## 1. Introduction

[2] The evaluation of extreme snowfalls is a challenging issue for risk management in mountainous regions. In particular, extreme snowfall constitutes one of the critical parameters for road viability analysis and avalanche risk management [e.g., *Schweizer et al*., 2009]. For instance, the systematic implementation of avalanche propagation models requires the precise evaluation of the snow input distribution [e.g., *Ancey et al*., 2004; *Eckert et al*., 2010b]. In France, the 100 year quantile (quantile corresponding to a return period *T* = 100 years) is widely used for hazard mapping or for the conception of defense structures. However, in practice, the evaluation of this input turns out to be difficult for several reasons:

[3] 1. In mountainous areas, available data are sparse with generally incomplete and short time series very rarely longer than 100 years. Consequently, extrapolating beyond the highest observed values is necessary. In this context, extreme value theory (EVT) is an adequate formalism to deal with since it provides a solid theoretical basis for extrapolation, namely, the convergence of block maxima to the GEV (generalized extreme value) distribution via Fisher-Tippett theorem [*Fisher and Tippett*, 1928] and of peaks over thresholds to the generalized Pareto distribution via Pickands theorem [*Pickands*, 1975].

[4] 2. The data consist mostly in chronicles of precipitation measured in water equivalent (w.e.). The distinction rain/snow is not always done, which requires the joint analysis of temperature series.

[5] 3. Measurement stations are usually located far from avalanche release zones. It is therefore necessary to use spatial interpolation methods adapted to the specificity of extreme values.

[6] 4. These stations are usually located in the valleys rather than at high altitudes, which requires using an orographic snowfall gradient for the quantification of the water equivalent in avalanche release zones.

[7] In the current engineering practice, all these difficulties are often circumvented by oversimplifying assumptions. The problems of spatial interpolation and orographic effects are sometimes treated by defining “homogeneous zones by altitude band” [*Salm et al*., 1990; *Bocchiola et al*., 2006]. Besides the difficulty of zones definition, this method introduces discontinuities at the zone borders that are incompatible with the natural phenomenon. Simple kriging interpolation techniques have also been used by *Prudhomme and Reed* [1999] for extreme rainfall mapping in Scotland using a Gaussian field. This method has the advantage of allowing smooth spatial prediction but without acknowledging the specificity of extreme values. *Weisse and Bois* [2001] also used kriging for rainfall quantile mapping. In this case, the previous limitation is thus partially overcome by smoothing directly the quantile of interest obtained by fitting adapted EVT-like distributions rather than the process. However, this method has the main drawback of separating two estimation procedures (local GEV distributions and spatial fields) without reporting the local error on the spatial model. Furthermore, it loses estimation power by using one single quantile value per location instead of the full series of maxima. Finally, for extrapolation, the current engineering methods use almost systematically Gumbel laws rather than a more general model of the GEV type. This can result in systematic underestimation of most extreme snowfalls [*Parent and Bernier*, 2003; *Bacro and Chaouche*, 2006].

[8] An important point in spatial extreme approaches is that covariates are often used to infer the spatial dependence of the GEV parameters, the rationale being the capability to predict high quantiles at any point and to reduce the dimension of the problem. *Blanchet and Lehning* [2010] used both altitude and the mean snow depth (“climate space”) [*Cooley et al*., 2007] to characterize extreme snow depths at the ground level using smooth spatial models for the GEV parameters. These authors have shown the superiority of such an approach over quantile smoothing even without modeling the extremal dependence. Many other studies have already considered the GEV parameters as spatial fields [*Naveau et al*., 2009], especially in the context of gridded data resulting from climate modeling [*Rust et al*., 2009; *Maraun et al*., 2010].

[9] Recently, a solid formalism based on the multivariate EVT has been proposed to characterize the spatial dependence of block maxima extreme values. Applied to a set of data series, maps of extremal dependence can be obtained [*Coles et al*., 1999]. Furthermore, the definition of an extremal function can extend to spatial fields of extreme values the notions of variogram and range, i.e., the distance up to which the different series are dependent [*Cooley et al*., 2006]. This formalism is now beginning to be successfully applied in hydrology. For instance, *Bel et al*. [2008] compared different spatial models on extreme temperature and rainfall data, and *Blanchet et al*. [2009] analyzed the spatial dependence of extreme snowfalls in the Swiss alpine region using the *χ* and statistics [*Joe*, 1993; *Coles et al*., 1999].

[10] To model spatial dependence in extreme values consistently with EVT, max-stable process (MSP) is an approach based on the pioneering work of *Brown and Resnick* [1977] and *DeHaan* [1984] and further developed by R. Smith (Max-stable processes and spatial extremes, unpublished manuscript, 1991), *Schlather* [2002], *Schlather and Tawn* [2003], and *Kabluchko et al*. [2009]. The practical use of MSPs in environmental sciences, however, is very recent. This formalism has been applied with success by *Padoan et al*. [2009] combined with the use of latitude, longitude, and altitude as covariates to model rainfall data in the Appalachian mountains. These authors also propose a practical inferential method for the fitting of MSPs to spatial data by maximization of a composite likelihood [*Lindsay*, 1988; *Xu and Reid*, 2011] since the full likelihood is out of reach. A similar approach has been applied by *Blanchet and Davison* [2011] for snow depth data with a modified anisotropic Schlather MSP, selected among large classes of the Smith and Schlather MSPs.

[11] The aim of this article is the modeling of extreme snowfalls in the French Alps by MSPs, a crucial ingredient to evaluate avalanche depth distributions in all the potential release areas [*Gaume et al*., 2012]. Snowfalls are measured in water equivalent and are thus independent of density effects. A simple method is proposed to apply the spatial extreme formalism at a constant altitude in order to infer “true” spatial effects. With regard to existing approaches, we bridge the work of *Blanchet and Lehning* [2010] and *Blanchet and Davison* [2011] by estimating the GEV parameters as continuous functions of space within the max-stable framework. Furthermore, in addition to the more classical Smith and Schlather MSPs, we also implement the more flexible Brown-Resnick MSP, more adapted to snowfall, which is a less spatially dependent variable than snow depth. We also take into account the directional effects related to the local alpine geography. Finally, we show how quantile maps can be obtained and demonstrate the prediction ability of our approach using cross-validation for the used data sets but also for other stations.

[12] As stated in *Segers* [2012] and *Ribatet and Sedki* [2012], max-stable copulas would have been another option to reach similar goals (other copulas, often used in hydrology, would fail in providing a fair representation of the dependence structure). However, such an approach would have implied fitting first the margins at each station and the extremal dependence structure in a second time. Our work is one of the firsts that performs the two steps simultaneously. This is for us theoretically preferable, in order to take into account the estimation error on the margin parameters in the estimation of the parameters of the extremal model.

[13] This article is organized as follows: section 2 provides the theoretical elements about extreme value statistics in the spatial case and more precisely about MSP. The studied data are presented in section 3 in which an empirical analysis is performed. In section 4, a criterion for model selection is defined, and the results using linear models and penalized smoothing splines are compared. Finally, in section 5, a discussion is dedicated to the comparison of our results to previous approaches, to the study of the influence of the accumulation period on the results and to a joint analysis that uses the available bivariate distributions and conditional levels which can be useful for operational purposes.