The performances of different modeled time series of snow depth are compared using the root mean squared error (RMSE) statistic. This metric is used because it measures how well the estimates from the model covary with observations, while it additionally reflects whether or not there is any systematic mean bias between the two time series. This metric is computed using time series of both absolute (which are the raw modeled and observed values expressed in centimeters and should not be confused with the mathematical absolute value) and anomalous values. The modeled RMSE is calculated as follows: 1RMSEMOD=1n∑i=1n(ymod,i-yobs,i)2, where ymod,i and yobs,i are, respectively, seasonal time series of modeled and observed snow depths at station i. Likewise, RMSECLIM is defined as 2RMSECLIM=1n∑i=1n(yclim,i-yobs,i)2, reflecting the error associated with estimates using climatology, where yclim,i is the reference mean climatological snow depth at station i. One can think about yclim,i as either a single value (i.e., the climatological mean) or a time series array, with all of the values being the same. The values of ymod,i and yclim,i change depending on whether we are computing the RMSE skill score using absolute values or anomalies. For the absolute values evaluated over the period of 1972–2021, yobs,i, ymod,i, and yclim,i would all contain values expressed in centimeters, where yclim,i is the mean seasonal snow depths, computed station by station, over the 1902–1971 calibration period. When computing the skill of the model estimated anomalies, yobs,i, ymod,i, and yclim,i all contain values expressed as a percentage of normal (e.g., 120 % of normal, which is 20 % above normal), where yclim,i is the mean seasonal snow depth at station i over the 1902–1971 calibration period (i.e., 100 %). Additionally, an RMSE skill score is also used to evaluate the performance of the models in terms of their ability to capture observed trends. In that case, a trend of 0 % per decade is treated as the climatological reference. Then, the RMSE skill score, SSRMSE, is defined as follows: 3SSRMSE=1-RMSEMODRMSECLIM, where an SSRMSE value of 1.0 would perfectly estimate the observations, while values between 0.0 and 1.0 reflect model estimates that perform better than the climatology, and values below 0.0 indicate that the model is less skillful than the climatology.

In this paper, we focus on snow depth averaged over the November–March season. Our first goal is to investigate historical empirical relationships between mean seasonal temperature, precipitation, and snow depth. Second, we use these historically derived empirical relationships to estimate changes in snow depth driven by changes in mean seasonal temperature and precipitation. During warmer months, and particularly at stations at lower elevations, an observable amount of precipitation will not always translate to a measured snow depth. This would result in trying to fit a predictor time series (i.e., precipitation) that varies with our predictand time series that does not (i.e., snow depth). We try to minimize the number of cases where there is zero measured snow depth by excluding the months of April and May from our seasonal average since those months often contain more stations with zero recorded snow depth. That way, we can have consistency in the lengths of our seasons for both the predictors, temperature (T) and precipitation (P), and our predictand, snow depth (SD). Figure 2a shows the percentage of data for which the 291 stations in this study measured an average monthly snow depth greater than 0.0 cm. For example, when considering all of the Januaries between 1901–2020, there were 20 263 station months with full data coverage. Of those, 500 recorded 0.0 cm for every day throughout January at a station. The percentage of zero measured snow depth for January is then 2.5 %, which is equal to 500/20 263 (this quantity is also equal to 100 % minus the percentage shown in the left-most bar of Fig. 2a). We used the season November–March because each of those 5 months contained less than 20 % zero measured snow depth. While the November–March season is somewhat shorter than what some other studies have used e.g.,, the average anomalous November–March snow depth (anomalies are represented as a percentage of normal; see Eq. 6) varies strongly with the average anomalous snow depth over the longer November–May season (see Fig. 2b). Thus, if one can skillfully estimate November–March snow depth then these will also be skillful for a longer season such as November–May. In Fig. 2c, average seasonal snow depth can be seen to vary with elevation, with higher elevations generally receiving more snow.

By itself, snow depth is not a measure of how much melted water is contained in the snow mass. Many different meteorological conditions can affect the density of the snowpack or the melting of snow. As a result, we cannot directly infer how much precipitation has fallen based on the snow depth measurements themselves. The precipitation could have fallen as rain, while precipitation falling as snow can accumulate at varying densities. Ultimately, we want to quantify historical changes in mean temperature and precipitation and how these have translated into changes in snow depth. Therefore, since many of the snow depth stations have neither temperature nor precipitation measurements, we construct these time series using monthly homogenized values of temperature and precipitation using nearby stations over the period of 1901–2021.

First, we obtain November–March sums of precipitation and averages of mean temperatures at all of the homogenized stations (see Fig. 1) over the years 1901–1902 to 2020–2021. Additionally, time series of standardized anomalies (i.e., z scores) are computed for each homogenized station (the mean and standard deviations are computed using the calibration period of 1902–1971). Then, for each snow depth station, we find the nearest five homogenized stations for temperature and, separately, for precipitation. Next, we compute two time series. The first contains the averaged standardized anomalies of temperature or precipitation as a function of the inverse distance of the nearest five stations. The second time series contains the averaged absolute values of temperature or precipitation as a function of the inverse distance of the nearest five stations. The inverse-distance-weighted standardized anomalies of the first time series are then adjusted to match the mean and standard deviation of the second time series. This is performed by simply reversing the steps of computing a z score, where the weighted standardized anomalies at each station are first multiplied by the standard deviation of the second time series at the same station, following which we add the mean of the second time series (also at the same station). This provides us with time series of seasonal (i.e., November–March) mean temperature and precipitation located at each of our 291 snow depth stations.

Information concerning the bivariate correlations between mean temperature and snow depth and/or precipitation and snow depth can be useful. However, correlations by themselves do not provide information about the steepness of the slope between the two variables. For example, given a 1.0 °C increase in mean temperature or a 20 % increase in precipitation, what would the expected impact on snow depth be? Additionally, how might these expected changes be affected by elevation? Furthermore, what would the multivariate impact on snow depth be given some combination of mean temperature and precipitation changes? And finally, can we apply methods that do not have an underlying assumption of linearity?

In Fig. 4a and b, one can observe the spatial distribution of the bivariate correlations between snow depth and temperature and, similarly, the correlations between snow depth and precipitation. In Fig. 4c and d, these correlations are shown to vary as a function of elevation. Generally, we find the largest correlations (either positive or negative) at lower elevations for temperature and at higher elevations for precipitation. For example, a station below 500 m is more likely to see increases or decreases in temperature translate more strongly to decreases or increases in snow depth than stations at higher elevations. The opposite influence is observed for snow depth and precipitation. Therefore, stations (or regions) at lower elevations are primarily driven by changes in temperature, while stations at higher elevations are primarily driven by changes in precipitation. These relationships support prior findings, such as those by and .

The utility of the information from Fig. 4 can be improved in the following ways: (1) instead of only considering the bivariate statistical relationship between either temperature with snow depth or precipitation with snow depth, we can consider all three variables in a non-linear, multivariate framework, and (2) by computing anomalies in the data, we can leverage information across multiple stations to provide a more robust empirical–statistical relationship.

In Fig. 5, we illustrate why it can be important to use anomalous values with our methodology. Figure 5a–c shows bar plots for the distribution of values using absolute temperature, precipitation, and snow depth, respectively, for the Austrian and Swiss stations between 500–1000 m. The average station elevation of the Austrian stations used is 745 m, while it is 742 m for the Swiss stations. Hence, the average station elevations are not very different between the two regions. One can observe in Fig. 5a and b that the Swiss stations are generally warmer and wetter than their Austrian counterparts. At the same time, the Swiss stations have lower seasonal averages of snow depth. In order to stress our point regarding the usefulness of anomalies versus absolute values in our methodological context, we can take a further subset of these Austrian and Swiss data points over this 500–1000 m elevation band. The individual values of these subsets of data are shown as the scatterplots in Fig. 5d and e. A Student's t test shows that the means (for temperature, precipitation, and snow depth) of the subset of Austrian data points (Fig. 5d) are all statistically significantly different compared to the subset of Swiss data points (Fig. 5e). We find that, while this subset of data points in Austria has a greater absolute temperature and less absolute precipitation than the Swiss subset, the Austrian stations have significantly more absolute snow depth than the Swiss stations. As we decrease temperature and increase precipitation, we should expect snow depth to increase. However, this is exactly the opposite of what the absolute data are telling us. By simply using the absolute data alone, we can potentially get the wrong signal when comparing one region to another. This spatial climatological difference can be addressed through computing anomalies, station by station, for the different variables. After computing anomalies, we can then leverage the information across a larger region.

To compute anomalies in the data, we begin with absolute (or raw) values of November–March seasonal temperature, precipitation, and snow depth data. Then, November–March average temperature anomalies, Tx,t∗, for station x∈(1,…,291) and year t∈(1902,…,2021) are computed as 4Tx,t∗=Tx,t-T‾x, where Tx,t denotes the seasonal averages of absolute temperature, and T‾x is the time-averaged mean temperature over the calibration period of 1902–1971 at station x. Anomalies of November–March precipitation accumulations, P∗, are computed as 5Px,t∗=Px,tP‾x, where Px,t denotes the seasonal accumulations of absolute precipitation, and P‾x is the time-averaged precipitation over the period of 1902–1971 at station x. Similarly, anomalies of November–March average snow depths, HS∗, are obtained using 6HSx,t∗=HSx,tHS‾x, where HSx,t denotes the seasonal averages of absolute snow depth, and HS‾x is the time-averaged snow depth over the period of 1902–1971 at station x.

Once we have computed our data anomalies, we can plot in Fig. 6 the observed historical anomalous snow depths (i.e., HS∗) along with temperature and precipitation anomalies (i.e., T∗ and P∗, respectively). The larger squares, bounded by the black lines, correspond to the anomalous measurements for one example station. That station, named “Feldkirch” with the number “11110”, has the coordinates of 47.27 ° latitude and 9.60 ° longitude and is situated at an elevation of 439 m. As one would expect, the average snow depth anomalies increase as the temperature anomaly decreases and as the precipitation anomaly increases. This figure can also be used to show that, for some increase or decrease in temperature, there can be a corresponding increase or decrease in precipitation that will yield approximately the same snow depth anomaly. Consider, for example, the snow depth anomalies in Fig. 6 between 2–3 °C below normal temperature and between 100 %–150 % of normal precipitation. Similar snow depth anomalies can also be observed between 0–1 ° below normal temperature and between 200 %–250 % of normal precipitation. This gives us a general idea of how sensitive snow depth anomalies are to temperature and precipitation anomalies. Even though Fig. 6 gives us a first look at the multivariate sensitivity of snow depth to temperature and precipitation anomalies, we can refine the approach by adding in a third variable. The sensitivities will change as elevation changes. Therefore, we can break up the multivariate sensitivities shown in Fig. 6 into different elevation bands. We have chosen to use four elevation bands: 0–500 m (containing 52 stations), 500–1000 m (75 stations), 1000–1500 m (91 stations), and >1500 m (73 stations).

The points along the left column in Fig. 7 are like those from Fig. 6, except that the data are broken up by the four elevation bands, and the data are now only plotted for the calibration period of 1902–1971. These are the data that we will use to fit a model and to make model estimates for our 1972–2021 validation period.

To begin, we calculate the averages of snow depth anomalies across two-dimensional bins of mean temperature and precipitation for different elevation bands. In our effort to construct the sensitivity diagrams, let us first consider the elevation band 0–500 m. In the lowest elevation band over the calibration period, the arrays T∗, P∗, and HS∗ all have a maximum possible number of measurements of 3640 (which is 70 seasons over 1902–1971 multiplied by the 52 stations from that band). Next, we find all of the values in HS0–500 m∗ that fall within a 0.8 °C window centered about a given temperature anomaly (with 0.2 °C increments of the binning window), T0–500 m∗, and a 40 % window centered about a given precipitation anomaly (with 10 % increments), P0–500 m∗. We did experiment with using different bin sizes, but we found that the choice of bin size does not strongly affect model performance (please refer to our comments in the paper discussion at 10.5194/egusphere-2024-1172-AC2). We calculate the average of all of the station snow depth anomalies that fall within this two-dimensional window of temperature and precipitation anomalies, given that there were at least 50 observed snow depth measurements that fall within that two-dimensional window. Then, we move the center of the window in order to perform the same set of operations across the range of temperature and precipitation anomalies. And lastly, we repeat the process for the other three elevation bands. The resulting averages using these two-dimensional bins are shown in Fig. 7e–h.

Next, the snow depth anomalies, resulting from the multivariate binning (Fig. 7e–h), are used to construct a smoothed and extrapolated sensitivity surface that will then allow us to estimate new values outside of what has been seen in the calibration record. For all binned values across the two-dimensional temperature and precipitation space, distances (in data space) are computed between each specific bin center and the center of all of the bins where we computed averages in the previous section (i.e., those are the grid cells which are colored in Fig. 7e–h). Multiple linear regression is then fit to the nearest quartile of values, where the bin centers of mean temperature and precipitation anomalies are the predictors and where the binned averages of snow depth anomalies are our predictands. To find the nearest quartile of values, we use a Euclidean distance measure which essentially equates the distances of a 10 % precipitation anomaly to a 0.2 °C temperature anomaly. Thus, a data point that is 0.4 °C warmer with 0 % of normal precipitation with respect to a point of interest and another data point that is 0.0 °C with 20 % of normal precipitation would be treated as the same distance. We did not find the model to be overly sensitive to providing more or less weight to the temperature or precipitation axes. Then, the regression coefficients are used along with the center point of the bin to obtain a snow depth anomaly. Our application of localized linear regression is only fit to the nearest quartile of data points for each bin center; therefore, it can accommodate a non-linear response surface across most of the data domain. At the same time, it smooths out the sensitivity surface (Fig. 7i–l) and also provides extrapolated values beyond what was observed in the calibration period.

The multivariate sensitivity plots shown in Fig. 7i–l, which are constructed using only data in the 1902–1971 calibration period, are now used to estimate snow depth anomalies at all stations and for all NDJFM seasons over the period of 1972–2021. For each station, we first determine the elevation band in which it falls (e.g., 0–500 m, 500–1000 m). Second, we locate the nearest bin center given each season's (e.g., November 1971–March 1972) anomalies of mean temperature and precipitation. The snow depth anomaly corresponding to that bin is then used as the estimated value for that season for that station. For example, consider a station in the elevation band 0–500 m which experienced a mean seasonal temperature at 2.0 °C above normal (where normal is defined with respect to 1902–1971) and a mean seasonal precipitation at 100 % of normal. The snow depth anomaly in Fig. 7i at 2.0 °C along the x axis and 100 % of normal along the y axis corresponds to 37.6 % (indicated by the color in that bin). So, given those meteorological anomalies at that station and in that season, the historical sensitivities would lead to an estimation of the snow depth being 37.6 % of normal. This procedure is repeated to produce the model estimates for all 291 stations and for all 50 seasons over the 1972–2021 validation period. The model estimates derived in this manner and driven by the historical sensitivities are referred to as the SnowSens model for the duration of the paper.

To be clear, we use known values of concurrent seasonal temperature and precipitation anomalies to estimate snow depth using the SnowSens model. As a result, our modeled values are not really forecasts. While we refer to the values produced by the SnowSens model as “estimates”, our broader aim is for the model to be capable of producing actual forecasts of future snow depth conditions given future projections of temperature and precipitation. True long-term forecasts of snow depth over some future period can be thought of as a modeling chain with two chain links. The first link in the modeling chain is made up of the projections of temperature and precipitation, while the second link is made up of the projections of snow depth driven by the temperature and precipitation projections from the first link. In this paper, our model validation is done by quantifying the skillfullness or the uncertainty associated with the second link of that modeling chain. With knowledge of how skillful the SnowSens model is, the model can then be forced or driven with a plausible range of future projections of temperature and precipitation. In this way, the SnowSens model can be run to produce actual forecasts of snow depth, which also contain the uncertainty of future projections from the first link of the modeling chain.

When evaluating the performance of the SnowSens and the SNOWGRID-CL models, we use both the absolute and anomalous values. The methodology outlined above provides SnowSens model estimates as anomalies for all stations and seasons. At the same time, the SNOWGRID-CL model provides estimates of absolute snow depths. Therefore, we must produce absolute estimated values for the SnowSens model and estimated anomalies for the SNOWGRID-CL model. Let us begin with the estimates of the SnowSens model, where the absolute snow depths, HSMOD, are computed as 7HSMOD,x,t=HSMOD,x,t∗×HS‾OBS,x, where HSMOD,x,t∗ denotes the SnowSens- estimated anomalies at station x and time t, and HS‾OBS,x is the observed mean seasonal snow depth at station x over the calibration period of 1902–1971. Now, we have absolute estimates for both models. Next, we want to address any mean biases present in the models while simultaneously computing anomalies in the data. To do this, a common period of record is used. The SNOWGRID-CL model data begin in 1962, and, therefore, we use the common reference period of 1962–1971 to compute the anomalies and bias correction: 8HSMODBC,x,t∗=HSMOD,x,tHS‾MOD1962-1971,x×HS‾OBS1962-1971,x∗, where HSMODBC,x,t∗ denotes our estimated anomalies at station x and time t, which have been bias-corrected to remove any mean biases present over the common period of 1962–1971. Equation (8) is applied to both models to provide the estimated anomalies which have been mean-bias-corrected for all of the stations (i.e., all Austrian stations for SNOWGRID-CL and all Austrian and Swiss stations for SnowSens).