## 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.

*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

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

*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

*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

*k*th 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

*k*th station corresponding to (2). With the assumption of spatially independent stations, the log-likelihood function then reads as

*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

*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

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.

*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:

*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

### 4.2 Selection of the smooth model

#### 4.2.1 GEV parameters estimated at the stations

*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.

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.

### 4.3 Selection of the max-stable model

*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.

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

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.

*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.

*k*th 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

*k*th station.

*F*is the cumulative distribution function of the

*k*th station.

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 |

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.

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.

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

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.

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).

## 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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