The description and analysis of compound extremes affecting mid- and high latitudes in the winter requires an accurate estimation of snowfall. This variable is often missing from in situ observations and biased in climate model outputs, both in the magnitude and number of events. While climate models can be adjusted using bias correction (BC), snowfall presents additional challenges compared to other variables, preventing one from applying traditional univariate BC methods. We extend the existing literature on the estimation of the snowfall fraction from near-surface temperature, which usually involves binary thresholds or nonlinear least square fitting of sigmoidal functions. We show that, considering methods such as segmented and spline regressions and nonlinear least squares fitting, it is possible to obtain accurate out-of-sample estimates of snowfall over Europe in ERA5 reanalysis and to perform effective BC on the IPSL

Despite the expectations of less frequent snow events in a warming climate, there are still several motivations to study trends in future snowfall. First of all, snowfall extremes can still have a great impact on economy and society. Recent snowfall over large, populated areas of France in February 2018 and in Italy in January 2017 caused transport disruption, several casualties, and economic damage. Snow is also an important hydrological quantity and crucial for the tourism industry of some countries. Although climate models predict a general reduction in snowfall amounts due to global warming, accurate estimates of this decline heavily depend on the considered model. Moreover, while this prediction is valid at the global scale, there may be regional exceptions, with northern areas receiving more snowfall during the winter. Large discrepancies in snowfall amounts indeed exist for observational or reanalysis datasets: in detecting recent trends in extreme snowfall events,

Climate models are the primary tool to simulate multi-decadal climate dynamics and to generate and understand global climate change projections under different future emission scenarios. Both regional and global climate models have coarse resolution and contain several physical and mathematical simplifications that make the simulation of the climate system computationally feasible but also introduce a certain level of approximation. This results in biases that can be easily observed when comparing the simulated climate to observations or reanalysis datasets. Therefore, they provide limited actionable information at the regional and local spatial scales. To circumvent this problem, it is of crucial importance to correct these biases for impact and adaptation studies and for the assessment of meteorological extreme events in a climate perspective

In order to mitigate the aforementioned biases, a bias correction (BC) step is usually performed. This step usually consists of a methodology designed to adjust specific statistical properties of the simulated climate variables towards a validated reference dataset in the historical period. The chosen statistics can be very simple, e.g., mean and variance, or it may include dynamical features in time, such as a certain number of lags of the autocorrelation function for time series data, it can be constructed using a limited number of moments or aim at correcting the entire probability distribution of the variable, and the correction can also be carried out in the frequency domain so that the entire time dependence structure is preserved. For an overview of various BC methodologies applied to climate models see, for example,

Despite the effort devoted to correcting precipitation bias, only a few studies propose specific BC methods for snowfall data from climate projections. For instance,

Indeed, snowfall presents additional challenges compared to other variables, precluding the use of traditional univariate BC methods. Besides the intermittent and non-smooth nature of snowfall fields – a feature in common with total precipitation – snowfall is the result of complex processes which involve not only the formation of precipitation but also the existence and persistence of thermal and hygrometric conditions that allow the precipitation to reach the ground in the solid state. As a result, snow is often observed in a mixed phase with rain, especially when considering daily data. This phase transition poses additional challenges to the bias correction of snowfall, namely the need of separating the snow fraction, using the available meteorological information. Ideally, methods to perform such a separation, also known as precipitation-phase partitioning methods, should be based on wet-bulb temperature, to which the snow fraction is particularly sensitive

In the following,

The fact that

In a hydrological modeling context,

Slightly more complex methods aim at reproducing the quasi-smooth shape of the precipitation-phase transition by fitting S-shaped functions to the relationship between

We stress that other studies highlight a dependency of the snow fraction on additional variables. For example,

We aim at finding a feasible method that allows for the accurate estimation of

The rest of the paper is organized as follows: in Sect.

Most of the hydrological and climatological studies cited in Sect.

To match the goals defined in the previous section, we decide to rely on a gridded reanalysis dataset at the European scale to specify and validate our snowfall method, rather than on observational data from limited areas.

In particular, we use the fifth generation reanalysis product (ERA5) provided by the European Centre for Medium-Range Weather Forecast (ECMWF). This dataset has a high (0.25

In most reanalysis and climate simulation models, snowfall is represented as snowfall flux (SF) in

Snowfall in ERA5 consists of snow produced by both large-scale atmospheric flow and convective precipitations. It measures the total amount of water accumulated during the considered time step as the depth of the water resulting if all the snow melted and was spread evenly over the grid box.

We aggregate the hourly ERA5 data into daily values to match the time step of the climate simulations described in the following. We choose ERA5 as the dataset for our study because of its physical consistency and the use of advanced assimilation techniques for its compilation

In this paper, we use outputs of a climate projection model from the EURO-CORDEX project obtained by nesting the regional circulation model (RCM) IPSL-WRF381P within the r1i1p1 variant of the IPSL-IPSL-CM5A-MR global circulation model (GCM) from CMIP5. The RCM results are presented at a 0.11

The relevant variables are

In the following, we show how statistical modeling of SF based on bias-adjusted

In this section, we describe a set of candidate statistical models for the snow fraction

First, we consider the single threshold method (STM) as our baseline method. Given the spatial extent of our dataset and the relatively fine grid resolution, we anticipate that more refined methods could be better suited to the purpose of climatological analysis of snowfall. In particular, we aim at finding parsimonious models that can be easily fitted, pointwise, on the grid, producing location-specific parameter estimates that we may exploit to obtain an accurate approximation of snowfall using

In order to do so, we explore different statistical models. We extend the STM to a more flexible framework, consisting of two steps. First, for each grid point we analyze the relationship between

As a final remark, given their ready availability in bias-adjusted form, we estimated the presented regression models including also 850 hPa air temperature and total precipitation as explanatory variables of the snow fraction. However, including these additional explanatory variables did not significantly improve the goodness of fit or prediction skills of the regression models. For this reason, we only present and discuss results of one-dimensional methods considering near-surface temperature.

First of all, we assess the results obtained applying the STM, introduced by

Despite its simplicity, this technique presents some advantages. First, if

However, this method is also naive, as it gives a binary representation of a quantity continuously varying in [0, 1]. This makes it impossible for the results to provide insights on snowfall features in case of more in-depth climatological analysis or more refined hydrological models. Furthermore, the search for the optimal threshold should not be complex or computationally expensive, otherwise it invalidates the advantage of using such simplified assumptions. This does not prevent us from detecting a representative value of

Estimates reported in the literature for the single threshold range across quite different values. For example,

However, it is worth mentioning that the results by

In the following, we discuss a methodology that encompasses the case of a locally selected threshold temperature, while enabling us to determine the optimal number of thresholds and their respective values for each point of the considered domain.

In order to overcome the limitations of the STM of

determine the optimal number

in each of the

While segmented logit-linear regression allows for more flexible functional forms compared to a STM, it is based on the specific assumption of a piecewise logit-linear relationship between

Finally, following

We assess the performance of each method in recovering

As a preparatory step, we transform the data so that they are included in (0,1) without assuming the boundary values. In fact, the segmented regression on logit-transformed data is ill-behaved in case the variable assumes the limiting values of 0 and 1. To circumvent this problem,

For the statistical model selection and validation steps, we use the entire available period from 1979 to 2005. For the STM, we test two threshold temperatures

The performance of the method is assessed by comparing true and predicted

We consider the best method to be the one providing the best performance in terms of minimum MAE and RMSE over the considered domain. We repeat the estimation for each grid point using the entire sample size

The capability of the chosen methods to perform a BC-like task can be evaluated in terms of similarity between the distribution of the estimated daily snowfall and the reanalysis values. We will use three measures of dissimilarity between the distributions, i.e., the Kolmogorov–Smirnov (KS) statistics, the Kullback–Leibler (KL) divergence, and the

The breakpoint analysis for the search of optimal threshold temperatures is described in detail in Sect.

It is worth noting that at least one threshold temperature is found for each grid point, even though having no breakpoint is an admissible outcome from the search algorithm. This corroborates the idea that some form of transition between two regimes is to be expected concerning the relationship between

As mentioned in Sect.

The

Threshold temperatures optimized for segmented logit-linear regression, according to Eq. (

Figure

Forecasting performance of the five considered methods applied to ERA5 reanalysis in terms of mean absolute error (orange) and root mean squared error (green).

We assess the performance of each method in terms of snow fraction prediction, as described in Sect.

We show results for both the snow fraction, directly derived from the statistical prediction (Fig.

From a visual inspection, the STM produces the poorest performance in terms of median and variability of the errors, for both

In order to choose the best possible methodology based on quantitative considerations, we test for significant differences among groups using the rank test proposed by

We perform a total of four Kruskal–Wallis tests on snow fraction RMSE, snow fraction MAE, snowfall RMSE, and snowfall MAE. In all cases, the null hypothesis must be rejected, with virtually null

For

The close similarity of results from the segmented logit-linear and the spline regression is also evident from the summary statistics of the distributions of the two error measures for the two variables, as shown in Table

Summary statistics of the distributions of the RMSE and MAE for snow fraction and snowfall in ERA5. The mean, standard deviations, median, and interquartile range are shown.

Summary statistics of the distributions of correlation between ERA5 reanalysis and predicted snow fraction. Note: IQR indicates the amplitude of the interquartile range.

Box plots of correlation coefficients between the ERA5 reanalysis and predicted snow fraction, using the five selected methods, for all grid points with at least 30 snowfall events

As a further criterion to choose the best-performing method, we consider the Pearson's correlation coefficient computed, at each grid point, between the snow fraction observed in ERA5 and predicted using the five methods under investigation. Notice that the factor linking

Figure

Snow fraction as a function of temperature for the five ERA5 grid points closest to some European cities (gray circles). Solid lines represent fitted values for the segmented logit-linear regression (blue), spline regression (orange), sigmoid fit (dark red), and constrained sigmoid fit (green).

Snow fraction as a function of temperature for the five ERA5 grid points closest to some European cities (gray circles). Solid lines represent fitted values for the segmented logit-linear regression (blue), spline regression (orange), sigmoid fit (dark red), and constrained sigmoid fit (green).

Figures

It is clear that the unconstrained sigmoid fit has some issues with extreme values of

As expected, fitted values from segmented and spline regressions are quite close, with the splines giving the smoothest result, since they are continuous at the knots by construction.

It is clear that, overall, the segmented logit-linear regression and the spline regression perform significantly better at reconstructing the snowfall compared to the STM. However, these results do not constitute strong evidence towards a better performance in reconstructing snowfall in practical cases, i.e., when it is unobserved or severely biased. To this purpose, we will apply both methods to reconstruct the snowfall in a climate projection model, and we assess which one produces the least biased snowfall using bias-adjusted temperature and precipitation as an input.

As an additional element to evaluate the performance of the identified methods, we assess if they can produce robust snowfall estimates in a pseudo climate change scenario. In order to do so, we repeat the validation procedure described in Sect.

Performance of the statistical methods for the prediction of snow fraction, trained on the

Figure

Similar to the case of randomly drawn train and test sets, the STM shows the lowest correlations, followed by the segmented logit-linear and spline regressions, as shown by the box plot in Fig.

Overall, assuming that separating cold and warm years can be a proxy of climate change to assess statistical model performance, the two regressions perform very similarly to the general case in terms of forecasting error, without any visible improvement or decrease in accuracy. As expected, the STM has the worst performance, while the sigmoid fit could present advantages over some areas but more pronounced convergence problems where DJF precipitation mainly falls as snow. However, we observe an improvement in the correlation between predicted and true forecasting values. We argue that this effect is likely due to precipitation patterns in years characterized by extreme temperatures in the historical period, and it should not be expected to happen under future climate change.

We now assess the performance of the considered methods on the output of the IPSL

Difference in meters between the statistical approximation and reanalysis average of DJF snowfall for the years 1979–2005. The IPSL

Figure

It is also worth noting the different spatial distribution of the bias sign, depending on the method. The two versions of the sigmoid fit are the only methods that avoid the positive bias patch over Norway; however, the unconstrained sigmoid is characterized by larger areas of negative bias between Scandinavia and Russia, while the constrained version produces more positively biased values over central Europe.

To assess the overall performance of the models, we compute the mean absolute bias as the spatial mean of the absolute differences between modeled and ERA5 DJF snowfall differences. The cubic splines produce the smallest bias (0.008 m), followed by segmented regression (0.0096 m), constrained sigmoid (0.0118 m), STM (0.0122 m), unconstrained sigmoid (0.0136 m), and non-bias-corrected IPSL

Performance of the statistical methods as bias correction methods of snowfall in terms of statistical divergences. Divergence measures box plots for statistical methods and raw IPSL

We evaluate the performance of the STM, cubic spline regression, and the two versions of the sigmoid fit and compare it to the values produced by the IPSL

The results are summarized in Fig.

The spline regression appears to be the best method in terms of both median and variability, as it consistently displays the smallest interquartile range (IQR) for all the three measures. However, while

The results discussed so far show that the IPSL

Maps of average 1979–2005 DJF snow differences over the Alps region between raw IPSL

The IPSL

Overall, the median bias of the STM and spline regression remains slightly positive, but the IQR of the differences with respect to ERA5 is reduced by around an order of magnitude, as evident from Fig.

Since comparing IPSL

The upper left quadrant of Table

In case of an ideal bias correction method, differences between the reference dataset and corrected model output would be a sequence of uncorrelated zero-mean Gaussian random variables. This implies a linear relationship with zero (nonsignificant) intercept, a significant slope close to 1 (i.e., with 1 included in the confidence interval), and an

Maps of average 1979–2005 DJF snow differences over Norway between raw IPSL

We repeat the same analysis for Norway, another area where DJF snowfall constitutes a large proportion of total DJF precipitation and the IPSL

As shown in Table

Maps of average 1979–2005 DJF snow differences over France between raw IPSL

Maps of average 1979–2005 DJF snow differences over Germany between raw IPSL

Both Norway and the Alps regions are characterized by mountain ranges subject to orographic forcing of moist air masses and by low DJF temperatures due to elevation (especially for the Alps) and latitude (Norway). Moreover, all methods show the lowest correlation between observed and reconstructed snowfall in the validation phase over these areas (see Fig.

Comparison between cumulative distribution function of daily snowfall in ERA5 reanalysis (red lines), raw IPSL

As already mentioned, bias adjustment usually aims at correcting as much of the distribution of the observable as possible. To assess if and how well our methods work in this sense, we compare the empirical cumulative distribution function (ECDF) of daily snowfall, once again considering each region as homogeneous. Figure

In summary, we have shown that combining a breakpoint search algorithm and a flexible regression method – be it a segmented logit-linear or a cubic spline regression – allows for a reliable snowfall reconstruction in climate simulations. This method proves to be effective both in correcting large mean biases and in preserving the shape of the entire probability distribution of (daily) snowfall rather than only long run totals. This result is crucial for studying the characteristics of future snowfall in a wide range of environments, encompassing regions characterized by frequent and abundant snowfall in cold climates and temperate areas where occasional snowstorms and heavy wet-snow events can cause serious loss and damage.

Summary statistics of the linear relationship between reconstructed and reanalysis snowfall over the Alps, Norway, France, and Germany.

We have presented four statistical methods to estimate the snowfall fraction of total DJF precipitation over Europe, provided that a reliable measure of near-surface temperature is available. This is a relevant problem in both hydrology and climatology, since an accurate estimation of snowfall is challenging in case of both observed or simulated precipitation.

In case of observational data, especially over large areas where a single weather station is not representative, snowfall is often unobserved due to difficulties in making its measurement an automated procedure. On the other hand, climate model outputs often include snowfall, but this is affected by a bias arising from the physical and mathematical approximations contained in the model scheme. For other variables such as temperature and total precipitation, we can rely on well-established and relatively simple univariate bias correction methods that can be applied pointwise in the case of gridded data. However, snowfall presents more challenges, since not only the magnitude but also the number of events is biased as a consequence of biases in the temperature. Thus, its correction would require conditioning on temperature and precipitation, and possibly including a stochastic generator of snowfall events to correct snowfall frequency, other than snowfall magnitude. Therefore, the availability of simple methods to reconstruct the snow fraction of total precipitation is a great advantage in contexts where much more complex and computationally costly procedures should be created and applied in order to obtain an accurate snowfall measurement.

The techniques applied in the existing literature mainly consist of a binary representation based on a threshold temperature, linear interpolations between two thresholds, and a binary representation outside of the inter-threshold interval or fitting parametric S-shaped functions with nonlinear least squares. The simple binary description is effective in its simplicity when the researcher is interested in particular in extreme events or long run total climatologies, but it cannot provide a reliable reproduction of the entire snowfall distribution. Moreover, the thresholds in the first two methods are often established simply by visual inspection of the plot of snow fraction against temperature, or analyzing the entire available dataset at once. However, in the case of gridded data over large areas, the optimal threshold values may vary, depending on the location.

The first considered method consists of a binary partition (STM), testing two possible thresholds at 1 and

The second method is a segmented linear regression on the logit of the snowfall fraction, informed about the location-specific optimal number of thresholds (between 0 and 2) via a breakpoint search algorithm.

As a more flexible alternative to segmented logit-linear regression, we introduced a nonlinear regression based on cubic splines, where the spline knots are taken as the deciles of the location-specific temperature distribution. This allows us to construct a flexible statistical model without requiring computationally intensive additional step such as the breakpoint search in the case of the segmented logit-linear regression.

Finally, we adopt a parametric nonlinear statistical model, in which the link between near-surface temperature and snow fraction is given by a hyperbolic tangent function, whose parameters are estimated via nonlinear least squares.

We used ERA5 reanalysis over Europe for the period 1979–2005 for validation, by estimating each statistical model at each grid point over a training set and comparing the performances in terms of prediction of out-of-sample values. In this validation phase, the STM provides much less accurate prediction compared to the more complex methods. The results obtained with the two regression models are very similar; however, the longer, computationally intense, and more complex procedure required to inform the segmented logit-linear regression makes it less advantageous compared to the spline regression. Finally, the constrained sigmoid fit produces results comparable overall to the spline regression over most grid points. However, it seems to be slightly less flexible than its competitors, as the fit can fail to converge over areas where DJF precipitation mainly falls as snow, such as Scandinavia and the Alps, or be negatively affected by outliers, as seen in the transition curve for Oslo. Based on our result, we conclude that this method could be superior to others for studies conducted over limited areas characterized by very smooth snow transitions. However, it does not seem to be adequate for studies over very large domains or when using already validated and published datasets that could still contain some noisy data, as is the case for snowfall in ERA5.

Results hold when using the 25 % coldest and warmest years as training and test sets, respectively. This observation is encouraging in view of the application of the analyzed techniques to study snowfall under different climate projection scenarios, provided that the performance can be reproduced when considering climate simulations instead of reanalysis.

To tackle this question, we consider the historical period of 1979–2005 in the IPSL

We validate our results by both considering the entire domain and specific regions. We find that, in general, the reconstructed snowfall improves remarkably in terms of long run statistics and similarly between probability distributions with all methods, proving that they can be used in place of more complex multivariate bias correction schemes. However, it is clear that the best method depends on both the geographical area and the objective. For example, over the Alps, the best performance is given by the spline regression both in terms of average DJF snowfall bias and distribution of daily snowfall events but without dramatic differences among the three methods; over Norway, another region characterized by large DJF snowfall, the STM performs discernibly worse than the other two methods in terms of average snowfall, but it produces the best bias correction of the daily snowfall distribution, even though with no dramatic differences compared to the spline regression and sigmoid fit. For areas such as France and Germany, the STM is competitive against more complex methods in terms of both snowfall averages and daily snowfall distribution.

Overall, among the tested methods, we discard the segmented logit-linear regression, which requires a time-consuming step to perform breakpoint analysis over the grid and without performing better than its competitors. We remark that, to reproduce average DJF snowfall, the simple STM can be applied with success; however, unlike what is reported in

We also clarify some of the limitations of our analysis. The nature of climate datasets makes multiple comparisons among methods and BC techniques very demanding in terms of data storage and computational time. For this reason, we limited our analysis to one reanalysis dataset (ERA5), one marginal bias correction technique (CDF-t), one climate projection model (IPSL

We do not consider the choice of ERA5 problematic with respect to other gridded datasets that could be observational (e.g., E-OBSv20) or other reanalysis (e.g., NCEP/NCAR). While the actual values could change between datasets, we do not foresee this directly affecting the performance of the methodology we presented in terms of improvement of the raw simulations with respect the chosen reference dataset.

On the other hand, the choice of the BC may influence the outcome of our statistical modeling procedure. The CDF-t is applied marginally to each variable, so that there is no guarantee that the inter-variable correlations are correctly reproduced in the bias-corrected climate model output. Indeed,

On the same note, we remark that prediction accuracy may vary across different climate models, due to the different physical approximations and parameterizations, which are likely to affect the relationship between near-surface temperature and precipitation. Due to these differences, even other RCMs from the EURO-CORDEX project may exhibit variability in the performance of the snowfall reconstruction. This holds true for all statistical methods cited in Sect.

We have presented four statistical methods to estimate the snowfall fraction of total precipitation, provided that a reliable measure of near-surface temperature is available. This is a relevant problem in both hydrology and climatology, since an accurate estimation of snowfall is challenging in the case of both observed or simulated precipitation.

There are two methods, namely the segmented logit-linear and the spline regressions, that are an extension of traditional precipitation-phase partitioning methods based on estimating the snowfall fraction of total precipitation on the base of one or multiple threshold temperatures. For the segmented logit-linear regression, we estimate the number of such thresholds by means of a breakpoint search algorithm, while splines do not require the specification of physically meaningful thresholds, and quantiles of the temperature can be assumed as knots. The other two methods, i.e., STM and sigmoid fit, are used in studies already present in the literature, even though not with the scope of eliminating biases in climate simulations.

The two regression models perform the best in terms of prediction error and correlation between real and reconstructed values in a train–test sets validation framework, based on the ERA5 reanalysis dataset. These two methods also show robustness with respect to possible non-stationarity, when choosing the 25 % coldest years as training set and the 25 % warmest years for testing. In this context, the sigmoid fit also produces good results, but it could fail in case the snow fraction transition curve is not well represented by an S-shaped function.

When applied to reconstructing snowfall in a regional circulation climate model, all techniques produce results with a markedly reduced bias respect to ERA5, when compared to raw climate model simulations.

We conclude that statistical methods based on spline regression and sigmoid fit, informed by bias-corrected temperature and precipitation, are capable of providing a reliable reconstruction of snowfall that can replace more complex bias correction techniques, with better performances than similar methods based on parametric assumptions or binary phase separation. For limited areas, and depending on the task, simpler single threshold methods can perform equally well and could be advantageous in case fast or computationally light procedures are needed.

In order to estimate the temperature thresholds, we rely on breakpoint analysis. This method was originally developed by

The method can be summarized as follows. Let us first consider the case of a univariate response variable

It is crucial to remark that, while the method was originally developed to detect structural breaks in time series,

In principle, imposing two threshold temperatures is not necessarily the best assumption for every point on the grid. In fact, the EURO-CORDEX domain clearly includes areas where, even if only considering DJF precipitation, snowfall is infrequent and usually happens at positive temperatures and areas (such as Scandinavia and continental Eastern Europe) where most of the winter precipitation is likely to fall as snow. For this reason, we admit

Once the number of thresholds and their values are obtained, we estimate a segmented logit-linear regression of the general form presented in Eq. (

In our case, the dependent variable is the snow fraction

Given

A statistical model of the type shown in Eqs. (

The number of threshold temperatures are found using the

A threshold-free model of the form Eq. (

If the number of breakpoints estimated in step (i.) are larger than 0, then the object containing the result from

The methodology presented thus far has a few drawbacks. The relationship between

However, if

Similarly, in our setting, the use of the logit transform requires the following two assumptions to enable us to state that statistical models, as in Eqs. (

The segmented logit-linear regression previously described presents some drawbacks. First, a piecewise linear function can only be a coarse approximation of the underlying nonlinear relationship. Second, segmented regression does not theoretically guarantee that the regression lines in different regimes match at the threshold points. In order to fit functions that are more flexible in shape and smooth at the threshold temperatures, we rely on splines. Spline regression can be thought of as an improvement of the traditional power transform and polynomial regression, where the regression splines are piecewise polynomials constrained to meet smoothly at the knots, i.e., the transition

Let us consider the general case of a problem with

In our case, we choose

For a more comprehensive overview on spline regression, see, e.g.,

The capability of the chosen method(s) to perform a BC-like task can be evaluated in terms of the similarity between the distribution of the reconstructed daily snowfall and the reanalysis values. We will use three measures of dissimilarity between the distributions, i.e., the Kolmogorov–Smirnov (KS) statistics, the Kullback–Leibler (KL) divergence, and the

While the KS statistic can be seen as a distance, since

In order to choose the best possible methodology based on quantitative considerations, we test for significant differences among groups using the rank test proposed by

For the purpose of detecting the stochastically dominant methods, we rely on post hoc testing using the pairwise Wilcoxon rank sum test

The following table shows the NUTS 3 codes of the provinces included in the definition of the Alps region analyzed in Sect.

NUTS 3 units used for the definition of the Alps region and divided by country.

Scripts and data files to recreate our analyses are available for direct download from Figshare (

DF conceived the study within the frame of the ANR project BOREAS, provided expertise on the topic of snowfall modeling in climate studies, and contributed to writing the article. FMEP collected the datasets, reviewed the existing literature, conceived the statistical methodology and the experimental design, wrote the R scripts to analyze the data, and contributed to writing the article.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors thank Saverio Ranciati, for the useful discussion.

This research has been supported by the ANR-TERC (grant no. BOREAS).

This paper was edited by Sarah Perkins-Kirkpatrick and reviewed by two anonymous referees.