In a context of climate change, trends in extreme snow loads need to be determined to minimize the risk of structure collapse.
We study trends in 50-year return levels of ground snow load (GSL) using non-stationary extreme value models. These trends are assessed at a mountain massif scale from GSL data, provided for the French Alps from 1959 to 2019 by a meteorological reanalysis and a snowpack model.
Our results indicate a temporal decrease in 50-year return levels from 900 to 4200 m, significant in the northwest of the French Alps up to 2100 m.
We detect the most important decrease at 900 m with an average of

Extreme snow loads can generate economic damages and casualties. For instance, more than USD 200 million in roof damages occurred
during the Great Blizzard of 1993

Ground snow load (GSL) is defined as the pressure exerted by accumulated snow on the ground, which can be directly associated with accumulated snow on structures, e.g. on roofs

Snowpack variables related to GSL (snow depth, SWE) evolve with climate change. As shown in Table

Past trends in snowpack variables, snow depth (HS) and snow water equivalent (SWE), according to existing studies in the Western Alps, i.e. in Italy (IT), France (FR) and Switzerland (CH). In the Trend column, “north” and “south” refer to the considered country.

The impact of climate change on GSL was not taken into account in current European standards for structural design, a.k.a Eurocodes

We fill these gaps by studying annual maxima of GSL provided every 300 m of altitude at a mountain massif scale for the 23 French Alps massifs. We rely on the SAFRAN–Crocus reanalysis

Our statistical methodology consists in applying stationary and non-stationary extreme value models to annual maxima time series. We select one model by massif and altitude with the Akaike information criterion (AIC) statistical criterion, validate the selected model with the Anderson–Darling test, and assess its significance with the likelihood ratio statistical test. Finally, for each massif and altitude, we compute the relative change of 50-year return levels of GSL between 1960 and 2010, and we compare the non-stationary return level in 2019 with the stationary return level designed for French building standards.

This paper is organized as follows. Section 2 presents our data. Section 3 describes standards for ground snow load. Then, Sect. 4 explains our methodology. Results, discussion and conclusions are introduced in Sects. 5, 6 and 7, respectively.

The study area covers the French Alps which are located between Lake Geneva to the north and the Mediterranean Sea to the south (Fig.

To sum up, GSL equals SWE from the SAFRAN–Crocus reanalysis times the gravitational acceleration. We study time series of annual maxima of GSL for each massif from 1959 to 2019 every 300 m of altitude from 300 to 4800 m (Fig.

The SAFRAN–Crocus reanalysis is produced by a chain of two models. First, SAFRAN meteorological reanalysis

The SAFRAN–Crocus reanalysis has been evaluated against various observation datasets, as reported in previous publications

GSL French standards

French standards were elaborated with annual maxima time series of snow depth on the ground measured at stations from 1945 to 1992. GSL data were approximated from annual maxima of snow depth and by assuming that snow density
equals

Following extreme value theory, we employ two stationary models and six non-stationary models for time series of annual maxima of GSL (Sect.

Climate extremes are generally studied with statistics.
As underlined in the IPCC special report on climate extremes,
a large amount of
statistical literature
builds on extreme indices to examine moderate extremes

EVT offers a suitable framework to analyse extreme values, i.e. to model the form of the tail for almost any probability distribution.
Asymptotically, as the central limit theorem motivates sample means modelling with the normal distribution, the Fisher–Tippett–Gnedenko theorem

In a context of climate change, a large amount of hydrological literature builds on non-stationary modelling

We consider non-stationarity for both the Gumbel distribution and the more general GEV distribution,
since they represent natural extensions of the Gumbel distribution which was used for French building standards (Sect.

Statistical models considered for annual maxima of GSL are based on the Gumbel or the GEV distribution
and are extensions of the stationary Gumbel model.
For non-stationary models, the location and/or the scale vary linearly with years

Then, for each

The selected model

In a stationary context, the

In a non-stationary context, return level and return period concepts

For the stationary Gumbel model

For the selected model

For any considered model, the time derivative of the return level is constant, as

In the context of maximum likelihood estimation, uncertainty related to return levels can be derived by the delta method, which quickly provides confidence intervals both in the stationary and non-stationary case

First, we exclude four time series of annual maxima with more than

Figure

Distribution of selected models. Frequency of selected model (in %)
with respect to all time series, i.e. for all massifs and altitudes.
For the selection procedure and the definition of significance, we refer to Sect.

Figure

Shape parameter values for the selected models at low (900 m), mid (1800 m) or high (2700 m) altitude. Markers show selected model

In Fig.

Figure

Trends in 50-year return levels of ground snow load (GSL) between 1960 and 2010 at low (900 m), mid (1800 m) or high (2700 m) altitude. Markers show selected model

Figure

Temporal decreasing trend of 50-year return levels of ground snow load (GSL) between 900 and 4800 m of altitude.

Figure

To sum up trends in return levels of ground snow load, from 900 to 4800 m, either no trends or decreasing trends of 50-year return levels of GSL are found (Figs.

We compare 50-year return levels of GSL and their uncertainty (Sect.

50-year return levels of ground snow load (GSL) from altitude 300 to 1800 m for Vercors

Figure

Figure

Comparison of 50-year return levels of ground snow load (GSL) with French standards between 300 and 1800 m. We show the percentage of massifs (green histogram) whose return levels exceed French standards and the mean relative difference (blue line) between return levels and standards.

First, if we estimate return levels from data with the French standards method (Fig.

However, if we consider the actual GSL, i.e. computed with SWE from the reanalysis, then French standards drastically underestimate return levels.
Indeed, with a stationary Gumbel model

Furthermore, despite the fact that uncertainty intervals (black bars) can be large, it does not impact the main conclusions of this article.
Indeed, in Fig.

We discuss in depth the statistical models chosen for this study.
It is well-known that an annual-maximum-based approach can be wasteful in terms of data

For the non-stationary models, we focus on simple deterministic functions of time (

We decided to consider non-stationarity only for the location and scale parameter. Indeed, in the literature, a linear non-stationarity is considered sometimes only for the location parameter

For time series containing zeros, French standards rely on a mixed discrete–continuous distribution.
They fit both a Gumbel distribution on non-zero annual maxima and the probability of having a non-zero annual maxima.
However, with our reanalysis data, this approach sometimes leads to fitting non-stationary extreme value models with fewer than 40 non-zero annual maxima.
Therefore, we rather decided to exclude any time series with more than 10 % of zeros (Sect.

Limitation of approximating annual maxima of ground snow load (GSL) from annual maxima of snow depth (HS).

SWE times the gravitational constant equals GSL.
However, most countries do not measure SWE but only have access to snow depth (HS)

We find that annual maxima of GSL are always underestimated by French standards' approximation (Fig.

Based on both a reanalysis and a snowpack model, we detect an overall temporal decreasing trend of 50-year return levels of ground snow load (GSL) between 900 and 4200 m, which is significant up to 2100 m in the northwest of the French Alps.
This confirms other studies in the Western Alps which also found overall decreasing trends in linked snowpack variables: SWE and snow depth.
The largest decrease is found at 900 m with

We hypothesize that this number of exceedances might be due to an underestimation of GSL by French standards. Indeed, these standards were devised with GSL estimated from snow depth maxima and constant snow density equal to

Many potential extensions of this work could be considered.
First, our methodology could be extended with more advanced definitions of non-stationary return levels

Finally, even if, according to our analysis, GSL exceeds French standards return levels in the French Alps, (Fig.

In this section, we report, for every 300 m of altitude from 900 to 4200 m, the map of the relative change of 50-year return levels of GSL between 1960 and 2010 (Fig.

Trends in return levels of ground snow load (GSL) between 900 and 4200 m of altitude. Markers show selected model

Quantile–quantile (

In Fig.

In practice, with this test, we assess whether the transformed annual maxima

We apply this test on the transformed data using

The dataset can be downloaded from the AERIS website:

ELR performed the analysis and drafted the first version of the manuscript. All authors discussed the results and edited the manuscript.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Advances in extreme value analysis and application to natural hazards”. It is a result of the Advances in Extreme Value Analysis and application to Natural Hazard (EVAN), Paris, France, 17–19 September 2019.

The S2M data are provided by Météo-France – CNRS, CNRM Centre d'Etudes de la Neige, through AERIS. We are grateful to Eric Gilleland for his “extRemes” R package. Finally, we are indebted to Jacques Biétry for providing us the report on French standards with respect to ground snow load and for his explanations on their methodology.

Erwan Le Roux holds a PhD grant from INRAE.

This paper was edited by Ivan Haigh and reviewed by two anonymous referees.