# A hybrid statistical-dynamical method to translate past extreme temperature days into the future climate

Julien Boé, Margot Bador, Laurent Terray

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Julien Boé, Margot Bador, Laurent Terray. A hybrid statistical-dynamical method to translate past extreme temperature days into the future climate. Weather and Climate Extremes, 2025, 49, pp.100785. 10.1016/j.wace.2025.100785. hal-05131988

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# A hybrid statistical-dynamical method to translate past extreme temperature days into the future climate

Julien Boé* , Margot Bador, Laurent Terray

CECI, Université de Toulouse, CNRS, CERFACS, 42 Avenue Gaspard Coriolis, cedex 1, 31057, Toulouse, France

## ARTICLE INFO

**Keywords:**

Temperature

Heatwave

Cold spell

Attribution

Futurization

Climate change

Europe

Large scale circulation

## ABSTRACT

This study presents a novel hybrid statistical-dynamical method intended to translate past observed weather events into the future climate, and applies it to warm and cold extreme temperature days over western Europe. The method estimates the temperature anomalies that would result if an observed event of the 1940–2023 period, defined in terms of atmospheric circulation, were to occur at the end of the 21st century, under new climatological conditions. In practice, constructed analogues of observed extreme events are built using data from regional climate projections. Three regional climate projections under the RCP8.5 emissions pathway are used in order to assess the role of model uncertainties in this context. The same approach is also used beforehand to assess the role of large-scale circulation in the observed extreme temperature days, and the ability of regional climate models to capture it is evaluated. The study finds significant variability in the role of atmospheric dynamics in extreme temperature days, contributing 35–80 % of the temperature anomaly for warm days and 20–90 % for cold days, with other factors such as land-atmosphere interactions playing an amplifying role. Regional climate models generally capture the dynamical part of temperature anomalies quite correctly. Not surprisingly extreme temperature days become more intense in the future climate, but a large inter-event spread exists. Some of the events could become much warmer, while others would not change much. Moreover, this intensification varies widely between regional climate models, and not necessarily in line with the average warming.

## 1. Introduction

In the aftermath of an extreme weather event, two questions are often asked: what role did climate change play in this event? What would be the impacts of the same event if it occurred in a warmer future climate, i.e. should we expect something worse?

The first question pertains to the issue of attribution. One common approach to addressing this question is through the use of a conditional or storyline framework (Trenberth et al., 2015; Stott et al., 2016). Within this framework, the event is typically defined by the large-scale atmospheric circulation pattern leading to it (Trenberth et al., 2015). The question is therefore whether climate change has increased or decreased the climate anomalies (e.g., in temperature) associated with the circulation pattern responsible for the event. To assess this, the climate anomalies associated with the circulation pattern in the actual (factual) world, i.e. under current climate conditions, are compared with the climate anomalies associated with the same circulation pattern in a hypothetical (counterfactual) world without anthropogenic climate

change.

This conditional approach is motivated by the idea that the signal-to-noise ratio regarding anthropogenic changes in large-scale atmospheric circulation is generally small (Trenberth et al., 2015), and that these changes may not even be very robust (Shepherd, 2014). Working with a given fixed large-scale circulation pattern avoids these potential issues. However, it does not address the question of the probability of this event occurring given climate change.

The second question "What would be the impact of the same event if it occurred in a warmer future climate", of "futurization", i.e. of translating a present-day event into a future and warmer climate, is in fact very similar to the first one, within the conditional framework. The only difference is that for futurization, the counterfactual world is a distant future rather than a pre-industrial past.

Translating a present-day event into a future climate is an interesting scientific question in itself, but it is also of considerable practical interest. Translating past events into the future climate is a powerful way to engage policy makers or the general public, who usually have vivid

<footnote>* Corresponding author.
E-mail address: boe@cerfacs.fr (J. Boé).</footnote>
memories of catastrophic extreme events they have experienced, their vulnerability during these events, and practical ideas about how the consequences could have been mitigated for them. This may be a good starting point for practical thinking about adaptation to climate change. It is even more true since the recorded impacts of specific past extreme weather events are sometimes used as benchmark in the planification against climate-related risks, for example for floods. Translating these specific past events into the future climate would be a simple way to bring climate change within a classical risk-planification framework in an adaptation context.

Futurization and conditional attribution studies can purely rely on climate models. Global models with large-scale circulation nudged towards observations of a particular weather event can be used to perform factual and counter-factual experiments by perturbing sea surface temperature (SST), external forcing (e.g. Hawkins et al., 2023; Van Garderen et al., 2021; Van Garderen and Mindlin, 2022; Wehrli et al., 2019). Regional climate models can be forced by the boundary conditions corresponding to the event of interest, with additional perturbations to account for the counter-factual world (Matte et al., 2022) such as in pseudo-global warming experiments (Imada et al., 2020; Brogli et al., 2023). Forecast-based methods are also used (Ermis et al., 2024; Leach et al., 2021). All of these approaches require to run specific numerical experiments, astutely designed to capture the climate signals correctly. These dedicated experiments can therefore be complex to set up, and although they are only performed for a specific event, they can still be numerically expensive, as it may be necessary to run multiple members to account for the influence of internal variability, or to use a very high resolution to correctly capture the event of interest (e.g. km-scale resolution for extreme precipitation events). Results may also be model dependent, and it may be preferable to perform these experiments with different models.

In this study, we describe a simple hybrid statistical-dynamical approach for futurization. The first step of such an approach is to characterize the role of large-scale circulation in a given event using a statistical method. A natural way to characterize the dynamical contribution to an extreme event is the analogue method (Lorenz, 1969), two analogues being defined as two days with similar large-scale atmospheric circulation patterns. While the concept of analogues has been initially used to study predictability (Lorenz, 1969) or for downscaling (Zorita and von Storch, 1999), they can also be used to separate the role of large-scale circulation and other processes, often grouped under the umbrella of “thermodynamic” factors, on the anomalies of a variable of interest. Boé et al. (2009) describe a method based on analogues to assess the part of future changes in summer precipitation over Europe that can be explained by changes in large-scale circulation, in a large multi-model ensemble of global climate projections. This method looks for circulation analogues of future climate projections in historical simulations. Conceptually similar approaches have been widely used to study the role of large-scale circulation on specific events (Cattiaux et al., 2010) and for conditional attribution studies (Yiou et al., 2017; Faranda et al., 2022; Ginesta et al., 2023).

One problem with the standard analogue method, as already pointed out by van Den Dool (1994), is that it is difficult to find a good analogue with a few decades of data, let alone a large number of good analogues. Indeed, in order to estimate correctly the pure role of atmospheric circulation for a given event, one would indeed need a large number of near-perfect analogues, for example occurring under different sea surface, soil moisture or snow cover conditions, in order to average out the influence of non-dynamical processes. When assessing the role of atmospheric circulation on climatological averages as in Boé et al. (2009), this issue is mitigated, as computing climatological means on a large number of days allows the influence of other processes to be averaged-out more effectively. This is not the case when focusing on a single event: the number of good analogues is small, making it difficult to average-out correctly non-dynamical processes while maintaining very similar circulation patterns. The standard analogue method

therefore comes down to looking at a class of events rather than looking at the specific event itself.

The constructed analogue approach (van den Dool, 1994; Maurer and Hidalgo, 2008), in which analogues are constructed by combining existing circulation patterns can be used to overcome these difficulties. Constructed analogues have been used for example for downscaling (Maurer and Hidalgo, 2008; Werner and Cannon, 2016) or emulation of Regional Climate Models (RCMs) (Boé et al., 2023). Deser et al. (2016) have used constructed analogues for “dynamical adjustment” i.e. to remove the role of large-scale circulation on the observed evolution of a climate variable of interest. Terray (2021) have introduced this method for conditional attribution of extreme events, and applied it to the 2021 northwestern North-American heatwave (Terray, 2023), with an extension to futurization in the latter study.

This hybrid statistical dynamical approach for futurization is developed in this work. We present a constructed analogue method for futurization and apply it to futurize all observed extreme temperature days, cold in winter, warm in summer, of the 1940–2023 period over western Europe. We assess how these events could change in the future climate, using climate simulations from multiple RCMs as base for constructing analogues. Conditional attribution or futurization, when the event of interest is defined by its large-scale circulation as it is generally the case, makes the implicit assumption that large-scale circulation is a major driver of the event. Indeed, if large-scale atmospheric circulation does not play a significant role in an event, there is no point to define this event by its large-scale circulation. A secondary objective of this work is therefore to assess the role of large-scale circulation in extreme warm and cold days over western Europe since the middle of the 20th century.

In section 2, we describe the data used and how our events of interest are defined. The constructed analogue method for futurization is described in section 3. After having documented the observed evolution of the occurrence of extreme warm and cold days in section 4, we characterize the role of large-scale circulation in observed extreme temperature days in section 5. Then, we evaluate in this context the regional climate models used for futurization (section 6). In section 7, we describe the results of futurization with different regional climate projections. Conclusions and perspectives are presented in the last section.

## 2. Data and definition of events of interest

Daily sea level pressure and maximum temperature from ERA5 (Hersbach et al., 2018, 2020) available on the 1940–2023 period are analyzed. Note that hourly 2-m temperature from ERA5 has been used to construct what is considered ERA5 daily maximum temperature in this study. This estimate does not correspond to instantaneous maximum temperature during the day, but to the maximum of the 24 hourly values during the day, given by the time step on the hour. It is the best estimate of daily maximum temperature that can be derived from ERA5 for most applications according to the producer of the dataset (https://confluence.ecmwf.int/display/CKB/ERA5%20%3A+2+metre+temperat ure).

In order to focus on large-scale circulation, and to reduce computational time, sea level pressure from ERA5 is first aggregated on a regular 2° × 2° grid before the constructed analogue step. Maximum temperature is analyzed on the original distribution grid of 0.25° × 0.25°.

Daily sea level pressure and maximum temperature from three regional climate projections with the RCP8.5 scenario from the EURO-CORDEX project at a 12 km resolution (Jacob et al., 2014) are used to translate past weather events into the future climate: ALADIN63 (Nabat et al., 2020) forced by CNRM-CM5, HadREM3-GA7-05 (Walters et al., 2019) forced by MPI-ESM-LR and CNRM-CM5. These simulations will be referred to as CNRM-CM5_ALADIN, MPI-ESM-LR_HadREM3 and CNRM-CM5_HadREM3.

These regional climate projections have been chosen to sample a
large range of future regional climate changes, as will be shown later. In addition, two simulations share the same forcing model, while two other simulations share the same regional model. This may help to disentangle the respective role of boundary conditions versus regional model for futurization.

Maximum temperatures from the RCMs are interpolated on the original 0.25° × 0.25° distribution grid of ERA5. Sea level pressure is interpolated on the same 2° × 2° regular grid used to interpolate ERA5 sea level pressure.

A first step of this work consists in defining the extreme temperature days studied. These days are defined as days with large cold temperature anomalies in winter and large warm temperature in summer over a substantial part of western Europe. An extreme warm (cold) day (EWD and ECD, respectively) occurs when the area where the maximum temperature anomaly is greater than the 95th percentile during summer (smaller than the 5th percentile during winter) covers more than 1/3 of the land area of the domain of interest (which corresponds to the domain shown in Fig. 2). Note that the maximum temperature is used to define both EWD in summer and ECD in winter. The detailed methodology is described below:

- First, at each grid point, the daily anomalies of maximum temperature are computed by removing the daily annual cycle. The daily maximum temperature is first smoothed using a 31-day running average, and then the daily annual cycle on the 1961–1990 period is computed and removed from the maximum temperature of each day.

- The 95th percentile of daily maximum temperature anomalies during summer (JJA) on the entire 1941–2023 period is then computed at each grid point. For ECD, the 5th percentile of daily maximum temperature anomalies is computed for winter (DJF) months.

- For each day, all the land grid points over the domain of interest where the local maximum temperature anomaly in summer is greater than the local 95th percentile are selected, and the corresponding surface is calculated. An EWD is defined when the surface is greater than 1/3 of the total land area of the domain of interest. A similar procedure, considering the area where the maximum temperature anomaly is smaller than the 5th percentile in winter is used to defined ECD.

- The intensity of an EWD (ECD) is defined as the averaged maximum temperature anomalies over the area where temperature anomalies are greater (smaller) than the 95th (5th) percentile. The intensity of an extreme temperature day is therefore calculated over a domain tailored for each specific event.

The extreme warm and cold days previously defined often happen during heatwaves and cold spells, and therefore many EWD and ECD previously defined are not independent. Among all EWD et ECD, we therefore also select the day of each year with the greatest intensity. These days are called EWDx and ECDx in the following.

### 3. Constructed analogues

The constructed analogue method (e.g. van den Dool, 1994; Maurer and Hidalgo, 2008; Terray, 2021; Patterson et al., 2024) is used to estimate the maximum temperature anomaly associated with the observed large-scale circulation patterns of ECD and EWD under different climates (present climate as captured by ERA5, present and future climates as simulated by three regional climate projections).

In this study sea level pressure is used to characterize large-scale circulation, rather than other variables that are also commonly used to study the links between atmospheric circulation and temperature, such as geopolitical height at 500 hPa. Geopolitical height requires an additional preprocessing step intended to remove the effects of thermal expansion in order to focus on atmospheric dynamics, and there is no unique way to do this, introducing additional methodological uncertainties. We have therefore chosen to use sea level pressure. Note that

in this study "dynamical anomalies" implicitly refer only to dynamical anomalies as captured by sea level pressure.

First, the daily anomalies of the necessary variables (sea level pressure and maximum temperature) of the different datasets (ERA5 and regional climate simulations) are computed, by removing the annual cycle following the same approach described in Section 2, using the same climatological period (1961–1990).

We want to estimate the anomaly of maximum temperature of a day $t_1$ due to atmospheric circulation only (i.e. “dynamical anomaly” in the following, $Tx_{dyn}(t_1)$), knowing the corresponding map of sea level pressure anomaly on the domain of interest i.e. SLP($t_1$). SLP comes from ERA5 on the 1940–2023 period (Section 2).

First, we search for the $N_a$ best analogues of SLP($t_1$) in a base dataset (SLP$_r$), by minimizing the Euclidean distance between SLP($t_1$) and SLP$_r(t)$ i.e. the daily maps of sea level pressure anomaly of the base dataset on a given period. SLP($t_1$) corresponds to an observed event and always comes from ERA5, but SLP$_r$ can come from ERA5 or from climate projections, and different periods (e.g. CNRM-CM5_ALADIN on 1991–2020 or CNRM-CM5_HadREM3 on 2070–2099).

The analogues are searched within a 31-day calendar window centered on day $t_1$. When the base dataset is ERA5, the year of day $t_1$, as well as the year before and the year after are removed from the potential pool of analogues in order to avoid artificial skill.

Let's call these $N_a$ analogues SLP$_Ai$, where $i$ varies between 1 and $N_a$. Note that when $N_a$ is large, many of these “analogues” obtained may be far (in terms of Euclidean distance) from SLP($t_1$) and therefore very poor analogues. The name “analogues” is kept here for simplicity. This is not a problem thanks to the next step.

Then, $N_s$ analogues are randomly selected among the $N_a$ analogues, and the best linear combination, i.e. the best set of $\alpha_i$ coefficients, is searched so that:

$$SLP(t_1) \approx \sum_{i=1}^{N_s} \alpha_i SLP_{Ai} \quad (1)$$

The $\alpha_i$ coefficients are obtained using the Moore-Penrose pseudo-inverse. The right-hand side of equation (1) is a constructed analogue of day $t_1$. Thanks to this approach, it is possible to obtain very good constructed analogues (see Appendix A) even if the individual analogues are not very good.

Finally, the corresponding dynamical temperature anomaly, $Tx_{dyn}(t_1)$, is obtained by applying the same $\alpha_i$ coefficients to the maximum temperature anomalies $Tx_{Ai}$ corresponding to the sea level pressure anomalies of the analogues SLP$_Ai$ (i.e. $Tx_{Ai}$ is the maximum temperature anomaly of the day of SLP$_Ai$, in the same base dataset.)

$$Tx_{dyn}(t_1) = \sum_{i=1}^{N_s} \alpha_i Tx_{Ai} \quad (2)$$

For each day $t_1$, these steps are repeated $N_r$ times, and the mean of $Tx_{dyn}$ over the $N_r$ realizations is used as the best estimate of the dynamical temperature anomaly for the day $t_1$. Using a large number of constructed analogues ($N_r$) based on many analogues ($N_s$) allows to sample a large diversity of surface conditions (SST, soil moisture, snow) for a given large-scale circulation pattern, in order to average-out the impact of non-dynamical processes.

The following parameters are used in this study for the constructed analogue method: $N_a = 500$; $N_s = 100$; $N_r = 200$ (see Appendix A).

In this study, different base datasets for the construction of analogues are used: ERA5 on 1991–2020, and 1951–1970 for sensitivity tests, CNRM-CM5_ALADIN, MPI-ESM-LR_HadREM3 and CNRM-CM5_HadREM3 on 1991–2020 and 2070–2099.

When estimating dynamical anomalies with ERA5 and the reference 1991–2020 period, what is obtained is the dynamical anomaly of the event of interest, if it had occurred on the period 1991–2020, which is not necessarily the period during which it actually occurred. For the sole purpose of characterizing the dynamical anomalies of past events, it may be preferable to use a running reference period for the calculation of the dynamical anomalies, e.g. a 30-year window centered around the event.
For ERA5, such a running reference period is also used for a test discussed in Section 5: the constructed analogues for a day $t_1$ are constructed using data within a 30-year window centered on the year of $t_1$ (the year of day $t_1$, as well as the year before and the year after, are removed from the potential pool of analogues).

Note that with a running window, the dynamical anomalies on the first 15 years and last 15 years of ERA5 are calculated on a window actually smaller than 30 years. This is not ideal because, as shown later in Fig. 1, many events of interest occur at the beginning (for cold days) or at the end (for warm days) of the period. Moreover, in the context of translating an event into the future climate, the main objective of this work, it is preferable to use a fixed present-day window, identical for all events, in order to be able to compare the impact of climate change consistently across events, whenever they occurred. A sensitivity analysis consisting in also calculating the dynamical anomalies for ERA5 using the 1951–1970 period as reference is performed in Section 5 to document the impact of the present-day reference period.

The dynamical temperature anomalies are averaged each day over the same area used to define the intensity of EWD and ECD (Section 2). This is done to allow a meaningful comparison of total anomalies and dynamical anomalies derived with the different simulations and periods.

Note that in this study, we use the term “dynamical anomalies” to refer to the component of the temperature anomaly that is directly or indirectly related to atmospheric circulation. These dynamical anomalies depend on the time period used to construct the analogues: the same atmospheric circulation pattern is not associated with the same temperature anomalies in the future as in the present climate. The difference between the dynamical anomalies associated with the same circulation pattern in the present and future climates is likely to be due to other processes: e.g. soil-moisture-atmosphere interactions or snow albedo feedback, advection of warmer maritime air due to increased sea surface temperature, radiative effects of greenhouse gases or aerosols.

## 4. Evolution of the occurrence of extreme warm and cold days

The number of extreme cold days per winter and extreme warm days per summer over the 1941–2023 period, as defined in Section 2, is

shown in Fig. 1. Not surprisingly, a large inter-annual variability is noted. Some years have no extreme temperature days, while others have more than 15. Note that not all extreme temperature days in a given season are independent, as they may occur during the same heat wave or cold spell.

Years known for major heatwaves and cold spells stand out clearly, such as 1941–1942 (Lejenäs, 1989), and 1955–1956 (Dizerens et al., 2017) for cold spells, or 2003 (García-Herrera et al., 2010), 2019 (Xu et al., 2020) or 2022 (Kim et al., 2024) for heatwaves.

As expected in the context of climate change, years with a large number of ECD are much more frequent in the early period, while the opposite is true for years with many EWDs. Summers with more than 5 EWDs never occur before 2000, while since 2012 several EWDs are observed over western Europe every summer. This is consistent with a large body of work showing that climate change has already reduced the frequency of cold spells and increased the frequency of heatwaves over Europe (Seneviratne et al., 2021).

## 5. Role of atmospheric circulation in observed extreme temperature days

An extreme cold day (26 February 1986), which allows for interesting discussions on the role of surface processes in the following, is used as a case study throughout the article to illustrate the approach taken in this work, before generalizing the analyses to all extreme temperature days. This day is characterized by strong negative temperature anomalies over central and northwestern Europe (Fig. 2). The anomalies are lower than the 5th percentile over a large area (shown as dashed lines in Fig. 2b). Note that this area is used to compute the total and dynamical intensity of this specific event, both in the present and future climate.

The atmospheric circulation of this day is characterized by south-eastern advection (Fig. 2a). The temperature anomalies due to atmospheric circulation alone obtained using as base ERA5 on 1991–2020 (“dynamical anomalies”) are shown in Fig. 2b. Because continental air is generally colder than oceanic air in winter, this circulation pattern leads to moderate negative temperature anomalies over eastern Europe. This circulation pattern is not the one typically associated with very cold conditions over western and central Europe. Indeed, they are typically due to northeastern advection (Quesada et al., 2023). It is therefore not surprising that the dynamical negative anomalies have a weak amplitude (Fig. 2b). As discussed in Section 3, the choice of the reference period for constructed analogues may be important. Indeed, using the earlier 1951–1970 reference period (for the same February 26, 1986 event), the dynamical anomalies are colder than with the latter period (Fig. 2c versus Fig. 2b). Consistent with the impact of global warming, the same circulation pattern leads to colder anomalies in the mid 20th century than in the late 20th century.

The dynamical anomaly (Fig. 2b) is substantially smaller than the total anomaly (Fig. 2a), especially over a latitudinal band over northern France (Fig. 2d). This residual anomaly (difference between the total anomaly and the dynamical anomaly), not explained by atmospheric circulation, is associated with a very strong positive anomaly in upward shortwave radiation at surface (Fig. 2e). Such large anomalies in winter are often associated with strong positive snow-cover anomalies (not shown). Two days before the 26th February, a heavy snowfall indeed occurs over this area, with the same very distinctive latitudinal band over northern France as shortwave radiation and residual temperature anomalies (Fig. 2f). A large snowfall also occurs over the Alps, but since this region is generally snow-covered in winter, this has much less effect on upward shortwave radiation at surface and temperature than over northern France, where snowfall is rare. Interestingly, the snowfall occurring the 24th February is probably due to a large extent to the large-scale atmospheric circulation, with a front over this region, between a cold eastern flow over northern Europe and a warm western flow over southern Europe (Fig. 2f).

<figure><img src="image_2.png" /><figcaption>Fig. 1. (a) Number of Extreme Warm Days per summer (JJA) and (b) number of Extreme Cold Days per winter (DJF). Winter of year y corresponds to December of year y-1, and January and February of year y.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** (a) Maximum temperature anomaly (K) of 26 February 1986 and corresponding sea level pressure (red contours, hPa). (b) Dynamical temperature anomaly (K) with the 1991–2020 reference period for constructed analogues. Hatchings show the area where the total maximum temperature anomaly (shown in (a)) is smaller than the 5th percentile in winter, used to calculate the total and dynamical anomalies characteristic of the events. (c) Dynamical temperature anomaly (K) with the 1951–1970 reference period for constructed analogues. (d) Difference between the total anomaly shown in (a) and the dynamical anomaly shown in (b). (e) Anomaly of upward shortwave radiation at surface (W.m⁻²) (f) Snowfall (mm of liquid water) of the 24 February 1986 and corresponding sea level pressure (red contours, hPa). All anomalies are relative to 1961–1990.</figcaption></figure>

The role of atmospheric circulation is now investigated for all ECDx and EWDx (see Section 2). The total temperature anomalies of EWDx and ECDx in ERA5 and the dynamical temperature anomalies obtained with the constructed analogue method using ERA5 on the 1991–2020 period or with a 30-year running period are compared (Fig. 3). Note that these anomalies are calculated on an area that is specific to each event, and which is the same for the total and dynamical anomaly (see Section 2).

By construction, EWDx are characterized by strong daily maximum temperature anomalies, between 9 and 11 K. The associated dynamical anomalies are always positive, between 4 K and 11 K, with the fixed 1991–2020 reference period. The total and dynamical anomalies and their ratio have a larger spread in winter than in summer. The total anomalies range from –7 to –16 K and the dynamical anomalies range from 0 to –8 K with the fixed 1991–2020 reference period.

The use of a running 30-year reference period leads to less intense dynamical anomalies for EWD and more intense anomalies for ECD. This

is expected as many ECD occur before 1991–2020 (Fig. 1b), therefore under colder climatological conditions. With the running reference period, the dynamical anomaly accounts for between 35 % and 80 % of the total temperature anomaly for EWDx and between 20 % and 90 % for ECDx (Fig. 3c). On average, the role of circulation is therefore somewhat smaller for ECDx in winter than for EWDx in summer (median ratio close to 0.6 versus 0.7), but the inter-event variations are larger for ECDx.

Favorable large-scale circulation conditions are therefore necessary for EWDx and ECDx to occur, even if the role of large-scale circulation varies from event to event. The ratio of dynamical to total anomalies is always smaller than one, which means that during EWDx and ECDx other processes amplify the temperature anomaly initially caused by large scale circulation. In summer, high sea surface temperature (SST) over adjacent sea, or dry soil leading to low evaporative cooling, as during the summer 2003 heatwave for example (Fennessy and Kinter, 2011) could play this amplifying role. In winter, as shown in Fig. 2, snow can play a major role in amplifying dynamical anomalies. Other
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** (a) Box-and-whisker plots for total, dynamical using the 1991–2020 reference period (Dyn. fix.) and dynamical using a 30-year running window (Dyn. run.) temperature anomalies (K) of EWDx in summer. (b) Same as (a) for ECDx in winter. (c) Ratio of dynamical to total temperature anomalies (no units) for EWDx and ECDx. The anomalies are computed relative to the 1961–1990 climatology. The box shows the 1st and 3rd quartiles, and the median. The whiskers extend to the minimum or maximum of the distribution, or to 1.5 times the interquartile range if there are values outside this range. The values outside this range are shown with circles.</figcaption></figure>

processes can also be involved, such as those related to SST over adja-
cent seas, as in summer.

As discussed in Section 3, and suggested by Fig. 3, a later reference
period can lead to warmer dynamical temperature anomalies and
therefore to an increase of dynamical anomalies for EWD and to a
decrease of dynamical anomalies for ECD. The sensitivity to the choice
of the reference period is now fully explored in Fig. 4.

When the 1950–1979 period is used as a reference for constructed
analogues, on average, the dynamical anomalies are less intense for
EWD in summer (by 1–5 °C, Fig. 4C) and more intense for ECD in winter
(by 0.7 °C, Fig. 4c). This is consistent with the idea that global warming
may have reduced the intensity of cold days and increased the intensity
of warm days in the second half of the 20th century. However, there are
large variations from event to event, especially in winter. In roughly 25
% of the winter cases, the dynamical anomaly is smaller when using the
1991–2020 period as a reference. It never happens in summer, even if
for some events, a small impact of the reference period can be noted.

**6. Evaluation of regional climate models**

To translate an observed weather event into the future climate, it is
first necessary to translate the weather event into the present-day
climate of a regional climate model. As an example, the dynamical
anomalies corresponding to the 26 February 1986 using three RCMs and
the 1991–2020 period as reference are shown in Fig. 5a–c. The
dynamical anomalies obtained with the three RCMs are generally

similar to the observed one (Fig. 5a–c versus Fig. 2b), with negative
anomalies over central Europe, positive anomalies over the north of
Scandinavia and southwestern France. The consistency of the dynamical
anomalies across different datasets increases the confidence that the
dynamical anomalies obtained by the constructed analogue method are
robust features.

Still, some differences between models are seen: MPI-ESM-LR_HADREM3 shows stronger negative anomalies over central Europe for example. Interestingly, common biases are also seen, with too warm dynamical anomalies over Spain, and too cold anomalies over eastern Europe compared to ERA5.

Considering all ECD and EWD (not shown), the dynamical anomalies
obtained with regional simulations are generally more similar when
either the forcing GCM or the RCM is shared. In winter, sharing the same
GCM leads to closer dynamical anomalies, while in summer the picture
is less clear, as the results depend on the similarity measure (not shown).

The ability of the RCMs to capture dynamical anomalies is now
investigated more systematically by comparing, for each EWD and ECD,
the dynamical anomalies obtained thanks to constructed analogues
using the three RCMs with those obtained using ERA5. For all extreme
days in winter and summer, we compute the spatial correlation, the
spatial mean error (ME) and the spatial mean absolute error (MAE) for
the dynamical anomalies obtained with the three RCMs with respect to
the dynamical anomalies obtained with ERA5, over the entire domain
considering only land points (Table 1).

In general, the spatial patterns of temperature anomalies are well

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** (a) Box-and-whisker plot of dynamical temperature anomalies (K) for (a) EWDx and (b) ECDx, using two reference periods for the constructed analogues (1950–1979 and 1991–2020). (c) Difference of the dynamical anomalies calculated with these two reference periods (1991–2020 minus 1950–1979). All the anomalies are computed relative to the 1961–1990 climatology. The box shows the 1st and 3rd quartiles, and the median. The whiskers extend to the minimum or maximum of the distribution, or to 1.5 times the interquartile range if there are values outside this range. The values outside this range are shown with circles.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Dynamical temperature anomalies (K) corresponding to the 26 February 1986 with (a) CNRM-CM5_ALADIN, (b) CNRM-CM5_HADREM3 and (c) MPI-ESM-LR_HADREM3. (d-e-f) Differences in dynamical temperature anomalies (K) between (d) CNRM-CM5_ALADIN, (e) CNRM-CM5_HADREM3 and (f) MPI-ESM-LR_HADREM3 and ERA5 (Fig. 2b). The reference period for constructed analogues is 1991–2020. All anomalies are relative to 1961–1990.</figcaption></figure>

<caption>**Table 1**</caption>

Evaluation of the dynamical anomalies of EWD and ECD estimated with the three regional climate models with respect to the dynamical anomalies estimated with ERA5. The dynamical anomalies are estimated using the 1991–2020 reference period for the construction of the analogues in the three regional climate simulations and in ERA5. Three scores are analyzed: the mean absolute error (MAE), the mean error (ME) and the spatial correlation, in winter and in summer. The scores are calculated for land points within the domain in Fig. 2 for all EWD in summer and ECD in winter, and then the median and the 5–95 percentiles (between brackets) are calculated across the events.

<table><thead><tr><th></th><th>CNRM-CM5_ALADIN</th><th>CNRM-CM5_HADREM3</th><th>MPI-ESM-LR_HADREM3</th></tr></thead><tbody><tr><td>MAE winter</td><td>1.72 [1.07–3.02]</td><td>1.88 [1.21–3.56]</td><td>1.72 [1.13–3.35]</td></tr><tr><td>MAE summer</td><td>1.53 [1.01–2.66]</td><td>1.43 [1.04–2.52]</td><td>1.46 [0.94–2.32]</td></tr><tr><td>ME winter</td><td>0.23 [-1.97–2.52]</td><td>0.46 [-1.66–2.78]</td><td>0.10 [-2.23]</td></tr><tr><td>ME summer</td><td>-0.84 [-2.35–0.41]</td><td>-0.28 [-2.07–1.14]</td><td>0.12 [-1.52–1.83]</td></tr><tr><td>Correlation DJF</td><td>0.82 [0.57–0.94]</td><td>0.79 [0.39–0.93]</td><td>0.80 [0.50–0.93]</td></tr><tr><td>Correlation JJA</td><td>0.86 [0.65–0.93]</td><td>0.86 [0.73–0.93]</td><td>0.88 [0.75–0.93]</td></tr></tbody></table>

captured by all RCMs in both winter and summer, with median correlations around 0.8 in winter and 0.85 in summer. The 5–95 % range of correlations is larger in winter, mainly due to the presence of days with mediocre correlations.

The median MAE is about 1.8 K in winter and 1.5 K in summer, with also a smaller range in summer. Differences between models are seen, with CNRM-CM5_ALADIN performing slightly better in winter and worse in summer than the two other models. CNRM-CM5_ALADIN in particular tends to underestimate the dynamical anomalies in summer (median error of -0.84) while CNRM-CM5_HADREM3 tends to overestimate the dynamical anomalies in winter.

Overall, the RCMs do a good job in capturing dynamical temperature anomalies, which increases our confidence in their use for translating past events into future climate conditions. More generally, independently of futurization, the evaluation of dynamical temperature

anomalies could be an interesting diagnostic when using climate models to study extreme events and their changes. Overly biased dynamical anomalies in a model would limit our confidence in its ability to capture future changes in extreme events driven by atmospheric circulation.

## 7. Futurization of observed extreme temperature days

The futurization method developed in this study consists in constructing large-circulation analogues of observed events using regional climate projections as base dataset. In the previous section, the base dataset comes from the 1991–2020 period in three RCM historical simulations. The same analysis is now done using a base dataset from the 2070–2099 simulated period with the same RCMs. The dynamical anomalies obtained correspond to the temperature anomalies that would be obtained should the observed circulation pattern occurred in the future climate projections. For each RCM, they can be compared to the anomalies obtained using the 1991–2020 reference period, in order to assess how the temperature anomalies associated with the same atmospheric circulation pattern change between the present and future climates.

As an illustration, the dynamical anomalies obtained for the three RCMs with the 2070–2099 reference period and their changes with respect to the 1991–2020 reference period are first compared for the 26 February 1986 (Fig. 6).

While, as noted earlier, the dynamical anomalies are very similar for the three RCMs on the present climate (Fig. 5), this is less true for the same event in the future climate, except over Scandinavia (Fig. 6). In CNRM-CM5_ALADIN and CNRM-CM5_HADREM3, strong positive dynamical anomalies are observed over a large part of France and Spain, while it is much less the case for MPI-ESM-LR_HADREM3. The dynamical anomalies are still negative over central Europe in MPI-ESM-LR_HADREM3, but not anymore in CNRM-CM5_ALADIN. This is because the changes in dynamical anomalies (Fig. 6d–f) are very different between models. For CNRM-CM5_ALADIN a very large increase in temperature occurs over central Europe. For MPI-ESM-LR_HADREM3, the warming is generally largely smaller, with no
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Dynamical temperature anomalies (K) corresponding to the 26 February 1986 in (a) CNRM-CM5_ALADIN, (b) CNRM-CM5_HADREM3 and (c) MPI-ESM-LR_HADREM3 in the future climate: the reference period for constructed analogues is 2070–2099. Changes in dynamical temperature anomalies (K) corresponding to the 26 February 1986 between 2070–2099 and 1991–2020 for (d) CNRM-CM5_ALADIN, (e) CNRM-CM5_HADREM3 and (f) MPI-ESM-LR_HADREM3. All anomalies are relative to 1961–1990.</figcaption></figure>

warming over the north of Spain and southwestern France.

Table 2 shows the inter-event correlations calculated for each pair of RCMs of the dynamical anomalies of EWD and ECD, both in the present climate, and for their changes. The correlations are generally small for the changes, much smaller than the correlations obtained for the present-day dynamical anomalies (Table 2). This is consistent with what was noted in Fig. 6 for a particular case. The change in intensity of extreme temperature events is therefore very model dependent.

Interestingly, the inter-event correlations for present-day dynamical anomalies and their changes are greater for the pair CNRM-CM5_ALADIN/CNRM-CM5_HADREM3, i.e. the pair that shares the same forcing GCM, and in winter rather in summer. In summer, no relationship is found between the changes for the pairs forced by different GCMs. Although a larger sample of regional climate simulations would be needed to draw definitive conclusions, the forcing GCM appears to be more important than the regional climate model in

<caption>**Table 2**</caption>

Inter-event correlations of the present-day dynamical anomalies and their future changes estimated with different regional climate models for ECD in winter (DJF) and EWD in summer (JJA), calculated for each pair of RCMs.

<table><thead><tr><th></th><th>CNRM-CM5_ALADIN/MPI-ESM-LR_HADREM3</th><th>CNRM-CM5_ALADIN/CNRM-CM5_HADREM3</th><th>MPI-ESM-LR_HADREM3/CNRM-CM5_HADREM3</th></tr></thead><tbody><tr><td>Present-day DJF</td><td>0.63</td><td>0.79</td><td>0.61</td></tr><tr><td>Present-day JJA</td><td>0.47</td><td>0.56</td><td>0.55</td></tr><tr><td>Future change DJF</td><td>0.22</td><td>0.40</td><td>0.12</td></tr><tr><td>Future change JJA</td><td>0.05</td><td>0.36</td><td>0.00</td></tr></tbody></table>

determining the dynamical anomalies and their changes, in term of inter-event correlation. This is probably because large-scale temperature gradients play an important role in establishing the temperature anomalies associated with a given circulation pattern, and these large-scale temperature gradients in RCMs are strongly influenced by the boundary forcing from the GCMs.

We now look at the changes in dynamical anomalies of all ECDx and EWDx. Note that these changes are computed as previously over the area where the total temperature anomalies of the observed event are greater than the 95th percentile (for EWDx) or smaller than the 5th percentile (for ECDx).

A positive shift in dynamical temperature anomalies is noted for both extreme warm days and extreme cold days in the future climate. Not surprisingly, extreme temperature events, as defined by their atmospheric circulation pattern, become more intense in the future climate, due to global warming, although in some cases not by much. In fact, perhaps less expected, is the very large inter-event spread regarding the intensification of extreme days in the future climate. In winter for example, some extreme days are 6 K or more warmer in the future climate, while others are barely warmer.

Even if the intensity of dynamical changes with CNRM-CM5_ALADIN and CNRM-CM5_HADREM3 are somewhat correlated between events in summer (Table 2), there are some large mean differences between the models, especially in summer, with a small warming in CNRM-CM5_ALADIN, but a much larger one in CNRM-CM5_HADREM3 (Fig. 7). This is consistent with a much larger average warming in summer in CNRM-CM5_HADREM3 (3.9 versus 5.3, Fig. 7). Even if they are uncorrelated from event to event (Table 2), the mean dynamical changes in CNRM-CM5_HADREM3 and MPI-ESM-LR_HADREM3 are more comparable, probably because of similar mean warming (4.9 K versus 5.3 K, Fig. 7).

In the end, the mean warming of extreme temperature days in summer seems to be strongly influenced by the mean regional warming. However, there is a large inter-event spread, probably because large-scale temperature gradients, influenced by boundary forcings, are
<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Box-and-whisker plot of changes between 2070-2099 and 1991-2020 in dynamical temperature anomalies for EWDx (orange) and ECDx (blue) using (C_A) CNRM-CM5_ALADIN, (C_H) CNRM-CM5_HADREM3 and (M_H) MPI-ESM-LR_HADREM3 as base for constructed analogues. The mean maximum temperature change calculated over the land points of the domain in Fig. 2 in summer and winter are shown with dark red X symbols. The annual global temperature change on the same period from the forcing GCMs is shown with green dots. The box shows the 1st and 3rd quartiles, and the median. The whiskers extend to the minimum or maximum of the distribution.</figcaption></figure>

important in that context. The role of global temperature change in the forcing GCM does not seem to be important, as it is very close for the two forcing GCMs.

Interestingly, while the mean warming in winter (3.5 K and 3.1 K respectively in CNRM-CM5_ALADIN and MPI-ESM-LR_HADREM3) is smaller than the mean warming in summer (3.9 K and 4.9 K for the same models) over western Europe, extreme cold days can intensify more in the future climate than extreme warm days in summer (in CNRM-CM5_ALADIN) or as much (in MPI-ESM-LR_HADREM3). This likely partly explained by the fact that cold events over western Europe in winter are often associated with northeasterlies advecting very cold air, from the Arctic or northeastern Europe. A strong mean warming is generally noted over northeastern Europe in winter (Lee et al., 2021), linked to the snow-albedo feedback (Groisman et al., 1994), and over the Arctic because of the Arctic amplification, which is likely to influence the intensity of extreme cold days in western Europe (Tamarin-Brodsky et al., 2020). More generally, Tamarin-Brodsky et al. (2020) show the important role of advection and warming patterns in shaping the variance and skewness of temperature changes. For the circulation pattern corresponding to a given EWD or ECD, the regional warming pattern is also likely be important in determining how the associated dynamical anomalies will change with climate change.

## 8. Conclusion

In this study, a hybrid statistical dynamical method to translate observed historical weather events into a future warmer climate is presented, and applied to extreme temperatures. The method is intended to estimate the temperature anomalies over western Europe that would be obtained should an observed event of the 1940–2023 period, defined in terms of atmospheric circulation, happened at the end of the 21st century, under new climatological conditions. In practice, constructed analogues of historical extreme events are built using as base dataset three regional climate projections, from two RCMs and two GCMs. The same approach is also used beforehand to estimate the part of the observed temperature anomaly during extreme events that is due to atmospheric circulation alone. Events of interests are defined as extreme warm days in summer and extreme cold days in winter on the 1940–2023 period, characterized by large climate anomalies over a large part of western and central Europe.

The role of large-scale circulation in observed extreme temperatures varies greatly from event to event, ranging from 35 % to 80 % in summer

and from 20 % to 90 % in winter. Other processes, such as land-atmosphere interactions, therefore play an amplifying role. An example from the winter of 1986 is shown, where snow anomalies from previous snowfall strongly amplify the temperature anomaly during a cold spell.

The intensity of temperature anomalies increases in the future climate for the three RCMs used for futurization with the RCP8.5 scenario, with a large inter-event spread, between 1 K and 7 K in summer and 0.5 K and 7 K in winter. Some historical severe heatwaves could therefore become much more intense in the future climate, while some historical cold spells could remain almost as cold relative to the present-day climatology.

While all RCMs generally capture the dynamical part of the magnitude of extreme temperature days correctly, and are close to each other in the present climate, their changes can be largely different in terms of intensity and spatial pattern. This shows, that even for an identical circulation pattern, major differences may exist between the dynamical contribution response of different models to climate change.

The exact same method described in this work could be used for conditional attribution, using a pre-industrial regional climate simulation as counter-factual world rather than a future projection. Regional simulations long enough to capture pre-industrial climate are needed in this context, which is generally not the case, as they tend to be initialized in the second half of the 20th century. Alternatively, it is possible to use a different counter-factual world, for example corresponding to global warming level of 1 °C.

As noted in the introduction, purely dynamical methods for futurization or conditional attribution exist. They are based on global climate simulations where the atmospheric circulation is nudged towards the circulation of an event of interest, or on regional climate simulations with boundary forcing from the event of interest. It would be interesting to compare the results of the hybrid-dynamic statistical method described in this study with those of purely dynamic approaches. The two approaches could be highly complementary.

Dynamical methods based on a nudged GCM or an RCM, which are numerically expensive, typically use a single model, with a single way of perturbing the simulation to represent the influence of climate change on the event of interest. They therefore do not capture the large inter-model uncertainties highlighted in this work. Our hybrid approach could be used to explore these uncertainties, while purely dynamical methods would allow to better study the mechanisms at play, thanks to the simulation of all necessary variables.

Dynamical studies based on specific numerical experiments are generally expensive, for example if many members are needed to extract the climate change signal and/or if a high resolution is needed to better represent the weather event of interest. The constructed analogue approach described here can be used to confirm that the event of interest is largely circulation driven. If it not the case, there is no point in defining the event in terms of large-scale atmospheric circulation, and the event of interest would be rather difficult to reproduce with GCM or RCM simulations (e.g. Matte et al., 2022).

There is a debate about conditional attribution (e.g. García-Portela and Maraun, 2023). Critics of conditional attribution based on large-scale circulation point out that it can lead to an overestimation of the role of global warming (Stott et al., 2017), because it does not take into account how large-scale atmospheric circulation conditions are affected by global warming. Note that there is actually no reason why not considering potential changes in large-scale atmospheric circulation would necessarily lead to an overestimation of the role of global warming. It could also lead to an underestimation (Shepherd, 2021).

Like conditional attribution based on large-scale circulation, futurization does not take into account how large-scale atmospheric circulation may change. We translate an observed event defined by its atmospheric circulation pattern into the future climate, but it is possible that this specific type of circulation becomes very unlikely in the future climate (note that a robust demonstration of this statement would
require a very large initial-condition multi-model ensemble). Such an extreme case is unlikely because the signal-to-noise ratio of large-scale circulation changes is generally small, but the probability of occurrence of the event could change with climate change. The constructed analogue method cannot be used to study changes in the occurrence of extreme events.

In our view, as conditional attribution, futurization remains a valuable approach. As discussed in the introduction, focusing on specific events translated into the future climate could be a useful complementary way of approaching adaptation to climate change, and understanding how, given a particular large-scale atmospheric circulation pattern, other remote and local processes might change, potentially leading to different impacts.

As a proof of concept, three regional climate simulations are used in this paper for futurization. Our analyses show that there are large intermodel differences in the futurization results. Obviously, a larger multi-model ensemble of regional climate projections could be used to better capture model uncertainties in this context. The use of convection permitting RCM instead of RCM could also be interesting, especially when dealing with extreme precipitation events, given their added value in that context (Caillaud et al., 2021). The approach developed in this study could also be used simply as an evaluation tool to assess whether climate models are able to correctly capture the dynamical part of extreme events.

The futurized events derived with our method correspond to the same event under different (i.e. future) climatological conditions. Although some of them are already much more severe in the future climate, other mechanisms may further amplify (or dampen) these dynamical anomalies corresponding to climatological conditions. For example, unusual SST anomalies or soil moisture conditions could play that role. It might be possible by extending the approach discussed in this article to explore worst-case scenarios, by adding additional constraints in the search for analogues, for example, using only days with dry soils or positive SST anomalies rather than all days for the construction of analogues. This will be explored in a future study.

## Appendix A. Choice of parameters for constructed analogues

Three parameters of the constructed analogue method have to be fixed: $N_a$, $N_s$, $N_r$. The constructed analogue method is applied using ERA5 on the 1991–2020 reference period for different values of $N_a$, $N_s$, $N_r$. The dynamical and original sea level pressure and maximum temperature anomalies are then compared (Table A.1). The daily spatial correlations and root mean square errors (RMSE) are computed on the domain shown in Fig. 2. Only land points are considered for maximum temperature; land and sea points are used for sea level pressure. Ideally, the dynamical and original sea level pressure anomalies should be virtually identical (RMSE = 0 and spatial correlation of 1). This is not the case for maximum temperature, because large-scale circulation explains only part of the temperature variability, but lower RMSEs and higher correlations are desirable.

Arguably, the larger the value of $N_r$ is, the more robust the estimate of dynamical anomalies is, with smaller sampling uncertainties. Larger $N_r$ however increases the necessary computing resources and time. We choose a value of $N_r$ = 200 as a trade-off, and we show that in any case there is virtually no improvement from $N_r$ = 100 to $N_r$ = 200 (Table A.1). For the two other parameters, $N_a$ and $N_s$, multiple values are tested (Table A.1).

<caption>Table A.1</caption>

Median and [5-95th] percentile of the distribution of daily spatial correlations and RMSEs between sea level pressure (hPa) and (K) maximum temperature obtained with the constructed analogues method, calculated for each day of the 1940–2023 period for different triplets of parameters ($N_a$; $N_s$; $N_r$)

<table><thead><tr><th>N<sub>a</sub>;N<sub>s</sub>;N<sub>r</sub></th><th>Corr. SLP</th><th>RMSE SLP</th><th>Corr. max temp.</th><th>RMSE max temp.</th></tr></thead><tbody><tr><td>100; 20; 200</td><td>0.99 [0.97-1]</td><td>1.62 [0.89-4.87]</td><td>0.65 [0.14-0.87]</td><td>0.028 [0.017-0.047]</td></tr><tr><td>500; 20; 200</td><td>0.99 [0.98-1]</td><td>1.91 [0.82-5.78]</td><td>0.66 [0.17-0.87]</td><td>0.028 [0.017-0.047]</td></tr><tr><td>400; 80; 200</td><td>0.99 [0.99-1]</td><td>1.61 [0.39-5.14]</td><td>0.74 [0.33-0.90]</td><td>0.025 [0.016-0.043]</td></tr><tr><td>500; 80; 200</td><td>1.00 [0.99-1]</td><td>1.61 [0.39-5.17]</td><td>0.74 [0.35-0.90]</td><td>0.024 [0.016-0.043]</td></tr><tr><td>900; 200; 200</td><td>1.00 [0.99-1]</td><td>1.67 [0.34-5.36]</td><td>0.75 [0.37-0.90]</td><td>0.024 [0.016-0.041]</td></tr><tr><td>500; 100; 200</td><td>1.00 [0.99-1]</td><td>1.61 [0.37-5.23]</td><td>0.74 [0.35-0.90]</td><td>0.024 [0.016-0.042]</td></tr><tr><td>500; 100; 100</td><td>1.00 [0.99-1]</td><td>1.62 [0.36-5.22]</td><td>0.74 [0.34-0.90]</td><td>0.024 [0.016-0.042]</td></tr></tbody></table>

The daily spatial correlations for sea level pressure are excellent for all combinations of parameters, with somewhat better RMSE for ($N_a$ = 400; $N_s$ = 80), ($N_a$ = 500; $N_s$ = 80) and ($N_a$ = 500; $N_s$ = 100). The constructed analogues are therefore near perfect analogues, which shows the interest of the approach. For maximum temperature ($N_a$ = 400; $N_s$ = 80), ($N_a$ = 500; $N_s$ = 80), ($N_a$ = 500; $N_s$ = 100) and ($N_a$ = 900; $N_s$ = 200) lead to higher correlations and smaller RMSEs. For the remainder of the study, the parameters ($N_a$ = 500; $N_s$ = 100; $N_r$ = 200) are used.

## CRediT authorship contribution statement

Julien Boé: Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Mar-got Bador: Writing – review & editing, Conceptualization. Laurent Terray: Writing – original draft, Conceptualization.

## Data statement

Hersbach, H. et al. (2018) was downloaded from the Copernicus Climate Change Service (2023): ERA5 hourly data on single levels from 1940 to present. Copernicus Climate Change Service (C3S) Climate Data Store (CDS), https://doi.org/10.24381/cds.adbb2d47.

The results contain modified Copernicus Climate Change Service information 2020. Neither the European Commission nor ECMWF is responsible for any use that may be made of the Copernicus information or data it contains.

## Funding

This work was supported by the European Union through the HO-RIZON-EUROPE IMPETUS4CHANGE project under Grant Agreement 101081555. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them. This study has received funding from Agence Nationale de la Recherche - France 2030 as part of the PEPR TRACCS programme under grant number ANR-22-EXTR-0011.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
# Data availability

Data will be made available on request.

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