We develop an extension of the statistical approach by

In the context of climate change, extreme events such as heat waves can happen more frequently due to the shift of temperature to higher values. More generally, climate change signals alter the probability distribution of many climate variables, with impacts on the frequency of rare events. More frequent and/or more intense extreme events such as heat waves, extreme rainfall or storms have been shown to have critical impacts on human health, human activities and the broader environment. Over the last decade, there has been extensive research on the attribution of extreme weather and climate events to human influence.

In order to model extremes of a physical variable in a statistical
sense, generalized extreme value (GEV) distributions are commonly used.
Attribution analysis requires deriving the probability of an event occurring in the factual world (our world) and in a counterfactual world (without
anthropogenic signal), which can be done from their respective GEV
distributions

Recently,

Here, we overcome two limitations of their approach. First, we extend
the procedure to nonstationary GEV distributions. Second, while only the
stationary parameters and the covariate were constrained by observations in

In Sect.

To present our methodology, we propose implementing an attribution
analysis of the extreme heat wave of July 2019. We focus on France (42–51

The time series of observations of

List of CMIP5 models used in the literature

For climate models, we extract a simulated time series of

We use the summer mean temperature anomaly with respect to period 1961–1990 over Europe as a covariate, noted as

To summarize, we have the following:

The observation

The observation

A total of 26 time series of

A total of 26 time series of

A goal, in attribution, is to find the probability of the realization of the 2019 event, in the factual world (our world) and in the counterfactual world (without human influence). Noting

Classically, for the maximum temperature, the random variable

In the literature, the covariate

Our goal is to infer the probability distribution of

Here, we estimate the covariate

We have represented, in Fig. S1, the two covariates and their respective uncertainty for three climate models. We can see different behaviors, with the model MIROC–ESM–CHEM increasing to

This procedure (decomposition and perturbed realizations) is applied to each of the 26 realizations

We start by defining four submodels of
Eq. (

Tree representing the likelihood ratio test (LRT) performed in
Sect.

The parameters of all GEV model are fitted from the realizations of

Results of likelihood ratio test (LRT) to find which GEV model can be used for each CMIP5 model (

Results of our GEV model selection procedure are shown in Fig.

First, all models agree with the consideration that

Overall, these results suggest that we have to take into account a nonstationarity in the trend and in the scale or the shape. Because the MLE can swap information between the scale and the shape, and a small error of
estimation in the shape can have a big impact on the fit, we choose to select
the model

At this stage, we have

The goal is to synthesize our 26 distributions of

The effect of the multimodel synthesis is shown in Fig.

So, with this approach, we obtain a good candidate

To derive the posterior of

The posterior of

The last step is to draw from the posterior of the Eq. (

Finally, we obtain set of nonstationary parameters of a GEV distribution from a multimodel synthesis. It is fully constrained via Bayesian techniques. We have a synthetic description of the variable

Once the parameters of the GEV distributions have been inferred, we can compute the cumulative distribution function at any time

In the literature

First, the main argument assumes that

GEV parameters

In this section, we discuss the effect of the Bayesian constraint on GEV model parameters. The multimodel parameters and their confidence intervals are summarized in Fig.

For the location parameter

For the scale parameter in the factual world (Fig.

Figure

We have represented, in Fig.

Figure

Statistics of multimodel synthesis after Bayesian constraint in the years 2019 and 2040. The first column shows the statistics, the second the best estimate, the third the quantile

In this section, we discuss the attribution results derived from our methodology, based on the two indicators defined above. The first is the
probability ratio

The probability ratio can be undetermined, e.g., if

Before 1990, the PR is not well defined, indicating no evidence of human influence on such an event. After 1990, the lower bound of the PR confidence interval increases quickly above

Estimated changes in intensity are consistent with the above picture. No change is detected before the year 1980, and then a sharp increase is found after that date, reaching

We finish with a comparison to the fast attribution paper by

V19 reports a return period of around

Attribution results, i.e., PR and

Another noticeable difference between these two studies involves the method of combining climate models and observations. For V19, the parameters of the GEV distribution are fitted from observations, and all models that are in disagreement (e.g., scale parameter too far) are rejected. They assume that the climate models rejected are lacking some physical process vital for the generation of extremes in the real world. We, instead, are assuming that the physical processes are there but potentially misrepresented in some way. After exploring the model uncertainty comprehensively, we find no inconsistency between models and observations. We therefore derive parameter ranges which are consistent with these two sources of information. In the end, model data do not play the same role in these two studies.

In this paper, we propose an extension of the method of

This method illustrates how CMIP5 models can be used to estimate the human influence on extremely hot events. A key point is the nonstationarity of the scale parameters, which is often assumed to be constant. Some CMIP5 models exhibit a strong change in this parameter. For a small subset of these models, the shape parameters could also be nonstationary. Over the observed period, such changes are limited and hidden by internal variability, so they cannot be ruled out by the observational constraint.

Potentially, our new method can be applied to any variable by just considering the maxima. Illustrating the potential of this technique on another type of extreme event, such as extreme rainfall, is an important area of exploration for future work. Examining the response of hot extremes to climate change in the new generation of climate models (CMIP6) would also be an attractive approach for assessing the nonstationarity of the scale and shape parameters from a broader ensemble.

The goal of this section is to explain how the coefficients

We propose using the following modification of the initialization. We
perform a quantile regression

The main hypothesis of this multimodel synthesis is the paradigm that models are statistically indistinguishable from the truth. Following

In this section, we describe a Markov chain Monte Carlo method, namely the
Metropolis–Hastings algorithm

The CMIP5 database is freely available. Source codes realizing the GEV fit are freely available in the R Python package of SDFC under the CeCILL-C license (

The supplement related to this article is available online at:

YR performed the analyses. The experiments were codesigned by YR and AR. All the authors contributed to the writing of the paper.

The authors declare that they have no conflict of interest.

We acknowledge the World Climate Research Program (WCRP) Working Group on Coupled Modelling (WGCM), which is responsible for CMIP, and we thank the climate modeling groups (listed in Table

Part of this work was supported by the French Ministère de la Transition Énergétique et Solidaire, through the “Convention pour les Services Climatiques” grant, and the EUPHEME project, which is part of ERA4CS, an ERA-NET initiative by JPI Climate and cofunded by the European Union (grant no. 690462).

This paper was edited by Dan Cooley and reviewed by Daithi Stone and two anonymous referees.