Anticipating risks related to climate extremes often relies on the quantification of large return levels (values exceeded with small probability) from climate projection ensembles. Current approaches based on multi-model ensembles (MMEs) usually estimate return levels separately for each climate simulation of the MME. In contrast, using MME obtained with different combinations of general circulation model (GCM) and regional climate model (RCM), our approach estimates return levels together from the past observations and all GCM–RCM pairs, considering both historical and future periods. The proposed methodology seeks to provide estimates of projected return levels accounting for the variability of individual GCM–RCM trajectories, with a robust quantification of uncertainties.
To this aim, we introduce a flexible non-stationary generalized extreme value (GEV) distribution that includes (i) piecewise linear functions to model the changes in the three GEV parameters and (ii) adjustment coefficients for the location and scale parameters to adjust the GEV distributions of the GCM–RCM pairs with respect to the GEV distribution of the past observations. Our application focuses on snow load at 1500 m elevation for the 23 massifs of the French Alps. Annual maxima are available for 20 adjusted GCM–RCM pairs from the EURO-CORDEX experiment under the scenario Representative Concentration Pathway (RCP) 8.5. Our results show with a model-as-truth experiment that at least two linear pieces should be considered for the piecewise linear functions. We also show, with a split-sample experiment, that eight massifs should consider adjustment coefficients. These two experiments help us select the GEV parameterizations for each massif. Finally, using these selected parameterizations, we find that the 50-year return level of snow load is projected to decrease in all massifs by

The use of climate model simulations is the main scientific paradigm to anticipate extreme climate events. In particular, multi-model general circulation model (GCM) and regional climate model (RCM) ensembles are widely used to quantify the changes in climate extremes and their uncertainties

Temporal non-stationary GEV-based approaches
for GCM ensembles and GCM–RCM ensembles. An asterisk (

Climate extremes are often assessed within the statistical framework of extreme value theory (EVT) by focusing on either annual maxima or values exceeding a high threshold

Most approaches using EVT to study climate extremes from multi-model ensembles (MMEs) rely on stationary generalized extreme value (GEV) distributions estimated separately on each climate simulation of the MME, i.e., with each ensemble member

Temporal non-stationary GEV approaches address these limitations by taking into account all the available annual maxima for each ensemble member, i.e., all the historical and future annual maxima are fitted with a single statistical model

A majority of temporal non-stationary approaches for MMEs rely on the GEV distributions estimated separately with each climate simulation of the MME (Table

The present study introduces an alternative approach that relies on a temporal non-stationary GEV model fitted to all ensemble members. This approach enables us to quantify uncertainties using standard tools from non-stationary extreme value analysis. Such an approach has mainly been proposed for initial condition ensembles (Table

Besides, non-stationary GEV-based approaches for climate projection ensembles usually consider linear functions for the non-stationary functions, with the exception of the study by

We illustrate the proposed methodology with an application to snow load data, which corresponds to the pressure exerted by accumulated snow on the ground (proportional to the snow water equivalent). The probabilistic assessment of ground snow load is of major interest for the structural design of buildings

Section

Our application focuses on snow loads at 1500 m elevation in the 23 massifs of the French Alps, i.e., between Lake Geneva to the north and the Mediterranean Sea to the south (Fig.

The S2M reanalysis

ADAMONT

The anomaly of global mean surface temperature (GMST) with respect to the pre-industrial period (1850–1900) is chosen as the temporal covariate for our statistical methodology. In practice, we smooth this anomaly with cubic splines to obtain a covariate that does not depend on the internal variability of GMST (Fig.

Raw output (dotted lines) and smoothed output (plain lines) for the anomaly of global mean annual temperature with respect to industrial levels (1850–1900). For the six GCMs, we show the anomaly of global mean surface temperature using historical emissions until 2005 and projected emissions (RCP8.5). Years correspond to periods centered on each winter (August–July).

Following the block maxima approach of extreme value theory

In theory, the GEV distribution is adequate when maxima are computed over blocks of infinite size. In practice, the GEV distribution is usually applied to annual maximal values and has been shown to provide reliable estimates of return levels in many hydrometeorological applications

Let

For a GCM–RCM pair

Illustration of the evolution of the location parameter

The five parameterizations of the adjustment coefficients

Finally, the size of the entire vector of parameters

We did not consider adjustment coefficients on the shape parameter because it sometimes leads to prediction failures. This situation can happen when

For each massif, a temporal non-stationary GEV distribution, parameterized by a vector of coefficients

Our first evaluation experiment is a model-as-truth experiment, i.e., a perfect model experiment, which evaluates long-term predictive performances using future projections

Our second evaluation experiment is a split-sample experiment, i.e., a calibration–validation experiment, which enables us to estimate the short-term predictive performance of each parameterization. Specifically, for the calibration of the non-stationary GEV distribution, we rely on the oldest observations from the S2M reanalysis (Sect.

In these two evaluation experiments for GCM–RCM ensembles, we calculate the mean logarithmic score (

First, for a set of past and projected annual maxima, we select one parameterization of the GEV distribution (number of linear pieces, parameterization of the adjustment coefficients) using a two-step selection method: (i) we select the number of linear pieces with a model-as-truth experiment using zero adjustment coefficients for the GEV parameters, and
(ii) we then select the parameterization of the adjustment coefficients with three split-sample experiments where the calibration set is composed of 60 %, 70 % and 80 % of the observations, using the number of linear pieces selected in the model-as-truth experiment. Following this, we study trends in the 50-year return level of snow load. For each massif we rely on the parameterization of the GEV distribution selected using the two-step selection method. We report RL50, the 50-year return level that corresponds to Eq. (

In Fig.

In the first step, we select the number of linear pieces that minimizes the mean logarithmic score of a model-as-truth experiment using zero adjustment coefficients for the GEV parameters. The mean logarithmic score is averaged on the hold-out pseudo-observations (2020–2100) for each of the 12 GCM–RCM pairs (which are set as pseudo-observations; see Sect.

A detailed analysis of the mean logarithmic scores of each parameterization for each massif is provided the Supplement.

Estimated 50-year return levels between

In this section, we rely on the parameterization of the GEV distribution selected in Sect.

The 50-year return levels (RL50) of snow load at 1500 m for

All 50-year return levels (for the non-stationary GEV distribution fitted on the observations, on each GCM–RCM pair, and on the observations and all GCM–RCM pairs) are decreasing with the anomaly of global mean temperature.
We observe that RL50 with adjustment coefficients (shown in a warm color) is closer to the 50-year return level of the observation (in dark gray) than RL50 without adjustment coefficients (in cyan). This figure also shows how adjustment coefficients adjust the distribution toward the distribution of the observations by illustrating the probability density functions (with and without adjustment) at

Figure

Figure

Relative changes in 50-year return levels (RL50) of snow load at 1500 m for

For each massif, it is also possible to compute the average 50-year return level for several time periods: 1986–2005, 2031–2050 and 2080–2099.
For instance, for the time period 1986–2005, the average return level equals the average of the return level found for the years 1986, 1987, …, 2005. In order to compute the return level of a given year, e.g., 1986, we rely on the relationship between the anomaly of global mean surface temperature (GMST) and the years (Fig.

In Table

Projected trends in snow water equivalent (SWE) and snow load under the scenario RCP8.5 using the EURO-CORDEX experiment. In the first four rows of the table, we specify that the result is approximated because the trend was read from the Fig. 2.3 of

For the 23 massifs, the average return level for several time periods (1986–2005, 2031–2050, 2080–2099) can be obtained as explained in Sect.

Figure 2.3 of

For the non-stationarity of the GEV parameters, we choose piecewise linear functions because they can approximate more complex functions with only a few parameters. This makes our methodology widely applicable. One limitation is that the nodes of the piecewise linear functions are fixed. However, we are confident that these functions are well estimated, owing to the large number of maxima: each of the

For the two-step selection method, we first rely on a model-as-truth experiment to select the number of linear pieces. It assesses the optimal number of linear pieces to predict annual maxima of the pseudo-observations for the evaluation set (2020–2100), i.e., to find a good trade-off between underfitting and overfitting for the calibration set. In this first step, adjustment coefficients are not considered, such that this experiment does not depend on a specific parameterization.

Following this, the best parameterization of the adjustment coefficients is selected with a split-sample experiment. It assesses whether applying adjustment coefficients helps to predict observations of the evaluation set, i.e., whether it is reasonable to assume that the observations do not follow the same distribution as the GCM–RCM pairs. The evaluation score is average for three split-sample experiments where the evaluation set corresponds to the last 40 %, 30 % and 20 % of the observations (Sect.

The 90 % confidence intervals of return levels (Fig.

The goodness of fit of the selected models has been tested with the application of the Anderson–Darling test (see Appendix

First, our methodology based on adjustment coefficients can be seen as an extension of

Thus, our approach based on piecewise linear functions for the non-stationarity of the GEV parameters can be viewed as using linear splines. In the literature, there are many extreme value theory approaches using splines. For instance, linear splines have been applied to model the temporal non-stationarity

Following statistical methods that constrain climate projections using past observations

In order to select the best parameterization of the non-stationary GEV model (number of linear pieces, parameterization of the adjustment coefficients), we design a two-step selection procedure based on two evaluation experiments for GCM–RCM ensembles: a model-as-truth experiment and a split-sample experiment. The model-as-truth experiment is first applied to select the number of nodes that are required to adequately represent the evolution of the GEV parameters. The split-sample experiment evaluates the added value provided by the adjustment coefficients for the different possible parameterizations.

In this article, as a case study the proposed approach is applied to snow load in the French Alps at 1500 m elevation using 20 GCM–RCM pairs that are statistically adjusted from the EURO-CORDEX experiment under the scenario RCP8.5. More generally, the proposed approach could also be applied to other scenarios, climate variables, and climate projection ensembles. In contrast with most applications of non-stationary GEV models in the literature (which consider linear trends), the piecewise linear functions proposed in our approach are well suited to non-monotonic trends.

Many extensions of this work could be considered. First, if adjustment coefficients are not included, our parameterization of the GEV model considers the same non-stationary GEV distribution for the different GCM–RCM pairs. Even in the case where adjustment coefficients are selected, the distributions corresponding to the GCM–RCM pairs are still constrained to have the same changes with global warming because adjustment coefficients are constant. In future work, we could imagine adjustment coefficients that vary with global warming being used to better account for different changes of distributions among the GCM–RCM pairs. A second potential extension of this work could be to improve the parameterization of the GEV distribution by adding weights for each GCM–RCM pair. In our methodology, GCM–RCM pairs are currently considered as equally plausible even though it is known that for each application some of them can have a better agreement with the past observations. Following the intuition of weighting schemes for climate ensemble

Finally, further work is needed to obtain a better agreement between the non-stationary GEV model representing the ensemble of maxima from climate projections and past observed maxima. Indeed, observed maxima are mainly used to identify and correct strong disagreements between the observed and simulated maxima using adjustment coefficients, and the fitted non-stationary GEV model is not really constrained to represent observed maxima. A Bayesian approach representing the predictive distribution of climate projections conditional on historical observations

We estimate the uncertainties resulting from in-sample variability
with a semi-parametric bootstrap resampling method adapted to non-stationary extreme distributions

We generate

In addition to the sampling uncertainty, we also assess the goodness of fit of the fitted GEV models with the Anderson–Darling statistical test

The code is publicly available at the following link:

The full S2M reanalysis on which this study is based is freely available from AERIS

The supplement related to this article is available online at:

ELR, GE and NE designed the research. ELR performed the analysis and drafted the first version of the manuscript. All authors discussed the results and edited the manuscript.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We are grateful to Ben Youngman for his “evgam” R package. INRAE, CNRM and IGE are members of Labex OSUG. We are indebted to Raphaëlle Samacoïts from Météo France for providing us the latest version of the climate projection data. The authors also thank the editor, Tamas Bodai, Antonio Speranza and an anonymous referee, who provided constructive and useful comments.

This research has been supported by the Institut National de Recherche en Sciences et Technologies pour l'Environnement et l'Agriculture, IRSTEA, which became INRAE in 2020 (PhD grant).

This paper was edited by Valerio Lucarini and reviewed by Tamas Bodai, Antonio Speranza, and one anonymous referee.