To present our methodology, we propose implementing an attribution analysis of the extreme heat wave of July 2019. We focus on France (42–51∘ N, 5∘ W–10∘ E), and we consider the random variable of annual maxima of a 3 d average of the mean temperature anomaly with respect to the period 1961–1990, noted as T.

The time series of observations of T, denoted as To (note the difference between the random variable, which is a mathematical abstraction, and the realization, which is the observations), comes from the Météo-France thermal index. This index is built as an average of observations from 30 ground stations, showing data available between 1947 and 2019. The time series of To is represented in Fig. a. The annual 2019 maximum occurred in July during the heat wave. The anomaly of this event is equal to 4.98 K, slightly higher than the 2003 maximum (4.93 K; second highest).

For climate models, we extract a simulated time series of T from the CMIP5 data set . We select all models for which simulations were produced between 1850 and 2100 and merge their historical runs with their representative concentration pathway 8.5 RCP8.5; scenario simulations. All models selected are summarized in Table . In all, we have 26 climate models over France representing the random variable T.

We use the summer mean temperature anomaly with respect to period 1961–1990 over Europe as a covariate, noted as Xt, consistent with previous attribution studies . For observations of the covariate, the data set HadCRUT4 is used and noted as Xto. For each CMIP5 model included in our analysis, we also extract this covariate by taking the mean temperature anomaly over Europe.

To summarize, we have the following:

The observation To of T, given by the Météo-France thermal index.

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 PtF and PtC, the probability distribution of T in the factual and counterfactual world, respectively, we are looking for the following probabilities: ptF:=PtF(T≥4.98),ptC:=PtC(T≥4.98). Observe the dependence in time of PtF and PtC. It is justified by the anthropic and natural forcings (such as volcanic eruptions) which alter the probability distribution with time. From the two probability distributions, we can also derive the intensity of the event. Noting QtF and QtC, the quantile functions, the intensity ItF and ItC in the factual/counterfactual world are defined by the following: ItF:=QtF(1-p2019F),ItC:=QtC(1-p2019F). Finally, the attribution is performed with the two following indicators, namely the probability ratio and the change in intensity, which measure the influence of anthropic forcing as follows: PRt:=ptFptC,ΔIt:=ItF-ItC. Consequently, an attribution study requires knowing the probability distribution PtF and PtC. The next sections describe the statistical model and how to use the climate models to infer these distribution with a Bayesian approach.

Classically, for the maximum temperature, the random variable T is assumed following a generalized extreme value distribution, in which parameters vary with a covariate Xt (either linearly or through a link function), depending on time and describing the response to external forcings. Mathematically we write that T∼PtF=GEV(μ(t),σ(t),ξ(t)), as follows: 1Mμ,σ,ξμ(t)=μ0+Xtμ1σ(t)=exp⁡(σ0+Xtσ1)ξ(t)=ξ0+Xtξ1. The parameters μ(t), σ(t) and ξ(t) are, respectively, the location (similar to the mean), scale (similar to the variance) and shape parameters. μ0, σ0 and ξ0 are called the stationary parameters, while μ1, σ1 and ξ1 are called the nonstationary parameters. The exponential link function is used for the scale parameter to ensure its positivity. We note this model Mμ,σ,ξ. Note that several studies assume that σ1≡ξ1≡0 see e.g.,, and we will test this hypothesis in the next section.

In the literature, the covariate Xt is assumed given by a 4-year moving average of the global mean surface temperature GMST; see e.g.,. Here, we take the forced component of the European summer mean temperature, which takes into account the regional and seasonal effect of aerosols. Then, we estimate the value of this covariate in the factual world, i.e., in response to all external forcings, denoted as XtF, and in the counterfactual world, in response to natural forcings only, denoted as XtC. have shown a large uncertainty in the inference of Xt through the climate models and have proposed a method for considering uncertainty in the covariate Xt. Following their strategy in the GEV case, we define the random variable θ as follows: θ:=(XtF,XtC,μ0,μ1,σ0,σ1,ξ0,ξ1). The inference of the probability distribution of θ allows us to know the probability distributions PtF and PtC just by replacing the covariate Xt with XtF and XtC, respectively, in Eq. (). Note that we assume that the coefficients μ0,μ1,σ0,σ1,ξ0 and ξ1 are the same in the factual and counterfactual world; only the value of Xt differs between these two worlds.

Our goal is to infer the probability distribution of θ and to constrain it with Bayesian techniques by the observations Xto and To. Mathematically, we want to derive the posterior of the following: 2Pθ|T=To∩XtF=Xto=Pθ|To∩Xto. Our strategy is as follows. First, we estimate θ in each individual climate model considered. Second, we use the ensemble of climate models, in particular the spread in the estimated values, to derive a prior for the real-world value of θ. Third, we apply Bayesian techniques to derive the posterior of θ given observations Xto of XtF and observations To of T.

Here, we estimate the covariate Xt, i.e., the forced response of European summer mean temperature, for each single climate model over the period 1850–2100. This approach implies some uncertainty in the value of Xt. Indeed, the value of Xt is noise by the internal variability, and there is no link between the internal variability of Xt and that of T. This is done following the  approach closely, and it provides estimates of XtF and XtC. Our goal is to find the forced response from Xt, which requires a smoothing procedure. A generalized additive model GAM; see e.g., is used to decompose the time series of Xt as follows: 3Xt=X0+XtN+XtA+ε, where

X0 is a constant,

We start by defining four submodels of Eq. (), namely Mμ,σ, Mμ,ξ, Mμ and M0, which assume, respectively, that ξ1≡0, σ1≡0, σ1≡ξ1≡0 and μ1≡σ1≡ξ1≡0. We want to determine which GEV model out of Mμ,σ,ξ, Mμ,σ, Mμ,ξ, Mμ and M0 is the most relevant. We propose using a likelihood ratio test (LRT) from theorem 2.7 of . We start from two models, Ma and Mb, such that Ma is a submodel of Mb, and we note k as being the number of supplementary parameters in Mb with respect to Ma. For example, if Ma=Mμ, Mb=Mμ,σ then k=1. The likelihood ratio test rejects Ma in favor of Mb at the significance level α if, in the following: 5D:=2(l(Mb)-l(Ma))>cα, where l is the maximum of the log-likelihood function, and cα is the 1-α quantile of the k degree of freedom χk2 distribution. We propose identifying the best common GEV model by increasing the complexity by 1 degree of freedom at a time, i.e., k≡1. The pairs of models tested are represented as a tree in Fig. . Each edge of the tree is a test in which the left model represents Ma, while the right model represents Mb.

The parameters of all GEV model are fitted from the realizations of T for each CMIP5 model, with the associated covariate XtF from the GAM decomposition, using the maximum likelihood estimation (MLE). More details about the fit are given in Appendix .

Results of our GEV model selection procedure are shown in Fig.  in the form of LRT p values. Low p values indicate that Ma is rejected in favor of Mb (green), i.e., a parameter has to be added. High p values suggest Ma is suitable (red). p values between 0.05 and 0.15 (blue) are considered to give limited evidence.

First, all models agree with the consideration that μ must be nonstationary (the first line is very close to 0). This result is expected because the climate change signal in the RCP8.5 scenario is very high and directly alters the trend. Results are less obvious for other parameters. A total of seven CMIP5 models do not show evidence of any other form of nonstationarity, while 14 models suggest that either the scale or the shape are nonstationary. A total of eight (respectively, four) of these models suggest that the scale (respectively, shape) is nonstationary. For the two other models, it is not clear if the scale and shape must be simultaneously nonstationary or if we have a transfer of information between these two parameters.

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 Mμ,σ. So, in the rest of this paper, we assume that ξ1≡0, and our nonstationary GEV model is written as follows: Mμ,σμ(t)=μ0+Xtμ1,σ(t)=exp⁡(σ0+Xtσ1),ξ(t)≡ξ0. Consequently, θ is now written as follows: θ=(XtF,XtC,μ0,μ1,σ0,σ1,ξ0). We can now infer the probability distribution of θ for each CMIP5 model. A bootstrapping procedure is used for each climate model in which the variable T is resampled 1000 times, corresponding to the 1000 estimates of XtF derived from the previous section. Thus, 1000 perturbed μ^0, μ^1, σ^0, σ^1 and ξ^0 are estimated per CMIP5 model. The goodness of fit of this model is assessed using a Kolmogorov–Smirnov test (see Fig. S2 and Table S1 in the Supplement) and shows a good agreement with the CMIP5 models. The best fit is shown in Fig. b for the CNRM–CM5 model (Fig. S3 in the Supplement shows the distribution of all fitted parameters and exhibits a large variability). First, in that case, the shape parameter is negative, equal to -0.23, implying the existence of an upper bound. Second, the upper bound follows the trend, and an extreme event occurring in the future might have been impossible in the past.

At this stage, we have 26 distributions of θ – one for each CMIP5 model – modeling the uncertainty of the covariates and the GEV parameters. Uncertainty of each distribution is quantified through a sample of 1000 bootstrapped parameters. We now focus on how to merge them into a single distribution representing multimodel uncertainty.

The goal is to synthesize our 26 distributions of θ into one single multimodel distribution. In a nutshell, we assume that each single model θ is a realization of this multimodel distribution, which is a multivariate Gaussian law in the dimension of 251+251+5=507 (251 years for factual/counterfactual world and 5 parameters for the GEV). This distribution is therefore representative of the model spread – which is usually much larger than the uncertainty resulting from estimating θ in a single model. We subsequently use the “models are statistically indistinguishable from the truth” paradigm  and assume that this multimodel distribution is a good prior for the real-world value. In practice, this synthesis is made following ; details are given in Appendix .

The effect of the multimodel synthesis is shown in Fig. . Panels (a) and (b) depict the synthesis of the covariates XtF and XtC, respectively. The dotted blue line is the best estimate, and the light blue shading is the 95 % confidence interval. These two panels show a large uncertainty across models for the covariates; note that the uncertainty is almost null between 1961 and 1990 because the models are given in anomalies with respect to this period. Panel (c) shows the covariance matrix of the distribution of (σ0,ξ) as ellipsis of the 95 % confidence interval. The dotted red ellipsis are the individual models, depicting many different behaviors. The solid red ellipsis is the multimodel synthesis, which includes almost all the best estimates from individual models.

So, with this approach, we obtain a good candidate θ for our Bayesian prior, described in Eq. (), inferred from the climate models.

To derive the posterior of θ|To∩Xto, we start by applying the Bayesian theorem, as follows: 6Pθ|To∩Xto=P(θ∩To∩Xto)P(To∩Xto)=P(θ∩Xto)∩ToP(To)×P(To)P(To∩Xto)(conditioning w.r.tTo)=P(θ∩Xto)|ToPXto|To(conditioning w.r.tXto)=P(θ|Xto)|To(Bayes theorem)=PTo|(θ|Xto)P(θ|Xto)P(To). In other words, to constrain θ by To and Xto, we can, first, constrain θ by Xto, and, second, constrain the new random variable (θ|Xto) by To. Note the two steps; the prior of θ derived in the previous section is used to infer (θ|Xto). This new random variable is then used as a prior to derive the posterior of the constraint by To.

The posterior of (θ|Xto) is already computed and used by , under the name of CX constraint. This distribution is easy to derive because both θ and XtF are assumed to follow Gaussian distributions, and XtF is a subvector of θ, so the Gaussian conditioning theorem see e.g., applies, and (θ|Xto) also follows an explicit Normal law. The effect of the constraint is represented in Fig. a and b. The blue line is the constrained best estimate, and the blue shading is the new confidence interval. We can see a significant decrease in the confidence interval. In the historical period, the covariate is closer to 0, except during volcanic episodes. For the projection period, the signal is increased by removing the lowest covariates.

The last step is to draw from the posterior of the Eq. (). To draw a θ fully constrained from the posterior, we use a Markov chain Monte Carlo (MCMC) method, namely the Metropolis–Hastings algorithm . We propose, for each sampling, performing 10000 iterations of the MCMC algorithm (see Appendix ), and we assume that the last 5000 samples are drawn from the posterior Pθ|To∩Xto. We uniformly draw one θ among these 5000 values, and we consider it as our new parameter. We use the median of these posterior drawings as a best estimate (e.g., for the final indicators ΔI and PR). As example of the effect of the constraint, we have added in Fig. c the covariance matrix of the pairs of parameters (σ0,ξ) in blue. We can see the significant decrease in the uncertainty and the shift of the parameters.

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 T as a GEV law, with best estimates of the parameters of that law and 1000 bootstrapped samples characterizing uncertainty on these parameters. Furthermore, using the decomposition of the covariate Xt, we can switch between the factual and counterfactual worlds.

Once the parameters of the GEV distributions have been inferred, we can compute the cumulative distribution function at any time t and derive the instantaneous probability of our event in the factual world, denoted as ptF, and in the counterfactual world, denoted as ptC. The usual attribution indicators, such as the probability (or risk) ratio PRt:=ptF/ptC, the fraction of attributable risk FARt:=1-1/PRt and also the change in intensity ΔIt, can be derived consistently from our description of T.

In the literature see e.g.,, it is assumed (i) that σ1≡ξ1≡0 (forcings do no affect the variability and shape), (ii) the covariate Xt is given by the observations of the 4-year moving average of the global mean surface temperature or the European mean temperature, (iii) the GEV distribution is directly fitted with the observations To, and (iv) the GEV distribution fitted describes only the factual world, while the counterfactual world is defined as a particular time in the past (e.g., the year 1900). In other words, PtC≡P1900F.

First, the main argument assumes that σ1=ξ1=0 is the large uncertainty during the fit, but we have seen that the climate models can exhibit at least a σ1 that is significantly different from 0. The uncertainty can be due to the small size of To (73 realizations currently). Second, the uncertainty of the covariate Xt is not taken into account, while we were able to see that it was not negligible, including over the period when the observations are known. Third, the definition of the counterfactual world depends on the date selected as being representative of preindustrial conditions, e.g., 1900 by . In particular, the counterfactual world can be affected by some anthropogenic forcings already present at that time. Our approach allows us to take this into account.

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 μ(t) in the factual world (Fig. a), we can see the trend of RCP8.5 simulations for each set of parameters. The effect of natural forcings is discernible over the period 1850–2000. The constraint tends to slightly reduce the confidence interval to between 40 % and 60 %. The best estimate of the change in μ(t) is slightly increased by the constraint, by about 0.5 K at the end of the 21st century. Note that in the counterfactual world (Fig. d) the trend is flat during the 21st century due to the absence of anthropogenic forcings in RCP scenarios over that period.

For the scale parameter in the factual world (Fig. b), application of the constraint leads to a large reduction in uncertainty by a factor of 2. A large trend in the scale becomes visible in the confidence interval over the 21st century. Note that the trend can be negative, or positive, corresponding to different climate models. No trend is visible in the counterfactual world (Fig. e). The Bayesian constraint suggests a much higher upward trend in σ(t).

Figure c shows the shape parameter, which is the same in the factual and the counterfactual worlds (the shape does not depend on the covariate). The shape, as estimated in the multimodel distribution, is always negative and is in the interval [-0.35,-0.15]. Applying the Bayesian constraint, the uncertainty range is reduced to [-0.3,-0.1], with a best estimate equal to -0.2. An upper bound always exists in this case.

We have represented, in Fig. , the parameters σ0,σ1 and ξ for the prior (i.e., the multimodel), the Bayesian constraints and for observations To. The parameters fitted for observations are derived from the same GEV model, and the covariate used is a spline smoothing of Xto. The observed time series is bootstrapped 5000 times to sample the joint distribution between all three parameters. The Bayesian constraint is locked at the intersection between the prior and the observations. The prior and the distribution derived from observations often exhibit some overlap. This suggests that there is no clear evidence of observations being inconsistent with model-simulated parameters given the sampling uncertainty in observations, i.e., the difficulty in directly fitting a nonstationary GEV distribution to a very small sample (71 observations). We can also deduce that the prior has a relatively strong influence on the posterior; in particular, the range of σ0 is narrowed.

Figure f shows the upper bound of the fitted GEV distribution in the factual world. This bound is given by μ(t)-σ(t)/ξ(t) if the shape is negative and is infinite if the shape is positive. Here, the effects of the constraint are critical. In the unconstrained multimodel distribution, the estimated upper bound is sometimes lower than observed events; such events should be impossible, which suggests that this estimate is unreliable. After applying the constraint, the estimated upper bound is always higher than observations, consistent with expectations. Furthermore, we notice that the observed 2019 value is above plausible values of the GEV upper bound over the 19th and 20th centuries (i.e., within the confidence range), suggesting that the 2019 event might have had a null probability of occurring at that time.

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 PRt:=ptF/ptC, shown in Fig. a and b. The second is the change in intensity ΔIt between the factual and counterfactual worlds, shown in Fig. c and d. The statistics for the years 2019 and 2040 are summarized in Table , where RTtF:=1/ptF is the factual world return time.

The probability ratio can be undetermined, e.g., if ptF=ptC=0. In this case, if the proportion of undetermined values is greater than 0.05, we assume the confidence interval to be [0,∞] (∞ included). Specifically, the upper (respectively, lower) bound of the confidence interval is computed by assuming that all undetermined values are equal to ∞ (respectively, 0), corresponding to the least confident case. The proportion of undetermined values is indicated in gray along the timeline below each panel. After the 1990s, the ratio of undetermined values is lower than 0.05. We note that, to a certain extent, undetermined values indicate that the odds of the event have not changed (i.e., similar to PR=1) because the event is impossible in both the factual and the counterfactual worlds. The key point is that the heat wave is assessed as impossible in the factual and counterfactual worlds over the 19th and 20th centuries, with a probability between 5 % and 50 % particularly when volcanic eruptions occur.

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 1. The probability of an event like the 2019 event is found to have increased by a factor in the range 19–∞ in 2019 and by a factor in the range 75–∞ in 2040. The value ∞ comes from realizations in which the event has a null probability in the counterfactual world. The estimated return time is in the range 13–153 years in 2019 and 3–20 years in 2040.

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 ΔI to +2 K in the year 2019 (95 % confidence intervals of 1.5–2.7), which is consistent with the long-term climate change signal estimated over France in summer currently nearing +2 K; see e.g.,. According to the RCP8.5 scenario, the ΔI will continue to increase to 3.6 in 2040 (2.6–4.6). We note that the upper bound in the year 2100 is approaching +12 K, which corresponds to an annual maxima of mean temperature around 39 ∘C (29 ∘C in 2019).

We finish with a comparison to the fast attribution paper by hereafter V19, who investigated the same event. V19 considers a T time series extracted from E-OBS observations and fits a GEV model with a nonstationary location parameter (a smoothed global mean temperature is used as a covariate) and a constant scale and shape. The method used to build confidence intervals has not been published, so the discussion below focuses on the best estimates.

V19 reports a return period of around 134 years (i.e., p2019F=7×10-3), with a probability ratio at least equal to 10 and a change in intensity between 1.5 and 3 K from E-OBS. Results from CMIP5 models were rejected due to a scale parameter in models (=1.78, with uncertainty range [1.74, 1.83]) being in sharp disagreement with that estimated from E-OBS observations (1.08, with uncertainty range [0.81, 1.25]). Here, by making the best use of all available information, we find a return period between 13 and 153 years, a probability ratio at least equal to 19 and a change in intensity between 1.5 and 2.6.

Attribution results, i.e., PR and ΔI, agree reasonably well between the two studies. However, there is a substantial discrepancy in the estimated return time. We can propose some explanation for this discrepancy. (i) Our GEV model assumes a nonstationary scale parameter. In practice, we find a significant change in that parameter after the end of the 20th century, contributing to a shorter return time of the event. (ii) We use the entire historical record to apply our Bayesian constraint, while V19 excluded the observed 2019 value from their estimation procedure.

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.