cropped to the area of BC. The corresponding surface heights for the grid weredownloaded using the download URL http://www.esrl.noaa.gov/psd/thredds/fileServer/Datasets/NARR/time_invariant/hgt.sfc.nc, and similarly reprojected and cropped.2.3.2.3 InterpolationBoth the ERA5 and NARR reanalysis air temperature data sets were interpolated to thelocations of the ECCC climate stations for the length of each station’s air temperaturerecord. They were also interpolated to the locations of the MWAT stations for the lengthof the reference period (1990-2003) as well as the length of each station’s stream temperaturerecord. Only the ERA5 data set was interpolated to the locations of the WSC hydrometricstations for the length of each station’s stream temperature record.The first step to interpolate the reanalysis air temperature data to a station location wasto identify the four nearest grid points, which are the coordinates at the center of the cells,to the station using the distance formula:𝑑𝑖𝑗 = √(𝑥𝑖 − 𝑥𝑗)2 + (𝑦𝑖 − 𝑦𝑗)2 (2.3)where 𝑑𝑖𝑗 is the distance between the grid point 𝑗 and station 𝑖 (𝑚), 𝑥𝑗 and 𝑦𝑗 are thecoordinates of the grid point 𝑗 in BC Albers Equal Area Conic projection, and 𝑥𝑖 and 𝑦𝑖are the coordinates of station 𝑖 also in BC Albers projection.Once the four grid points with the smallest distances to the station were selected, a lapserate calculation was applied to the time series for the grid points to adjust each to the sameelevation as the station. This calculation was based on monthly lapse rates for daily maxi-mum and minimum temperatures computed by Stahl et al. (2006), which were averaged forthe daily mean air temperature as shown in Figure 2.4. These lapse rates were computedbased on 6 pairs of low-elevation/high-elevation meteorological stations across southern BC(Stahl et al., 2006). The four elevation-adjusted time series were calculated as follows:𝑇𝑎,𝑖𝑗 = 𝑇𝑎,𝑗 + 𝜆𝑚(ℎ𝑖 − ℎ𝑗) (2.4)17where 𝑇𝑎,𝑖𝑗 is the adjusted air temperature at grid point 𝑗 (oC), 𝑇𝑎,𝑗 is the original airtemperature at grid point 𝑗 (oC), 𝜆𝑚 is the lapse rate for the daily mean air temperaturefor month 𝑚 based on the purple line in Figure 2.4 (𝑜𝐶𝑚−1), ℎ𝑖 is the elevation of station𝑖 (𝑚), and ℎ𝑗 is the elevation of grid point 𝑗 (𝑚).The final step involved bilinear interpolation from the four grid points to the locationof the station (see Appendix A for details). In the case of the ECCC climate stations,these interpolated air temperatures were compared to observed air temperatures. Theinterpolated air temperatures at the MWAT and WSC stations were used to fit statisticalmodels of stream temperature.The interpolated air temperature data were used to calculate the 42-year air temperatureanomaly for each WSC station during the heat wave period of June 25, 2021, to July 2,2021. For each location, the average air temperature over June 25 to July 2 was calculatedfor each year from 1979-2020, then averaged over the entire period. This long-term averagewas subtracted from the average air temperature over the heat wave period to compute theanomaly.Figure 2.4: Monthly lapse rates from Stahl et al. (2006) for daily maximum, minimum, andmean air temperatures.182.4 Streamflow dataFor each of the 23 WSC stations chosen with stream temperature data, WSC stream-flow data were also obtained for the same period of record. Data from before2022 were obtained using functions in the tidyhydat package in R, and prelimi-nary data for 2022 were downloaded manually from the ECCC website at the URLhttps://wateroffice.ec.gc.ca/services/links_e.html (Albers, 2017). Data for the two timeperiods were joined to form one time series. For days with missing streamflow data, linearinterpolation was performed with the ‘na.approx’ function in R (R Core Team, 2021).Additionally, these streamflow data were used to calculate the 25-year streamflow anomalyfor eachWSC station during the autumn drought period of September 1, 2022 to October 31,2022. For each stream, the average streamflow over September and October was calculatedfor each year from 1997-2021, then averaged over the entire period. This long-term averagewas subtracted from the average streamflow over the autumn drought period to computethe anomaly. The relative streamflow anomaly was calculated by dividing the anomaly bythe average streamflow over 1997-2021.2.5 Statistical models2.5.1 Spatial model for MWATTo evaluate the utility of reanalysis products in spatial statistical models, they were usedto compute the air temperature deviations used in the statistical model fit by Moore et al.(2013), which can be expressed as𝑇𝑀𝑊𝐴𝑇 = 𝑏1 + 𝑏2𝑇𝑎 + 𝑏3𝑙𝑜𝑔(𝐴) + 𝑏4𝑧𝑚 + 𝑏5𝑠𝑙𝑜𝑝𝑒 + 𝑏6𝑓0.5𝑔 + 𝑏7𝑓0.5𝑙 + 𝑏8𝑘2 + 𝑏9𝛿𝑇𝑎 (2.5)where 𝑇𝑀𝑊𝐴𝑇 is the predicted value of MWAT (oC), 𝑏1 to 𝑏9 are the coefficients of themodel, 𝑇𝑎 is the mean July-August air temperature for a climatic normal period (oC), 𝐴 iscatchment area (𝑘𝑚2), 𝑧𝑚 is mean catchment elevation (𝑚), slope is the channel slope, 𝑓𝑔is fractional glacier coverage, 𝑓𝑙 is fractional lake coverage, and 𝑘2 is an index of the 2-yearflood as interpolated from a map generated by Eaton et al. (2002). Values of MWATand all predictors except for 𝛿𝑇𝑎 were the same as used in the original analysis by Mooreet al. (2013). In the original analysis, the 𝛿𝑇𝑎 term was computed using station-basedinterpolation, as described in Section 2.3.1. In this analysis, the model was re-fit using theNARR- and ERA5-based values of 𝛿𝑇𝑎.192.5.2 Temporal models for daily stream temperature2.5.2.1 Linear regressionLinear regression models were fit between daily stream temperature and daily air temper-ature for each MWAT station using the corresponding air temperatures interpolated fromERA5, NARR, and SBI. For each station, separate regression models were fit for eachwarm month (April-October) with at least 21 days of data; the regression models had thefollowing form:𝑇𝑤 = 𝑏0 + 𝑏1𝑇𝑎 + 𝑒 (2.6)where 𝑏0 and 𝑏1 are the y-intercept and slope, respectively, and 𝑒 is the error term.2.5.2.2 Logistic curveWhile non-linearity can be addressed by fitting separate models by month, as describedin the preceding section, that approach only works for the warm months when streamtemperatures remain above 0 oC. An alternative is to fit an S-shaped curve (Mohseni andStefan, 1999) of the following form:𝑇𝑤 =𝑏1 + 𝑒𝑥𝑝[𝑐(𝑇𝑎 − 𝑑)](2.7)where 𝑏, 𝑐, and 𝑑 are the parameters to be estimated; 𝑏 represents the upper asymptote, 𝑑 isthe air temperature corresponding to the inflection point of the fitted curve, and 𝑐 is relatedto the slope of the curve at the inflection point. Because the relation between stream and airtemperatures exhibits hysteresis, separate models were fit for the warming (March-August)and cooling (September-February) seasons for a subset of 53 MWAT stations based on eachstation having a minimum of 100 days of data in each season.2.5.3 Air2streamThe air2stream model is based on the following heat budget equation, which governs thevariation of stream temperature, 𝑇𝑤 (oC), within a volume, 𝑉 (𝑚3), of water in a riverreach and its tributaries:𝜌𝑐𝑝𝑉𝑑𝑇𝑤𝑑𝑡 = 𝐴𝐻 + 𝜌𝑐𝑝(∑𝑖𝑄𝑖𝑇𝑤,𝑖 −𝑄𝑇𝑤) (2.8)where 𝜌 is water density (𝑘𝑔𝑚−3), 𝑐𝑝 is the specific heat capacity of water at constantpressure (𝐽𝐾−1𝑘𝑔−1), 𝑡 is time (expressed in days), 𝐴 is the surface area of the reach (𝑚2),𝐻 is the net heat flux between the river and the atmosphere (𝑊𝑚−2), 𝑄 is the streamflowdownstream (𝑚3𝑠−1), 𝑄𝑖 is the discharge associated with the 𝑖th contributing upstreamwater flux (𝑚3𝑠−1), and 𝑇𝑤,𝑖 is the temperature of the 𝑖th contributing upstream water flux(oC).20The net heat flux at the water surface is parameterized as a linear function of air andwater temperatures. Many simplifications are made as shown in Appendix B, including aTaylor series expansion for 𝐻 and an assumption that stream temperature annually can bedescribed as a sinusoidal variation. The final 8-parameter model equation takes the form𝑑𝑇𝑤𝑑𝑡 =1𝛿 (𝑎1 + 𝑎2𝑇𝑎 − 𝑎3𝑇𝑤 + 𝜃[𝑎5 + 𝑎6𝑐𝑜𝑠(2𝜋(𝑡𝑡𝑦− 𝑎7)) − 𝑎8𝑇𝑤]) (2.9)where 𝑇𝑤 is stream temperature (oC), 𝑡 is time (days), 𝛿 = 𝜃𝑎4 , 𝑎1 to 𝑎8 are the parameters ofthe model, 𝑇𝑎 is air temperature (oC), 𝜃 = 𝑄𝑄−1, Q is the streamflow (𝑚3𝑠−1), and 𝑡𝑦 is theduration of a year in the same units as 𝑡. The model was written in Fortran and is availablefor download as a compiled executable via https://github.com/spiccolroaz/air2stream.The model was calibrated using stream temperature, air temperature, and streamflow timeseries to the end of 2020. Data for 2021 and 2022 were used as validation data. Additionally,to assess the model performance during the 2021 heat wave and 2022 drought, separateerrors were calculated for the heat wave period (June 25, 2021 to July 2, 2021) and theautumn drought period (September 1, 2022 to October 31, 2022).To compare the performance of air2stream and a statistical model under extreme conditions,linear regression models of the form of Equation 2.6 were fit using data for June and July(for prediction during the heat dome) and September and October (for prediction duringthe autumn drought). Model comparisons were based on visual assessment of time seriesgraphs and scatterplots, as well as root-mean-square error (RMSE), mean bias error (MBE),and mean absolute error (MAE), which were computed as𝑅𝑀𝑆𝐸 = √1𝑛𝑛∑𝑖=1(𝑆𝑖 −𝑂𝑖)2 (2.10)𝑀𝐵𝐸 = 1𝑛𝑛∑𝑖=1(𝑆𝑖 −𝑂𝑖) (2.11)and𝑀𝐴𝐸 = 1𝑛𝑛∑𝑖=1|𝑆𝑖 −𝑂𝑖| (2.12)where 𝑆𝑖 and 𝑂𝑖 represent the simulated and observed water temperatures on day 𝑖 (oC).213 Results3.1 Evaluation of ERA5 and NARR air temperature data3.1.1 Comparison to climate station observationsAs seen in Table 3.1, ERA5 air temperatures generally had better agreement with observedair temperatures in all seasons than did NARR air temperatures in terms of r2, RMSE,and MAE. Both products had better agreement in summer, with RMSE values of 1.58 and1.99 oC for ERA5 and NARR, respectively, and the worst agreement in winter, with RMSEvalues of 2.26 and 2.88 oC. Mean bias errors were less than 0.7 oC in magnitude for bothproducts and all seasons, with ERA5 having lower bias than NARR in all seasons exceptsummer and spring, when the differences were minor.Figure 3.1 presents the spatial distribution of RMSE in the summer, when values variedfrom 0 to 5 oC. For ERA5, 538 stations had RMSE values lower than 2 oC, whereas NARRhad only 355, as can be seen by the decrease in purple dots in the bottom NARR panel ofFigure 3.1. There is also a gradient between areas with high concentrations of stations to lowconcentrations, where higher concentrations appear to be associated with lower RMSE.3.1.2 Comparison of reanalysis products and station-based interpolation as airtemperature predictors in the regression model for MWATWhen used to compute the temporal air temperature covariate, 𝛿𝑇𝐴, in the MWAT modeldeveloped by Moore et al. (2013), both ERA5 and NARR air temperatures yielded similarTable 3.1: Seasonal errors between climate station air temperatures and NARR and ERA5air temperatures.ERA5 NARRSeason r2 RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)r2 RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)Summer 0.88 1.58 0.29 1.26 0.80 1.99 0.24 1.57Autumn 0.91 1.66 0.14 1.32 0.86 2.01 0.20 1.59Winter 0.87 2.26 -0.34 1.78 0.81 2.68 0.62 2.07Spring 0.89 1.75 -0.13 1.36 0.82 2.20 0.09 1.7022Figure 3.1: Spatial pattern of RMSE between observed air temperatures and interpolatedERA5 (top row) and NARR (bottom row) air temperatures for each climatestation during the summer.23Table 3.2: Comparison of MWAT models using the temporal air temperature covariatescomputed from the ERA5, NARR, and SBI air temperatures. Error statisticsare based on a leave-one-out cross-validation. 𝑏9 is the model coefficient forthe temporal air temperature covariate, 𝛿𝑇𝐴, and df is the residual degrees offreedom, both based on fitting the model to the full data set.r2 RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)𝑏9 dfERA5 0.63 2.18 0.00 1.63 0.32 409NARR 0.64 2.14 0.00 1.62 0.60 409SBI 0.64 2.13 -0.01 1.59 0.64 409Figure 3.2: Predicted versus observed MWAT for all stations for models fit using the tempo-ral air temperature covariate, 𝛿𝑇𝐴, computed from ERA5 (left column), NARR(middle column), and SBI (right column) air temperatures. Data points arebased on a leave-one-out cross-validation. Symbol colours indicate the July-August air temperature deviation for the monitoring year from the July-Augustaverage over the reference period (1990-2003). The black line represents perfectagreement24accuracy to that generated by using the covariate computed by station-based interpolation,with 𝑟2 values of 0.63 or 0.64 and RMSE values from 2.13 to 2.18 oC for all three products asshown in Table 3.2. The NARR and SBI covariates generated slightly better performancethan those based on ERA5. Interestingly, using the NARR-based covariate produced avalue of the coefficient for the 𝛿𝑇𝐴 covariate, 𝑏9, equal to 0.60, which is similar to that forthe SBI-based covariate, equal to 0.64. In contrast, the ERA5-based covariate produceda lower value for 𝑏9, equal to 0.32. Inspection of scatterplots of predicted versus observedMWAT reveals no substantial differences in agreement (Figure 3.2).3.1.3 Performance of ERA5, NARR, and SBI data in linear regressions of dailystream temperatureAs seen in Figure 3.3, relations between daily stream and air temperatures by month werevisually similar for the three air temperature sources. However, differences in the relationsamong stations are apparent. For example, for station 170, the slope appears much moreshallow than for the other two sites during the month of April when water temperatureswere the lowest, possibly due to the influence of snowmelt.Regressions between daily stream and air temperatures based on the three air temperatureproducts produced broadly similar ranges of 𝑟2 for the months of July to October (Fig-ure 3.4). The regressions based on ERA5 and SBI provided better fits than that based onNARR for all months, and ERA5 generated slightly higher values of 𝑟2 than both for May,June, and October.The variation of the fitted intercepts and slopes by month were similar across the three airtemperature data sources (Figures 3.5 and 3.6). The fitted intercepts tended to increasefrom April through July, then remain constant, although the ranges of values varied amongstations. The sample of stations shown in Figure 3.6 indicates differences among stationsin the seasonal pattern of the regression slopes, with stations 100 and 442 exhibiting sub-stantial variability from month to month while the slope for station 170 remained roughlyconstant.3.1.4 Performance of ERA5, NARR, and SBI data in logistic curve relations ofdaily stream temperatureFigure 3.7 shows the warming and cooling logistic curve fits using each temperature sourcefor the same three example stations used for the linear regressions by month. The fittedlogistic curves vary between the cooling and warming seasons, with station 170 exhibitingthe weakest hysteresis. Residual standard errors from the regression fits had similar rangesamong the three air temperature sources for the cooling season (Figure 3.8). However,ERA5 produced substantially lower residual standard errors than the other sources for thewarming season.25Figure 3.3: Scatterplots of daily stream temperature versus daily air temperature by monthusing each air temperature source for three example stations. The lines representlinear regressions by month.26Figure 3.4: Boxplots of the 𝑟2 for each monthly linear regression using each air temperaturesource.27Figure 3.5: Variation of the fitted y-intercept by month for the linear regressions of dailystream temperature using each air temperature source.28Figure 3.6: Variation of the fitted slope by month for the linear regressions of daily streamtemperature using each air temperature source.29Figure 3.7: Scatterplots of daily stream temperature versus daily air temperature with fit-ted logistic curves for the cooling (September-February) and warming seasons(March-August).30Figure 3.8: Boxplots of the RSE of the cooling (left column) and warming (right column)fitted logistic curves for each air temperature source.313.2 Evaluation of air2stream for predicting stream temperature3.2.1 Calibration and validation periodsThe fitted coefficients for all 23 stations are shown in Table 3.3. While some parameterswere similar among stations (e.g. 𝑎7), others exhibited greater variability (e.g. 𝑎5). Ascan be seen in Figures 3.9, 3.10, and 3.11, air2stream was able to reproduce the seasonalpatterns, shorter-term weather-related variability, and interannual variability for all threeexample streams during both the calibration and validation periods. One notable systematicerror is visible for the Fraser River at Shelley station (Figure 3.11), where simulated wintertemperatures dropped to 0 oC whereas, in reality, they remained above-freezing throughthe winter seasons.The generally good fits are also apparent by examination of scatterplots (Figure 3.12) andin the error statistics shown in Table 3.4. For the calibration period, air2stream produceda maximum RMSE of 1.15 oC, and most were less than 1 oC (Table 3.4). Values of MBEfor the calibration period were all less than or equal to 0.22 oC in magnitude. The modelperformed slightly worse for the validation period for most stations, with a maximum RMSEof 1.41 oC and a maximum MBE of 0.59 oC. As seen in Figure 3.13, performance was similarbetween regulated and unregulated stations, although regulated stations had slightly higherRMSE for both the calibration and validation periods.3.2.2 Heat wave and autumn drought periodsExample time series during the heat wave are shown in Figure 3.14 for each example station.For the date range associated with the 2021 heat wave (June 25 to July 2), air2streampredictions for the calibration period had RMSE values ranging up to 1.74 oC and MBEranging from –0.39 to 0.68 oC (Table 3.5). Model performance for the heat wave in 2021deteriorated compared to the calibration period, with RMSE up to 3.28 oC and MBEranging from -3.11 to 2.03 oC. This deterioration in model performance is also illustratedin Figure 3.15 where unregulated streams appear to again have slightly lower RMSE andMAE values. The spatial distribution of RMSE in Figure 3.16 shows an increase in errorduring the heat wave as compared to the calibration period among all hydrologic regimes,with the stations located in the southern Rocky Mountains having less of an increase. Airtemperature anomalies during the heat wave ranged up to 12.47 oC, and stream temperaturechanges ranged up to 4.03 oC.Figure 3.17 shows a scatter plot of the calibration RMSE for the heat wave dates vs the2021 heat wave RMSE for unregulated stations. For glacial, hybrid, and rainfed catchments,air2stream generated greater errors during the heat wave as compared to the same daterange in the calibration period, indicated by the corresponding symbols falling below the 1-1line. For snowfed catchments, the symbols follow the 1-1 line relatively well, meaning thatmodel performance did not deteriorate as significantly for unregulated, snowmelt-dominatedstreams.32Table 3.3: Fitted parameters for each station’s air2stream model.Station ID a1 a2 a3 a4 a5 a6 a7 a807EA004 0.30 0.53 0.48 -1.00 1.17 6.01 0.56 0.7407EB002 -0.10 0.21 0.20 -1.00 1.89 4.84 0.60 0.7308CG001 -0.65 0.47 0.55 -1.00 8.77 4.74 0.52 1.5408FF001 -1.45 0.79 0.66 -0.84 9.55 6.27 0.60 1.5508FF003 -1.14 0.98 1.03 -0.70 11.26 8.36 0.65 2.3508GA077 1.90 0.69 1.12 -1.00 7.44 5.15 0.84 1.3808HA002 1.53 0.28 0.27 -0.48 17.01 9.83 0.62 1.4708HA011 2.32 0.62 0.68 0.00 17.88 10.00 0.61 1.5808HA069 0.22 0.44 0.53 -0.39 17.18 7.33 0.67 2.3808HA070 1.31 0.79 1.09 -0.78 18.76 8.65 0.65 2.4108HB023 -0.20 0.79 0.65 -0.55 12.70 7.86 0.59 1.3008JA017 2.42 0.30 0.42 -1.00 4.30 8.01 0.62 0.7308JB008 2.20 0.52 0.59 -0.97 2.90 7.49 0.61 0.5108KB001 -1.87 0.38 0.15 -1.00 3.34 4.06 0.59 0.6508KH006 1.08 0.33 0.34 -1.00 4.77 6.76 0.63 0.8008LB078 3.29 0.55 1.08 -0.46 4.58 8.31 0.59 0.7508LE031 0.90 0.35 0.34 -1.00 6.34 6.71 0.65 0.7808LF051 1.10 0.26 0.32 -1.00 9.29 8.68 0.64 1.0008MF040 -1.09 0.29 0.21 -1.00 3.68 4.01 0.59 0.5108NE074 2.96 0.91 1.41 -1.00 5.55 4.39 0.62 1.1708NG065 3.25 0.35 0.52 -0.97 0.50 4.89 0.53 0.5008NH118 1.84 0.25 0.37 -1.00 16.01 8.47 0.64 1.8408NK002 2.40 0.51 0.79 -1.00 3.24 3.65 0.57 0.7033Figure 3.9: Time series of discharge (top row), air temperature (middle row), and streamtemperature (bottom row) for the air2stream model for Ingenika River aboveSwannel River (Station ID 07EA004). Observed stream temperature is in black,and simulated stream temperature is in red. The vertical dashed line separatesthe calibration and validation periods.34Figure 3.10: Time series of discharge, air temperature, and stream temperature for theair2stream model for Renfrew Creek near Port Renfrew (Station ID 08HA069).See caption for Figure 3.9 for details.35Figure 3.11: Time series of discharge, air temperature, and stream temperature for theair2stream model for Fraser River at Shelley (Station ID 08MF040). See cap-tion for Figure 3.9 for details.36Figure 3.12: Simulated versus observed stream temperaturea for the calibration (left col-umn) and validation (right column) periods for each example station. Symbolcolours indicate month. The black line represents perfect agreement.37Table 3.4: RMSE, MBE, and MAE for each station during the calibration and validationperiods.Calibration ValidationStation ID RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)07EA004 0.86 -0.03 0.54 0.90 0.03 0.5907EB002 0.76 -0.22 0.61 0.81 -0.03 0.7008CG001 0.67 0.00 0.50 1.15 -0.44 0.9008FF001 0.85 -0.03 0.64 1.10 -0.28 0.8608FF003 0.75 -0.02 0.58 0.95 0.33 0.7608GA077 0.86 -0.01 0.67 1.40 0.22 1.1408HA002 0.87 -0.01 0.67 1.08 0.18 0.8208HA011 0.94 0.01 0.72 1.32 0.48 1.0508HA069 0.70 0.00 0.54 0.74 0.48 0.5908HA070 0.61 0.00 0.50 0.85 0.27 0.6508HB023 0.88 0.00 0.71 0.96 -0.45 0.7708JA017 1.26 -0.20 1.01 1.19 -0.24 0.9608JB008 1.28 -0.22 1.02 1.41 -0.27 1.1208KB001 0.90 -0.06 0.62 0.93 -0.07 0.7008KH006 1.15 -0.06 0.88 1.16 0.17 0.8408LB078 0.87 -0.05 0.69 0.97 -0.24 0.7808LE031 1.16 -0.04 0.96 1.13 0.07 0.9208LF051 1.11 -0.05 0.90 1.27 -0.67 1.0308MF040 1.11 -0.25 0.91 0.92 -0.16 0.7408NE074 0.84 -0.03 0.65 0.88 0.31 0.6608NG065 1.68 -0.08 1.30 1.21 0.59 0.9408NH118 0.83 0.00 0.62 1.12 0.61 0.9108NK002 1.03 -0.05 0.77 0.84 0.29 0.6138Figure 3.13: Boxplots of error indices across all stations for the calibration and validationperiods, separated by regulation status.39For the date range associated with the 2022 autumn drought (September and October),air2stream predictions for the calibration period generated RMSE values ranging up to 1.12oC and MBE ranging from –0.12 to 0.28 oC (Table 3.6). Model performance for the autumndrought in 2022 deteriorated slightly compared to the calibration period, with RMSE up to1.89 oC and MBE ranging from -1.43 to 1.67 oC; this deterioration in model performancewas less marked than for the 2021 heat wave. As seen in Figure 3.18, the unregulatedsystems appear to have a slightly worse performance with larger spreads of RMSE andMAE. Example time series during the autumn drought are shown in Figure 3.14 for eachexample station.The regression models fit for June and July during the calibration period performed poorlycompared to air2stream for the heat wave. As seen in Table 3.7, RMSE ranged up to6.68 oC and MBE ranged from -0.59 oC to 6.35 oC. That is, the regression model not onlyhad greater errors than air2stream, but also more consistently over-predicted the responseto the heat wave. When the regression models fit for September and October duringthe calibration period were applied to the 2022 autumn drought period, they generatedRMSE values up to 1.88 oC and MBE ranging from -1.50 oC to 0.94 oC (Table 3.7). Thisperformance is similar to that achieved by air2stream, as shown in Table 3.6.Figure 3.19 shows scatterplots of the regression errors vs the air2stream errors for boththe heat wave and the autumn drought. For all plots except the autumn drought MBE,the majority of stations fall above the 1-1 line, indicating the superior performance byair2stream.40Figure 3.14: Stream temperatures during the 2021 heat wave (left column) and 2022 au-tumn drought (right column) from each example station. Simulated streamtemperatures from the air2stream model are shown in red, and observed areshown in black.41Table 3.5: RMSE, MBE, and MAE for each station during the calibration and heat waveperiods. Also included are the air temperature anomalies compared to the 1979-2020 reference period and water temperature changes compared to the precedingfive days during the heat wave.Calibration𝐻𝑊 Heat WaveStation ID RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)𝑇𝑎anomaly(𝑜C)𝑇𝑤change(𝑜C)07EA004 1.04 0.04 0.82 1.56 -1.29 1.44 8.57 2.8307EB002 0.80 0.41 0.65 0.99 0.80 0.82 9.08 1.4408CG001 0.83 -0.12 0.70 0.72 0.47 0.47 4.78 0.3208FF001 0.76 -0.19 0.62 0.97 0.84 0.84 8.70 1.1808FF003 0.60 -0.01 0.54 1.50 1.47 1.47 8.57 1.0808GA077 0.64 -0.27 0.56 3.13 -3.11 3.11 8.96 2.8708HA002 0.88 0.05 0.64 1.32 1.04 1.04 9.12 2.4508HA011 0.99 -0.39 0.83 1.93 1.32 1.64 8.36 2.8208HA069 0.68 -0.19 0.54 2.13 2.03 2.03 7.66 2.3108HA070 0.57 -0.12 0.44 0.81 0.41 0.69 8.63 2.6308HB023 0.81 0.22 0.58 0.84 -0.28 0.70 8.38 4.0308JA017 1.74 -0.32 1.47 1.52 -0.78 1.18 9.59 3.2708JB008 0.96 -0.14 0.77 3.28 -3.00 3.00 9.18 4.4708KB001 1.16 0.15 1.03 1.31 0.53 1.17 9.26 1.7508KH006 1.98 0.05 1.64 2.54 1.33 2.26 10.34 2.3408LB078 0.56 0.14 0.45 0.84 0.18 0.78 11.78 2.6108LE031 1.17 -0.21 0.90 1.22 -0.58 0.98 12.47 1.9208LF051 1.40 0.57 1.14 1.26 -1.05 1.06 11.86 2.2208MF040 1.17 0.08 0.96 1.61 1.05 1.49 11.68 1.9508NE074 0.59 -0.34 0.49 0.40 0.05 0.37 11.33 3.6908NG065 1.26 0.68 1.08 1.20 -1.01 1.02 9.52 2.9508NH118 1.49 0.64 1.14 0.37 -0.25 0.32 10.70 1.3308NK002 0.76 0.38 0.61 0.51 -0.40 0.44 8.89 2.7842Figure 3.15: Boxplots of error statistics for the heat wave date range across all stations forthe calibration period and the 2021 heat wave, separated by regulation status.43Figure 3.16: Spatial pattern of air2stream RMSE for the calibration period over the heatwave dates (top row) and the heat wave period (bottom row). Shapes representthe hydrologic regime.44Figure 3.17: Scatterplot of RMSE during the calibration period for the heat wave date rangevs RMSE during the 2021 heat wave for unregulated stations. Symbol coloursrepresent the air temperature anomaly compared to the 1979-2020 referenceperiod and shapes represent the hydrologic regime. The black line representsperfect agreement.45Table 3.6: RMSE, MBE, and MAE for each station during the calibration and autumndrought periods. Also included are the discharge anomalies compared to the1997-2021 reference period during the autumn drought.Calibration𝐴𝐷 Autumn DroughtStation ID RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)Qanomaly(% Q𝑎𝑣𝑔)07EA004 0.84 0.14 0.64 1.03 0.88 0.89 -42.7607EB002 0.76 0.24 0.58 0.82 0.73 0.73 -49.7108CG001 0.60 -0.08 0.47 1.35 -0.98 0.99 32.1308FF001 0.79 0.03 0.62 0.84 -0.28 0.65 -38.5908FF003 0.85 -0.02 0.66 0.75 0.08 0.60 -27.5808GA077 0.95 -0.12 0.78 1.21 0.56 1.01 -59.1808HA002 0.73 -0.02 0.57 0.54 -0.27 0.45 -57.3608HA011 1.06 0.07 0.78 1.01 0.53 0.79 -67.3608HA069 0.62 -0.01 0.49 0.83 0.67 0.69 -54.3108HA070 0.66 0.00 0.55 1.76 1.67 1.67 -76.3008HB023 0.86 -0.05 0.69 0.92 -0.56 0.78 -55.3208JA017 0.82 0.21 0.63 1.09 -0.92 0.92 9.1608JB008 0.85 0.28 0.64 1.72 0.86 1.23 -70.1408KB001 0.72 0.07 0.58 0.56 0.11 0.45 -41.3508KH006 0.97 0.20 0.73 0.96 -0.36 0.68 -19.6608LB078 0.99 -0.08 0.78 0.85 -0.28 0.71 -64.9308LE031 1.12 0.01 0.91 0.61 -0.19 0.49 -5.3808LF051 0.91 -0.07 0.73 1.52 -1.43 1.43 -20.3808MF040 0.87 0.22 0.71 0.87 0.30 0.71 -20.0408NE074 1.00 0.03 0.80 1.66 1.24 1.32 -47.5908NG065 1.52 0.03 1.21 1.89 1.49 1.53 -17.5008NH118 0.68 0.10 0.54 1.05 0.83 0.87 -11.5208NK002 0.95 0.13 0.76 1.02 0.72 0.83 -15.7546Figure 3.18: Boxplots of error statistics for the autumn drought date range across all sta-tions for the calibration period and the 2022 autumn drought, separated byregulation status.47Table 3.7: RMSE, MBE, and MAE for each station during the heat wave and autumndrought for the regression model.Heat Wave𝑅 Autumn Drought𝑅Station ID RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)RMSE(𝑜C)MBE(𝑜C)MAE(𝑜C)07EA004 3.39 2.78 3.05 1.35 -0.16 1.1107EB002 4.16 3.79 3.79 1.08 -0.16 0.8908CG001 1.76 1.55 1.55 1.27 -0.74 0.9608FF001 5.58 5.21 5.21 1.29 -1.00 1.0608FF003 6.68 6.35 6.35 0.87 0.00 0.7108GA077 2.54 1.80 2.12 1.16 0.06 0.9708HA002 2.85 1.36 2.63 1.38 -0.32 1.1408HA011 1.70 0.53 1.53 1.14 -0.04 0.9708HA069 2.48 1.29 2.09 0.93 -0.23 0.7508HA070 1.66 -0.59 1.52 1.26 0.78 1.0208HB023 2.79 -0.32 2.71 1.53 -0.45 1.3008JA017 3.12 2.23 2.71 1.66 -0.99 1.3008JB008 2.31 -0.50 2.08 1.55 0.09 1.2608KB001 3.97 3.64 3.64 1.88 -1.50 1.6208KH006 4.95 4.17 4.17 1.60 -0.58 1.2408LB078 1.09 -0.02 0.93 1.16 0.22 0.9508LE031 3.28 2.70 2.92 1.20 -0.41 0.8808LF051 2.33 2.05 2.05 1.47 -1.06 1.1308MF040 3.48 3.08 3.09 1.56 -0.35 1.1808NE074 1.46 0.59 1.35 1.44 0.80 1.1908NG065 1.49 1.30 1.31 1.52 0.65 1.2208NH118 2.11 1.99 1.99 1.20 0.94 1.0008NK002 1.40 1.32 1.32 1.30 0.41 1.0848Figure 3.19: Scatterplots of error statistics generated by the regression models versus meanabsolute error generated by air2stream for the 2021 heat wave (left column)and the 2022 autumn drought (right column). The black line represents perfectagreement. Axis scales vary among plots.494 Discussion4.1 Assessment of ERA5 and NARR air temperature products foruse in empirical stream temperature modelsThe production of reanalysis products involves using observed data to update state vari-ables in a numerical weather model, so that reanalysis products are not truly independentof observations. Indeed, each product had lower accuracy in locations with fewer climatestations, such as the northern part of the province. Consequently, the comparison of re-analysis air temperatures and observations demonstrates the extent to which observed datacan be reproduced from reanalysis products through spatial interpolation.The analyses demonstrated that ERA5 provided similar results to station-based interpola-tion when used as a predictor in both spatial and temporal empirical stream temperaturemodels, and was slightly superior to NARR.A major advantage of the reanalysis products is that they are available with a lag of a coupleof months, whereas ECCC observations can be multiple years behind. Another advantageof reanalysis products is that they have higher spatial coverage, especially in mountainousregions and in the northern part of the province where climate stations are more sparse.4.2 Air2stream performanceThe air2stream model predicted stream temperature in the validation period with a maxi-mum RMSE of 1.4 oC, and with RMSE less than 1.0 oC for about half the stations. Thisperformance was substantially better than for the empirical logistic curve, for which themedian RSE was over 1.0 oC for both the cooling and warming periods, with maximumRSE approaching 4 oC for the warming period.The performance of air2stream during the heat wave was worse than in the calibration pe-riod, with maximum RMSE exceeding 3 oC, compared to just under 2 oC in the calibrationperiod for the same date range. Mean bias errors for the heat wave ranged from -3.11 to2.03 oC, compared to -0.39 to 0.68 oC for the calibration period. Notwithstanding thisdecline in performance during the heat wave, air2stream performed notably better thanthe regression model, which generated RMSE values of up to 6.68 oC and mean bias errorsranging from -0.59 to 6.35 oC.This superior performance of air2stream during the heat wave is likely due, in part, tothe timing of the event, which occurred while snow was still present at higher elevations.50The rapid melting of snow during the heat wave would have increased the inputs of coldwater to the stream and also increased streamflow, both of which tend to suppress streamtemperature response (Piccolroaz et al., 2018). This influence would have been capturedby air2stream due to the model’s explicit dependence on streamflow. The regression model,on the other hand, was unable to account for hydrological influences.While the air2stream model performance did deteriorate during the heat wave for glacier-ized, hybrid, and rain-dominated catchments, it did not deteriorate for unregulated, snow-dominated catchments, especially those found in the southern Rocky Mountains, even forlocations with large air temperature anomalies. This can also likely be attributed to theair2stream model capturing the snowmelt processes well during the heat wave.Similar to the situation during the heat wave, the range of errors during the autumn droughtwas also higher than during the calibration period as well, but not as high as during theheat wave, likely due to the fact that the heat wave coincided with the annual peak in solarradiation, whereas incident solar radiation would have substantially lower during autumn.Also similar to the heat wave, air2stream generally had lower RMSE and reduced biascompared to the regression model.The air2stream model was more accurate for unregulated streams than for regulated streams.This difference in performance is likely due to the fact that thermal processes in reservoirsand the effects of flow management cannot be explained by the physics coded into the model.While air2stream did not overall perform as well during the extreme events, it still providedsuperior predictive capability compared to empirical models, and would be the preferredapproach for locations for which the required data are available. The main downside isthat air2stream requires streamflow data and thus can only be applied at stream gauginglocations.515 Conclusions5.1 Summary of key findingsThe analysis found that ERA5 outperformed NARR in all seasons when used to reproduceobserved air temperature at weather stations. Performance was best in summer, whenRMSE was 1.58 and 1.99 oC for ERA5 and NARR, respectively, and worst in winter, withRMSE of 2.26 and 2.68 oC. From the perspective of predicting stream temperature, thesummer results are the most important since that is the hottest and driest time of theyear.When the reanalysis air temperatures were used to compute the temporal air temperaturecovariate in the multiple linear regression derived by Moore et al. (2013) to predict MWAT,the NARR product performed almost as well as the station-based interpolation used in theoriginal analysis, and yielded a similar coefficient for the air temperature covariate (0.60for NARR versus 0.64 for ERA5). The ERA5 product had a slightly higher RMSE (2.18oC versus 2.13 oC for SBI) but yielded a lower coefficient of 0.32.When used as predictors in temporal stream temperature models, ERA5 and NARR pro-vided similar performance to air temperatures generated by station-based interpolation,with ERA5 often being slightly better. These results support the validity of using ERA5air temperature in empirical stream temperature models. A major advantage of the reanal-ysis products is their better spatial coverage compared to observations, and their greatertimeliness.The ERA5 data was additionally used to predict stream temperatures in the air2streammodel, which resulted in RMSE of 0.61 to 1.68 oC for the calibration period and 0.74 to1.41 oC for the validation period. The air2stream model generally outperformed regressionmodels under “normal” weather conditions. Performance deteriorated when air2stream wasapplied to the heat wave and autumn drought events, but not as dramatically as for theregression models, and not significantly for unregulated, snow-dominated catchments.5.2 Avenues for future researchThe air2stream model was superior to empirical models under “normal” weather conditions,and was notably better when applied to extreme events outside the range used for calibra-tion. These results suggest that models that explicitly account for hydrological processesshould be preferred for making projections under future climatic conditions. However, a52major limitation is the need for streamflow data. Further work should explore whetherpredicted streamflow from a hydrological model can be used in place of observations. Forexample, a number of studies have explored the use of mechanistic catchment-scale hydro-logical models with regionally calibrated parameter for predicting streamflow at ungaugedlocations (e.g., Burn and Boorman, 1993; Bárdossy, 2007).Another limitation is that air2stream, like empirical models, requires sufficiently long timeseries of stream temperature to support model calibration. However, stream temperature isnot well monitored in many regions, and multi-year time series are relatively rare. Furtherresearch should explore the sensitivity of model performance to the length and/or char-acteristics of the calibration period to provide guidelines for future applications. 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River Research and Applications 25 (6):675–686 DOI: 10.1002/rra.117959Appendix ABilinear interpolation is common approach to use with gridded data sets. It begins witha prediction point with coordinates (𝑥𝑝, 𝑦𝑝). In order to find the four closest grid pointsthat the prediction point falls between, 𝑖 is used to index position in the 𝑥 direction and 𝑗is used to index position in the 𝑦 direction. 𝑖 and 𝑗 are determined such that𝑥𝑖 ≤ 𝑥𝑝 ≤ 𝑥𝑖+1 (.1)and𝑦𝑗 ≤ 𝑦𝑝 ≤ 𝑦𝑗+1. (.2)The final goal is to estimate the value of a variable, 𝑧, at the prediction point, 𝑧𝑝, usingthe values of 𝑧 at the four closest grid points. These are denoted as 𝑧𝑖𝑗, 𝑧𝑖+1𝑗, 𝑧𝑖𝑗+1, and𝑧𝑖+1𝑗+1.The first step is to interpolate linearly in the 𝑦 direction between (𝑥𝑖, 𝑦𝑗) and (𝑥𝑖, 𝑦𝑗+1):𝑧𝑝,𝑖 = 𝑧𝑖𝑗 +𝑧𝑖𝑗+1 − 𝑧𝑖𝑗𝑦𝑗+1 − 𝑦𝑗(𝑦𝑝 − 𝑦𝑗). (.3)The next step is to interpolate linearly in the 𝑦 direction between (𝑥𝑖+1, 𝑦𝑗) and (𝑥𝑖+1, 𝑦𝑗+1):𝑧𝑝,𝑖+1 = 𝑧𝑖𝑗 +𝑧𝑖+1𝑗+1 − 𝑧𝑖+1𝑗𝑦𝑗+1 − 𝑦𝑗(𝑦𝑝 − 𝑦𝑖). (.4)The final step is to interpolate linearly in the 𝑥 direction between (𝑥𝑖, 𝑦𝑝) and (𝑥𝑖+1, 𝑦𝑝):𝑧𝑝 = 𝑧𝑝,𝑖 +𝑧𝑝,𝑖+1 − 𝑧𝑝,𝑖𝑥𝑖+1 − 𝑥𝑖(𝑥𝑝 − 𝑥𝑖). (.5)60Appendix BThe air2stream model is based on the following heat budget equation, which governs thevariation of stream temperature, 𝑇𝑤 (oC), within a volume, 𝑉 (𝑚3), of water in a riverreach and its tributaries:𝜌𝑐𝑝𝑉𝑑𝑇𝑤𝑑𝑡 = 𝐴𝐻 + 𝜌𝑐𝑝(∑𝑖𝑄𝑖𝑇𝑤,𝑖 −𝑄𝑇𝑤) (.6)where 𝜌 is water density (𝑘𝑔𝑚−3), 𝑐𝑝 is the specific heat capacity of water at constantpressure (𝐽𝐾−1𝑘𝑔−1), 𝑡 is time (expressed in days), 𝐴 is the surface area of the reach (𝑚2),𝐻 is the net heat flux between the river and the atmosphere (𝑊𝑚−2), 𝑄 is the streamflowdownstream (𝑚3𝑠−1), 𝑄𝑖 is the discharge associated with the 𝑖th contributing upstreamwater flux (𝑚3𝑠−1), and 𝑇𝑤,𝑖 is the temperature of the 𝑖th contributing upstream water flux(oC). Figure .1 shows a schematic of the river reach.It is assumed that air temperature, 𝑇𝑎 (o), can be used as a proxy for the short- and long-wave radiation and latent and sensible heat fluxes in the net heat flux term, H (Caissie(2006)). As a result, the following Taylor series expansion of H is made:𝐻 = 𝐻∣𝑇𝑎,𝑇𝑤 +𝛿𝐻𝛿𝑇𝑎∣𝑇𝑎,𝑇𝑤× (𝑇𝑎 − 𝑇 𝑎) +𝛿𝐻𝛿𝑇𝑤∣𝑇𝑎,𝑇𝑤× (𝑇𝑤 − 𝑇𝑤) (.7)where 𝑇 𝑎 and 𝑇𝑤 are the long-term averaged values of air and stream temperatures. Byderiving parameters ℎ𝑜, ℎ𝑎, and ℎ𝑤 from Equation .7, it can be rewritten as:𝐻 = 𝜌𝑐𝑝(ℎ0 + ℎ𝑎𝑇𝑎 − ℎ𝑤𝑇𝑤) (.8)A fictitious mean inflow temperature ?̂? = 𝑄−1∑𝑖𝑄𝑖𝑇𝑤𝑖 is introduced based on the prin-ciple that 𝑄 = ∑𝑖𝑄𝑖 in order to simplify the second term on the right side of Equation .6which represents the difference between heat fluxes leaving and entering the river reach.Because the reference temperature ?̂? oscillates with the seasons, it can be expressed as asinusoidal annual variation:?̂? = ?̂?0 + ?̂?1𝑐𝑜𝑠[2𝜋(𝑡𝑡𝑦− 𝜙)] (.9)where ?̂?0 is the reference value, ?̂?1 is the amplitude of the oscillation, and 𝜙 ∈ [0, 1] isthe phase, all left unspecified. 𝑡𝑦 is the duration of a year in days, the same units as 𝑡.61Figure .1: Schematic of the river reach considered in Equation .6 from Toffolon and Piccol-roaz (2015).62Equation .6 can now be rewritten as𝑑𝑇𝑤𝑑𝑡 =𝐴𝑉 (ℎ0 + ℎ𝑎𝑇𝑎 − ℎ𝑤𝑇𝑤) +𝑄𝑉 ( ?̂? − 𝑇𝑤). (.10)The two ratios in Equation .10, 𝐴𝑉 and 𝑄𝑉 , both depend on the discharge as well as quantitiesthat are difficult to measure. In order to make these simpler to measure, it is assumed that𝛿, a dimensionless reference volume, can be written as a function of the discharge in theform of a power law:𝛿 = 𝑉𝑉 = 𝜃1/𝑚 (.11)where 𝜃 = 𝑄𝑄−1 and 𝑚 = 5/3.After substituting Equation .11 into Equation .10, the final air2stream equation for streamtemperature, 𝑇𝑤 (o), is formed:𝑑𝑇𝑤𝑑𝑡 =1𝛿 (𝑎1 + 𝑎2𝑇𝑎 − 𝑎3𝑇𝑤 + 𝜃[𝑎5 + 𝑎6𝑐𝑜𝑠(2𝜋(𝑡𝑡𝑦− 𝑎7)) − 𝑎8𝑇𝑤]) (.12)where 𝛿 = 𝜃𝑎4 and 𝑎1 to 𝑎8 are the parameters of the model. The model waswritten in Fortran and is available for download as a compiled executable viahttps://github.com/spiccolroaz/air2stream.63
Thesis/Dissertation
eng
Geography
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Moore, R. Daniel
Predicting daily stream temperature in British Columbia, Canada, using reanalysis air temperature in statistical models and the hybrid air2stream model
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