In situ snow survey data are retrieved from a dataset created by the Climate Research Division of Environment and Climate Change Canada (ECCC), which has been updated to include observations up to the end of 2017 . This dataset includes snow survey measurements from approximately 33 Conservation Authorities (CAs) at 383 locations throughout Ontario between 41 and 49.5∘ N and -87.875 and -73.375∘ E. The locations of these survey sites are marked in Fig. a along with an outline of our Ontario study region. Survey site locations are selected based on recommendations set forth by the Conservation Authorities and Water Management Branch which suggest that measurement sites should be selected based on their representativeness of the surrounding region, in easily accessible locations free from the effects of excess wind drifting . The snow surveys provide an estimate of SWE calculated as an average of 10 individual snow core SWE samples taken over 10 m at each survey site . These observations are recorded bi-weekly around the 1st and 15th of each month from November to May and have been recorded since 1933, but for this study only data from January 2011 until December 2017 have been considered. Snow survey density is higher in the southernmost portion of Ontario (below 44.5∘ N), with 189 survey locations closely grouped near the United States (US) border between Lake Huron and Lake Ontario. A similar survey count (n=194) exists above 44.5∘ N but with sparser spatial coverage due to the region's larger size and lower population density.

The measurement tools used for retrieving in situ observations vary between locations over the time span of our study . Common strategies for collecting SWE measurements by hand include the use of snow corers, which are portable, handheld tubes that are inserted into the snowpack down to the soil layer and weighed to retrieve a SWE estimate at that point within the snowpack . An Ontario snow inventory summary completed by provided a questionnaire to 265 of the snow survey sites (with a 67 % response rate) and found that the Federal Snow Sampler (also known as the Mount Rose sampler) was used at 94 % of sites and that the ESC-30 was used at the remaining 6 % of locations. Methods of SWE measurement also varied, with 62 % using a calibrated spring balance in the field, 30 % using a digital balance in the field, 6 % grouping snow samples into a container and weighing in the field, and 2 % bagging the samples and weighing them later . Although an important observational metric, in situ measurements of SWE take, on average, 20 times longer than snow depth measurements, and due to the additional time investment this often results in poor spatial and temporal data coverage of SWE measurements across large regions .

A set of statistical methods that have previously been applied to bias correction in different contexts is analysed in this study to identify the method which displays the highest performance in reducing the bias between SNODAS SWE and in situ observations over our study period. The methods examined include mean bias subtraction (MBS), simple linear regression (SLR), decision trees (DT) and random forest (RF). All models (excluding MBS) are implemented using the scikit-learn Python package which includes built-in linear regression and machine learning modules . For MBS, the average difference in SWE between SNODAS and in situ is calculated and then subtracted from each SNODAS estimate to produce a bias-corrected dataset. More formally, mean bias (MB) is defined as MB=1n∑i=1n(xi-zi), where xi and zi are the respective daily SNODAS and in situ SWE measurements, and n is the number of measurements over the study period. The linear regression techniques used in this study conform to the least squares general regression model which relates a response variable y to a linear combination of n explanatory x-variable predictors y^=b0+b1x1+b2x2+…+bnxn, where b0 is the model intercept and b1 to bn are predictor coefficients. Both SLR using a single explanatory variable and multiple linear regression (MLR) with numerous explanatory variables are considered in this study. A single SLR model is applied to all stations in our study region using daily SWE from SNODAS as the sole predictor and in situ snow survey SWE estimates as the response variable. MLR, DT and RF methods all use the full list of predictors outlined in Table  to predict in situ SWE.

A decision tree is a flowchart-like data structure, wherein the decision making process begins at the root node (the top of the tree) followed by a series of cascading decisions based on the included model predictors until the terminal leaves are reached at the bottom of the tree, which represent the regression estimate of the response variable. As implied by its name, the random forest regression model is an ensemble of decision trees that are generated during model training . Each tree included in the RF model ensemble is generated from a randomized subset of the available training data, coupled with a randomized subset of the predictor variables . This inherent randomness improves the learning process of this technique but also contributes to uncertainty in the accuracy of an individual tree . The ensemble approach used in RF accounts for and minimizes the uncertainty present in an individual decision tree by calculating the mean prediction from all trees in the ensemble. Our RF model is run with a forest size of 100 trees in its ensemble, both RF and DT methods use a maximum tree depth of 15 (the maximum number of decisions before determining an estimate for the response variable) and each tree was allowed to grow to its full extent, with no set number of maximal terminal nodes (no set maximum number of leaf nodes). These model parameters were obtained through a brute-force grid search hyperparamaterization of the RF model, which nudged each parameter value and examined changes in model accuracy and impacts on computational efficiency to select optimal values for running the model. A variety of regression predictors were considered for use in this study, including land use parameters, elevation, and indicators of general climate. The final set of predictors for all methods (shown in Table ) was selected based on the (non-zero) model importance score for each variable in the RF model summary output.

To mitigate against model overfitting, the data for the RF and DT models are randomly split into a training set composed of 75 % of the values and a testing set which comprises the remaining 25 %. Additionally, a separate 10-fold cross-validation (CV) resampling procedure was applied to further evaluate model performance on unseen data. The CV K-fold splits the full dataset in time into 10 consecutive groups of samples which are held constant for the full CV procedure. We then train the model on each combination of k-1 folds and their performance is calculated as the average of all training and testing scores for each K-fold split. The fold value of k=10 was selected as a compromise between the size of the training sample and the computational overhead and is a typical choice for similar applications .

A range of metrics has been considered in order to determine whether additional performance can be gained from using more sophisticated statistical methods over traditional approaches. To assess model skill, we have selected absolute mean bias (accuracy) and RMSE (precision) as our model performance criteria, as these properties have demonstrated effectiveness in previous studies for assessing the capabilities of competing bias-correction methods in the geosciences .

In addition to applying each model to the full set of in situ survey sites for the full time span, we also run spatially and temporally partitioned models to assess changes in performance over specific regions and periods. Partitioning is applied spatially by separating the set of in situ measurement locations into northern and southern regions at 44.5∘ N to help account for snow survey density differences between the two regions of Ontario as described in Sect. . Model performance is also analysed temporally with training restricted to different portions of the snow season: December to February (DJF), March to May (MAM) and the combined period: December to May (DJFMAM). All models and partitioned datasets were trained on 75 % of the data and tested on the remaining 25 % (excluding MBS, which does not include a model training step).

Initial comparisons between current SNODAS SWE estimates and in situ observations throughout Ontario describe, on average, a positive absolute mean bias of 50 % in the SNODAS estimates from 2011 to 2017. Additionally, the snow survey sites in Fig. a display a pattern of strong relative mean bias present in the SNODAS estimates at the majority of survey locations. Relative mean bias (RMB) is defined as RMB=1n∑i=1nxi-zizi⋅100, where xi and zi are the respective daily SNODAS and in situ SWE measurements and n is the observation count. This relative bias is positive at 212 of the 383 measurement sites and rises above +100 % relative bias at 67 locations. These sites with a strong relative bias also generally exhibit a strongly positive absolute mean bias, with SNODAS overestimating SWE by over 100 mm SWE at many of these locations. The sites with the strongest relative and absolute mean biases are typically grouped together in the northern portion of the study region above 44.5∘ N as well as in areas east of both Lake Huron and Lake Superior.

There also exists a strong temporal bias in the bi-weekly SWE estimates from SNODAS when compared with in situ ones (Fig. b). This bias is strongest during the first half of our study period until the beginning of 2015 where, although SNODAS estimates are generally still higher on average (by approximately 5 mm SWE), the overall absolute mean bias is reduced. If we consider the full temporal domain, the absolute mean bias in the SNODAS estimates is approximately 16 mm SWE, which corresponds to a 50 % increase compared to that of the in situ SWE observations. The change in bias between the first and second halves of our study period implies a change in the data assimilation system used by SNODAS, wherein new datasets are assimilated into the system to further reduce model error.

In the following section, the performance of four bias-correction techniques will be discussed. The progression of mean bias and RMSE over the two study regions and three time periods is summarized in Fig. , time-series summary metrics for the full region are shown in Fig. , and the spatial pattern of (remaining) absolute mean biases at snow survey sites is shown in Fig. ; the time series of corrected and uncorrected, domain-averaged SWE (along with the 95 % confidence intervals based on each sample) is shown in Fig.  for our full study period.

The DT method displays further improvements over MBS and SLR in terms of model skill, with the second lowest overall RMSE between 3 and 8 mm SWE on average, coupled with near-zero absolute mean bias when partitioned spatially and temporally (Fig. ). Differences in RMSE are quite small between each region and time period, and the resulting RMSE between DT and the in situ observations is substantially lower, on average, than that of uncorrected SNODAS (an 80 % improvement). We note similar large improvements in model performance using the DT method across the full region for all months, with an overall RMSE of 4.03 mm SWE and absolute mean bias of 0.6 mm SWE.

Building on the improvements from DT, we find that RF displays the best overall skill of all tested models by producing SWE estimates with low absolute mean bias and the lowest overall RMSE when compared with in situ SWE. As noted in the predictor importance scores of Table , RF incorporates information from a suite of predictor variables which allows the model to better understand how SWE biases change in both time and space. RF was found to consistently outperform the other models for all time periods for both northern and southern regions of our study area, as shown in the partitioned model run summary statistics in Fig. , with absolute bias reduced below 1 mm SWE and RMSE between 3 and 5 mm SWE. Furthermore, RF continues to outperform other methods of bias correction when the model is trained and run over the full spatio-temporal domain, resulting in an RMSE of 3.64 mm SWE and absolute mean bias of only 0.2 mm SWE as shown in Fig. . This is an 86 % reduction in RMSE compared to the uncorrected SNODAS RMSE and a significant improvement over the 45 % reduction achieved by SLR and the 20 % reduction in RMSE from MBS. Since the RF is composed of an ensemble of DT models, it is not surprising that both methods perform similarly when run with the same predictor set, with RF slightly outperforming a single DT, because the ensemble is more robust and reduces systematic model error caused by overfitting.

As the bias in SNODAS is nonstationary (Fig. b), we next evaluate the bias-correction methods separately for a sub-period of high bias (2011–2013) and one of low bias (2014–2017). This test is performed using the same predictor variables in Table , excluding Year Id; i.e., we implicitly assume stationarity within each sub-period. During the high-bias period (with a default bias of 27.9 mm SWE and default RMSE of 38.5), we find similar results to the full period. MBS, SLR and RF all reduce the absolute mean bias to less than 1 mm SWE, and RF reduces RMSE to the lowest value of 2.7 mm SWE, compared to 5.7 and 26.6 for SLR and MBS, respectively. The low-bias period (with a default bias of 9.28 mm SWE and default RMSE of 16.5) again exhibits a similar pattern in model performance to the full period, with all models reducing the absolute mean bias to less than 1 mm SWE and RF again showing the lowest RMSE of 4.9 mm SWE, compared to 14.0 and 13.6 for SLR and MBS. In summary, the sub-period analysis shows consistent performance from the RF model but improved performance of the SLR model during the high-bias period, when the bias in SNODAS appears more uniform from year to year (Fig. b).

The RF model displays the best overall performance in terms of reducing bias and RMSE, and this skill is demonstrated spatially in Fig. d. Compared to the other bias-correction methods, the RF model is the most effective at reducing the spatial bias in SNODAS, with only small differences between model-corrected SWE values and in situ SWE across the majority of the region. This accuracy is also evident in the time series of domain-averaged SWE values shown in Fig. , with the RF-corrected SWE estimates closely tracking the in situ observations across all years. Comparisons of interannual correlations further emphasize the strengths of the nonlinear techniques over traditional bias-correction methods at capturing changes in bias over time. Interannual correlations between RF-corrected SWE and in situ are the highest at approximately 0.99, with correlations of approximately 0.93 for linear regression and of approximately 0.90 between the default SNODAS and in situ SWE. The RF model is therefore selected as the best-performing candidate model to perform bias correction on the Ontario-wide SNODAS data.

The difference in snow water volume between uncorrected and bias-corrected SNODAS SWE has important implications for understanding the regional water balance of Ontario. The reduction in mean SWE resulting from the bias correction should reduce the regional melt estimates, which we estimate using hydrographs of area-normalized discharge from three Ontario river gauges for the period 2011 to 2018: Pic River (48.77∘ N) near Marathon, the South Branch Muskoka River (45.14∘ N) at Baysville, and the Thames River (42.54∘ N) at Thamesville (basins outlined in Fig. a). Monthly melt water amounts are estimated as the (negative) SWE differences between consecutive monthly means from the SNODAS and RF SWE datasets . To compare the melt volumes with normalized discharge values, the melt is averaged over the drainage area associated with each stream gauge. We argue that this provides a reasonable estimate of the amount of water being released from the snowpack during the spring freshet period in each watershed. Note, however, that this does not include losses of water due to evapotranspiration or additional water input from rainfall.

Figure  shows the time series of area-normalized discharge and estimated melt rate over the study period for the three drainage areas. The timing of observed peak streamflow closely aligns with melt rate peaks during the spring freshet at the northern gauges of the Pic River and the South Muskoka River. Since the melt water estimates do not include rainfall, they should be considered a conservative estimate of potential spring discharge. The melt volumes derived from the corrected dataset are close to, but mostly below, observed discharge values in the two northern catchments, while the estimates based on the uncorrected SNODAS SWE data significantly exceed the observed discharge and can thus be considered unphysical. This serves as an independent validation of the physical plausibility of the bias-correction method proposed here.

We further note that the differences between the corrected and uncorrected melt estimates are most apparent during the period of high bias prior to 2015. In this context it is also interesting to note that the accumulation of SWE in the uncorrected SNODAS dataset exceeds the total amount of precipitation (based on the NRCan dataset) for most winter months prior to 2015 and for isolated winter months after 2015. In the southern Thames River watershed, on the other hand, there exists a much lower bias between SNODAS and RF-corrected melt compared to the two northern watersheds, which is consistent with the previously discussed spatial pattern of biases in Fig. a. In addition, the Thames River watershed is not snowmelt dominated, so the biases do not affect streamflow in the same way as they do in the two northern watersheds. The changes in the magnitude of snowmelt shown in Fig.  suggest that the RF bias-corrected SWE constitutes an improvement over the uncorrected SNODAS-derived melt estimates throughout the study region and that the RF-corrected dataset could provide a valuable new resource for hydrologic modeling and flood risk forecasting.

Linear regression and machine learning techniques have previously been used effectively across the geosciences for bias correction of global and regional climate model output . Previous studies on the estimation of North American SWE using artificial neural networks and support vector machines also exhibit similar results, with machine learning techniques outperforming general linear models . However, recent work by and suggests that bias-correction methods have their own associated uncertainties which must be considered when applied to datasets like SNODAS. These studies suggest that potential inconsistencies can exist between real-world and model dynamics and their interactions with bias-correction techniques. This can lead to unphysical changes in the relationships between variables and model dynamics and even violate basic physical principles. This last point is relevant to this study, as some models (like MBS) overcorrect SWE on the ground to negative values, which are physically meaningless. Our research has found that more sophisticated nonlinear statistical techniques like DT and RF produce bias-corrected SWE values that adhere more closely to these physical principles.

We must also consider uncertainties in the in situ snow survey data record. Hand-measured SWE observations are generally considered to be of high accuracy; however, measurement error can still occur. Common issues include snow sticking to the inside of the measurement device or falling out of the bottom of the device due to improper soil capping . Issues like these can lead to underestimations in SWE when measurements are being recorded. Furthermore, from the available documentation by , not all CAs use the same snow coring device and measurement techniques when retrieving SWE samples, and this may result in systematic differences in their reported SWE estimates. Errors in the reference dataset can propagate through into the bias-correction model during training and negatively impact the reliability of the model, even away from the snow survey locations. Additional error also arises in our comparison of point to grid data since our analysis assumes that the snow survey data are generally representative of the surrounding area in the containing 1 km SNODAS grid cell. While snow survey locations are selected to be representative of their surrounding landscapes , snow density varies drastically over small spatial scales, and this assumption of homogeneity contributes to further uncertainty in our analysis .

An additional source of uncertainty in the water balance analysis (Sect. ) arises from the lack of sublimation and evapotranspiration estimates as components of our melt calculation. Improving our estimates of melt and runoff in these basins requires additional high-frequency precipitation data (including phase estimation), temperature data along with a series of non-obvious judgements to estimate sublimation . Furthermore, the inclusion of a hydrologic or land surface model, which would be necessary to properly account for sublimation and evapotranspiration, would also not be helpful for this purpose, as these models compute snowpack internally and one would be left with a comparison against modeled snowpack SWE. We also note that average liquid precipitation during the spring freshet exceeds the influence of potential evapotranspiration during this region at this time , and it can therefore be argued that snowmelt places a lower bound on the spring freshet volume . While the inclusion of these additional components in the water balance equation would improve melt estimates, through our comparison of SNODAS SWE differences with those estimated from NRCan climate normals and streamflow, it is clear that the uncorrected SNODAS values are unphysical, while the bias-corrected values appear reasonable.

Any additional uncertainties that exist in SNODAS and the NRCan gridded precipitation product (which are ingested as predictors into the bias-correction models) will further contribute to the overall error in the bias-corrected SWE dataset . Uncertainties in the SNODAS numerical forecast model along with measurement error from the datasets being assimilated by each product add to the total uncertainty of the system. Furthermore, we note again that the reference period of the climate normals that have been used to characterize the climate in the RF model is 1981 to 2010, while the study period is 2011 to 2018. This may introduce additional uncertainty due to decadal variability and transient shifts in climate; however, since only long-term averages (monthly normals) have been employed for this purpose, the error is likely small. Additionally, a study by suggests that RF is sensitive to the spacial representativeness of the bias throughout the region and that spatio autocorrelation in the training and testing datasets used in the RF may also negatively influence the accuracy of our model fit.

While there is no clearly documented reason behind the SNODAS SWE bias that occurred after 2015, we believe this may be the result of new datasets being inserted into the data assimilation scheme used by SNODAS, although one may argue that since the general magnitude of the SNODAS bias is reduced post 2014, a bias correction of SNODAS SWE in this region is unnecessary. We suggest that the bias correction is still a valuable contribution, since the SNODAS bias remains non-zero (approximately 5 mm SWE on average, and even higher throughout the northern region) during this period when compared with in situ, and the extended bias-corrected data record allows us to better calibrate current hydrologic models. Another area of potential interest for other groups using SNODAS in Canada exists in the latitudinal gradient of bias we note in Fig. a, which suggests that the mean SNODAS bias increases in magnitude as we move further away from the US border (SNODAS is a US product which mainly ingests US data).

Each of the bias-correction methods examined here shows skill in reducing the absolute bias present between SNODAS and in situ SWE observations, from the default 16 mm SWE in SNODAS to less than 1.5 mm SWE across all techniques. MBS and SLR exhibit an inability to capture year-to-year variability present in the bias and often overcorrect or undercorrect the amount of SWE on ground, resulting in high RMSE between their corrected estimates and in situ observations. The more sophisticated machine learning techniques display further improvements in skill, with RF reducing RMSE by approximately 86 % compared to that of the uncorrected SNODAS RMSE, and a reduction in absolute bias throughout the region to 0.2 mm SWE. The additional predictors combined with the ability of the model to capture nonlinear behavior, allows the RF model to closely reproduce observed SWE values and remain within physically plausible limits. The RF model also provides insights into the strengths of the relationships between biases and various model predictors, suggesting a connection between SNODAS biases and elevation, total precipitation and air temperature. Furthermore, it is also evident that the bias diminishes over time, even though this may not be adequately reflected in the predictor importance ranking of the calendar year variable. Unfortunately, due to lack of documentation regarding changes in the assimilation system of SNODAS, it is not possible to identify the reasons behind these changes.

In this study we have only employed simple linear regression and decision tree-based methods of bias correction. Nevertheless, we have demonstrated that nonlinear techniques can be used very effectively for bias correction and are far superior to linear methods. Neural networks and support vector machines have also been effectively implemented for the purpose of bias correction in the geosciences . A paper by also found that machine learning methods can act as effective operators at estimating North American snow mass. It is possible that these other machine learning techniques may offer further improvements to the methods examined here, and should be considered in additional follow-up work. Furthermore, it has been suggested by and that deep learning methods can provide powerful new perspectives in addressing common challenges in information extraction for water resource research. However, the region that was considered in this study is relatively small and climatologically homogeneous, and the number of in situ observations is likely insufficient to justify the use of more complex techniques that typically require very large training datasets. If, on the other hand, bias correction were to be attempted on a larger scale, for example the entire SNODAS domain, a more complex technique should be considered: likely a deep neural network, potentially with recurrent properties or convolutional layers, so as to account for memory effects and spatial structure. In this scenario, it would also be possible to make use of significantly more in situ observations across North America (e.g., SNOTEL sites), that could be used to train such a model.