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Natural Hazards

, Volume 94, Issue 3, pp 1367–1389 | Cite as

Modeling snow depth extremes in Austria

  • Harald SchellanderEmail author
  • Tobias Hell
Original Paper
  • 136 Downloads

Abstract

Maps of extreme snow depths are important for structural design and general risk assessment in mountainous countries like Austria. The smooth modeling approach is commonly accepted to provide more accurate margins than max-stable processes. In contrast, max-stable models allow for risk estimation due to explicitly available spatial extremal dependencies, in particular when anisotropy is accounted for. However, the difference in return levels is unclear, when modeled smoothly or with max-stable processes. The objective of this study is twofold: first, to investigate that question and to provide snow depth return level maps for Austria; and second, to investigate spatial dependencies of extreme snow depths in Austria in detail and to find a suitable model for risk estimation. Therefore, a model selection procedure was used to define a marginal model for the GEV parameters. This model was fitted to 210 snow depth series comprising a length of 42 years using the smooth model approach and different max-stable models allowing for anisotropy. Despite relatively clear advantages for the Extremal-t max-stable process based on two scores compared to the smooth model as well as the Brown–Resnick, Geometric Gaussian and Schlather processes, the difference in 100-year snow depth return levels is too small, to decide which approach works better. Spatial dependencies of snow depth extremes between the regions north and south of the Austrian Alps are almost independent. Dependencies are stronger in the south for small distances between station pairs up to 120 km and become stronger in the north for larger distances. For risk modeling the Austrian Alps could be separated into regions north and south of the Alps. Fitting an anisotropic Extremal-t max-stable process to either side of the Alps can improve modeling of joint exceedance probabilities compared to one single model for the whole of Austria, especially for small station distances.

Keywords

Extreme values Max-stable process Smooth model Snow depth Spatial modeling Risk estimate Spatial dependence 

1 Introduction

A spatial representation of extreme snow depth in the alpine region of Austria is of crucial importance for numerous purposes such as the planning and construction of buildings, for avalanche simulation (Rudolf-Miklau and Sauermoser 2011) or in general risk assessment. As foreseen in the European Standard (e.g., EN 1991-1-3 2003), buildings and particularly roofs have to withstand a snow load occurring with a maximum probability of 0.02, defining a return period of 50 years. Snow load is defined as the weight of a snow pack on a roof and is measured in terms of snow water equivalent. As there do not exist reasonable snow load measurements in Austria, expertises of snow load return levels rely on a more or less notional interpolation of snow depth to the location of interest followed by the multiplication with an estimated bulk snow density.

Extreme value theory builds the well-established foundation of modeling extremes (Coles 2001). In a univariate setting it has been widely used in geosciences (e.g., Palutikof et al. 1999; Naveau et al. 2005) and to a smaller extent with snow-related parameters (Bocchiola et al. 2006, 2008; Blanchet et al. 2009; Marty and Blanchet 2012). However, all those studies infer their spatial findings from univariate extreme value modeling, where the spatial dependency across different observation sites is not accounted for. A natural step forward and an intuitive way to bring local estimates of extremes into space would be a spatial interpolation. Unfortunately, this approach has some disadvantages, as, e.g., Blanchet and Lehning (2010) showed for extreme snow depths in Switzerland. Uncertainties may be hard to assess, and quantiles for more complex (joint) events cannot be mapped at all. Despite those constraints kriging variants were used to interpolate snow depth or snow load extremes for snow-related hazard mapping in Canada (Hong and Ye 2014) and China (Mo et al. 2016). As an improvement Blanchet and Lehning (2010) suggested a direct estimation of a spatially smooth generalized extreme value (GEV) distribution, called smooth spatial modeling. With smooth modeling the GEV parameters are modeled as smooth functions of spatial covariates. Spatially varying marginal distributions are achieved by maximizing the sum of the log-likelihood function over all stations. Compared to several interpolation methods, smooth modeling for swiss snow depth led to more accurate marginal distributions, especially in data sparse regions. The key feature of smooth modeling, permitting to approximate the likelihood as a sum of GEV likelihoods at the stations, is the simplifying assumption that annual snow depth maxima are approximately independent in space and time.

As a natural way to account for spatial dependence of extremes, max-stable processes as an extension of multivariate extreme value theory to infinite dimensions can be used (de Haan 1984). With max-stable processes, the margins and their spatial dependency can be modeled simultaneously but independently. In recent years an increasing number of studies have investigated the ability of different max-stable processes to model geophysical extreme values in a spatial context. Due to the unavailability of the multivariate density function, Padoan et al. (2010) developed a composite likelihood-based method for fitting max-stable processes to rainfall data in the Appalachian Mountains on the US east coast. Blanchet and Davison (2011) picked up that idea to model the margins and spatial dependence of extreme snowfall in Switzerland. They extended the max-stable representations of Smith (1990) and Schlather (2002) to explicitly account for the direction of the dependence and the dependence between different climatic regions. As the focus of their work laid on the spatial dependence of extremes rather than on the marginal distributions, they did not use full models for the three GEV parameters. This was done by Gaume et al. (2013), thus bridging the work of Blanchet and Lehning (2010) and Blanchet and Davison (2011). For extreme snowfall in France Gaume et al. (2013) showed that the Brown–Resnick max-stable representation (Brown and Resnick 1977; Kabluchko et al. 2009) provides a more flexible spatial dependence than the Schlather and Smith processes. The spatial dependence of extreme snowfall can strongly depend on the orientation of large valleys and mountain ranges. They showed, for instance, that the dependence range in the direction of large French alpine valleys is more than twice as high as along the orthogonal direction. To account for that anisotropy, a space transformation similar to that used by Blanchet and Davison (2011) was carried out. Due to a dense station network, rough topography and a high spatial variability in snowfall Gaume et al. (2013) not only used the topographical variables longitude, latitude and altitude, but also mean snow depth as a climatological covariate. Newer max-stable representations like Extremal-t (Opitz 2013) and Geometric Gaussian (Davison et al. 2012) were compared recently on the basis of a composite likelihood approach for modeling French extreme snowfall by Nicolet et al. (2015). By transferring their data into unit Fréchet margins, they focused mainly on the spatial dependence structures of the Smith, Schlather, Brown–Resnick, Extremal-t and Geometric Gaussian max-stable processes, also accounting for spatial anisotropy. They found that the dependence structures of the Smith and Schlather max-stable models were not flexible enough, whereas an anisotropic Extremal-t model represented the spatial structures best according to the composite likelihood information criterion (CLIC). Nevertheless, the differences to the Geometric Gaussian and Brown–Resnick models were negligible. In a recent study, Sebille et al. (2017) compared the Schlather, Brown–Resnick and Extremal-t max-stable processes with the latent variable model of Davison et al. (2012) and the Hierarchical Kernel Extreme Value Process of Reich and Shaby (2012) for their ability to spatially reproduce precipitation extremes in France. They recommend the Extremal-t process for modeling spatial extremes, when the interest lies in the estimation of the marginal distribution, and the Brown–Resnick process when joint probabilities are the main goal.

The aims of this study are twofold: a model for Austrian snow depth extremes is needed, that (1) is best suited for spatial modeling of return levels and (2) is appropriate for modeling spatial bivariate dependencies. To achieve the first goal, the smooth modeling approach is compared with Schlather, Brown–Resnick, Extremal-t and Geometric Gaussian max-stable processes. For the second goal, the best max-stable model from the previous step is used.

As extreme value theory builds the basis of modeling extremes, a short overview of the matter tailored to extreme snow depth is given at the beginning of Sect. 2. This is followed by the description of the smooth modeling approach and the different max-stable processes and their covariance functions. In Sect. 3, the methodology for achieving the two goals is described. In Sect. 4, the methods are applied to Austrian snow depth data, and the obtained snow depth return levels are investigated via a validation procedure. Also the performance of the different max-stable models for risk estimation is examined. Conclusions are drawn in Sect. 5.

2 Spatial statistics of extreme snow depth

In the planning of construction projects in Austria, incorporating estimates for extreme snow load is of crucial importance. Therefore, respective expertises have to be compiled on a regular basis. Spatially consistent models for extreme snow depth can be used to approximate the expected snow load. For instance, as determined in the corresponding norms (EN 1991-1-3 2003; ÖNORM B 1991-1-3:2006 2006), a roof has to withstand a 50-year return level of maximum annual snow load. Consequently, respective measures such as quantiles and especially return levels for the maximum snow depth have to be estimated.

Extreme value theory is the main framework for modeling the tail behavior of a probability distribution. For this study, observed daily snow depths are required to be independent, which is of course not the case, due to a strong temporal dependence. However, after Leadbetter et al. (1983), extreme value theory can still be applied, if daily observations are only near-independent (\(D(u_n)\) condition), provided the block size is large enough. As Blanchet and Lehning (2010) already showed that both mentioned conditions of short-range time dependence and large block size holds for Swiss snow depths, the same is assumed for Austria.

Assume that \(S_x\) is the random variable describing the maximum snow depth within a year at a location x in Austria. Then \(S_x\) approximately follows a generalized extreme value (GEV) distribution, see, for instance, Coles (2001), which we denote by \({{\rm GEV}}(\mu _x,\sigma _x,\xi _x)\). Therefore, \(\mu _x\in {\mathbb {R}}\) denotes the so-called location parameter, \(\sigma _x>0\) the scale parameter and \(\xi _x\in {\mathbb {R}}\) the shape parameter. The corresponding distribution function is given by
$$\begin{aligned} F(s;\mu _x,\sigma _x,\xi _x) = \exp \left( -\left( 1+\xi _x\cdot \tfrac{s-\mu _x}{\sigma _x}\right) ^{-1/\xi _x}_+\right) \quad {\text{for }} \,s\in {\mathbb {R}}, \end{aligned}$$
(1)
i.e.,  \(\mathbb {P}(S_x\le s) = F(s;\mu _x,\sigma _x,\xi _x)\) for all \(s\in {\mathbb {R}}\), and \(y_+ = \hbox{max}(y,0)\) for \(y\in {\mathbb {R}}\).

Maximum snow depth is given as an annual time series of length \(N\in {\mathbb {N}}\) only at finitely many locations, namely at \(K\in {\mathbb {N}}\) stations with coordinates \(x_1,\ldots ,x_K\). By standard techniques, the GEV parameters can be estimated at a station. However, one is usually interested in the law of the maximum annual snow depth \(S_x\) at a location x where there is no station. One approach is to interpolate the point estimations for the GEV parameters computed at the stations which generally leads to unsatisfying results (Blanchet and Lehning 2010).

2.1 Smooth modeling

The maximum annual snow depth can be interpreted as a time-space stochastic process \(\{S_{x}^{(t)}\}\), where t denotes the corresponding year and \(x\in {\mathscr {A}}\) the location in Austria. However, we assume that the distribution of \(S_x^{(t)}\) does not depend on the time t and therefore for each of the GEV parameters, we consider a linear model, i.e., a model of the form
$$\begin{aligned} \eta (x) = \alpha _0 + \sum _{k=1}^m \alpha _k y_k(x) + f\left( y_{m+1}(x),\ldots ,y_{n}(x)\right) \quad \end{aligned}$$
(2)
at location x, where \(\eta\) denotes one of the GEV parameters, \(y_1,\ldots ,y_n\) are the considered covariates as functions of the location, \(\alpha _0,\ldots ,\alpha _m\in {\mathbb {R}}\) are the coefficients of the linear part and f is a P-spline with a certain number of knots, evenly distributed across the spatial domain. For \(k=1,\ldots ,K\) the kth station is given by the location \(x_k\), and therefore, we have a realization \(s_{x_k}^{(1)},\ldots ,s_{x_k}^{(N)}\) of the random sample \(S_{x_k}^{(1)},\ldots ,S_{x_k}^{(N)}\) given as measurements. Note that \(S_{x_k}\sim {\rm GEV}(\mu _{x_k},\sigma _{x_k},\xi _{x_k})\) and \(\mu (x_k),\sigma (x_k),\xi (x_k)\) are the GEV parameters given by the linear models in (2). By \(\ell _k\left( \mu (x_k),\sigma (x_k),\xi (x_k)\right)\) we denote the log-likelihood function at the kth station corresponding to (2). With the assumption of spatially independent stations, the log-likelihood function then reads as
$$\begin{aligned} l = \sum _{k=1}^K \ell _k\left( \mu (x_k),\sigma (x_k),\xi (x_k)\right) , \end{aligned}$$
where l only depends on the coefficients of the linear models for the GEV parameters, cf. (2). As in Blanchet and Lehning (2010), we call this approach smooth modeling.

The advantage of maximizing the sum of the log-likelihood functions at the stations compared to maximizing the log-likelihood function at each station lies in the following fact: A good fit at a single station leading to worse fits at several other stations will be penalized. As a consequence, the stations become intertwined in terms of the fitting. As the smooth model does not provide any spatial dependence, it is generally assumed to be less suited to spatially model extremes, compared to other approaches as fitting a max-stable process. Max-stable processes and their ability to account for spatial dependencies by choosing an appropriate covariate function are described in the following section.

2.2 Max-stable processes

We are interested in the distribution of the stochastic process \(\{S_x\}_{x\in {\mathscr {A}}}\), where \({\mathscr {A}}\) denotes the set of all locations in Austria and \(S_x\) is the maximum annual snow depth at location \(x\in {\mathscr {A}}\). As we want to investigate the maximum snow depth in \(n\in {\mathbb {N}}\) consecutive years, we consider a random sample \(S_x^{(1)},\ldots ,S_x^{(n)}\) for \(S_x\). Then the process \(\{S_x\}_{x\in {\mathscr {A}}}\) is called max-stable, if for all \(n\in {\mathbb {N}}\) there exist continuous functions \(a_n:{\mathscr {A}}\rightarrow (0,\infty )\) and \(b_n:{\mathscr {A}}\rightarrow {\mathbb {R}}\) such that
$$\begin{aligned} \max _{i=1,\ldots ,n}\frac{ S_x^{(i)}-b_n(x)}{a_n(x)} \sim S_x \end{aligned}$$
for all \(x\in {\mathscr {A}}\). From this stability property of the maxima in terms of distributions the benefit for spatial modeling of extremes is derived. In fact, for a max-stable process \(\{S_x\}_{x\in {\mathscr {A}}}\) one can show that \(S_x\) follows a GEV distribution for all \(x\in {\mathscr {A}}\).

There exists a variety of max-stable processes in the literature (see, e.g., Smith 1990; Schlather 2002; Opitz 2013; Davison et al. 2012; Kabluchko et al. 2009; Xu and Genton 2016). Most of them have already been used for spatial modeling of extreme snow depth (Blanchet and Davison 2011) or snowfall (Gaume et al. 2013; Nicolet et al. 2015). One drawback of max-stable processes is that they are usually fitted by minimizing the respective sum of pairwise log-likelihood functions, as the full likelihood for high-dimensional data is excessively difficult to compute (Padoan et al. 2010). More efficient, higher-order composite likelihood approaches have been developed in recent years by Genton et al. (2011), Huser and Davison (2013) and Castruccio et al. (2016). Very recently Dombry et al. (2017) showed how to perform full likelihood inference for max-stable processes. However, as the latter expect to handle up dimensions of only 50–100 for Brown–Resnick-like models in a reasonable amount of time, we stick to the conventional pairwise likelihood for this study.

Provided that extremes exhibit an anisotropy, i.e., a directional dependence, this would be valuable to account for in the spatial model. The max-stable process of Smith (1990) is the only one naturally accounting for anisotropy. Using a space transformation Blanchet and Davison (2011), Gaume et al. (2013), Nicolet et al. (2015) and Blanchet and Creutin (2017) have incorporated anisotropy in their spatial models for snow depth, snowfall and precipitation extremes in the Alps.

A common measure of the spatial dependence between the maximum annual snow depths \(S_x\) and \(S_y\) at two given locations x and y is the so-called extremal coefficient (Schlather and Tawn 2003). For the sake of simplicity, assume in the following that \(\{S_x\}\) is a stationary max-stable process with unit Fréchet margins \(S_x^{\rm frech}\), which can easily be transformed from GEV margins \(S_x^{\rm gev}\) at location x with the following transformation:
$$\begin{aligned} S_x^{\rm frech} = \left( 1 + \xi _x \cdot \left( S_x^{\rm gev} - \mu _x \right) / \sigma _x \right) _+^{1/\xi _x} \quad {\text {for}} \,x\in {\mathscr {A}}. \end{aligned}$$
Then for two locations \(x,y\in {\mathscr {A}}\), the extremal coefficient \(\theta (x,y)\) is implicitly given by
$$\begin{aligned} \mathbb {P}\left( S_x\le s,S_y\le s\right) = \exp \left( -\frac{\theta (x,y)}{s}\right) \quad {\text {for }}\, s>0 \end{aligned}$$
and therefore describes the probability that both annual maxima \(S_x\) and \(S_y\) do not exceed the threshold s. Note that \(\theta (x,y)\in [1,2]\) holds and that \(\theta (x,y)=1\) corresponds to complete dependence of \(S_x\) and \(S_y\) whereas \(\theta (x,y)=2\) complies with independence. Four estimators for the extremal coefficient have been compared by Bel et al. (2008). The madogram-based estimators emerged as slightly better than a nonparametric and a maximum-likelihood estimator, which was used to describe the extremal dependence of extreme snowfall in Gaume et al. (2013) and Nicolet et al. (2015). In this study the madogram-based estimator of Cooley et al. (2006) was used to assess empirical extremal coefficients.

For Smith, Schlather, Brown–Resnick, Extremal-t and Geometric Gaussian processes, the extremal coefficient can be calculated explicitly and depends only on the distance \(|h |= ||x-y ||\) between two locations x and y (see, e.g., Nicolet et al. 2016, for a summary). For the Schlather process also a correlation function (e.g., Whittle–Matèrn, Cauchy, Powered Exponential, Bessel and Generalized Cauchy correlation families) has been chosen. It is worth noting that Schlather’s extremal coefficient is restricted to a maximum of 1.7 for large distances \(|h |\rightarrow \infty\). This could be an advantage for parameters with strong dependence at large distances, like, e.g., snow depth or temperature.

3 Methodology

This section exposes the general workflow used for achieving the two goals of this study, which are (1) finding the best model for return levels of Austrian snow depths by comparing a smooth model with max-stable processes, and (2) finding a suitable model for risk estimation.

For the first goal, at the beginning the smooth GEV model was selected. Therefore, the GEV parameters were examined at a number of fitting stations, to see which covariates could be used for fitting. Using an AIC (Akaike information criterion, Akaike (1974))-based model selection procedure the most suitable models for the GEV parameters were fixed. Secondly, an anisotropic max-stable model was selected for comparison with the smooth model. For that purpose, the already fixed parameter models of the smooth model were used to fit Schlather, Brown–Resnick, Extremal-t and Geometric Gaussian max-stable processes to the same fitting data. To account for anisotropy, spatial dependencies of snow depth extremes were probed, to get an idea about the main direction and strength of spatial dependencies. Then the four max-stable models with fixed marginal models were fitted using different space transformations, spanning the ranges and directions of the anisotropy observed in the previous step. From all fitted max-stable models with different space transformations the one with the smallest TIC (Takeuchi information criterion, Takeuchi 1976) value was chosen. The smooth GEV model and the best anisotropic max-stable model were then compared, to find the best model for estimating snow depth return levels.

To fulfill the second goal, the best anisotropic max-stable model of the last step was used for an examination of spatial dependencies of Austrian snow depth extremes.

4 Results

At first the dataset used in this study is introduced. Then, a smooth and a max-stable model are selected, as outlined in Sect. 3. Return levels computed with the best smooth and the best max-stable models are compared in Paragraph 4.4. Finally, spatial dependencies of Austrian snow depth extremes are discussed in detail in Paragraph 4.5, to find a suitable model for risk estimation of snow depths extremes in Austria.

4.1 Data

Daily snow depth measurements of 421 stations from meteorological networks of the National Weather Service and the Hydrological Service of Austria (ZAMG and HD, respectively) were used in this study. All measurements have been quality controlled by the owning institutions (e.g., Lipa and Jurkovic 2012). Although most of the available stations offer time series way longer than 50 years, the introduced methods are only stable in terms of convergence, when time series at all used stations are of equal length and span the same period of time. This is also physically reasonable, because otherwise a model would be calibrated with data from different periods of time. In that case the model would collect the distribution of extremes, for instance, from the 1920s in one region and from the 1990s in another region. In order to maximize the length of data series and the number of stations simultaneously, while not violating the constraint mentioned above, a cut-set of available stations with same series length encompassing the same years was calculated. Sensitivity tests were performed to find a reasonable series length while trying not to neglect too many stations, and to keep a minimum horizontal radial distance between neighboring sites. In that way 421 measurement sites sharing the length of 42 winter seasons without gaps from 1970 to 2012 have been defined as dataset. Although it was not investigated in detail, it was assumed that the data series do not exhibit a temporal trend, in which case the assumption mentioned in Sect. 2 holds. The stations cover the whole domain of Austria more or less homogeneously as well as a height range between 118 and 2045 m above sea level, most of them laying in flatlands or valleys below 1000 m. Only six observation sites are situated higher than 1500 m. 210 out of 421 stations have been randomly chosen as validation stations, by taking into account a reasonable representation of climate regions and altitude coverage. The remaining 211 sites have been used for model calibration. The highest station used for fitting is situated at 2045 m, and the highest station used for validation at 1620 m (Fig. 1).
Fig. 1

Topographical map of Austria. Colors represent elevation. White circles denote 211 stations used for model fitting, and gray squares show 210 validation stations. The black dashed line represents the main alpine crest, separating the northern and southern side of the Alps

4.2 Selection of the smooth model

4.2.1 GEV parameters estimated at the stations

In order to estimate extreme snow depths at an arbitrary location x, the three GEV parameters \(\mu _x\), \(\sigma _x\) and \(\xi _x\) determining the underlying GEV distribution have to be modeled as functions of the position given by longitude and latitude. Moreover, the dependence of the GEV parameters on altitude is of particular interest. As in Blanchet and Lehning (2010), mean snow depth \(\overline{HS_x}\) at a location x (\({\overline{HS}}\) henceforth) was also used as covariate. At the stations, the GEV parameters can be approximated by point estimation, i.e., by standard maximum-likelihood fitting. Note that only 42 measurements at each station are available leading to a non reliable estimation of the GEV parameters. Hence in Sect. 4 we use a modified Anderson–Darling Score in addition to a normalized root-mean-squared error for validation. However, the point estimates reveal certain trends. Plotting the estimated GEV parameters against the corresponding covariates longitude, latitude, elevation and mean snow depth \({\overline{HS}}\) gives an impression of their relation at the measurement sites (Fig. 2). The location and the scale parameter increase almost linearly with height and \({\overline{HS}}\) as they describe the center and the variation of the corresponding GEV distribution, respectively. This is in accordance to the findings in Blanchet et al. (2009) and Gaume et al. (2013) for data from the Swiss and French Alps. The shape parameter as a function of altitude slightly decreases below zero at around 1000 m. As a result, higher stations feature a negative shape parameter. Similar observations were made in Blanchet et al. (2009) and Blanchet and Lehning (2010) reflecting the physical nature of extreme snow depth: A negative shape parameter corresponds to a bounded distribution, i.e., a small increase in snow depth may lead to a large increase of the return level. Consequently, at higher situated stations with a snow cover most of the year outlying extreme snow depths are not very likely to occur.
Fig. 2

Location parameter \(\mu\), scale parameter \(\sigma\) and shape parameter \(\xi\) (from top to bottom) estimated locally at the model fitting stations, plotted against longitude, latitude, elevation and mean snow depth \({\overline{HS}}\) (from left to right). A significant positive trend in elevation and \({\overline{HS}}\) can be observed for \(\mu\) and \(\sigma\), a less distinct negative trend for \(\xi\)

The location and scale parameters decrease with longitude. This reflects the fact that the mean annual maximum snow depth is usually larger in western Austria, because the GEV expectation is largely correlated with the location parameter. This may be explained by the significant difference in altitude between western and eastern Austria (Fig. 1). In addition, higher values of the location and scale parameters at stations more to the south are observed. Consequently, extreme snow depths appear with greater variation in the mountainous areas of Austria which might be due to higher absolute snow depths and the complex terrain.

Whether considered as a function of longitude, latitude, elevation or mean snow depth, the shape parameter estimated at the model fitting stations wildly spreads between − 0.5 and 0.4 with a mean of approximately − 0.05. This arises the question whether the shape parameter significantly depends on the geographical position at all? Ribatet (2013) and Padoan et al. (2010) suggest to keep the shape parameter constant. In contrast, Blanchet and Lehning (2010) give the opposite answer in the context of snow depth extremes in Switzerland. Linear regression between the shape parameter and the predictors longitude, latitude, altitude and \({\overline{HS}}\) at Austrian stations reveals a weak dependency to a significance level of 0.05. The weaker dependence of the shape parameter on altitude compared to the study of Blanchet and Lehning (2010) might at least partly be due to the smaller number of stations above 1500 m used in this study.

Using longitude, latitude, altitude and mean snow depth as regressors, a model selection was performed via AIC. The best marginal models are then
$$\begin{aligned} \mu&\sim \hbox {altitude} + {\overline{HS}} + \hbox {Spline(longitude,latitude)}\nonumber \\ \sigma&\sim \hbox {latitude} + \hbox {altitude} + {\overline{HS}}\nonumber \\ \xi&\sim \hbox {longitude} + \hbox {latitude} + {\overline{HS}} \end{aligned}$$
(3)
The models nicely reflect the above findings. Obviously all parameters benefit from height dependent covariates, which are altitude itself or mean snow depth \({\overline{HS}}\). A cubic P-spline for longitude and latitude appears in the model for the location parameter. The marginal models (3) were fitted to the fitting data described in Sect. 4.1. Some sensitivity tests have been performed with an increasing number of knots, but the model with 6 knots achieved the smallest TIC. In addition, as Gaume et al. (2013) already experienced, the fitting became impossible with more than 8 knots. Fitting of the smooth model was performed in R (R Development Core Team 2008).

4.3 Selection of the max-stable model

To see whether anisotropy should be taken into account for the max-stable models, at first spatial dependencies of Austrian snow depth extremes are shortly probed. Figure 3 shows pairwise empirical dependencies, calculated by applying the model-independent F-madogram of Cooley et al. (2006) to the stations used for calibration. Blue lines depict strong dependencies with \(\theta <1.4\), occurring mainly in two regions, north and south of the main alpine crest, which is sketched with a dashed line. In both regions strong dependencies have a pronounced direction along the Alps between \(0^{\circ }\) and \(40^{\circ }\) (visual inspection), i.e., roughly from west to northeast. Moreover, some station pairs north of the main alpine barrier exhibit strong dependence (\(\theta <1.4\)) of their snow depth extremes over distances larger than 260 km (red lines in Fig. 3). No such dependencies can be spotted in the south. Figure 3 allows two conclusions: (1) Anisotropy of the extremal dependence should be taken into account. This has already been shown to be relevant for an alpine country in Blanchet and Davison (2011), Gaume et al. (2013) and Nicolet et al. (2015) for snowfall, and in Sebille et al. (2017) and Blanchet and Creutin (2017) for rainfall. (2) Considering one single model for the whole of Austria might wrongly estimate the probability of joint exceedances in the two regions. (This is discussed in more detail in Sect. 4.5.)
Fig. 3

Dependence of snow depth extremes between pairs of fitting stations. Blue lines connect stations with \(\theta <1.4\), and red lines show the same dependence over distances above 260 km. The dashed line follows the main alpine crest and denotes the border between the northern and the southern side of the Alps

To account for anisotropy, the max-stable processes of Schlather, Brown–Resnick, Extremal-t and Geometric Gaussian were fitted with the fixed parameter models (3) and different space transformations (4). Anisotropy for the max-stable models was incorporated via a space transformation, which transforms a location \((x_1,x_2)^T\) in Austria to \(({x'}_1,{x'}_2)^T\) via
$$\begin{aligned} \left( {\begin{array}{l} {x'}_1\\ {x'}_2\\ \end{array} } \right) = \left( {\begin{array}{ll} \cos (\varphi ) &{} -\sin (\varphi ) \\ r\sin (\varphi ) &{} r\cos (\varphi ) \\ \end{array} } \right) \left( {\begin{array}{l} x_1\\ x_2\\ \end{array} } \right) \quad \hbox {for } \varphi \in [0,2\pi ], r>0 \end{aligned}$$
(4)
with range r and directional angle \(\varphi\). The range of directions and distances for different space transformations was motivated from visual inspection of Fig. 3. The standard space was incrementally transformed for \(\varphi\) varying from \(0.5^{\circ }\) to \(40^{\circ }\) and for ranges r from 70 to 400 km (note that the zonal elongation of Austria is about 750 km). All four max-stable models with the fixed marginal parameter models were fitted with the different space transformations. For the Schlather max-stable model the Powered exponential correlation function was chosen, as other available choices made no noticeable difference. As for the smooth model, also for the max-stable models different numbers of spline knots were taken into account. The best max-stable model was then chosen by the smallest TIC value. Max-stable models were fitted in R (R Development Core Team 2008) with the package SpatialExtremes (http://spatialextremes.r-forge.r-project.org). Table 1 shows the best max-stable models after fitting the fixed models (3) with different space transformations.
Table 1

Smallest TIC values and corresponding range and direction at which the TIC values were smallest for the four max-stable processes in ascending order

Max-stable process

Range r (km)

\(\varphi \,(^{\circ })\)

TIC

Extremal-t

315

1.5

16,199,477

Geometric Gaussian

315

1.5

16,200,763

Brown–Resnick

315

1.5

16,202,329

Schlather

315

1.5

16,225,414

Lower scores are better

All models exhibited the smallest TIC value with 5 spline knots for longitude and latitude at a range of 315 km and a directional angle of \(1.5^{\circ }\). The Extremal-t max-stable process remains as the best max-stable model. The small angle and the large range nicely fit the expectations gained at the beginning of this Section. The Geometric Gaussian model ranked second, closely followed by the Brown–Resnick and far behind the Schlather processes. The ranking coincides with results of Gaume et al. (2013) and Nicolet et al. (2015) for French snowfall or Sebille et al. (2017) for rainfall in France.

4.4 Return levels

After the model selection procedure in Sects. 4.2 and 4.3, the smooth GEV model and the anisotropic Extremal-t model are fixed. Both models use the parameter models (3). The smooth model achieved the smallest TIC with 6, the Extremal-t model with 5 knots for the P-spline in the location model. In addition, the anisotropic Extremal-t model uses a space which is transformed by the transformation (4) with range 315 km and direction \(1.5^{\circ }\). In this section, the two models are compared on the basis of return level estimates at the validation sites. To our knowledge, return levels gained with the smooth model were never compared against return levels from a max-stable model.

To assess the accuracy of the smooth and the Extremal-t models they are compared on the basis of return level estimates. The normalized root-mean-squared error (NRMSE) as given in (5) adapted from Gaume et al. (2013), is directly calculated from the resulting return levels. As noted in Sect. 4.2.1, the validation was not only based on point estimations, but also on a test statistic being sensitive to deviations between the fitted and the true distribution in the upper tail. For that purpose the modified Anderson–Darling test (mAD) as shown in (6) taken from Sinclair et al. (1990) as mean over all validation stations was applied. It seems more appropriate to interpret the mAD as an error measure in this context. We believe, that providing p values, which are all less than 0.001 due to the comparably large sample size, would be misleading and does not generate more insight.
$$\begin{aligned} \hbox {NRMSE} = \sqrt{ \sum _{T=2}^N \sum _{k=1}^K \left[ s_T({\mathbf {x}}_k)-S_{T,k}\right] ^2 / \left[ S_{{\rm max},k}-S_{{\rm min},k} \right] ^2 / (N-1) / K } \end{aligned}$$
(5)
where \(s_T({\mathbf {x}}_k)\) is the modeled T-year return level at the kth station, \(S_{T,k}\) is the same quantity estimated locally from the GEV distribution at station k, \(K=210\) is the number of validation stations and \(N=100\) is the number of return periods used for validation. Moreover, \(S_{{\rm min}}\) and \(S_{{\rm max}}\) are the minima and maxima of all return levels at the kth station.
$$\begin{aligned} \hbox{mAD} = \frac{1}{K} \sum _{k=1}^K \left( \frac{n}{2} - 2 \sum _{i=1}^n F(x_{i,k}) - \sum _{i=1}^n \left\{ 2 - \frac{2i - 1}{n} \right\} \log \left\{ 1 - F(x_{i,k}) \right\} \right) \end{aligned}$$
(6)
with \(K=210\) is the number of validation stations, \(n=42\) is the number of years at the validation stations and F is the cumulative distribution function of the kth station.
Table 2 gives a summary of the achieved scores. While the Extremal-t model is clearly superior to the smooth model regarding the mAD, it is slightly inferior when considering the NRMSE. As the mAD considers the whole distribution and gives more weight to the tail, the Extremal-t model seems to perform better.
Table 2

Normalized root-mean-squared error NRMSE and modified Anderson–Darling mAD calculated from return levels at the validation stations

 

mAD

NRMSE

Extremal-t

0.67

0.23

Smooth model

1.53

0.20

Lower scores are better

The marginal models (3) can directly be used to compute return levels on a grid. A map of return levels for a return period of 100 years computed with the Extremal-t model is shown in Fig. 4a. The necessary covariates longitude, latitude, altitude and mean snow depth were taken from the SNOWGRID climate analysis (Olefs et al. 2013) with yearly mean snow depth from 1961 to 2016. The grid features a horizontal resolution of 1 km. Note that some minor pixels on that latter grid exhibit unrealistically large snow depth values, which arise from a problem with lateral snow redistribution at high altitudes.

Overall, the return level map is in very good agreement with a climatology for Tyrol and its surroundings (compare western part of Fig. 4a, see http://www.3pclim.eu/tpclim/klima.php), where return levels for a return period of 100 years were estimated with an alternative interpolation method (see Hiebl et al. 2011). The return values look similar to a topographical map as valleys and mountain ridges are clearly featured. This is of course no surprise as the location parameter was modeled as a function of altitude and the altitude-dependent mean snow depth. The largest return levels (red colors in Fig. 4a) appear according to two climatological features. First, the highest values can be found where the highest mountains in Austria are located, namely along the main alpine crest (compare Fig. 1). A value of 1334 cm was modeled in the Hohen Tauern near the highest mountain in Austria (Großglockner, 3797 m). This seems reasonable as at station Sonnblick (3109 m) near the Großglockner a snow depth of 12 m has already been measured in late spring 1944. Note that station Sonnblick itself was not used for model fitting due to some gaps in the time series. Only 0.4% of all pixels show snow depth extremes larger than 750 cm. Values between 500 and 750 cm can also be found in the high glacier basins of the Ötztal and Zillertal mountains along the Italian border in the west of Austria (see annotation Wildspitze in Fig. 4a). Second, high return levels can also be expected in regions prone to North-Stau and heavy snowfall. In fact, this is visible along the northern Alps, with peak values between 500 and 750 cm in the Bregenzerwald and Arlberg region, the area north of Innsbruck and south of Salzburg and the Dachstein mountains (from west to east). In addition to the reasonable performance at higher altitudes, snow depth extremes for lower situated areas in Austria seem to be well estimated. This can be seen by comparing the Austrian flatland in the north, east and south (clockwise around the Alps) with 100-year snow depths estimated from observations at the validation stations (colored circles in Fig. 4a). Only few stations show values which are one class larger or smaller than the modeled snow depths. Despite the difference in the mAD score, the smooth model shows very similar 100-year snow depth return levels (Fig. 4b), albeit they are generally a little smaller (only 0.2% of all pixels have values larger than 750 cm). The differences of the two models for the flatlands are very small, and up to − 28 cm for the highest mountains with a mean of − 8 cm for whole Austria as can be spotted in Fig. 4c.
Fig. 4

100-Year snow depth return levels computed with the Extremal-t max-stable model (a) and the smooth model (b). Differences between those two models are shown in Fig. 4c. a 100-year snow depth return levels computed with the Extremal-t model. Local return levels estimated from observations at the validation sites are shown as colored circles. The highest computed value of 1334 cm lies near the Großglockner, 3797 m which is the highest point in Austria. The Wildspitze is nearly as high and features a similar 100-year snow depth. Note that 18 pixels with values way larger than 15 m were discarded for the reason described in the text. b Same as (a) but for the smooth model. Note that the same 18 pixels as in (a) with unrealistically large values were discarded. c Difference of the smooth and the Extremal-t model for 100-year snow depth return levels. Negative values in blue colors indicate larger quantiles for the Extremal-t model. The largest difference of − 28 cm corresponds to the location of the highest values in (a) and (b)

In order to discuss regional variabilities of return levels, Blanchet and Lehning (2010) computed normalized return levels by removing the altitudinal dependence in the GEV parameter models. However, as in the parameter models (3) used in this study the dependence of the location and scale parameters on mean snow depth \({\overline{HS}}\) is much stronger than on altitude alone, normalized return level maps of Austrian snow depth extremes do not provide any insight in regional distinctions.

4.5 Spatial analysis

In this section a suitable model for risk estimation of Austrian snow depth extremes is detected. Therefore, spatial dependencies of extreme snow depths are examined in detail. As the smooth model does not provide any spatial dependence, the anisotropic Extremal-t model selected as best max-stable model in Sect. 4.3 was used.

Figure 3 suggests the spatial dependence of snow depth extremes in the north to have a longer range and a smaller angle, than in the regions south of the main alpine ridge. In fact, on average, independence (defined as \(\theta > 1.8\)) in the north is only reached at distances between stations of about 385 km, whereas in the south independence is reached already at 208 km. These long-range dependencies arise from the combination of a topographical and climatological fact. When moving roughly from northwest to southeast, the precipitation area of a typical winterly frontal system sweeps across the whole zonal extension of the Austrian Alps. The synoptical effect of enhanced snowfall amounts ends, where the Austrian Alps start to run out in flat hills (which is approximately at the eastern end of the black dashed line in Fig. 3). Similar considerations apply to the regions south of the main alpine ridge, where strong dependencies also follow the form of the Alps but with a slightly larger angle, and arise from southerly flows and Stau effects. In contrast, the flatlands in eastern Austria are climatologically intertwined, as cold air with snowfall approaching from the northwest (south) flows around the Alps in a clockwise (counterclockwise) manner. This leads to roughly north–south-oriented long-range dependencies in the far east of Austria. Furthermore, it can clearly be noticed in Fig. 3 that in western and central Austria, where the climate system is dominated by the topography of the Alps, snow depth extremes are independent across the alpine barrier. In the east of Austria, where the Alps are smoothly sloping down to the flatlands, the dependence increases and the direction turns into north–south.

In Fig. 5, pairwise empirical extremal coefficients from the calibration dataset are plotted according to their direction. The average spatial dependence along the main geographical axes of Austria from west to east is only slightly larger than in other directions (black dots in Fig. 5). The highest dependence (lowest extremal coefficients of around 1.64 can be found in the sector between \(0^{\circ }\) and \(10^{\circ }\). The dependence slowly decreases until it reaches 1.7 at an angle of \(50^{\circ }\). In all other directions the average extremal dependence is between 1.7 and 1.8. This coincides nicely with results for the French Alps (Gaume et al. 2013; Nicolet et al. 2015; Blanchet and Davison 2011), where extremal dependence of snowfall, snow depth and precipitation (Blanchet and Creutin 2017) is also linked to the direction of the main ridges. The dependence of snow depth extremes in Austria is strongest along the northern alpine boundary, which is prone to snowfall events from prevailing north-westerly to northerly directions. This climatologically reasonable finding corresponds well with Fig. 5 and is strengthened by the fact that only 10% of pairwise extremal coefficients along the strong dependence sector are superior to 1.9, which corresponds to independence (red asterisks in Fig. 5).
Fig. 5

Empirical extremal coefficients at the calibration stations according to their pairwise direction. Black dots depict class averages over 20\(^{\circ }\), and red asterisks show the percentile of extremal coefficients superior to 1.9 in the corresponding sectors. On average the strongest dependence is oriented along angles between \(0^{\circ }\) and \(10^{\circ }\)

As already pointed out in Sect. 4.3 and in the results above, spatial dependencies of the regions north and south of the main alpine crest should be modeled separately. According to Blanchet and Davison (2011), this can be done in two ways. On the one hand, assuming weak dependence across the border of the two regions, the space transformation (4) could be adapted to account for different regions, as it was done by Blanchet and Davison (2011). On the other hand, complete independence between the regions is assumed in this study, allowing to fit dedicated Extremal-t max-stable processes to either side of the Alps. To see whether a dedicated model for either side of the Alps outplays one single model for whole Austria when the interest lies in risk estimation, an AIC-based model selection similar to that described in Sect. 4.2.1 with the same covariates longitude, latitude, altitude and mean snow depth was also performed for the north and south regions separately. The same directions and ranges as in Sect. 4.3 for anisotropy of the spatial dependence were then tested for each region. Table 3 shows the results of this procedure.
Table 3

Summary of Extremal-t models providing the best fit (smallest TIC value) for snow depth extremes in Austria

Model fitted to …

(stations used for fitting)

Range (km)

\(\varphi\) (\(\circ\))

\(\mu \sim\)

\(\sigma \sim\)

\(\xi \sim\)

Whole Austria (210)

315

1.5

alt + \({\overline{HS}}\) + Spline(lon,lat)

lat + alt + \({\overline{HS}}\)

lon + lat + \({\overline{HS}}\)

North (141)

245

2

alt + \({\overline{HS}}\) + Spline(lon,lat)

alt + \({\overline{HS}}\)

alt

South (70)

175

4

alt + \({\overline{HS}}\) + Spline(lon,lat)

lat + alt + \({\overline{HS}}\)

lon + lat + \({\overline{HS}}\)

“Whole Austria” in the first column stands for the model already selected in Sect. 4.3, which was fitted to all stations on both sides of the Alps. “North” or “south” means that the model was fitted solely to stations north or south of the alpine barrier. In parenthesis the number of stations available for fitting in each region is shown. Range and direction angle \(\varphi\) mark the anisotropy parameters for the space transformation leading to the lowest TIC. The three rightmost columns denote the covariates for the GEV parameter models that resulted in the best fit. The model fitted to whole Austria had 5, and the models fitted to the two regions north and south separately had 2 Spline knots each for the covariates longitude and latitude

It comes at no surprise that the best marginal models for all three fitting regions (whole of Austria, only stations in the north and only stations in the south of the alpine barrier) are similar to each other. Probably due to the smaller meridional extension, the scale and shape parameter models for the north lack latitude as explaining covariate. Range and direction of the anisotropy reflect the results discussed at the beginning of this section.

The fit of the three models of Table 3 to empirical extremal coefficients can be seen in Fig. 6. The modeled extremal coefficient function is generally in good agreement when fitted to all calibration stations in Austria (black line and dots in Fig. 6). For (transformed) distances larger than roughly 400 km, extremes become independent with \(\theta > 1.8\). Nevertheless, the model then begins to underestimate empirical dependencies for those distances. Also the model which was fitted only to stations in the south (red colors in Fig. 6) fits spatial dependencies well up to distances of 180 km. By trying to adapt to the growing independence at large distances (400 km), the model underestimates dependencies for distances between 180 and roughly 380 km, and slightly overestimates the dependency for the largest distances up to 450 km. The worst fit is provided by the model fitted only to stations north of the main alpine ridge (blue colors). Similar to the model for the south, it tries to adapt to large distances, but consequently underestimates extremal dependence up to 300 km. The reason for the bad fit is unclear, but might be explained by the very rough surface which is more difficult to fit. The northern model has a smoothness parameter of 0.3 compared to 0.4 and 0.5 for the south and whole Austrian model.

In addition, the green bullets in Fig. 6 show binned extremal dependencies for station pairs across the alpine barrier. These dependencies quickly increase from \(\theta = 1.6\) for the smallest distances to 1.8 from 120 km on and eventually to 1.9 for distances of 450 km. This impressively confirms the impression of Fig. 3.

Figure 6 reveals also two distance regimes in the dependence structure of snow depth extremes in Austria. For distances between station pairs below about 120 km, dependencies seem to be a little stronger in the south, than in the north. For larger distances, the dependence structure changes, i.e., at the same distance extremes on the northern side are more dependent than in the south. Above roughly 400 km distance, snow depth extremes in the south quickly become nearly independent (\(\theta > 1.8\)), whereas in the north, extremes seem to remain dependent up to very large distances of 600 km, before also in the north independence is reached.
Fig. 6

Extremal coefficients \(\theta\) versus transformed pairwise distances for anisotropic Extremal-t models fitted to different regions, depicted by colors: Black circles and line show the class-averaged dependencies and the model fit for all stations, blue colors show the same when fitted only to stations north of the alpine barrier, and red colors are related to stations in the south. The class-averaged dependencies across the alpine barrier are shown in green. Gray circles are empirical dependencies computed with the F-madogram for all stations and the appropriate space transformation

A quantitative comparison of empirically estimated extremal coefficients using observations at the validation stations with extremal coefficients modeled from fitting stations in whole Austria, and stations only in the north and south shows, that spatial dependencies of Austrian snow depth extremes indeed should be modeled in two regions separately, rather than in one single region. In Fig. 7, bivariate extremal dependencies for the two distance regimes identified in Fig. 6 are compared for models that are fitted to the whole of Austria, and to the north and south separately. To avoid the dependence being biased by different distances, the distance ranges are kept small, namely 20–50 km for the first, and 250–300 km for the second regime. For completeness, the fits are also compared to the model which was fitted to all distances. Generally the empirical extremal coefficients nicely reflect the findings from Fig. 6, i.e., at small distances dependencies are slightly stronger in the south, and at larger distances dependencies are stronger in the north, albeit of course weaker than for smaller distances.

From visual inspection it is hard to tell, which of the models fitted to whole Austria, the north or the south should be used to model dependencies over small distances in the north or in the south. A closer look reveals that for the stations in the north, the model which was fitted solely to that stations has the smallest difference to the median of the empirical extremal coefficients (\(-0.01\)). The other models perform only marginally worse (\(-0.04\) for the model fitted to the whole of Austria, and \(-0.07\) for the model fitted to the south). Similar results are achieved for the southerly stations only, where the model fitted solely to the southern stations has a difference to the median of the empirical extremal coefficients of \(-0.02\), the model fitted to the whole of Austria 0.01, and the model fitted only the northern stations 0.04.

For the larger distances between 250 and 300 km and stations only in the north, again the model fitted to the north performs best, closely followed by the models fitted to the whole of Austria and only to the southern stations. For stations situated south of the alpine barrier, all three models model dependencies between those stations slightly more dependent than they are, although the models of whole Austria and the one fitted only to the south come closest.

If all distances between station pairs are taken into account the differences between the fitting regions almost vanish. For the stations in the north, the model fitted solely to those stations remains best (0.001), very closely followed by the two others (0.02 for both). For the stations in the south the dedicated southern model and the model fitted to whole Austria perform equally well (− 0.002 and 0.002), followed by the model fitted to the north (0.006).

Although differences are very small and might be slightly different for different numbers of stations used for fitting, for the northern region and both distance regimes the dedicated model does a better job. However, for the southern stations one can equally use the model fitted only to those stations, or to the whole of Austria. Given the really small differences to the median of the empirical extremal coefficients, it might be sufficient to use the model fitted to the whole of Austria for modeling extremal dependencies on either side of the Alps. None of the models fitted either to the northern or the southern side of the Alps is of course able to capture the very weak dependencies across the Alps, because they simply do not appear in the fitting process (not shown). For that purpose the model fitted to both regions simultaneously must be used.
Fig. 7

Empirical and modeled bivariate extremal coefficients \(\theta\) computed for station pairs only in the north (left) and south (right), and distances between 20 and 50 km (top), 250 and 300 km (middle) and all distances (bottom). Colors refer to empirically estimated extremal coefficients (emp.) and the models fitted to whole Austria, only to stations in the north (north) and only to stations in the south (south). For visual orientation horizontal lines are drawn at \(\theta = 1.4\) and \(\theta = 1.8\), which can roughly be interpreted as limits for dependence and independence

In risk analysis joint exceedance probabilities for arbitrary points in space are of interest. They can easily be computed from extremal coefficients for station pairs. Figure 8 shows examples of risk estimates for extremal dependence between station pairs north and south of the main alpine ridge. As expected from Fig. 7 and from the findings in the above paragraph, for smaller distances the model fitted to the stations south (north) of the Alps provides more accurate extremal dependencies than the models fitted to the northern (southern) stations or all stations (see, e.g., Fig. 8a, b with distances between stations of 37 and 21 km, respectively). For larger distances again the dedicated model fitted only to the northern stations on average achieves more accurate extremal dependencies than the other models (see Figs. 6, 8c).
Fig. 8

Joint exceedance probability of two stations with empirical dependence \(\theta _{\rm emp}=1.47\) on the southern side of the Austrian Alps (left), two near stations with empirical dependence of \(\theta _{\rm emp}=1.47\) on the northern side (center) and two very distant stations in the north with weak dependence \(\theta _{\rm emp}=1.68\) (right). Colors represent Extremal-t max-stable models fitted to whole Austria (black line), only to stations in the north (blue line) and only to stations in the south (red line). Full dependence and independence are shown as dashed light-blue and purple lines, dots depict empirical joint probabilities. The gray circles in the inset map refer to the station locations. a Near stations south of the Austrian Alps (distance 37 km). b Two stations in the north, 21 km apart. c Two northern stations with 404 km distance

5 Conclusions

The main goals of this study were to find the best models for the spatial prediction and for risk estimation of snow depth return levels in Austria. A set of 421 snow depth series with a length of 42 years each was split into 210 calibration and 211 validation sites. With the aim of comparing the different methodological approaches of smooth modeling and diverse representations of max-stable processes (Schlather, Brown–Resnick, Extremal-t and Geometric Gaussian), an AIC-based model selection with the covariates longitude, latitude, altitude and mean snow depth for the GEV distribution was performed. The best marginal models comprise altitude or altitude-dependent covariates for all three GEV parameters, as well as a cubic P-spline with longitude and latitude for the location parameter. The found model was then fitted to the calibration stations using the smooth modeling approach as well as the different max-stable processes, also accounting for anisotropy of the extremal dependencies by using a space transformation.

After the fitting procedure, the Extremal-t process with an anisotropy of \(1.5^{\circ }\) and a range of 315 km remained as best max-stable model, outperforming the Geometric Gaussian, Brown–Resnick and Schlather processes by TIC. Nevertheless, the differences between all max-stable models are small and reflect the results of Gaume et al. (2013), Nicolet et al. (2015) and Sebille et al. (2017). In a validation procedure, return levels computed from observations at the validation sites were compared with return levels modeled with the smooth model and the anisotropic Extremal-t model. Based on a modified Anderson–Darling score (mAD), which emphasizes the upper tail, the Extremal-t model performs better than the smooth model, whereas it delivered slightly worse NRMSE values.

100-year snow depths modeled with the anisotropic Extremal-t model were then investigated in detail. It should be noted, that generally estimates for higher altitudes are relatively uncertain, as already Sebille et al. (2017) pointed out. In addition, as the best marginal model 3 depends on mean snow depth for all GEV parameters, return level maps are highly sensitive to that covariate, which has to be provided on a grid. On the one hand, the highest return levels can be found along the main alpine crest, where the highest mountains in Austria are located. This comes as no surprise as the location parameter was modeled as a function of mean snow depth, which strongly depends on altitude. On the other hand, the regions along the northern Austrian border, prone to North-Stau and heavy snowfall exhibit high return levels as well. As only a very small number of pixels are modeled larger than 7.5 m, the majority of the high 100-year return levels is between 5 and 7.5 m. The highest value of nearly 14 m is located near the Großglockner, which is the highest mountain in Austria. Despite the worse mAD score, the smooth model delivers very similar but consistently slightly lower 100-year return values than the Extremal-t model. Given the very similar return level maps, it may be hard to decide which one of the smooth or Extremal-t models is better suited to the prediction of snow depth return levels, despite the relatively clear score ranking.

As in Switzerland and France, also dependencies of snow depth extremes in Austria have a pronounced direction. They are stronger at smaller angles between \(0^{\circ }\) and \(10^{\circ }\) which is roughly along the main alpine barrier. The Austrian alpine barrier, running roughly from west to northeast, separates two climate regimes. Not surprisingly, extreme snow depths exhibit strong dependencies along the main alpine crest over large distances, while extreme snow depths across the alpine border are virtually independent. In the far east of Austria, where the dominating effect of the Alps on the climate system diminishes, north–south-oriented dependencies are possible. Strong dependencies can be found on both sides of the Alps. The main direction is approximately \(2^{\circ }\) in the north and \(4^{\circ }\) in the south. At short distances up to 120 km, dependencies are stronger in the south. At larger distances above approximately 120 km the effect of snowfall systems sweeping along the Alps from northwest to east becomes prominent and makes for a stronger dependency over larger distances in the north. In the south on average snow depth extremes are getting nearly independent (\(\theta > 1.8\)) at (untransformed) distances of roughly 208 km, in the north near independence is reached only at 385 km. Extremes across the alpine border are virtually independent for distances above 100 km. For smaller distances across the Alps snow depth extremes are weakly dependent with extremal coefficients larger than 1.6.

A quantitative validation of empirical and modeled extremal dependencies with models fitted to the whole of Austria and solely to stations north and south of the alpine barrier in Austria reveals that it might be reasonable, to model the spatial dependence on either side of the Alps with separate models. The Extremal-t model fitted to stations in the northern Austrian Alps models spatial dependencies between northern stations slightly better than the models fitted to all stations or to stations in the south only, independent of pairwise distances. For dependencies between southern stations, both models fitted to the southern stations only and to whole Austria seem to work slightly better than the one fitted to the northern stations. However, the differences are generally quite small and diminish almost completely when all distances are considered.

Notes

Acknowledgements

The authors are grateful to ZAMG for data acquisition and supply. The computational results presented have been achieved (in part) using the HPC infrastructure LEO of the University of Innsbruck.

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Zentralanstalt für Meteorologie und GeodynamikInnsbruckAustria
  2. 2.Department of Atmospheric and Cryospheric Sciences (ACINN)University of InnsbruckInnsbruckAustria
  3. 3.Department of MathematicsUniversity of InnsbruckInnsbruckAustria

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