Mapping extreme snowfalls in the French Alps using max-stable processes

Authors


Corresponding author: J. Gaume, IRSTEA, UR ETGR, F-38400 St Martin d''Heres, France. (johan.gaume@cemagref.fr)

Abstract

[1] The evaluation of extreme snowfalls is an important challenge for hazard management in mountainous regions. In this paper, the extreme snowfall data acquired from 40 meteorological stations in the French Alps since 1966 are analyzed using spatial extreme statistics. They are then modeled within the formal framework of max-stable processes (MSPs) which are the generalization of the univariate extreme value theory to the spatial multivariate case. The three main MSPs now available are compared using composite likelihood maximization, and the most flexible Brown-Resnick one is retained on the basis of the Takeuchi information criterion, taking into account anisotropy by space transformation. Furthermore, different models with smooth trends (linear and splines) for the spatial evolution of the generalized extreme value (GEV) parameters are tested to allow snowfall maps for different return periods to be produced. After altitudinal correction that separates spatial and orographic effects, the different spatial models are fitted to the data within the max-stable framework, allowing inference of the GEV margins and the extremal dependence simultaneously. Finally, a nested model selection procedure is employed to select the best linear and spline models. Results show that the best linear model produces reasonable quantile maps (assessed by cross-validation using other stations), but it is outperformed by the best spline model which better captures the complex evolution of GEV parameters with space. For a given return period and at fixed elevation of 2000 m, extreme 3 day snowfalls are higher in the NE and SE of the French Alps. Maxima of the location parameter of the GEV margins are located in the north and south, while maxima of the scale parameter are located in the SE which corresponds to the Mediterranean influence that tends to bring more variability. Besides, the dependence of extreme snowfalls is shown to be stronger on the local orientation of the Alps, an important result for meteorological variables confirming previous studies. Computations are performed for different accumulation durations which enable obtaining magnitude-frequency curves and show that the intensity of the extremal directional dependence effect is all the more important when the duration is short. Finally, we show how the fitted model can be used to evaluate joint exceedence probabilities and conditional return level maps, which can be useful for operational risk management.

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 math formula 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.

2. Extreme Value Statistics in the Spatial Case

2.1. Max-Stable Process

[14] math formula is a MSP if there exist sequences math formula and math formula, such that if for all n, math formula are independent copies of Z, then math formula has the same distribution as

display math

[15] As a consequence, all finite dimensional marginal distributions are max stable, and in particular, the univariate marginal Z(x) distribution belongs to the GEV family:

display math(1)

with

display math(2)

where μ(x), σ(x), and math formula are the location, scale, and shape parameters, respectively, at location x. According to the sign of math formula, the math formula distribution belongs to three different families of distributions known as Fréchet (math formula), Weibull (math formula), and Gumbel (math formula).

[16] Usually, it is convenient to transform the univariate marginal setting:

display math(3)

[17]  Z* is thus a MSP with unit Fréchet margins whose cumulative distribution function is defined as math formula. Models of MSPs have been proposed by several authors. The most popular are those of Smith (unpublished manuscript, 1991; extremal Gaussian), a particular case of DeHaan [1984] construction, Schlather [2002], and Brown and Resnick [1977] generalized by Kabluchko et al. [2009]. We will focus in this work mainly on the Kabluchko model defined as

display math(4)

where ξi is a Poisson process on math formula of intensity math formula, and Wi are the independent Gaussian fields, with stationary increments, variance σ 2(x), and variogram math formula.

2.2. Extremal Coefficient

[18] Spatial dependence of maxima at two locations x and x′ is characterized by the extremal coefficient denoted θ(x, x′). If Z* is the limiting process of maxima with unit Fréchet margins, then [Brown and Resnick, 1977]

display math(5)

[19] Thus, if θ(x, x′) = 1, there is perfect dependence of the maxima at stations x and x′ and on the contrary, if θ(x, x′) = 2 the maxima are independent. For Smith, Schlather, and Brown-Resnick models, the extremal coefficient can be calculated explicitly. According to these models, the processes are stationary, and the related extremal coefficient only depends on math formula. The Smith model extremal coefficient θ Sm is given by

display math(6)

with math formula representing the Mahalanobis distance, Σ representing a Gaussian covariance matrix with three parameters σ 11, σ 12, σ 22, and Φ representing the standard normal cumulative distribution.

[20] The Schlather extremal coefficient θ Sc is given by

display math(7)

with math formula being a valid correlation function (Wittle-Matern, Cauchy, exponential, Bessel, etc.). We tested several forms of correlation functions and retained the exponential form math formula, where c 1 is a range parameter, which was found to provide the best fit to our data on the basis of the Takeuchi information criterion (TIC; see below).

[21] Finally, the Brown-Resnick extremal coefficient θ BR is given by

display math(8)

[22] The behavior of the extremal coefficient may give an indication for the choice of the model. For instance, the Schlather MSP cannot achieve full independence (θ = 2). This can be useful for the applications with extremal dependence that remains strong even at very important distances but is a major flaw in other cases. Instead, the Smith MSP imposes full independence at long distances (math formula) but is quite rigid at short distances. The Brown-Resnick MSP is more flexible as the variogram may take a great variety of shapes near zero and allows for full independence (math formula) at long distances (equation (8)).

[23] The Smith MSP can directly model the anisotropy in the extremal coefficient by using a modified distance (Mahalanobis), a function of the Gaussian covariance matrix Σ. This covariance matrix plays a very important role because it determines the elliptical shape of the extremal dependence. On the contrary, Schlather and Brown-Resnick MSPs are primarily isotropic as they involve the euclidean distance. Thus, to take into account the possible directional effects in extreme snowfalls in the case of Schlather and Brown-Resnick MSP, we must modify the standard space [Blanchet and Lehning, 2010] while inferring the extremal dependence. To do this, we set math formula with math formula the euclidean coordinates and V the rotation matrix defined below:

display math(9)

where ψ represents the anisotropy angle of the transformation and ρ its intensity.

2.3. Spatial Models for the GEV Parameters

[24] At a given location x, GEV parameters math formula can be estimated if there are observations available. If no data are available at x, these parameters must be inferred from data at nearby locations. This can be done essentially in two ways: (1) estimate first the pointwise GEV parameters at locations with observations and interpolate or (2) model the spatial evolution of the GEV parameters [Blanchet and Lehning, 2010]. We choose the second option, and we investigate within the max-stable formalism the two classes of spatial models for the GEV parameters. The first model links linearly the GEV parameters at location x with the spatial coordinates math formula. If η is one of the three GEV parameters (µ, σ, or ξ):

display math(10)

[25] The second model is nonlinear, and it decomposes η in an appropriate basis math formula:

display math(11)

[26] In the following, we will consider penalized splines with radial basis functions (pr-splines) [Ruppert et al., 2003] of order 3:

display math(12)

where math formula are the coordinates of the rth knot of the spline, and R is the number of knots.

3. Empirical Analysis

3.1. Data Presentation

[27] MeteoFrance, the French meteorological agency, provided us with daily controlled and homogenized snowfall measurements (in water equivalent) for 124 alpine weather stations with different temporal series lengths. We retained for the modeling 40 weather stations whose measurements were conducted continuously from 1966 to 2009 (i.e., 44 years of measurement). The 84 others with shorter time series were kept for model validation. Figure 1a shows the location of all the stations on the French alpine terrain. Figure 1b shows the altitude distribution of the 40 retained stations. One can notice that most weather stations are located around 1000 m of altitude and that very few stations are available at high altitudes (>2000 m). Figure 1c shows the partition of the French Alps into four main alpine zones: northern Alps, central Alps, Southern Alps, and extreme Southern Alps and also into 23 massifs. Details on the mountainous massifs with their highest peak and winter mean snowfall (WMS) [from Durand et al., 2009] are presented in Table 1.

Figure 1.

(a) Location of weather stations on the alpine terrain (coordinates in meters extended Lambert II). The circles represent the 40 retained weather stations, and the squares represent the 84 stations with shorter time series used for model validation. (b) Distribution of the altitudes of the 40 retained stations (in meters). (c) The French Alps divided into four alpine zones. The numbers are massif indices summarized in Table 1.

Table 1. Details of the Mountainous Massifs of the French Alps Presented in Figure 1c With Their Highest Peak and WMS [from Durand et al., 2009]
IndexMassifHighest Peak (m)WMS (mm w.e.)
1Chablais2466941
2Aravis27521006
3Bauges22171003
4Chartreuse20821015
5Vercors2341844
6Mont Blanc4810885
7Beaufortin2995874
8Haute-Tarentaise3747658
9Haute-Maurienne3751508
10Vanoise3855632
11Maurienne3779718
12Belledonne2977906
13Grandes-Rousses3465708
14Thabor3178499
15Oisans3983703
16Pelvoux4102658
17Champsaur3163628
18Devoluy2789631
19Queyras3385387
20Parpaillon3046448
21Ubaye3412446
22Alpes-Azureennes3050620
23Mercantour3143651

[28] We extracted from this database the annual snowfall maxima over different accumulation periods (1–7 days) for each weather station. Only the winter period (15 November to 15 May) and snow precipitations were considered. Thus, for the search of maxima, only the days when maximum temperature remains below 2°C at the measurement station were considered. This is sufficient to ensure that precipitations fall as snow in avalanche release areas. In the following, we will mainly focus on the accumulation period of 3 days since this duration is generally considered as the best avalanche predictor [Salm et al., 1990; Schweizer et al., 2003; Ancey et al., 2004; Schweizer et al., 2009] for high return period events in the Alps. In fact, this duration often corresponds to the most intense avalanche cycles (Eckert et al., 2010a). In accordance with section 2, Z(x) henceforth denotes the annual snowfall maximum over 3 days at the weather station of coordinate x.

3.2. Altitude Consideration

[29] As shown in Figure 1b, the number of measurement stations above 2000 m is very low, which complicates the interpolation at higher altitudes where most avalanche release zones are located. Hence, we used the orographic gradients δ(x) by alpine zone derived in the study of Durand et al. [2009] and represented in Figure 2 to transform the data at the altitude alt(x) to the same constant altitude level (2000 m):

display math(13)

where z(x, 2000) and z(x, alt(x)) are the snowfall annual maximum data in x at 2000 m and at altitude alt(x), respectively. This transformation involves the massif's WMS(x) (see Table 1) for weighing the local annual maximum. This consideration of the altitude is equivalent to assuming that the annual maxima vary with the altitude as a proportion of the annual accumulation. This assumption is reasonable because the annual accumulation can often be explained by a few extreme events only. We also take into account an attenuation of the gradient above a given threshold altitude s(x) (2700 m for the northern Alps, 3000 m in the central Alps, Southern, and extreme Southern Alps, see Figure 2) [Durand et al., 2009]. This amounts to replacing alt(x) by s(x) in equation (13) when alt(x)>s(x).

Figure 2.

Evolution of WMS as a function of altitude for the four alpine zones (constructed from Durand et al. [2009]).

[30] It is easily shown that the transformation of annual snowfall maxima using equation (13) is equivalent to transforming of the location and scale parameters by multiplying them by G, while the shape parameter remains constant. Hence, the marginal distribution of Z(x, 2000) is a GEV math formula. This simple way of handling orographic gradients thus enables a full analytical formulation of GEV marginals at the transformed altitude. From the point of view of the spatial analysis, it has the advantage of keeping only longitude-latitude effects in the spatial variation and avoids specifying a distance in the 3-D space, which is difficult since 1 m in altitude difference should certainly weigh differently than 1 m latitude/longitude. More practically, it would not have been possible to infer directly the vertical gradients from our data since the altitude range of the stations is too small.

[31] The results obtained in the Swiss Alps [Blanchet et al., 2009] on the evolution of extreme snowfall with elevation support our way of handling altitude. Indeed, with 247 stations in the Swiss Alps including automatic stations between 1600 and 3000 m, these authors were able to distinguish the trends of evolution of the GEV parameters with altitude: the location µ and scale σ parameters are increasing functions of altitude (with a gradient of approximately 0.015 mm/m for µ and 0.003 mm/m for σ), while the shape parameter ξ is almost not influenced by the altitude. In our case, the average location parameter gradient is equal to 0.02 mm/m, very close to the Swiss one, and the average scale parameter gradient is equal to 0.006 mm/m, slightly higher than the Swiss one.

3.3. Pointwise GEV Parameters

[32] The GEV parameters µ, σ, and ξ have been estimated pointwise on the transformed data (equation (13)) by maximization of the marginal likelihood for each station. They are plotted on maps in Figure 3 and versus longitude and latitude in Figure 4.

Figure 3.

Maps of the GEV parameters estimated pointwise on the transformed data (alt = 2000 m). (a) Location parameter µ, (b) scale parameter σ, and (c) shape parameter ξ.

Figure 4.

Evolution of the GEV parameters µ (a and d), σ (b and e), and ξ (c and f) determined pointwise on the transformed data (alt = 2000 m) as functions of longitude (a–c) and latitude (d–f) (in kilometer). Cubic pr-splines with two knots have been adjusted on each graph (solid line). The symbols/colors represent the four alpine zones: northern Alps, central Alps, Southern Alps, and the extreme Southern Alps (see Figure 1c).

[33] First, it can be noted that for the three GEV parameters there are strong disparities between zones, especially between the extreme Southern Alps and the rest of the Alps. Concerning the location parameter µ, there is a decrease with latitude from the extreme Southern Alps to the central Alps and then an increase from the central Alps to the northern Alps. Regarding the evolution of µ with the longitude, it is difficult to distinguish any significant trend, apart possibly from a slight increase. Thus, at first sight, latitude seems to be a good covariate to explain the location parameter µ. The scale parameter σ decreases generally from the extreme Southern Alps to the Southern Alps, before stabilizing in the central Alps and the northern Alps. As shown in Figures 4b and 4e, the longitude and latitude appear preliminarily as two good covariates to describe this scale parameter σ. Finally, from Figure 3, the shape parameter ξ appears to be generally positive (Fréchet domain) except in the extreme Southern Alps where it is negative (Weibull domain). It increases with latitude from the extreme Southern Alps to the Southern Alps, changes sign, and then decreases globally when moving to the northern Alps. It also generally decreases with longitude from the northern Alps to the extreme Southern Alps. Again, latitude and longitude thus appear to be the two good potential covariates for ξ (Figures 4c and 4f).

[34] These variations must nevertheless be considered with care since the GEV parameters (µ, σ, and ξ) are correlated, and compensation may thus occur. However, these preliminary results suggest that there is strong spatial variability of snowfall annual maxima over the whole French Alps. In particular, the extreme Southern Alps seem to behave differently than the rest of the Alps. This is probably due to the climatic differences between the alpine areas, which are submitted to different precipitation regimes. Indeed, northern and central Alps are generally affected mainly by westerly flows, whereas Southern and more particularly extreme Southern Alps are more often affected by easterly flow patterns strongly controlled by the Mediterranean influence.

3.4. Extremal Dependence

[35] The spatial dependence of the 3 day annual snowfall maxima was studied empirically, by calculating the extremal coefficient for each pair of stations (780 pairs) using likelihood maximization [Bel et al., 2008]. These values are plotted against the 2-D distance between the stations in Figure 5. Other estimators such as least squares and Cooley-Naveau-Poncet [Cooley et al., 2006] were also tested and gave similar results. Note that values higher than 2 were constrained to 2. We remark that full independence (θ = 2) is achieved only for a very small number of pairs. Distance classes (intervals) have been defined, and averages were then computed within these classes without taking into account the directional effects (Figure 5).

Figure 5.

Extremal coefficient estimated for all pairs of stations as a function of the distance between the stations using likelihood maximization estimation. The black dots represent the averaged extremal coefficient by distance classes. The Smith, Schlather, and Brown-Resnick extremal coefficients were adjusted to the class averages.

[36] Extremal coefficients of Smith (equation (6)), Schlather (equation (7)), and Brown-Resnick (equation (8)) were fitted to this empirical average extremal coefficient. For the Brown-Resnick MSP, a power variogram math formula was used. It can be noted that the Schlather and Smith extremal coefficients provide a poor fit. Indeed, with only one single parameter (dependence range) both models are very rigid, one imposing the asymptotic independence (math formula for math formula) at large distance and strongly constraining the shape at the origin (Smith), the other imposing a rather strong extremal dependence even at long distances (Schlather: math formula for math formula). Instead, the Brown-Resnick process has an additional smoothing parameter b, and thus, the shape of the extremal dependence is more flexible than in the case of the previous models, which leads to an excellent fit of the average extremal coefficient (Figure 5). If we define the range r as the distance corresponding to an extremal coefficient θ = 1.9, the Brown-Resnick model gives a range of r = 182 km, identical to the one provided by the empirical estimation. Note also that these results remain valid for different accumulation periods (1–7 days).

[37] The influence of a potential directional effect was also studied. Figure 6 shows that extreme snowfalls show a strong directional effect. Let us define α as the positive angle between the horizontal and the segment defined by two pairs of stations. In most cases, the independence (θ>1.9) is achieved only for pairs of stations whose direction is higher than α = 90°. Averages by angle classes were also computed showing that globally, the extremal coefficient values are lower if α<90°. In more detail, the number of pairs whose extremal coefficient is higher than 1.9 is the lowest in the 51°–77° range. The lowest average extremal coefficient is also found in this interval. Extremal dependence thus has a greater range along this interval of α than in other directions. Knowing that the main direction of the local alpine axis is around 60° due to the presence of large valleys in this direction (Isère, Rhône, and Durance), this suggests that annual snowfall maxima are very sensitive to the orientation of the mountains and the presence of valleys.

Figure 6.

Extremal coefficient estimated for all pairs of stations as a function of α defined as the positive angle between the horizontal and the segment linking the pairs. The black dots represent the averaged extremal coefficient by angle classes (the value of the extremal coefficient is given by the radius). The red bar-plot represents the percentage of extremal coefficient values θ higher than 1.9.

[38] Finally, to investigate the strength of the extremal dependence between regions, four stations corresponding to different alpine zones were chosen as references to compute maps of the interpolated extremal coefficient (Figures 7b–7e). We also selected from the data in Figure 5 all pairs of stations for which θ(h)<1.56 (arbitrary choice to get a good visual). These pairs are connected by a line in Figure 7a. From Figure 7, we can note a strong regionalization of the extremal dependence. There is almost no spatial dependence between the extreme Southern Alps and the rest of the Alps. Similarly, the Southern Alps have a strong internal dependence but only a slight dependence with the rest of the Alps, except with a few stations of the central Alps located at the border with the Southern Alps. On the contrary, there is a strong dependence between the northern Alps and central Alps accompanied by a significant internal dependence. Finally, in agreement with the previous paragraph (Figure 6), this dependence appears to have a preferred orientation along the local alpine axis.

Figure 7.

(a) Pairs of stations whose extremal coefficient is less than 1.56. Maps of the interpolated (simple kriging) extremal coefficients with reference to four stations belonging to the four alpine zones (Figure 1c). (b) Chamonix-Mont Blanc, (c) Lans en Vercors, (d) Saint Veran, and (e) Saint Etienne de Tinee.

[39] This preliminary study suggests at first that the spatial dependence in extreme snowfall in the French Alps may be better captured by a Brown-Resnick MSP, which is shown to be more flexible than those of Smith and Schlather. In addition, an important directional effect is exhibited by this empirical analysis. The orientation of the Alps and the presence of large valleys are most likely the cause of these directional trends.

4. Application: Adjustment of a MSP to Data

4.1. Composite Likelihood

[40] In order to estimate the various parameters of the model (β matrix representing the spatial evolution of the GEV parameters and the dependence parameters), likelihood maximization is used. However, we cannot calculate the complete likelihood since we only know analytically the expression of the different bivariate distributions according to equation (5). Padoan et al. [2009] showed that for MSP the full log likelihood can be advantageously replaced by a special case of composite likelihood: the pairwise log likelihood lp defined as

display math(14)

with N representing the number of years of measurements, K representing the number of measurement stations, β representing the matrix of parameters to estimate, and f representing the bivariate density of the MSP used (Smith, Schlather, or Brown-Resnick). In our case, math formula is the maximum annual precipitation for the year n and station i projected using equation (13) at a constant altitude level of 2000 m. One can then find the parameters math formula that maximize the composite likelihood by solving the partial differential equation:

display math(15)

and derive the associated standard errors from the Hessian and Jacobian information matrices H and J with

display math(16)

and

display math(17)

4.2. Model Selection: TIC

[41] To compare different MSPs and models of spatial evolution, a criterion weighing the value of the likelihood by the number of model parameters to estimate can be used. The classic Akaike information criterion (AIC) [Akaike, 1981] cannot be used in our composite case since the complete likelihood is not known. We therefore use a derivative of AIC suitable for composite likelihood, the TIC [Takeuchi, 1976]:

display math(18)

[42] The best model is the one that minimizes the TIC (equation (18)). Composite likelihood maximization and TIC computations are carried out under the SpatialExtremes R package [Ribatet, 2009] complemented with personal routines.

4.3. Linear Models

[43] We choose to describe the parameters math formula firstly through linear evolution models based on the coordinates math formula. The linear evolution models that we use can be written as

display math(19)

[44] For the 3 day maxima, we tested 18 different forms for the matrix β, with different numbers of nonzero coefficients. Three models of MSP have also been fitted to the data for these different models of spatial evolution of the GEV parameters. Figure 8 shows the values of the TIC (equation (18)) for these different models. These models and the different used covariates are presented in detail in Appendix A.

Figure 8.

TIC values as a function of the type of linear evolution model for different MSPs (Smith, Schlather, Brown-Resnick, and Brown-Resnick with a transformed space) and as a function of the type of pr-spline for different numbers of knots, using a Brown-Resnick MSP. Details about the different models used can be found in Appendix A.

[45] It can be noted that for all the cases the Brown-Resnick MSP (with a power variogram math formula) gives better results than the Smith MSP, itself better than the Schlather MSP. The superiority of the Brown-Resnick MSP compared to the Smith one is related to its greater flexibility (form of the extremal coefficient) in agreement with the preliminary empirical study. Schlather's MSP gives even worse results since it does not account for complete asymptotic independence (θ = 2 for large distances). We therefore retain the Brown-Resnick MSP. Initially taken as isotropic, we further improved it by transforming the standard space according to equation (9) in order to take into account the directional effects highlighted in the empirical analysis. This transformation allowed a nonnegligible reduction of the TIC of approximately 200.

4.3.1. GEV Parameters

[46] It appears that the linear models with evolution of µ with both longitude and latitude and of σ with latitude only give minimum TIC values (models 1 and 4 in Figure 8). Models of evolution with latitude and longitude for both the location and scale parameters also give low TIC values (models 1 and 13 in Figure 8). Note that models with only the longitude as covariate for the location parameter µ are the worst since they lead to the highest TIC values (peak values: models 2, 5, 8,…). The best model corresponds to model 4 in Figure 8 after transformation of the standard space (equation (9)). The resulting values of the different parameters of the matrix math formula (equation (19)) are summarized in Table 2.

Table 2. Parameters of the math formula Matrix With the Associated Standard Errors (in Parentheses) Evaluated for the Linear Model 4 in Figure 8 With a Brown-Resnick MSP and Taking Into Account Anisotropy by Space Transformation
math formula math formula math formula math formula
 −328.7 (49.56)0.0845 (0.0216)0.1598 (0.0213)
math formula math formula math formula math formula
 −38.39 (14.81)0.0726 (0.0163)0
math formula math formula math formula math formula
 0.0536 (0.0294)00

[47] For this best linear model, the location parameter µ is a function of both longitude and latitude, while the scale parameter σ depends on longitude only, and the shape parameter ξ is constant. At a constant altitude (2000 m), we obtain the highest location parameters µ in the northeast Alps (Mont Blanc) and the highest scale parameters σ in the southeast (extreme Southern Alps) which corresponds to the Mediterranean effect that tends to bring greater variability. The shape parameter is equal to 0.054 indicating most likely a Fréchet domain and hence an increase of the quantile with the return period stronger than predicted by a Gumbel model. However, the 95% confidence interval (CI) of ξ contains 0 (Table 2) and the Gumbel model could thus also constitute a sensible one. Globally, we note that these results do not fully account for the spatial patterns displayed in Figures 3 and 4, which justifies the further improvement using spline models (see section 4.4).

4.3.2. Directional Effect and Extremal Dependence

[48] Using the elliptic transformation matrix (equation (9)), the standard space was transformed iteratively for values of ρ in the 1–4 range and ψ in the 20°–80° range. With model 4, the lowest TIC values were found for transformation parameters ψ = 62.5° and ρ = 2.05, leading to extremal dependence parameters (equation (8)) equal to a = 15.5 and b = 0.8. The extremal coefficient obtained by this model is represented in Figure 9a with reference to Chamonix-Mont Blanc. A very important directional effect along the α = ψ = 62.5° axis is indeed observed. The range (distance corresponding to θ = 1.9) is maximum (about 185 km) along the direction of the local alpine axis (α = 62.5°) and is only about 85 km along the perpendicular axis (Figure 9b). This result is in agreement with the empirical analysis which showed that the main dependence direction was belonging to the 51°–77° range and thus confirms the importance of mountain barriers and valleys on extreme snowfalls.

Figure 9.

Extremal coefficient provided by the best linear model (anisotropic Brown-Resnick model 4). (a) Spatial evolution with reference station: Chamonix-Mont Blanc. (b) Evolution with the distance in the ψ direction (direction 1) and in the orthogonal direction (direction 2). The dots represent the empirical extremal coefficient of pairs whose direction belongs to a 90° cone around the ψ direction (cone 1: math formula) and around the orthogonal direction (cone 2: math formula).

4.3.3. Quantile Estimation

[49] The estimated GEV parameters (Table 2) allow us to compute in any location the quantile math formula for a return period T:

display math(20)
display math(21)

[50] Figure 10 shows the maps of the 3 day extreme snowfalls for a return period of 100 years at 2000 m and projected onto the local relief taking into account the actual altitude after application of the inverse gradient (equation (13)). We first note that the 3 day extreme snowfalls are the highest at the border with Switzerland and Italy. Moreover, even if Figure 10b seems to be mainly governed by the altitudinal effects, the strong regional patterns clearly visible in Figure 10a still significantly influence the 100 year quantile. For instance, the 100 year quantiles predicted in the Haute-Maurienne massif culminating at 3751 m are significantly higher (>400 mm w.e.) than those in the Pelvoux massif culminating at 4102 m (<300 mm w.e.).

Figure 10.

Maps of the 100 year quantile (a) at a fixed altitude of 2000 m and (b) projected on the relief using the best linear model from Table 2.

4.3.4. Standard Errors and Pointwise/Spatial Comparison

[51] The relative error of the 100 year quantile at a constant altitude (2000 m) was calculated from the standard errors on the GEV parameters math formula, and Δξ:

display math(22)

with math formula, and math formula. Matrices S µ , S σ , and S ξ are the asymptotic covariance matrices of the parameters math formula, and math formula, and X is the “design” vector (math formula).The 100 year quantile standard error at 2000 m is represented in Figure 11a. Note that the relative error does not exceed 26% where the data are available and is the highest in the extreme Southern Alps and in western regions where the spatial interpolation provides less information due to a lower spatial dependence (cf. Figure 7a) and/or to the proximity of the boundary of the studied domain. This error can be considered relatively low given all the assumptions made (linear evolution of the GEV parameters, orographic gradient, etc.). For instance, it is lower than in the case where dependence between stations would not have been accounted for, thus allowing more confident predictions of high quantiles.

Figure 11.

(a) Relative error on the 100 year quantile z 100 at 2000 m. (b) Comparison between the quantile estimated pointwise for a given station and the quantile predicted by the spatial linear model at the location of the station for different return periods. The red curve represents the average curve; the black dashed curves represent 95% CI around the average calculated from the different curves (average±2 × standard deviation).

[52] A pointwise/spatial comparison at a constant altitude of 2000 m is then used to demonstrate the accuracy of our model. In principle, fitting a MSP with spatial evolutions of the GEV parameters allows us improving the pointwise estimation by “sharing information between stations.” However, the results provided are not an exact interpolation of the pointwise quantiles, which constitutes a large difference with smoothed quantiles obtained using kriging techniques [Weisse and Bois, 2001]. Consequently, it must be checked that the spatial model is not “too far” from the pointwise prediction. Figure 11b shows for each station the comparison between the pointwise quantile projected at 2000 m using equation (13) and the quantile predicted by the spatial model at 2000 m for different return periods. We first note that the average curve is very close to the first bisector, which shows that the overall spatial model is not biased. In detail, however, the spatial model slightly overestimates the pointwise quantile on average. The most significant errors come from two stations, one in the northern Alps and one in the Southern Alps. Moreover, we note that the estimation for all extreme south alpine stations (four stations in black diamond) is locally biased. This is due to the constant shape parameter ξ = 0.056 in the spatial model which cannot account for the negative pointwise shape parameters found in this region (see section 3.3).

4.4. Spline Models

[53] Thirty-six different models (presented in detail in Appendix A) of spatial evolution of the GEV parameters following equation (12) were tested, using in this case only the Brown-Resnick MSP (which outperforms the Smith and Schlather MSPs regardless of the model used to estimate the spatial evolution of the GEV parameters) and different numbers of spline knots (Figure 8). The knots of the pr-spline are regularly distributed in the considered interval. It can be firstly noted that models with two knots in both coordinates have the lowest TIC values and therefore correspond to the most efficient models since they are more flexible. Beyond the two knots, the estimation has proved impossible because of too many parameters to be estimated compared to the amount of data available. Consequently, models with two knots in both directions were retained and further improved by transforming the standard space according to equation (9) in order to take into account the directional effects. Globally, the introduction of splines for the estimation of the evolution of GEV parameters induces a drastic improvement in the data fitting quality, well quantified by the significant TIC reduction (Figure 8) with regard to the linear models used before. In addition, as will be shown below, the covariates selected in the best spline model correspond better to the ones highlighted in the preliminary empirical analysis (Figure 4).

4.4.1. GEV Parameters

[54] The best model of spatial evolution is model 27 in Figure 8 after space transformation:

display math(23)

where the parameters of the matrix β and the knots parameters of the spline are summarized in Table 3.

Table 3. Parameters of Matrix math formula Evaluated for the Cubic Spline Model 27 on Figure 8 After Space Transformation
math formula math formula (mm)math formula (mm/km)math formula (mm/km3)math formula (mm/km3) 
 −2.705 × 102 1.668 × 10−1 8.536 × 10−6 1.05 × 10−5  
math formula math formula (mm)math formula (mm/km)math formula (mm/km)math formula (mm/km3)math formula (mm/km3)
 −3.596 × 102 9.562 × 10−2 1.472 × 10−1 −5.946 × 10−6 1.013 × 10−5
math formula math formula math formula math formula math formula  
 1.586−7.417 × 10−4 −1.078 × 10−8 −1.112 × 10−7  
Spline knotsmath formula (km)math formula (km)math formula (km)math formula (km) 
 903.9935.31986.32046.2 

[55] Using these pr-splines, the best covariates are the latitude only for the location parameter µ, both latitude and longitude for the scale parameter σ, and latitude for the shape parameter ξ. The scale parameter σ is thus a 2-D-cubic pr-spline depending on both longitude and latitude, while the location µ and shape ξ parameters are only 1-D-cubic pr-splines of latitude.

[56] At a constant altitude (2000 m), the spatial evolutions obtained using the linear model are improved. Figure 12a shows that we obtain the highest location parameters µ in the north (Mont Blanc), but the model also predicts the increase of µ in the extreme Southern Alps (see Figure 4d), which was not the case with linear models. Similarly, Figure 12b shows that the highest scale parameters σ are still in the southeast (extreme Southern Alps), but the attenuation of variations of σ in the northern Alps (Figures 4b and 4e) is better accounted for. The shape parameter ξ has an evolution with latitude (Figure 12c): it is negative in the extreme Southern Alps, positive in the rest of the Alps, and slightly negative again far north. This evolution enables to account for the different attraction domains empirically evaluated (Figure 4f) and controls the spatial variability of the quantile increase with the return period. For instance, this increase is stronger in Villard-de-Lans and Saint Veran (ξ>0) than in Chamonix-Mont Blanc (ξ ≈ 0) and stronger in Chamonix-Mont Blanc than in Tende (ξ<0; see Figure 17).

Figure 12.

Spatial evolution of the GEV parameters using the spline model (equation (12)). (a) µ, (b) σ, and (c) ξ at a constant altitude of 2000 m.

[57] Finally concerning the extremal dependence, there are no significant changes compared to the previous best fitted linear model. The directional effect found has the same direction ψ = 62.5° and intensity ρ = 2.05. The only difference is the value of the extremal dependence parameters which are slightly lower than for the linear model: a = 12.0 and b = 0.71. However, this does not significantly change the range of extremal dependence. Accordingly, the evolution of the extremal coefficient represented in Figure 9 remains nearly identical for the best spline model.

4.4.2. Quantile Estimation

[58] Equation (21) is again used to compute the values of the 100 year quantile at 2000 m (Figure 13a) and projected on the Alps relief (Figure 13b). One can note that the map of the 100 year quantile at 2000 m (Figure 13a) is close in terms of global evolution to the one obtained with the linear model (Figure 10a) but locally refined. In particular, low values of the quantile at the west of the Alps predicted by the linear model are increased by the introduction of splines. Figure 14 represents a map of the ratio between the 100 year quantile from the spline model math formula and the 100 year quantile from the linear model math formula. The “spline quantile” is generally higher compared to the “linear quantile” on the west side of the Alps (more than 25% in the Vercors massif, for instance). On the contrary, it is lower on the east side (approximately 10% lower in the Queyras and Parpaillon massifs, for instance).

Figure 13.

Maps of the 100 year quantile (a) at a fixed altitude of 2000 m and (b) projected on the relief using the cubic spline model (24).

Figure 14.

Map of the ratio between the 100 year quantile computed using pr-splines math formula and the 100 year quantile computed using linear models math formula.

4.4.3. Pointwise/Spatial Comparison and Improvement Compared to the Linear Model

[59] A comparison between the pointwise quantile at 2000 m and the quantile from the spline spatial model at 2000 m was performed (Figure 15a). We first note that the average curve is even closer to the first bisector than with the linear model. Hence, the spline model is more accurate on average than the linear one. In addition, the bias present in the Southern Alps with the linear model (Figure 11) has been removed by the introduction of pr-splines. Nevertheless, the two stations already highlighted before (one in the northern Alps and the other in the Southern Alps) still show a significant difference compared to the pointwise estimate.

Figure 15.

(a) Comparison between the quantile estimated pointwise for a given station and the quantile predicted by the spatial model with cubic pr-splines for different return periods. The red curve represents the average curve, and the dashed curves represent the 95% CI around this average (mean±2× standard deviation). (b) Comparison of the NRMSE obtained with the linear and spline models for all the French Alps and for each alpine zone.

[60] In order to further characterize the improvement of the model by the introduction of cubic pr-splines, we calculated the NRMSE (normalized root-mean-square estimator), both for all the French Alps and for each alpine region:

display math(24)

where math formula is the T year quantile for station i given by the spatial model, math formula is the same quantity estimated pointwise for the station i, math formula is the number of stations, and math formula years. One can notice (Figure 15b) that the spline model improves the results compared to the linear model for all the four alpine zones. In detail, the main source of error of the linear model comes from the extreme south of the Alps (math formula). This error is significantly reduced (math formula) by the introduction of pr-splines, and in particular by the spline model for the shape parameter ξ (now accounting for the Weibull domain in this zone). The error in other alpine areas is reduced by about 2%. Globally, for all the French Alps the NRMSE decreases from 13.2% for the linear model to 11.8% for the spline model. This global decrease seems relatively low compared to the drastic improvements noted in the extreme Southern Alps, but it is mainly due to the low number of weather stations in this zone (only 4 of the 40 total stations).

4.5. Prediction Accuracy: Validation on Nonused Stations

[61] The stations that were not retained for the modeling (squares in Figure 1a) have been used to demonstrate the prediction accuracy of the best spatial model with pr-splines (equation (24)). Figure 16a shows the result of this comparison with the 95% CI determined on the calibration sample at a constant altitude of 2000 m. Figure 16b shows the same result but projected onto the local altitude using equation (13). Only 5 stations on average are out of the 95% CI for a validation sample of 84 stations which is slightly more than 5% but still very reasonable given the many assumptions made (spatial evolution of the GEV parameters, orographic gradient, etc.). The value of the NRMSE at 2000 m is 13.15%, a little more than for the calibration sample, and 11% when projected onto the French Alps relief. This result is quite satisfactory and confirms the ability of our spatial model to predict high quantiles all over the French Alps. It also suggests that the chosen calibration sample was large enough to be representative of the main spatial patterns over the considered region. Furthermore, the fact that the NRMSE computed for the local altitude is lower than the one in the 2000 m case suggests that our simple way of handling altitude is appropriate. To explain this decrease of the NRMSE, one can note that for the 2000 m comparison both the data used for the pointwise fitting and the spatial modeling are transformed at 2000 m using equation (13). On the contrary, for the comparison projected onto the local relief, the pointwise fitting is done directly on the nontransformed data, while the results from the spatial model at 2000 m are then projected back on the local altitude using the inverse orographic gradient (equation (13)). Apparently, with this latter procedure, the error propagation is less important than in the 2000 m comparison.

Figure 16.

Cross-validation for nonused stations: comparison between the quantile estimated pointwise for a given station and the quantile predicted by the spatial model with cubic pr-splines for different return periods. The red curve represents the average curve, and the dashed curves represent the 95% CI computed from the calibration sample. (a) Results at 2000 m and (b) results projected onto local altitude.

Figure 17.

IDF curves for (a) Chamonix-Mont Blanc (Mont Blanc massif), (b) Villard-de-Lans (Vercors massif), (c) Saint Veran (Queyras massif), and (d) Tende (Mercantour massif). The curves are represented for different return period values (10, 30, and 100 years).

5. Discussion

5.1. Comparison With Previous Work

[62] MSPs have been only seldom used to characterize spatial variations of extreme hydrological quantiles. Thus, a comparison with the few existing applications is worthwhile.

[63] First, we have shown that for all tested models Brown-Resnick and Smith MSPs give better results than the Schlather one. On the contrary, Blanchet and Davison [2011] show that for extreme snow depth data in Switzerland, the Schlather MSP gives much lower TIC values than the Smith MSP, without testing the Brown-Resnick one. This shows that there is a significant difference in the spatial structure of extreme snowfalls (in water equivalent) and of extreme snow depths at the ground level. This difference is likely to be related to the fact that spatial evolution of snow depth is much smoother than that of snowfalls due to cumulative effects involved in the formation of snow cover (successive snowfalls and snow metamorphism). In contrast, the asymptotic independence is necessary for extreme snowfalls, i.e., θ → 2 for large distances, which is not the case with the Schlather MSP and explains its poor adjustment power in our case.

[64] Second, the anisotropy of the extremal dependence highlighted in our results is very similar to the one found by Blanchet and Davison [2011] in the Swiss Alps for extreme snow depths and to the one shown by Padoan et al. [2009] for U.S. precipitation data. Indeed, although the form of dependence highlighted is not the same for extreme snowfalls (Brown-Resnick), and for extreme snow depths and rainfall (Schlather), it is marked in all cases by an important and similar directional effect related to orography and its main direction. The local Alps direction in Switzerland is closer to E–W than in France, explaining why Blanchet and Davison [2011] found an anisotropy angle α Swiss = 20°. This angle corresponds to the direction of the widest valleys in Switzerland (Rhône and Rhine River valleys). Similarly, in Padoan et al. [2009], the main anisotropy corresponded to the orientation of the Appalachian mountains.

5.2. Influence of the Accumulation Period

[65] To investigate the influence of the accumulation period, the same procedure of fitting a MSP to data (section 4) has been repeated for annual snowfall maxima computed over 1, 5, and 7 days, using spline models for the GEV parameters and the anisotropic Brown-Resnick MSP.

5.2.1. Intensity-Duration-Frequency

[66] Intensity-duration-frequency (IDF) curves enable the synthesis of snowfall information at given station and thus constitute an interesting tool for risk management. Four stations in each alpine zone were selected from the maps to produce IDF curves and to compare the evolution of the quantile in these different areas with the duration of accumulation and return period (Figure 17):

[67] Northern Alps: Chamonix-Mont Blanc, altitude: 1042 m (Mont Blanc massif)

[68] Central Alps: Villard-de-Lans, altitude: 1050 m (Vercors massif)

[69] Southern Alps: Saint Veran, altitude 2010 m (Queyras massif)

[70] Extreme Southern Alps: Tende, altitude: 650 m (Mercantour massif).

[71] First, it can be noted in Figure 17 that the quantile strongly increases with the duration of accumulation in Chamonix-Mont Blanc (Figure 17a) and Tende (Figure 17d), reaching more than 370 mm w.e. for a return period of 100 years and 7 days of accumulation. The increase is weaker in Villard-de-Lans (Figure 17b) where the 100 year quantile reaches 290 mm w.e. for a 7 day accumulation duration. In Saint Veran (Figure 17c), the quantile evolution levels out at a value of about 250 mm w.e., this maximum being already almost attained for a 3 day accumulation duration. This result illustrates the longer persistence of heavy snowfalls in the northeast (Mont Blanc) and in the extreme southeast of the Alps. On the contrary, it indicates that intense episodes are much shorter in the Queyras massif and a bit shorter in the pre-Alps (low- and mid-altitude massifs).

5.2.2. Directional Effect

[72] To examine whether the directional effect highlighted in section 4 is influenced by the accumulation period, the ψ angle and the elongation parameter ρ of the space transformation matrix (equation (9)) are represented in Figure 18 as functions of the accumulation period. We can firstly note that the angle ψ of the transformation is almost constant, varying between 62.5° and 65.2° and hence corresponding to the local alpine axis for all the accumulation periods. This is presumably explained by the interaction between topography and predominant atmospheric flows and may therefore be a rather general result for hydrological variables.

Figure 18.

Influence of the accumulation period on the parameters of space transformation ψ (anisotropy angle) and ρ (elongation parameter).

[73] On the contrary, the elongation parameter of the matrix ρ and thus the intensity of the transformation are strongly influenced by the accumulation period. The directional effect is globally all the more important that the accumulation period is short (ρ = 3 for a 1 day accumulation period and ρ = 1.8 for a 7 day accumulation period). This greater sensitivity to the local relief (presence of valleys and mountainous barriers) of short extreme events with regard to more persistent ones complements and refines the preliminary results of Eckert et al. [2011] obtained with a first version of the model. Physically, it may indicate that persistent snow storms diffuse progressively in all directions while interacting durably with the topographic barriers, thus reducing the difference between the dependence ranges along the two principal anisotropy axes of the extremal coefficient.

5.3. Joint Analysis

[74] Quantifying the joint occurrence of extreme events in different locations can be very useful from an operational perspective. The joint probability of exceeding the T year quantiles math formula and math formula in two locations x and x′ can be computed by transforming the margin distributions in unit Fréchet and using equation (5):

display math(25)

[75] This probability was computed for four pairs of stations representative of different cases (Figure 19): low or large euclidean distances and angles α close to the main direction of dependence or almost orthogonal to it. Globally, we can note that the estimation lies within the case of complete dependence (θ = 1) and total independence (θ = 2) in all cases but is less close to total independence for low euclidean distances (Figures 19a and 19b) than for large distances (Figures 19c and 19d). The dependence is also more pronounced if the angle α between stations is close to the main dependence direction ψ = 62.5° (Figure 19a). For the same distance between stations (d ≈ 20 km), the extremal coefficient increases from θ = 1.6 to 1.7 for pairs of stations almost orthogonal one to each other (Figure 19b). In the case of large distances, the joint probability is very close to the one corresponding to the total independence between pairs of stations (Figures 19c and 19d), with θ → 2. However, the influence of the directional effect is still noticeable (dependence a bit stronger in Figure 19c than in Figure 19d). This again shows the need of using MSP enabling total asymptotic independence for modeling the extreme snowfalls, while extreme snow depth at ground level (much smoother than snowfalls) can be modeled with an asymptotic dependence θ<2 at large distance [Blanchet and Davison, 2011].

Figure 19.

Joint exceedence probability math formula for four couples of stations of the calibration sample. (a) Challes-les-Eaux (291 m, Bauges)-Lescheraines (590 m, Bauges), (b) Monestier de Clermont (800 m, Oisans)-Pellafol (930 m, Dévoluy), (c) Megève (1104 m, Mont Blanc)-Villard-de-Lans (1050 m, Vercors), and (d) La Mure (856 m, Oisans)-Guillaumes (620 m, Alpes-Azureennes). Black dots: empirical estimation; red curve: theoretical expression (equation (25)); green curve: case of perfect dependence (θ(x, x′)=1); blue curve: case of total independence (θ(x, x′)=2).

[76] Additionally, we compared the modeled joint exceedence probability P mod to the empirical one P emp (Figure 20) for return periods between 1 and 100 years. As well as for the quantile comparison, we remark that the model is only very slightly biased, with the mean curve nearly aligned with the first bisector. In detail, however, the model appears to slightly underestimate the empirical values. This can be explained by the fact that some empirical values of θ have been constrained to 2, while the modeled extremal coefficient is always strictly lower than 2. Beyond that, Figure 20 confirms the essential contribution of MSPs compared to previous approaches that usually assume independence between stations. It is clearly observed that on average a model with θ = 2 significantly underestimates the empirical values, especially for short distances between pairs of stations and/or high return periods.

Figure 20.

Comparison between the empirical joint exceedence probability P emp and the modeled joint exceedence probability P mod given by equation (25) for all the pairs of stations. The red curve represents the average curve, the green one is for a perfect dependence between stations (θ=1 in P mod; equation (25)), and the blue one for total independence (θ=2 in P mod; equation (25)). The scale is logarithmic in the main graph and linear in the inset.

5.4. Conditional Quantile Evaluation

[77] Quickly updating the unconditional quantile maps after an intense snowfall at one location can be very useful from an operational point of view. From the previous joint analysis, conditional return level maps can easily be obtained. We can define the conditional probability of exceeding the T year return level in location x knowing that the T′ year return level was exceeded in location x′ as

display math(26)

with math formula. Using the general expression of the bivariate probability [Brown and Resnick, 1977; Kabluchko et al., 2009; Davison et al., 2012],

display math(27)

with math formula, and math formula, we can determine the conditional quantile math formula by numerically solving the following equation:

display math(28)

[78] An example of this conditional return level for T=T′ = 30 years is plotted in Figure 21 for a reference station located in the Champsaur massif. Its spatial pattern is a combination between the shape of the extremal coefficient and of the 30 year quantile map. For instance, knowing that a snowfall has exceeded z 30 ≈ 150 mm w.e. in this station leads to a local increase of the 30 year return level of 35% at 2000 m (conditional quantile z 30 ≈ 220 mm w.e.). Moreover, the directional effect has a major influence on the spatial evolution of the conditional quantile. For instance, 100 km far from the reference station, in the main direction of dependence, an increase of the conditional 30 year return level is still noticeable, whereas almost no influence is observed in the perpendicular direction at 50 km only.

Figure 21.

Conditional return level maps for T=T′=30 years. (a) 2000 m and (b) local altitude. Conditional reference station: Chapelle-en-Valgaudemar (1270 m, Champsaur massif).

6. Conclusion

[79] In this study, extreme snowfalls have been evaluated in the French Alps by mapping snowfall water equivalent annual maxima at 40 measurement stations. The mathematical formalism of MSPs, which generalizes EVT to the multivariate spatial context, has been used. It has been shown in particular that the Brown-Resnick model provides an extremal coefficient that fits the data better than those of Smith and Schlather which are less flexible. Additionally, space transformation has been used to model anisotropy, which has further improved the adjustment. Hence, it appeared that the spatial extremal dependence depends strongly on the local orientation of the alpine axis and the presence of large valleys. For example, for a 3 day accumulation period, the dependence range is more than twice as high in this alpine axis direction in comparison to the orthogonal direction. In addition, we refined this important result, which was also obtained in other hydrological applications, by showing that the intensity of the directional effect is all the more important when the duration of accumulation is low.

[80] Linear models and pr-splines for the evolution of the GEV parameters with space were compared. Slightly lower NRMSE values were obtained with the retained spline model when considering the whole French Alps, the most significant improvement brought by the spline model being found in the extreme Southern Alps. More generally, spline models were shown to provide a better modeling of complex evolutions of GEV parameters with space. At a constant altitude (2000 m), the highest location parameters µ are very north (Mont Blanc, Aravis and Bauges massifs), although high values are also observed far south. The highest scale parameters σ are in the southeast (extreme Southern Alps) which corresponds to the Mediterranean effect that tends to bring variability. The shape parameter is mainly positive in the northern, central, and Southern Alps, showing a Fréchet attraction domain but becomes negative in the extreme Southern Alps (Weibull domain). From the (μ, σ, ξ) maps, the 100 year snowfall quantile could be determined at any point in the French Alps. In detail, it has also been shown that the 100 year quantile for a 3 day accumulation period is the highest in the central and Southern Alps at the boarder with Switzerland and Italy. This analysis was also performed for different periods of accumulations, showing the variability of the persistence of heavy snowfall events across the French Alps.

[81] These results, and more particularly quantile maps, constitute a powerful operational tool for long-term management of avalanche risk, especially to establish hazard maps or as inputs of propagation models [Naaim et al., 2003]. Besides, as shown in Gaume et al. [2012], these results can be rigorously coupled with a mechanical stability criterion to evaluate avalanche release depth distributions that can then be used to perform statistical-dynamical simulations to evaluate runout and pressure distributions [e.g., Eckert et al., 2008].

[82] The employed smooth modeling of GEV parameters associated with MSPs and a nested model selection procedure constitutes the methodological strong point of the work. We also studied how a joint analysis can be performed to evaluate the risk of obtaining an extreme event in two different locations within the same year, leading to conditional return level maps which can also be very useful from an operational perspective. Validation on other available data has shown the accuracy of the fitted model and of the simple way of handling altitudinal effects which has been proposed. This up-to-date framework could be put in use for various other applications in hydrology such as heavy rainfalls or flood extremes, as soon as a sufficient sample of long block maxima series is available.

[83] Finally, since the main objective of our work was prediction as a continuous function of space, we only used the geographical coordinates as covariates. However, additional physical quantities could be further introduced as covariates in the model, such as the WMS which has been used in some previous studies, in order to select the best physical drivers of the spatial trends we highlighted. Another improvement perspective of the model is the test of very recently developed MSPs which were not considered here. Additionally, a preselection of the data corresponding different meteorological flux types (NNW/SSE fluxes) would be interesting in order to investigate whether dependence patterns in extreme snowfall vary accordingly.

Appendix A: Covariates for the Spatial Evolution of the GEV Parameters

[84] Table A1 summarizes the different spatial evolution models used for the GEV parameters.

Table A1. Details on the Covariates Used for the Different Evolution Models of the GEV Parameters With Spacea
ModelTypeCovariates µ Covariates σ Covariates ξ
LatitudeLongitudeLatitudeLongitudeLatitudeLongitude
  1. a

    The numbers correspond to the model index in Figure 8. The last seven models are denoted “mixed” since they involve spline evolutions of µ and σ and a linear evolution of ξ with space.

1Linear××××  
2Linear ×××  
3Linear× ××  
4Linear×× ×  
5Linear × ×  
6Linear×  ×  
7Linear×××   
8Linear ××   
9Linear× ×   
10Linear××××××
11Linear ×××××
12Linear× ××××
13Linear×× ×××
14Linear × ×××
15Linear×  ×××
16Linear××× ××
17Linear ×× ××
18Linear× × ××
19Spline××××  
20Spline× ××  
21Spline×××   
22Spline× ×   
23Spline × ×  
24Spline ××   
25Spline×  ×  
26Spline××××× 
27Spline× ××× 
28Spline××× × 
29Spline× × × 
30Spline × ×× 
31Spline ×× × 
32Spline×  ×× 
33Spline×××× ×
34Spline× ×× ×
35Spline×××  ×
36Spline× ×  ×
37Spline × × ×
38Spline ××  ×
39Spline×  × ×
40Spline××××××
41Spline× ××××
42Spline××× ××
43Spline× × ××
44Spline × ×××
45Spline ×× ××
46Spline×  ×××
47Mixed×××××lin  
48Mixed× ×××lin  
49Mixed××× ×lin  
50Mixed× × ×lin  
51Mixed × ××lin  
52Mixed ×× ×lin  
53Mixed×  ××lin  

Acknowledgments

[85] Support from the European Interreg DYNAVAL and MAP3 projects and the ANR MOPERA is acknowledged. The authors are grateful to MeteoFrance and the French Environment Ministry for data acquisition and supply. We also wish to express our gratitude to M. Ribatet for advices while using the SpatialExtremes R package, to G. Toulemonde and J.-N. Bacro for comments about the text, and to three anonymous reviewers and the Associate Editor J. Montanari for their insightful and constructive comments.

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