Open access

Data-limited models to predict river temperatures for aquatic species at risk1

Publication: Canadian Journal of Fisheries and Aquatic Sciences
1 August 2021


In data-poor regions, modeled river temperatures are essential for predicting potential stressors for species at risk. With limited data from the Grand, Thames, and Sydenham rivers in southern Ontario, Canada, we evaluated simple mixed-effect regression models to predict water temperature using air temperature from nearby weather stations. Model performance was assessed for periods relevant to the fitness of the black redhorse (Moxostoma duquesni): June to August, when heat events may be likely; and May, when spawning occurs. All of the models performed better when trained on data from these periods, compared with using data from the entire growing season. The best model was a linear regression using 5 days of lagged air temperature. This model had a root mean square error for summer means of 1.5 °C. The differences in prediction error at different times of year highlight the importance of considering species ecology in model interpretation. However, the improvement in model fit when using only data from the relevant time of year suggests that relatively simple models can be used effectively in a management arena when applied appropriately.


Dans les régions pour lesquelles peu de données sont disponibles, les températures modélisées des rivières sont d’importance fondamentale pour la prédiction de facteurs de stress potentiels pour les espèces en péril. À partir de données limitées pour les rivières Grand, Thames et Sydenham, dans le sud de l’Ontario (Canada), nous avons évalué des modèles de régression à effets mixtes simples pour la prédiction des températures de l’eau à partir de températures de l’air mesurées à des stations météorologiques à proximité. La performance des modèles a été évaluée pour des périodes importantes pour l’aptitude du chevalier noir (Moxostoma duquesnii), soit de juin à août, quand des épisodes de chaleur sont probables, et en mai, quand le frai a lieu. Tous les modèles ont donné de meilleurs résultats quand ils étaient préalablement entraînés avec des données pour ces périodes que lorsque des données pour toute la saison de croissance étaient utilisées. Le meilleur modèle est une régression linéaire intégrant cinq jours de données de températures de l’air décalées (écart-type moyen estival 1,5 °C). Les différentes erreurs de prévision à différents moments de l’année soulignent l’importance de tenir compte de l’écologie de l’espèce pour l’interprétation des modèles. L’amélioration du calage sur les modèles que produit l’utilisation de données seulement pour les périodes importantes de l’année donne cependant à penser que, s’ils sont bien appliqués, des modèles relativement simples peuvent être utilisés efficacement en gestion. [Traduit par la Rédaction]


Water temperature is one of the primary factors that determine the survival, growth, and reproduction of aquatic organisms in flowing systems (Wismer and Christie 1987; Caissie 2006; Hasnain et al. 2010; Martin et al. 2017). Unfortunately, river temperature observations are often discontinuous at critical periods (e.g., summer maximum; Jones and Schmidt 2018) or absent where species at risk occur (Kaushal et al. 2010; Arismendi et al. 2012; Caldwell et al. 2015; Dugdale et al. 2017). Conversely, air temperature data are often available, continuously measured, and can be used to predict water temperature (e.g., Caissie 2006). We tested five data-minimal statistical models that could be used to predict future water temperature from current and past air temperatures on large rivers in southern Ontario.
Our model testing criteria were determined by time periods and characteristics of temperature that are related to organism fitness. We used an exemplar species, black redhorse (Moxostoma duquesni), to determine whether very simple statistical models would be effective at predicting water temperatures during critical periods, even if such models have poor performance over longer periods. This threatened fish is limited to a few large (25–130 m wide), fast flowing (14–20 m3·s−1), and warm water (mean July water temperature in exceedance of 20 °C) rivers in southwestern Ontario, Canada (Bowman 1970; Parker and Kott 1980; COSEWIC 2015). In southern Ontario, water temperature impacts may be associated with optimal spawning temperature (Wismer and Christie 1987) and the summer impairment threshold (also known as the critical thermal maximum; Hasnain et al. 2010). For example, the black redhorse spawns in water temperatures ranging from 15 to 21 °C (COSEWIC 2015). Although the species does not have a published summer impairment threshold, other congenerics in the Great Lakes region, such as the adult shorthead redhorse (Moxostoma macrolepidotum) and adult golden redhorse (Moxostoma erythurum) have a documented threshold of 35.1 °C (Reash et al. 2000).
There are many river temperature models that vary in complexity and ease of use (DeWeber and Wagner 2014; Piotrowski and Napiorkowski 2019), and several reviews of the different approaches (see Caissie 2006 and Benyahya et al. 2007). Caissie (2006) classified water temperature models into two groups: process-based models, which rely on an energy budget approach; and statistical models, which rely on air to water temperature correlations. Some process-based river temperature models (e.g., HEC-RAS and Mike 11) can forecast heat fluxes on a daily time-step and at a high spatial resolution, but this type of model is quite data intensive (Dugdale et al. 2017). Statistical models can be simpler than process-based models, although some are quite complex, and authors may include many parameters that account for the important effects of flow, land use, temperature lags, and seasonality (Benyahya et al. 2007). For example, van Vliet et al. (2011) found that adding a discharge parameter to a nonlinear regression model increased the predictive accuracy during heat waves. However, even when complex statistical models have high predictive power in one watershed, they may not transfer to data-poor regions. In addition, increasing the required number of predictors can create more gaps in already discontinuous time series, as these metrics may not be measured in different locations. Therefore, data intensive statistical models may also not be the optimal choice for many applications.
Data-minimal statistical models based on air and water relationships can be useful for aquatic species threat assessments when data deficits exist. Within the statistical model category, linear regressions based on air temperature are likely the simplest models to implement (e.g., Crisp and Howson 1982; Langan et al. 2001; Webb et al. 2003, 2008; Arismendi et al. 2014; Johnson et al. 2014). In addition, models incorporating data in the form of air temperature at various time lags can improve fits for larger systems (Stefan and Preud’homme 1993). However, while the relationship between daily and weekly air and water temperatures may be linear at most temperatures, it is nonlinear at air temperature extremes (e.g., approaching freezing: Mohseni and Stefan 1999). At the cold extremes, groundwater and ice cover reduce surface heat exchange, and at hot extremes, evaporative cooling enhances back radiation, such that water temperatures do not increase in step with air temperatures (Mohseni and Stefan 1999; van Vliet et al. 2011). To address these issues, Mohseni et al. (1998) calibrated and tested a nonlinear regression model on a weekly time step, which has been widely applied and modified for daily data (Koch and Grünewald 2010; van Vliet et al. 2011; Feng et al. 2018; Piotrowski and Napiorkowski 2019). Similarly, Maloney et al. (2016) used a logistic model to predict thermally stressful extreme events on the Delaware River. A slightly more complex approach is to combine seasonal oscillations in temperature with a linear model of any remaining variation in the residuals. This residual regression strategy is common for predicting daily mean and maximum river temperatures (Cluis 1972; Caissie et al. 1998; Benyahya et al. 2007; Nelson and Palmer 2007; Laanaya et al. 2017). Despite their potential simplicity, some of these residual regression models have a large number of parameters related to land use factors and time lags.
Other approaches that we have not considered, such as artificial neural networks or hybrid approaches, can have good predictive power (Benyahya et al. 2007; Piccolroaz et al. 2016; Zhu et al. 2018; Piotrowski and Napiorkowski 2019). Even when artificial neural network tools are used without flow data, predictive power can be greater than linear and nonlinear regression, especially at extreme temperatures (Zhu et al. 2018; Piotrowski and Napiorkowski 2019). However, it is unclear whether machine learning models trained in one watershed can be used to predict conditions in another, and these models are more complex to train and to interpret than simple statistical models such as linear regression.
We tested the performance of five simple models that use only air temperature to predict daily mean and maximum water temperatures: (1) linear regression; (2) linear regression with 5 days of lagged air temperature; (3) linear regression with a 7 day rolling mean of air temperature; (4) a nonlinear logistic regression; and (5) a linear regression of residuals from a fitted model of sinusoidal seasonal trend. We tested the performance of these models in predicting water temperatures during periods of time important to species like black redhorse (i.e., spawning season and months where heat impairment temperatures were likely). We compared the performance of models fit using all data from the entire growing season with models fit using only data from the period of interest. We hypothesized that a model which performs poorly in an annual evaluation could have still superior performance during important seasonal events, and thus be a useful tool for management.

Materials and methods

Study area

We modelled water temperatures from three rivers in southern Ontario: the Grand, the upper Thames, and the Sydenham (Fig. 1; see also the Supplementary data, Table S12). The Grand River flows along a gentle gradient (∼1.3 m·km–1) through an agricultural dominated landscape for approximately 290 km before it drains into Lake Erie (Fig. 1; Gardner 1977; Jyrkama and Sykes 2007). The Speed River, a tributary of the Grand River, is located within a similar agricultural dominated landscape in the northeastern portion of the Grand River watershed and flows 65 km before joining the Grand River near the city of Cambridge. In the upper 40 km it has a gradient of 2.5 m·km–1 (Bishop and Hynes 1969; Fig. 1). Including all tributaries, the drainage of this system is 6800 km2 (Lake Erie Source Protection Region Technical Team 2008). The watershed of the Upper Thames encompasses an area of 3421 km2, mostly dominated by farms and livestock operations (UTRCA 2020). Agriculture is also a primary land use within the 165 km-long Sydenham River watershed (SCRCA 2019), with a total drainage area of 2700 km2. These areas are in a continental humid climate zone, and the associated watercourses are characterized by periods of relatively low flow in the late summer and early fall, with large magnitude floods from ice jams, rainfall, and snowmelt in the spring, which has led to dam and levee installation, as well as channel dredging (Gardner 1977).
Fig. 1.
Fig. 1. Locations of the Environment and Climate Change Canada ( air temperature stations (closed circles), and water temperature sites (open triangles). Note that some water temperature collection sites overlap. Polygons for rivers were obtained from For coordinates, station names, and other details see the Supplementary data, Table S12.

Water temperature data

We obtained water temperature data from in-stream loggers (2011–2019) from the Grand River Conservation Authority ( Fisheries and Oceans Canada contributed data from loggers for the Sydenham River for 2017–2018, and the Upper Thames River Conservation Authority provided river temperature observations from loggers for the Thames River from 2012–2019 (Supplementary data, Table S12). Water temperature stations and sample years were then selected based on the relative completeness of the data (we required the station to have data for 60% of the days in a given period), and the lack of numerous artifacts, such as temperatures reading as constant for multiple sets of several days (see example of an acceptable level of artifacts in Fig. 2). In addition, one set of data immediately below the Shand reservoir dam was cooler than the other locations on the Grand that we considered, and was presumably heavily influenced by water release events, and so was excluded from this analysis. There were, however, data gaps in all of the selected series, such that not every series could be used in every evaluation (Table 1: Supplementary data, Table S12).
Fig. 2.
Fig. 2. Example of output from data quality checks. Included (gray) and excluded (red) water temperature data from 2019 at the Hanlon Water Quality Station, Ontario, Canada (lat. 43.53, long. –80.25, Fig. 1; Supplementary data, Table S12: site 170). Raw data were obtained from the Grand River Conservation Authority ( [Colour online.]
Table 1.
Table 1. Summary statistics and data availability for spring and summer water temperatures.
Before use, the hourly data from these stations were checked for probable errors, and then aggregated to daily averages and daily maximums for model fitting. Data quality errors included temperature readings that appeared to have been linearly imputed rather than measured, and other artifacts, possibly due to air exposure of the monitors or monitor malfunction, such as temperatures significantly above or below the expected values (Fig. 2). To remove temperatures that may have been imputed rather than measured, we completed a rolling linear regression over the hourly data at each site using a 12 h interval. Data points with a residual less than 1e–10 from the fitted line were then removed. To detect other potential artifacts (e.g., dramatically different values compared with other stations; see Fig. 2), we calculated the average water temperatures by Julian day for all stations on each river. We then removed any data outside ±3 SD of the mean daily value across all sites in the given region (for example, see Fig. 2). Finally, we also removed any water temperatures less than −0.1 °C.

Air temperature data

We downloaded previously collected daily maximum and daily mean air temperature data between 2011 and 2019 from the closest Environment and Climate Change Canada (ECCC) weather stations (; Fig. 1; Supplementary data, Table S12). Although the air temperature data at locations near to some river temperature stations were available from the Grand River Conservation Authority, there were fewer missing data and fewer presumed outliers in the ECCC data. Distances between air and river temperature stations range from 1.1 km to 102.8 km (Fig. 1; Supplementary data, Table S12). Each river temperature station was paired with one proximate air temperature station. In most cases, multiple river temperature stations were paired with the same air temperature station (Supplementary data, Table S12).

Models, fitting, and testing

We evaluated the fit and predictive power of five simple statistical models that used only air temperature as a predictor for daily mean or maximum water temperatures (Table 2). We compared model predictions when a given model was fit to growing season data (May–November) to predictions from the same model when fit using only temperature data from the season of interest (spring or summer), to determine whether these simple statistical models have better performance when their use is restricted to biologically important periods. We defined summer as June to August, because we observed some annual maximums in June. We predicted both means and maximums, because this information could be of use when evaluating the potential for heat stress in species of concern. In addition, we predicted spring water temperature means that can influence spawning events for the month of May.
Table 2.
Table 2. Models evaluated, where Twt,i refers to the water temperature on Julian day t, at water station i, and Tat–1,i gives the air temperature on Julian day t – 1 at the ECCC weather station that has been paired with that river site.
The simplest model we evaluated was a linear regression relating current water temperature to air temperature on the previous day (Table 2, eq. 1). In this, and all other linear models, we included a random effect for the particular water temperature station on the intercept (Table 2). In each case, we also corrected for autocorrelated errors using an autoregressive process at lag 1 (AR1). We also evaluated a linear model with 5 days of lagged air temperatures (Table 2, eq. 2; see also Stefan and Preud’homme 1993), and a linear model with a rolling 7 day mean for air as a temperature predictor (Table 2, eq. 3).
Using the same mixed-effect framework, we also fit a nonlinear logistic function relating water temperature to air temperature for daily temperatures (Table 2, eq. 4; see also Laanaya 2017), with a random effect of water station on parameter b. Finally, we evaluated a model where the residuals remaining after fitting a seasonal pattern were fit (Table 2, eq. 5; see also Cluis 1972; Nelson and Palmer 2007; Laanaya et al. 2017). After the nonlinear seasonal water and air temperature cycles were removed, we used a mixed effect linear regression with a random effect of water temperature station on the intercept to predict the residual errors (Table 2). In this case, we did not use an AR1 process for the regression errors because the seasonal pattern had previously eliminated such correlation. Models were fit with restricted maximum likelihood using the nlme package in R (Zuur et al. 2009; Pinheiro et al. 2017; R Core Team 2019).
Model fits were evaluated using fixed effects only, using metrics of: root mean square error (RMSE), mean absolute error (MAE), bias, and the Nash–Sutcliffe coefficient of efficiency (NSC). We calculated RMSE as (Janssen and Heuberger 1995; Hyndman and Athanasopoulos 2018), where Pt is the predicted temperature at time t, Ot is the observed temperature at time t, and n is the total number of observations. Mean absolute error is given as: . Bias is calculated as , and the Nash–Sutcliffe coefficient of efficiency (NSC), as (Nash and Sutcliffe 1970). This metric ranges from –∞ to 1, where 1 indicates perfect correspondence, and values greater than 0 indicate that the model predictions are performing better than a simple mean. Like RMSE, the NSC is heavily influenced by outliers, and so we also report the MAE. In addition, similar to R2, the NSC is not very sensitive to systematic model over- or under-estimation (Krause et al. 2005), and so we also give an estimate of model bias. We have provided the NSC, which is widely used in hydrological studies, because it is known that R2 has variable interpretations for mixed and nonlinear models (Nakagawa and Schielzeth 2013), is highly influenced by autocorrelation in regression errors, and in particular, cannot be directly compared between linear and nonlinear models (Kvålseth 1985). However, we are uncertain how robust the NSC is regarding these issues. We also compared models with an eye to parsimony by calculating the Akaike information criterion (AIC) as 2k + n ln, where k is the number of fixed-effect parameters. When comparing performance of models fit with different time periods (e.g., comparing the performance of models fit to annual data vs. seasonal data), only predictions within the season of interest were used to calculate the indices.
Models with reasonable in-sample fit may perform more poorly when used to forecast. In addition to estimating model fit, we estimated forecasting performance by training and testing each model using a repeated k-fold cross-validation approach with 5 folds and 10 repeats, where each year × station was used as a replicate.


Summer: mean and maximum daily temperatures

We are primarily interested in whether simple statistical models can successfully forecast water temperatures during critical periods. For daily summer means and maximums, the lowest errors and bias of forecasted temperatures from the repeated cross-validation procedure, using fixed effects only, were found for simple linear fits to the seasonal data (Figs. 3 and 4). In addition, we found that using data from the entire growing season (May–November) to train these models resulted in larger errors (Fig. 3), and greater bias (Fig. 4) in forecasted temperatures, than using only data from the season of interest.
Fig. 3.
Fig. 3. Root mean square error (RMSE) of test data (data withheld from model training) for forecasted summer daily mean (a) and maximum (b) water temperatures in a repeated cross-validation process with 10 repeats and 5 folds, for five different statistical models (see Table 2 for definitions) fitted using data from the Grand, Thames, and Sydenham rivers in southern Ontario, from 2011 to 2019. The models were fit using either data from the entire growing season of May 1 to November 1, or just the summer months (June 1 to August 31). In addition to the values from individual replicates of the cross-validation process, the boxplot gives the median (horizontal line), the 25th and 75th percentiles (lower and upper hinges of the box), and largest and smallest observation ≤ 1.5, multiplied by the upper or lower interquartile range (whiskers). [Colour online.]
Fig. 4.
Fig. 4. Bias of test data (data withheld from model training) for forecasted summer daily mean (a) and maximum (b) water temperatures in a repeated cross-validation process with 10 repeats and 5 folds, for five different statistical models (see Table 2 for definitions) fitted using data from the Grand, Thames, and Sydenham rivers in southern Ontario from 2011 to 2019. The models were fit using either data from the entire growing season of May 1 to November 1, or just the summer months (June 1 to August 31). In addition to the values from individual replicates of the cross-validation process, the boxplot gives the median (horizontal line), the 25th and 75th percentiles (lower and upper hinges of the box), and largest and smallest observation ≤ 1.5, multiplied by the upper or lower interquartile range (whiskers). [Colour online.]
Evaluations of model predictions over all the data suggested that for both summer maximums and means, the model with the lowest errors and bias was a linear regression using 5 days of lagged air temperature data (Table 2, eq. 2; linear lag5), and fit using data only from June to August (Tables 3 and 4) (note that all of the predictions reported were made using fixed effects only after fitting the mixed model to the data). Of course, we expect a model with more parameters to have lower error; however, the AIC calculation also confirms that the increased predictive power of this model is large enough to outweigh a penalty for greater model complexity (Tables 3 and 4). We briefly examined a linear regression model with a larger number of days of lagged air temperature as predictors, but did not find that lags beyond 5 days had slopes significantly different from zero.
Table 3.
Table 3. Model performance and coefficients for daily summer mean water temperatures, where models were fit to data from either the entire growing season (May 1 to November 1) or only the season of interest (June 1 to August 31).
Table 4.
Table 4. Model performance and coefficients for daily summer maximum water temperatures, where models were fit to data from either the entire growing season (May 1 to November 1) or only the season of interest (June 1 to August 31).
While overall metrics suggest that a linear model using a rolling mean of 7 days of air temperature (Table 2, eq. 3; linear mean) may have a similar fit (Table 3), this model (linear mean) had larger forecast error (Fig. 3). For summer means, the worst performance in forecasts and fit was the linear model with a single predictor of the air temperature on the previous day (Table 2, eq. 1; linear lag1) that was fit over the entire growing season. This model also had poor performance for summer maximums. In addition, the nonlinear model (Table 2, eq. 4; nonlinear) fit over the annual data had very poor performance (Tables 3 and 4; Figs. 3 and 4).
The performance of the best fit model, linear lag5, fit with summer data, was variable across sites and years. For summer means, the greatest error and bias was for the site on the Thames River in 2013 (RMSE 2.23 °C, bias 2.09 °C); however, the minimum RMSE for this same site was 1.37 °C in 2018. In comparison, the smallest RMSE was found for the Brant site on the Grand (Supplementary data, Table S12: site 59) in 2017 (0.84 °C), but this same site had relatively large error (RMSE 1.91 °C) in 2012. Across all sites on each river, the largest mean RMSE and bias was on the Thames (1.70 °C, 1.48 °C respectively), followed by the Grand (1.49 °C, −0.26 °C), and the Sydenham (1.22 °C, −0.12 °C). However, summer data for the Sydenham was only for 2018, whereas data from 2011–2019 was available for the Grand and 2012–2018 for the Thames.
The same variability was observed for maximum water temperatures predictions of this model, using air temperature maximums. Across individual years and stations, the largest error was found at the Glen Morris site on the Grand (Supplementary data, Table S12: site 65) in 2018 (2.56 °C), and the largest bias was at this same site in 2013 (−2.20 °C). The smallest error was at the road 32 site, also on the Grand (site 183), in 2013 (0.93 °C), whereas the smallest bias (−0.011 °C) was at the Brant site (site 59). Errors across the rivers were quite similar (Grand RMSE = 1.58 °C; Sydenham RMSE = 1.47 °C; and Thames RMSE = 1.63 °C), although greater bias was noted for the single Thames site (bias: Grand −0.084 °C; Sydenham −0.60 °C; Thames 1.28 °C).

Spring: daily mean temperatures

A similar pattern was found for both forecasts and overall fits for spring means. Forecasts based on models fit to data from May to November had larger errors and bias than those same models fit only to the daily data from May (Fig. 5). The model with the lowest errors and bias for forecast temperatures was also the linear regression with 5 days of lagged air temperature as predictors (Table 2, eq. 2; linear lag5). Model predictions, using only fixed effects from the best fit, also suggested that this model had the lowest overall error when fit to data from May, but it was also the best performing model when compared with others fit to data from the entire growing season (Table 5). The AIC also indicated that this model had the best performance when accounting for model complexity. The worst forecast performance and fit was observed for the linear lag1 and nonlinear models fit to annual data (Fig. 5; Table 5).
Fig. 5.
Fig. 5. Root mean square error (RMSE) (a) and bias (b) of test data (data withheld from model training) for forecasted spring daily mean water temperatures in a repeated cross-validation process with 10 repeats and 5 folds, for five different statistical models (see Table 2 for definitions) fitted using data from the Grand, Thames, and Sydenham rivers in southern Ontario from 2011 to 2019. The models were fit using either data from the entire growing season of May 1 to November 1, or just the month of May. In addition to the values from individual replicates of the cross-validation process, the boxplot gives the median (horizontal line), the 25th and 75th percentiles (lower and upper hinges of the box), and largest and smallest observation ≤ 1.5, multiplied by the upper or lower interquartile range (whiskers). [Colour online.]
Table 5.
Table 5. Model performance and coefficients for daily spring mean water temperatures, where models were fit to data from either the entire growing season (May 1 to November 1) or only the season of interest (May).
A closer examination of the predictions from the best model (i.e., linear lag 5 fit to data from May) also indicated that results were variable between station and year. The maximum RMSE was very large at 5.05 °C for Hanlon site (Supplementary data, Table S12: site 170) on the Grand River in 2016, and this same station had the largest absolute bias for the same year (−3.24 °C). However, in the previous year the site had a RMSE of 1.51 °C. Moreover, this error was also much larger than the next largest error of 3.16 °C at the Victoria site on the Grand (site 82), and we suspect there may be some monitor artifacts here (Supplementary data, Fig. S12). The lowest error was a RMSE of 0.83 °C for the site on the Thames in 2018, and minimum bias of 0.012 °C for the road 32 site on the Grand (site 183) in 2016. There were larger errors for the Grand River sites (mean RMSE = 2.04 °C, mean bias = −0.40 °C), compared with the Sydenham (1.39 °C, 0.22 °C) or the Thames (1.28 °C, 0.43 °C). However, the Sydenham sites were represented by only one year of data collection after data cleaning (Table 1), whereas the Grand River sites covered the period 2011–2019.


Water temperature is an important factor in the growth and persistence of aquatic populations (e.g., Caissie 2006). For example, Bjornn and Reiser (1991) showed that high stream temperatures in the range of 23–25 °C can increase the mortality rate of salmonid fish. In addition, the timing of temperature changes may trigger important events, such as spawning. However, temperature in rivers with species of interest may not be monitored, or data may be incomplete. As a result, prediction of the thermal regime of rivers may be necessary. Such predictions can be used to assess risks of impairment, or manage for biologically important events. For example, water temperature models can assist managers in determining optimum outflows that maintain adequate temperature ranges (e.g., Krause et al. 2005).
While very complex water temperature models have proliferated in the literature, these are not likely to be employed by management agencies when they require a large number of inputs to create a physical model of heat loading, or provide correlations related to diverse influences such as land use, riparian cover, and ground water flows, or have a complex formulation that requires extensive development (e.g., neural net models). Moreover, it may be that simpler models have similar prediction error as more complex approaches within a given region, or within a restricted time period.
We assessed the performance of five statistical models that used only air temperature for prediction across three watersheds in southern Ontario: the Grand River, the Upper Thames River, and the Sydenham River. Our evaluation was based the ability of the models to predict river temperatures during biologically sensitive periods, and so we focused on intervals when heat extremes might be expected (June to August), and when spring spawning might occur for an exemplar aquatic species at risk (e.g., May, for the black redhorse). We show that a quite simple regression model that uses only air temperature from a nearby weather station has good performance during these critical periods.
We found that a simple regression model using 5 days of lagged air temperature was the best fit to river temperature data, and provided forecasts of novel data with the smallest errors, compared with models using nonlinear relationships or seasonal fluctuations. In fact, this model had errors that were similar or lower than models of greater sophistication and complexity. For example, our lagged regression model had average RMSE of 1.5 °C (ranging from 0.8 to 2.2 °C for different sites and years) when used to predict daily summer mean temperature on several sites across eight years, whereas Laanaya et al. (2017) reported an average RMSE of 1.4 °C on one site across 9 years for a general additive model that included flow, and which was used to predict daily means on one site in the Sainte-Marguerite River in northern Canada. Zhu et al. (2018) reported a slightly higher range of errors (RMSE = 1.6–2.0 °C) for an artificial neural network used to predict across three sites on the Missouri River, USA, for daily temperatures in 2013. However, in both these cases, these authors evaluated performance throughout the year, rather than during a particular period, and so season-specific performance may be better or worse than our results. In addition, a thorough interpretation of prediction error to systems outside of our study requires accounting for the similarity of watershed characteristics. As we do not have this detailed information (e.g., extent of groundwater influence), we simply present RMSE for the purpose of understanding general predictive ability using different statistical techniques.
Summer maximums in this region are clearly of concern, owing to a prolonged summer period coupled with lower river flows. The data indicated that maximum temperatures greater than 31 °C can be observed, and the average summer maximums ranged from 23 to 25 °C. As expected, the errors for the best-fit model were larger when fit to these extrema rather than means (overall RMSE 1.58 °C, with a range from 0.93 to 2.56 °C for different sites and years). Maximum water temperatures are not as commonly predicted as means; however, our model performance was still within the range reported by other authors for other approaches. For example, Caissie et al. (2017) report a RMSE of 1.68 °C when an autoregressive model is used to simulate daily summer temperature maximums for the Little Southwest Miramichi River, New Brunswick, Canada, where water temperatures regularly exceeded 25–29 °C.
With respect to the other models evaluated, like other authors, we found the worst performance of the simple regression using air temperature lagged at one day, particularly when this model was fit to data from the entire growing season. Like Laanaya et al. (2017) for daily mean data, and Caissie et al. (2001) for daily maximum temperature, we also found poor performance relative to most other models of the nonlinear logistic regression model, with RMSE of 1.6 °C for summer means using a model fit with in-season data (1.8 °C for annual data), and 1.80 °C for maximums (2.0 °C for annual data fits), and larger forecast errors in each case. Piotrowski and Napiorkowski (2019) report a somewhat smaller range of error with a nonlinear variant tested on six sites in both Europe and the USA, averaged across 3 years of data (RMSE = 0.9–2.1 °C). Another study, on 14 globally distributed rivers with coefficients averaged across 5 years of data, reported larger errors (e.g., RMSE = 1.7–3.5 °C; van Vliet et al. 2011), and found that the model underestimated temperatures during summer and autumn and overestimated during spring, which appears to be the case for our models when they are fit with data from the entire growing season. It has been suggested that this nonlinear model is more suited for longer time steps, such as weekly data (Benyahya et al. 2007), and we do not recommend its use for daily data within a season.
The seasonal residual model had average performance (RMSE summer means = 1.7, and maximums = 1.85, when fit to in season data), and less difference between models fits using only in season data, vs. using all growing season data. For our region, the performance of this model was similar to that reported by others (RMSE = 1.3–2.5 °C; Laanaya et al. 2017; and RMSE = 1.7–2.1 °C; Zhu et al. 2018). However, a more complex model version had lower prediction error. Kamarianakis et al. (2016) fitted and tested a seasonal oscillation model for 43 rivers in Spain across one calibration year with forecasting accuracy close to 1 °C of the daily average temperature.
Performance of the best-fit model for the spring (daily means in May) was reduced compared with that for summer temperatures (RMSE = 1.88 °C, with a range from 0.9 to 5.05 °C for individual sites and years), although it is unclear whether the largest deviations may have resulted from monitoring artifacts. The performance of this linear regression model, however, was better than the nonlinear or seasonal residual models. The finding of lower predictive power in the spring is consistent with previous work, which suggests that there is “hysteresis” in the sense that water temperatures in the early part of the year are more strongly influenced by other factors, such as snow melt (Webb and Nobilis 1997), or even release of dam waters associated with flood management at this time of year, as well as the specific heat capacity of water. Given the larger errors for this time of year, we do not recommend simple regressions based on air temperature as a viable means of predicting water temperature in the spring. Instead, models that include easily available surrogates of flow, such as precipitation and winter snow pack, could possibly be employed for unmonitored rivers.
All of the models performed better when they were fit using data only from the period of interest rather than the entire growing season. When using all growing season data, the model coefficients are optimized to reduce prediction error on temperatures that do not occur frequently during the period of interest. These temperatures are outliers in our prediction period, and their influence can result in larger errors. Decreased performance when large amounts of data outside the prediction period are included in the data used for fitting is, of course, a reflection of the robustness of these various models. Some models were better able to accommodate such variability than others (e.g., compare the annual and seasonal fits of the seasonal residual model). While it may be expected that simple linear models would be unable to simultaneously accommodate seasonal variability and have low prediction error during a particular period of interest, this was even true for nonlinear models designed for annual patterns. We could improve the quality of linear regression results fit to the entire growing season by downweighing seasonal outliers (e.g., robust regression, Andersen 2008), but using data from only the target prediction period is a natural adjustment that accomplishes the same effect. We also used historical temperatures and fish behavior to choose our prediction time periods (e.g., we included June in our summer period since annual maximums sometimes occurred during this month). However, climate change is altering the timing and variability of critical temperature events, and we expect an earlier occurrence of spring spawning and summer maximums. In future work, a more rigorous selection of biologically important seasonal periods will likely be necessary, perhaps with reliance on regionally downscaled predictions (Zhang et al. 2020).
Overall, we find that simple approaches, such as linear regression, can outperform more complex statistical models in our region: information that may be of use to management agencies. While we did not directly compare our simple models with more complex physical or land-use models, we suspect that such models are not an option in many management scenarios, and the errors reported here for our simple models are certainly in the same range as some more complex models described in the literature. However, we also did not directly assess the impact of microhabitat, including factors such as groundwater discharge and dams, which likely influenced model performance. In some watersheds, both groundwater and air temperature variables regulate a diverse set of thermal regimes that support cold and warm water fish species (Chu et al. 2008). Groundwater plays a role in regulating temperature of the Grand River, and is an important habitat requirement for black redhorse (Bunt et al. 2013), but may have less influence in the other watersheds. In addition to groundwater influence, several large dams (e.g., Belwood) exist on the Grand River watershed and Upper Thames River, whereas the Sydenham River lacks dams in the main branch. Future air-to-river temperature modeling efforts may include multiple watersheds with and without groundwater influence or dams, and as a result, it may be necessary to include these factors to properly account for extreme heat events during low flows. Future work could evaluate the transferability of such more detailed models to determine what tradeoffs in application and accuracy may arise.
Importantly, the predictive power of these statistical air-to-river temperature models changed depending on the time frame of application, and the data used to fit the models. The differences in model performance metrics at different times of year highlight the key point that if river temperature models are to be useful for the management of species at risk, they must be evaluated for their performance during periods that have ecological relevance. In our study, we evaluated the mean temperature during the spawning period of the black redhorse, and the summer, when the impairment threshold could most likely be exceeded. All of the statistical models had a decreased error when fitted using data from the ecologically relevant periods, compared with fitting using all the data from the growing season, which might seem surprising, given the movement toward big-data approaches. However, larger errors were observed for spring mean temperatures. We conclude that, with caution, it is possible to use simple, data-minimal models for prediction during critical periods. Finally, although we did not test the models on areas outside of Ontario, given the minimal data requirements and empirical basis for the models, they could easily apply to other similarly sized, lowland river systems with the same degree of groundwater input, and the same general pattern of rapid spring warming and prolonged, stable, warm summer air temperatures.


The authors thank all of the technicians and engineers at the Grand River Conservation Authority, Thames River Conservation Authority, Fisheries and Oceans Canada, and Environment and Climate Change Canada for collecting and processing river and air temperature data that were used in this study. This research was funded through the Canadian Freshwater Species at Risk Research Network (SAR NET).


Supplementary data are available with the article at


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Supplementary Material

Supplementary data (cjfas-2020-0294suppla.docx)

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Published In

cover image Canadian Journal of Fisheries and Aquatic Sciences
Canadian Journal of Fisheries and Aquatic Sciences
Volume 78Number 9September 2021
Pages: 1268 - 1277


Received: 31 July 2020
Accepted: 27 July 2021
Published online: 1 August 2021


This paper is part of a Special Issue entitled “Science to support Canada’s SARA-listed freshwater species”, which brings together articles on threats and reintroductions for at-risk freshwater fishes and mussels in Canada.



Jordan Rosencranz
Department of Biology, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada.
Kim Cuddington
Department of Biology, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada.
Madison Brook
Department of Biology, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada.
Marten A. Koops*
Great Lakes Laboratory for Fisheries and Aquatic Sciences, Fisheries and Oceans Canada, 867 Lakeshore Road, Burlington, ON L7S 1A1, Canada.
D. Andrew Drake
Great Lakes Laboratory for Fisheries and Aquatic Sciences, Fisheries and Oceans Canada, 867 Lakeshore Road, Burlington, ON L7S 1A1, Canada.


Marten A. Koops served as an Associate Editor at the time of manuscript review and acceptance; peer review and editorial decisions regarding this manuscript were handled by Donald Jackson.
© 2021 Her Majesty the Queen in Right of Canada. © 2021 Brook, Cuddington, and Rosencranz. This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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