We use the “European Reanalysis of the Atmosphere, version 5” data ERA5 to characterize the historical observation-based extremes, and in the following, ERA5 will be referred to as “observations”. This atmospheric reanalysis combines data from weather forecasting model with observations using assimilation to produce a large number of atmospheric variables. The ERA5 reanalysis provides variables by pressure level at hourly time steps, with surface values calculated by interpolating between the lowest model level and the Earth's surface, taking account of the atmospheric conditions (https://cds.climate.copernicus.eu/datasets/reanalysis-era5-land-timeseries, last access: 19 March 2026). Note that ERA5 is a dataset that may be biased compared to actual observations, particularly for extreme events. Figure S1 in the Supplement shows the difference compared to E-OBS  – which is constructed by spatially interpolating surface observations – on average (Fig. S1a) and at maximum (Fig. S1b). The mean bias varies from 1 to 2 K, but locally can grow up to more than 10 K, particularly over North Africa. Despite such well-known limitations, ERA5 has decisive advantages in our context: global coverage (E-OBS is only available for Europe) for the entire time period, with some spatial consistency.

From this dataset, we retain temperatures over our European zone, aggregated on a daily time step, taking daily maxima between 00:00 and 23:00 (UTC), from 1940 to 2024, at the spatial resolution 0.25°. We only retain the land grid points (∼52 % of 185×271 = 50 135 grid points), see Fig. a (the Fig. b is used in the Sect. ).

Let us now take a look at how the variable representing a heatwave is constructed for each ERA5 grid point. Starting with daily maximum temperatures (TX), to account for a heatwave extending over several days, we work with the annual maximum of the 3 d moving average, noted TX3x. In general, mortality increases sharply with the duration of heatwaves , and a duration of three days allows us to capture this effect. To illustrate our methodology we zoom over the location of Paris (France). The bias of this time series compared to E-OBS has been represented on the Fig. S1c–d.

In the statistical method used, changes in extremes are assumed to scale with global or regional average temperature – which is used as a covariate. Changes in these spatially averaged temperatures are assumed to capture the response to external forcings. The temperature over Europe will be taken from HadCRUT5 , available from 1850 to the present day. We have chosen to use GISTEMP  for the global temperature.

Global Climate Models (GCMs) from the Climate Model Intercomparison Project phase 6 CMIP6 simulate climate evolution on a global scale, with a spatial resolution of the order of 100 to 200 km. The simulations feed into numerous scientific projects to understand physical mechanisms, evaluate models, lead multidisciplinary impact studies and serve as a reference for IPCC reports see, e.g., AR6 reports.

These simulations consist of a historical part, covering the period from 1850 to 2014, and several future emission scenarios ranging from 2015 to 2100. These scenarios are called Shared Socio-economic Pathways SSP, and describe climate evolution under assumptions of socio-economic evolution of human societies. Four scenarios will be used in this study, describing four levels of warming: the SSP1-2.6 (+1.8 K by the end of the century with respect to 1850/1900 period), the SSP2-4.5 (+2.8 K), the SSP3-7.0 (+4.1 K) and the SSP5-8.5 (+5 K) see, e.g. .

For each model (see Table ) we take the same variables as for the observations: TX3x on each European grid point, mean annual temperature over Europe, and over the world. These variables cover the period from 1850 to 2100, thus including the historical part as well as the future projections of the four SSPs scenarios described above.

The aim of ANKIALE is to enable the inference of a statistical model describing a climate variable (such as the annual maximum temperature) from either a climate model or observations (or similar product, such as reanalyses). The inference strategy, developed by  in the case of a normal distribution, then by  in the case of extremes, is based on a frequency analysis for climate models and a Bayesian analysis for observations. This difference in treatment stems from the idea, already exploited by , that a set of climate models can be used to construct an approximation of reality, called a prior, and that observations can be used to constrain this prior to what has been observed, allowing the construction of what is called the posterior. This posterior therefore incorporates information from climate models, constrained in such a way as to be made compatible with observations.

Formally, we have a climate variable Tt that follows a parametric probability distribution, whose parameters can be summarised in a vector θ. This vector θ can incorporate parameters that control a large number of elements, such as those parameterising the intensity of external forcings that apply at a time t, as well as the parameters of the underlying distribution. In this paper, we take as an example the annual maximum temperatures (over 3 d), which are assumed to follow a GEV (Generalised Extreme Value) distribution (see Appendix ), which is a standard choice in the literature for this variable see, e.g.,. It should be noted that our method can be adapted to other types of statistical models (such as Gaussian models). This distribution has three parameters: μ (location parameter, similar to the mean), σ (scale parameter, similar to the standard deviation) and ξ (shape parameter, controlling whether or not the extremes are bounded). The first two of these parameters are assumed to evolve over time scaling with a covariable Xt, which is representative of a mean climate change.

A statistical model typically used in attribution, as for example by or (assuming Xt is known), is given by: 1Tt∼GEV(μt,σt,ξt)μt=μ0+μ1Xtlog⁡σt≡σ0ξt≡ξ0 The idea is that climate change modifies the location parameter μt over time (which is similar to the increase of the mean), but that the variability and shape of the extremes remain unchanged. The indicator describing climate change Xt is generally given by a smoothing of the global temperature (e.g. a 15-year moving average). Then, the vector θ of the parameters of our statistical model can be written: 2θ:=(μ0,μ1,σ0,ξ0). θ can be estimated directly from observations or from climate simulations using maximum likelihood. Confidence intervals on θ are constructed using bootstrap. In this model, uncertainty on the climate change indicator Xt is not taken into account.

One further difficulty for practical application to climate data, is that the covariate Xt representing the response to climate change, is not fully well-known in general, and brings it's own uncertainty. This also applies to the breakdown between the response to natural forcings XtN and the response to anthropogenic forcings XtA. One possibility to include this additional uncertainty in our statistical model is to include Xt on the model parameters, thus extending the vector θ. This type of statistical model has been studied by  for the normal distribution and  for the GEV distribution, including a dependence on Xt for the scale parameter σt. It is written as follows: 3Tt∼GEV(μt,σt,ξt)μt=μ0+μ1Xtlog⁡σt=σ0+σ1Xtξt≡ξ0Xt=X0+XtN+XtA For this statistical model, θ therefore takes the following form: 4θ:=(X0,XtN,XtA,μ0,μ1,σ0,ξ0). This statistical model has several limitations:

It does not incorporate the work of , who worked on how to simultaneously account for global and regional covariates (which is important, for example, when the regional response differs significantly from the global response, due to aerosols).

Starting from several climate variables TtSSP, SSP∈{SSP1,…,SSPNSSP} (the term SSP here referring to one of the possible future scenarios, but the time series include also the historical period), this model can be written as: 5TtSSP1∼GEVμtSSP1,σtSSP1,ξtSSP1μtSSP1=μ0+XtR,SSP1μ1log⁡σtSSP1=σ0+XtR,SSP1σ1ξtSSP1≡ξ0XtR,SSP1=XR,0+XtR,N+XtR,SSP1,AXtG,SSP1=XG,0+XtG,N+XtG,SSP1,A⋮=⋮TtSSPNSSP∼GEVμtSSPNSSP,σtSSPNSSP,ξtSSPNSSPμtSSPNSSP=μ0+XtR,SSPNSSPμ1log⁡σtSSPNSSP=σ0+XtR,SSPNSSPσ1ξtSSPNSSP≡ξ0XtR,SSPNSSP=XR,0+XtR,N+XtR,SSPNSSP,AXtG,SSPNSSP=XG,0+XtG,N+XtG,SSPNSSP,A This model appears to be an extension of the one defined by Eq. () for each of the SSP scenarios, while incorporating the response to external forcings on σt in addition to μt. The two variables XtR,SSP and XtG,SSP correspond respectively to the Regional and Global forcings of historical and each SSP scenario. They are both decomposed as the sum of a constant (XR,0 and XG,0, respectively), a response to Natural forcings (XtR,N and XtG,N, respectively) and a response to Anthropogenic forcings (XtR,SSP,A and XtG,SSP,A, respectively), see Sect. S1.1 in the Supplement for the exact mathematical description and how the decomposition is performed. The vector θ, equivalent to Eq. (), and obtained after including all these terms, is given by: 6θ:=(XR,0,XtR,N,XtR,SSP1,A,…,XtR,SSPNSSP,A,XG,0,XtG,N,XtG,SSP1,A,…,XtG,SSPNSSP,A,μ0,μ1,σ0,σ1,ξ0). In this statistical model, only anthropogenic terms depend on the historical and SSP scenario. Compared to the model defined in Eq. (), the scale parameter σtSSP depends on XtR,SSP and is therefore no longer constant over time. Note that the parameters μtSSP and σtSSP depend directly only on the regional forcings XtR,SSP. They depend indirectly on the global forcings XtG,SSP through the knowledge of the dependence between the parameters in the vector θ (a vector that integrates all the information in the statistical model). This makes it possible to link global warming to local events. We also assume that the coefficients of the GEV distribution μ0, μ1, σ0, σ1 and ξ0 are independent of external forcings. This model is flexible and allows the underlying distribution to be easily modified. For example, it is possible to replace the GEV law with a Gaussian law (this statistical model is also proposed in ANKIALE). Other statistical models (with possibly others probability distributions), not necessarily implemented immediately, are proposed in Sect. S2.

In the following, in order to facilitate notation, we will break down θ as follows: θ:=(θR,θG,θGEV),θR:=XR,0,XtR,N,XtR,SSP1,A,…,XtR,SSPNSSP,A,θG:=XG,0,XtG,N,XtG,SSP1,A,…,XtG,SSPNSSP,A,θGEV:=(μ0,μ1,σ0,σ1,ξ0).

As mentioned at the beginning of the previous section, estimation in ANKIALE is carried out in three steps, which are summarised in Fig. . This strategy can be summarised as follows:

Inference in climate models. For each climate model m=1,…,NM, let θm be the value of θ for model m. We derive an estimate θ^m of θm, as well as the covariance matrix describing the estimation error, denoted Σθ^m, using a standard frequentist approach. Details are given in Appendix .

We will now illustrate these different steps using temperatures in Paris.

In this section, we illustrate our procedure using the annual maximum daily temperatures over 3 d (Tt:=TX3x, with a slight abuse of notation in the omission of time in TX3x). According to the block-maxima theorem see, e.g.,, we assume that the random variable TX3x follows a GEV distribution.

The ANKIALE package contains two main classes: ANKParams which contains the computer parameters (temporary directories, number of CPUs, amount of memory, etc.) and Climatology which describes the θ law. These two classes are instantiated when ANKIALE is launched. The first by the parameters of the user and the configuration of the system, the second either by a file passed by the user, or it is waiting to be built. The sub-module ANKIALE.stats then contains the classes and functions necessary for the estimations of θ:

Class ANKIALE.stats.MultiGAM: inference of the covariates,

Furthermore, the display functions are grouped in the sub-module ANKIALE.plot, the commands in the sub-module ANKIALE.cmd and the data in the sub-module ANKIALE.data.

Parallelization and memory are controlled by several parameters:

--n-workers: numbers of CPUs,

Parallelization occurs on several levels:

The samples to construct covariance matrices or confidence intervals, which are independent;

ank --help

Displays the documentation.

With ANKIALE, the entire procedure described in Sect.  can be performed in just a few lines of commands. For inference in climate models, this estimation can be done with two successive commands, one to estimate the parameters of the covariates (θ^mR,θ^mG), and the other to estimate θ^m (thus including the GEV part). Noting <file> as the input files and <climX>, <climY> as the files saving the estimates of θm, this gives: R, --save-clim ank fit Y --input --load-clim --save-clim ]]> For the multi-model synthesis, noting <climY1>, <climYNM> the files of the inferred θm for each of the climate models, the command is: ... --save-clim ]]> Constraints based on observations are applied using the two commands: --load-clim --save-clim ank constrain Y --input --load-clim --save-clim ]]>

Figure  shows the estimated return times of the maximum observed in between 2024 and 1940 in the counterfactual and factual world (Fig. a–b), as well as the change in intensity (Fig. c). The 95 % confidence intervals are given in Figs. S6 and S7. It should be noted that in 2024 we are in the projection period (2015 to 2100), and therefore we potentially have several choices. At this point, we consider the influence of the choice of scenario to be too weak (compared to the internal variability), and we have therefore represented the average of the four scenarios.

We can see that the counterfactual world shows return times (Fig. a) greater than 1000 years over almost the whole of Europe, showing that the maxima currently recorded are almost impossible without anthropogenic climate change. The 95 % confidence interval shows values down to 30 years over North Africa, Central Europe and Northern Europe, but almost the entire domain shows return periods of the order of at least 500 years.

In the factual world, North Africa shows return periods of 2 to 5 years (Fig. b), whereas in the counter-factual world they were in excess of 1000 years, showing that near-impossible events are currently becoming the new standard in this part of Europe. The same phenomenon can be seen over Western and Southern Asia, with equivalent values. The 95 % confidence intervals show the same phenomenon.

The temperature increase from the counter-factual to the factual world (Fig. c) is fairly uniform across the domain, with values around +2 K. The change is nevertheless marked in Northern Europe, with values of around +1.5 K. The signal remains clearly positive, with the low value of the confidence interval around +1.5 K, and falling to +0.6 K in Northern Europe. The high end of the confidence interval is closer to +3 K, with peaks at +3.7 K.

In line with all the studies on the attribution of extreme temperatures, it is clear that anthropogenic climate change implies a sharp increase in extreme temperatures. The sign of this change is unambiguous, as the low value of the confidence interval does not include a zero or negative change.

Let us finish with spatial variability, which appears to show breaks. Indeed, we can see that in Algeria, the return periods in Fig. b can vary very rapidly from a value greater than 1000 years to less than 30 years. To understand this phenomenon, we have shown three series extracted from ERA5 in Fig. S8: one in Paris and two in Algeria showing very different return periods. On these series (the black dots), we superimposed return periods of 2, 5, 10, 30, 50, 100, and 1000 years. In order to verify the quality of the fit for our three series, we have also displayed a histogram of the p-values of the Kolomogorov-Smirnov (KS) tests between 1000 draws of the GEV and ERA5 parameters. We can see that in at least 89 % of cases, the KS test gives a p-value greater than 5 %, showing that we cannot reject the hypothesis that the data are indeed derived from the inferred distribution. This therefore validates our fit. The difference between the series with a return period of >1000 years and the series with a return period of <30 years is the existence of an extreme event which does not change the adjustment but alters the value of the maximum observed. The spatial variability can therefore be explained by the existence or absence of an intense heatwave.

First and third rows of the Fig.  show the return times in 2040 in the counterfactual and factual world (Fig. a–e), as well as the change in intensity (Fig. k–n). The 95 % confidence intervals are given in Figs. S9a–e, k–n and S10a–e, k–n. Table S2 gives a summary of the statistics by scenario and country. The 95 % confidence interval is given in Tables S3 and S4.

For return times, the counterfactual world (Fig. a) is the same as that in 2024 (Fig. a), and shows return times of over 1000 years. The SSP2-4.5, SSP3-7.0 and SSP5-8.5 scenarios (Fig. c–e) are extremely close to each other, with values between 10 and 30 years over most of Europe, falling to 1 to 2 years over North Africa. Overall, the events are more likely than at present, reflecting the rise in temperatures over 16 years. However, these 3 scenarios have not yet been differentiated, unlike SSP1-2.6, which shows slightly longer return periods. Northern France, Belgium, Great Britain and Russia also show slightly longer return periods, between 50 and 500 years. The 95 % confidence interval (Figs. S9a–e and S10a–e) shows a similar message: the three scenarios SSP2-4.5, SSP3-7.0 and SSP5-8.5 are extremely close, and SSP1-2.6 has slightly longer return times.

The change in intensity in 2040, visible in Fig. k–n, shows, similarly to the return times, very close values – around +3 to +3 K – for the three scenarios SSP2-4.5, SSP3-7.0 and SSP5-8.5, and a scenario SSP1-2.6 with lower intensity changes of around 0.5 K. Northern Europe shows lower values, below +2 K, while Eastern and Southern Europe reach almost +4 K. The 95 % confidence intervals (Figs. S9k–n and S10k–n) show a similar spatial dispersion of values, with 1 K lower values for the lower bound, and 1 K higher values for the upper bound. Some countries even show changes of more than +5 K.

Figure f shows the return times in 2100 of the maximums observed in the counterfactual world, as the values are the same as for Fig. a, the conclusions of Sect.  apply.

Let us continue with the scenarios, represented on the Fig. g–j. Return times decrease with simulated climate change intensity. For SSP1-2.6, only 12 countries (on 55) show return times beyond 50 years, with 28 countries already having a return time of 10 years or less. From SSP2-4.5 onwards, the current maximums are almost commonplace, with only one country showing a return period in excess of 50 years. From SSP3-7.0 onwards, current maximums are the “normal” situation, with return times between 1 to 10 years and 1 to 2 years. The 95 % confidence interval is shown in Figs. S9g–d and S10g–d, and shows that return periods can fall below 10 years over the whole of Europe.

The scenarios for intensity change are represented on the Fig. o–r. For scenario SSP1-2.6, the change ranges from +1.7 K for Northern Europe to +4.1 K for Southern Europe, with the change relative to 2024 being almost the same everywhere, around +1 K. For the SSP2-4.5 scenario, the change ranges from +2.7 K for Northern Europe to +6.5 K for Southern Europe. The different regions of Europe show different changes compared to today, ranging from +1.5 to +3.8 K. The SSP3-7.0 and SSP5-8.5 scenarios show increasing intensity increases, from +5 to +9 K. The 95 % confidence interval (Figs. S9o–r and S10o–r) even shows changes in intensity of up to +18 K.

In this paper, we have presented an extension of the  method for estimating probabilities of extremes following a GEV law. Our new method allows us, on the one hand, to treat several scenarios simultaneously, and on the other, to force a counter-factual world that is common to all scenarios. We first applied this method to temperature extremes over Paris (France), and demonstrated not only its validity, but also that it drastically reduces differences in the counter-factual probabilities, reinforcing the inter-scenario consistency of our estimates. We have also verified that our estimates of current and future global climate change are consistent with current literature.

We also offer an open-source software that can be used to reproduce our results and easily applied to other fields. This tool is natively parallelized, with particular attention paid to the memory used. It can be deployed just as easily on a personal computer, a computing cluster or a supercomputer. This software is also extensible, and other probability distributions – such as the Normal or Generalized Pareto Distribution – may be integrated in the future.

We have applied this new approach and tool to the attribution of observed maxima over Europe, enabling us to analyze these statistics up to the end of the 21st century for four climate scenarios. In the future, the observed maxima will become the new norm for scenarios greater than the SSP2-4.5 and SSP3-7.0, and will be 2 to 3 K warmer even for a low-emission scenario like the SSP1-2.6. An increase in extreme temperatures of more than 10 K is conceivable within the 95 % confidence interval. We have focused on Europe here, but an extension to the rest of the world and to temperature-like variables such as the heat-index would enable a global map of future heat hazards to be drawn up.

Compared to other studies such as, the estimated return times and intensity changes can be different. This is due to different choices in the statistical model (regional covariate, smoothing method, counterfactual construction), as well as in the estimation method. As shown with the example of Paris, the use of direct observations gives a much stronger (and much more uncertain) trend, while the use of climate simulations as a prior allows for a stronger constraint on the signal. From this perspective, our approach appears much more robust. We should also note the strong influence of the choice of observation data (or similar), given the significant biases on extremes between E-OBS and ERA5. Even though datasets like ERA5 are available on a global scale, the calculation of return times (especially for impact studies) should probably be done from datasets closer to observations (such as E-OBS or stations). Several additional studies are needed to validate or refine the choices made, particularly with regard to the statistical model and the smoothing method.

Even improvements to the GEV model are possible. For example, the GEV model tends to overestimate return times see, e.g.. Recent work by  proposed a new GEV model where the upper bound on temperatures is imposed by physics . This approach would fit in naturally with the tools developed here. A similar method could be applied to precipitation by mixing an estimation of the upper bound  with a statistical model as the extended Generalized Pareto Distribution . Another interesting possibility would be to use external forcings directly as covariates rather than their responses (global and regional temperatures).

Further work is also needed to extend our model to other variables such as wind and precipitation. The tools and statistical model developed here were developed in the context of attribution, particularly in relation to heat waves. Several studies use other types of covariates such as CO₂ or Z500, see, e.g., or statistical models. Extending this to other variables would require work similar to that of , where the validity of the statistical model was verified in each climate model, as well as its quality after applying observational constraints.

It should also be noted that, on the one hand, we have remained in a univariate context, while the estimation of concurrent events increasing impacts appears increasingly necessary; and on the other hand, spatial structures are ignored. We also used only GCMs, which do not capture local specificities. The use of regional models (when those from CMIP6 become available) will allow us to refine the results obtained here work in this direction has already been carried out by , and ANKIALE will make this easy to do.

Finally, the analyses produced here provide local information on the worst possible future events, and show the need for rapid adaptation to extremes warming faster than global warming.