We use data from ERA5 , the fifth-generation atmospheric reanalysis from the European Center for Medium-Range Weather Forecasts (ECMWF), which provides globally complete, gridded data from 1940 to the present. The long-term availability and spatial completeness of reanalyses make them suitable for studying climate trends, variability, or extreme events despite substantial uncertainties across datasets . We use daily data using the spatial domain discussed in Sect. . We use atmospheric circulation fields that represent the state of the atmosphere and constitute the large-scale variables in the study: Geopotential at 500 hPa (g500) [m² s⁻²], Geopotential at 200 hPa (g200) [m² s⁻²], and Sea Level Pressure (psl) [hPa].

For land-surface variables, we use the ERA5-Land dataset. Soil moisture variability is more consistent in ERA5-Land because it does not assimilate screen-level observations to adjust soil moisture (unlike the ERA5 reanalysis) . The variables we use are daily maximum 2 m temperature (TX) to define our target events, and soil moisture at three depth levels (swvl1, swvl2, and swvl3) as predictors. TX from ERA5-Land is only used to define hot extremes, as the prediction target of the ML model. All land-surface variables are taken as the spatial average over a 100 km × 100 km region to avoid focusing on a single grid-point.

Lastly, we include seasonal atmospheric CO₂ concentrations to account for the trend of climate change. We employ U.S. National Oceanic and Atmospheric Administration (NOAA) measurements collected at the Mauna Loa Observatory in Hawaii. The concentrations used have a constant value for each summer and are a seasonal mean. By explicitly including CO₂, our framework controls for the long-term warming trend as a potential additional driver of heat extremes, allowing for a cleaner attribution of the physical drivers.

Circulation and land variables are used as lagged standardized anomalies (explained in Sect. ). For each prediction day, the model is trained on features derived from values recorded on preceding days. For the local-scale, soil moisture is averaged over the previous 7 d to account for its relatively low daily variability . The values of circulation variables are taken from the day before the prediction, corresponding to a 1 d lag in atmospheric features. The inclusion of up to three lags for the circulation features was tested, but we decided to keep a simpler model because the accuracy did not improve by including more lags (not shown).

Similar variable sets using atmospheric circulation and land features have been used in , where they developed a machine-learning framework combining spatial extreme-value theory and adapted loss functions to study large-scale heat extremes, or in , who predict the probability of occurrence of long-lasting 14 d heatwaves over France.

To test the robustness of our results to the data source, we supplement the analysis based on reanalysis with observational data for the daily maximum temperature and the land component. For the atmospheric circulation features, we keep the data from ERA5. For the local-scale daily maximum temperature used in the extreme event definition, we use directly the E-OBS gridded dataset (version 29.0) of TX .

Due to the lack of long-term, consistent observational soil moisture records, we employ the Standardized Precipitation Index (SPI) and the Standardized Precipitation Evapotranspiration Index (SPEI) as proxies for soil moisture anomalies, computed with the E-OBS data. These indices serve to validate the robustness of our ERA5-Land results against observational data, providing a reliable alternative for historical comparison. These indicators have already been assessed as surrogates of in situ and satellite soil moisture observations , with generally larger association when potential evapotranspiration is included. We use the Hargreaves method for computing the evapotranspiration, essential for the SPEI calculation, and the gamma distribution to standardize the data. Precipitation and temperature data are sourced from the E-OBS dataset. As for soil moisture, both SPI and SPEI grid-points are averaged spatially over a 100 km × 100 km region.

The timescale for the index computation was selected by computing the Spearman correlation of the drought indices with the three soil moisture levels of the original model (see Figs.  and ). Soil moisture is a variable with a memory that extends beyond a few days , so the drought index must be computed on a scale that reflects this persistence. Based on the correlation analysis, the selected timescale for the SPEI and SPI indices is 30 d (SPEI30d and SPI30d). This timescale shows the best correlation of the indices with swvl1 and swvl2, which we show to be in general the most relevant depth levels in our explainability analysis (see Sect. ). We also computed the correlation using a 7 d rolling mean of the soil moisture data – the same preprocessing applied before inputting soil moisture to the model – and obtained very similar results, with correlation values not differing more than 0.07. Additionally, we computed the drought indices using ERA5 data to validate the correlations computed with the E-OBS gridded data, with correlation between datasets not differing more than 0.1. In the correlation analysis, we did not lower the timescale of the drought indices computation (e.g. 15 d) to stay within the original monthly definition of the indices , and to avoid fitting the normalized distribution to data that may not be sufficiently smooth.

With the atmospheric pressure and geopotential height fields, we aim to capture the general circulation influencing the locations where we predict extreme temperature events. The large-scale domain must be large enough to cover the area of interest, but not too large to avoid including unrelated atmospheric variability . Figure  shows the selected domain marked as a red box. The full North Atlantic Ocean is included, since studies link this region to the development of extreme temperature events in various European regions . A large part of North Africa is included as well to capture warm and dry intrusions that may enhance the likelihood of extremes . The northern and eastern limits are extended enough to include synoptic-scale circulation patterns.

We study the roles of circulation, soil moisture and anthropogenic climate change during extreme temperature events in six regions around Córdoba, Marrakech, Lyon, Belgrade, Hannover, and Stockholm (see Fig. ). For each site, local-scale data is averaged over a surrounding 100 km × 100 km region, to smooth out unwanted variations across space. The chosen locations span different evaporative regimes: from water-limited (Marrakech) to energy-limited (Stockholm). Location choice is also partly based on literature: Córdoba and Marrakech are studied in , Stockholm's 2018 heatwave in , and Iberia and Central Europe heatwaves in . Belgrade was selected to provide a more eastern perspective, potentially influenced by sea surface temperatures and atmospheric anomalies in other regions discussed in .

Lastly, the robustness analysis using observational data (discussed in Sect. ) could not be carried out for Marrakech, due to missing data for several years (see Fig. ).

Both the climatology and 90th percentile are computed only for the reference period 1950–2000, within the model's training period (1950–2014). This avoids an indirect provision of future information to the model, which would leak evaluation data into training. The training period was extended to 2014 to include a period with a clear trend in the yearly count of extremes (see Fig. ).

First, we calculate the raw climatology by taking the daily average. Then, as in , we apply a local polynomial regression (LOESS) to obtain a smoothed profile. After a few tests with the window timescale, we selected a 30 d smoothing window for both climatology and standard deviation. Anomalies are computed daily by subtracting the smoothed climatology from the original data, and then standardized using the standard deviation. Subtracting the climatology removes the annual cycle, and applying the LOESS smoothing removes the short-term variability (see Fig.  example for Córdoba site). Extended detail on LOESS is given in Appendix .

Using the extreme definition in an extreme event is defined when the daily mean maximum temperature exceeds a specified percentile of the climatological distribution. Following , the percentile is calculated over a centered 5 d window using the reference period, and then we apply LOESS smoothing with a 30 d window. We use a 5 d rolling window prior to the LOESS smoothing to increase the number of samples available for estimating the 90th percentile for each calendar day. The window is centered in the day for which the percentile is computed. The primary analysis in this study uses the 90th percentile, but to test the robustness of our results to the extreme event definition, we conduct additional analyses using the 80th and 95th percentiles, and all are computed following the same procedure.

The model is trained on data from 1950 to 2013. Validation years are chosen by randomly selecting contiguous blocks of years (see Table ), while the testing period spans 2014 to 2024. Extreme days are defined using the 90th percentile threshold (i.e., approximately 10 % of all days), which results in a class imbalance between extreme and non-extreme days (see Table ). This imbalance represents a central challenge during model training, as machine learning algorithms may default to predicting the majority (non-extreme) class if no specific strategy is used to address it. At the same time, the relatively limited number of extreme cases increases the risk of overfitting in this binary classification problem when using models with many parameters. This constraint needs to be considered when designing the model architecture and training procedure. The imbalance issue becomes even more pronounced when higher percentile thresholds are used to define extremes, as the number of available training samples decreases further. (see Table ). In this study, we treat the class imbalance by training the model with a weighted loss function. By doing this, we penalize more the wrongly classified extreme samples.

The main contributor to high parameter dimensionality are the large-scale fields, which include spatial dimensions at 0.25° resolution, unlike local-scale data that is area-averaged and provides time-only vectors. Flattening large-scale data would result in nvar⋅nlags⋅nlat⋅nlon inputs, making the model overly complex. We first reduce dimensionality by re-gridding large-scale data to 1°, directly reducing computational cost and overfitting risk. We think this coarser resolution still captures atmospheric circulation features for extreme temperatures.

We use a model architecture designed to capture land–atmosphere coupling (see Fig. ). It includes a Multi Layer Perceptron (MLP) for land and CO₂ data, and a ConvNeXt convolutional neural network (CNN) for spatial features from circulation, which, to our knowledge, has not been used before in driver quantification studies. Their outputs are concatenated and passed to a final MLP for binary classification of extreme or non-extreme on a daily basis. This architecture is hereafter referred to as the CombinedModel.

The MLP for the Land State + CO₂ inputs receives an input vector of dimension din, followed by two hidden layers of 15 and 8 nodes. The final classification MLP then takes the concatenated outputs from the Local and Large-scale branches (with dimensions 8 and 16, respectively) as its input. These features are processed through three hidden layers of 64, 32, and 8 nodes before reaching the final output layer. For the spatial feature extraction, a custom ConvNeXt architecture is employed. The model accepts input tensors with C channels (corresponding to the number of Large Scale features) and utilizes a patch size of 4. The ConvNeXt here consists of four stages with depths of [1,2,2,1] and corresponding layer dimensions (channels) of [4,6,6,16]. A dropout rate of 0.05 is applied to prevent overfitting during training. The architecture of the ConvNeXt model is explained in detail in . ReLU is used as activation function in the MLPs, and softmax applied at the output to yield probabilities . A 0.5 threshold determines the predicted class, kept constant throughout the study. Additionally, we use a 20-member ensemble to filter out noise and handle part of the epistemic uncertainty of the model. These ensembles are generated by changing the initial random seed (detailed in Sect. ). Similar hybrid model architectures have been used in the past .

The optimizer used for the CombinedModel is the AdamW optimizer . We train the model with an 80 % and 20 % random split of the samples for the training and validation respectively, and apply a random shuffle. The Cross-Entropy Loss function is used (see Sect. ), with specific weights applied to the non-extreme and extreme classes to address the class imbalance. The Optuna Python package is used for the tuning of the following hyperparameters of the ML model: batch size, learning rate, weight decay and class weights of the predicted classes. The model is trained for 75 epochs unless validation loss increases for a total of five epochs. The score used in the hyperparameter tuning process is the balanced accuracy minus the final validation loss, which is the best validation loss found before early-stopping.

To quantify the contribution of each input feature to the model's predictions, we use SHapley Additive exPlanations (SHAP), a game-theoretic approach that explains the output of any machine learning model by attributing it to the input features .

Positive SHAP values mean a positive influence on the likelihood of the predicted class by the model, while a negative SHAP value means that the feature decreases the likelihood of predicting the class. The magnitude of the value quantifies the influence of the feature, and SHAP values are zero for irrelevant. SHAP values are an additive feature attribution method, meaning that the contribution of each feature can be computed independently, and all values summed up.

The sum of the SHAP values across all features equals the model output (prediction) minus the expected (baseline) prediction: 1f(x)=ϕ0+∑iNϕi, where f(x) is the model prediction for the input x, ϕ0 is the baseline (computed by averaging the model prediction over training and validation data), ϕi is the SHAP value for the feature i and N is the total number of features in the model. The additive nature of SHAP values allows us to compute a value per grid-point, highlighting the regions the model finds most important, and to spatially aggregate these values into global importance or combine them across features, as we do in Sect. .

We compute SHAP values in the testing period (years 2014–2024) using GradientExplainer , which is designed for differentiable models, such as deep neural networks, and estimates them by using an extension of the integrated gradients method . We believe this choice to be appropriate given the evaluation metrics results for the Integrated Gradients method shown in , though no XAI method evaluation was carried out for our study. Additionally, the method requires selecting a baseline to serve as a reference in the SHAP computation. Following best practices suggested by , the baseline was constructed by selecting the samples from the training and validation period that had non-extreme labels. Using only non-extreme days for the baseline is appropriate in this study, as our focus is on identifying the drivers of predicted extreme events with the aforementioned ML model.

Since our focus is to study the contribution of the atmospheric circulation, the soil moisture, and atmospheric CO₂ concentrations during extreme temperature events, we want to ensure that our explanations are done for predictions for which our model is confident. To do so, we take an ensemble of trained models with twenty members, adding robustness to our output probabilities and computed SHAP values. The members are created by setting deterministically twenty different seeds, initializing the random processes present in the framework. On top of that, we analyse the general explainability results for a certain percentage of top most confident predictions, following the approach of studies like .

We assess model performance using balanced accuracy (BA) and the Area Under the Receiver Operating Characteristic Curve (AUC), which are well-suited for imbalanced classification tasks.

For the primary analysis using the 90th percentile threshold (see Table ) gives relevant statistics of the trained CombinedModel on the test data using the 50 % most confident predictions. Figure  shows how we assess the CombinedModel model robustness by plotting the evolution of accuracies when varying the confidence threshold. For each member individually, the CombinedModel shows a smooth increase in both balanced and extreme accuracy across all locations. The improvement in both accuracy measures with increasing network confidence is an expected behavior. For some locations, low percentages of confidence (see Fig. ) show a trade-off between extreme and non-extreme accuracies, with the former increasing and the latter decreasing, until a minimum is reached and non-extreme accuracy starts improving again. We did not explore this behaviour in depth, but it may be related to how we handle class imbalance.

In Fig.  we show the ROC curve change when changing the percentage of confidence. When using the ensemble of 20 members, most locations show a displacement towards area under the ROC curve equal to 1.

The model trained on the 80th percentile definition shows a similar statistical robustness for the accuracies and ROC curves (Fig. ). The increased number of extreme events mitigates class imbalance, resulting in smooth and consistent improvements in balanced accuracy and AUC values as the confidence threshold increases. This indicates the model is well-trained, and its predictions are reliable for this definition. In contrast, the model performance for the 95th percentile definition is notably weaker for most locations, with the balanced accuracy behaving poorly when increasing the percentage of confidence in the predictions. The drastically reduced number of extreme events (see Table ) presents a significant challenge for training, leading to less stable behavior in accuracy metrics and ROC curves due to the strong class imbalance (see Fig. ). The class imbalance and lack of extreme events samples is a known issue in ML application for weather extremes , so the results using the 95th percentile are not surprising. Due to this lack of robust predictive skill, we do not consider the explainability results for the 95th percentile definition to be trustworthy.

Finally, the model trained with observational TX and SPI/SPEI indices for the 90th percentile also demonstrates robust predictive performance, providing confidence in the reliability of its explanatory outputs (not shown).

Taking the 50 % most confident extreme predictions, Fig.  shows the mean SHAP values across samples in the test dataset for each of the features in the model. The results are not sensitive to the confidence threshold. We calculate a global SHAP value for the Land and Atmosphere compounds by summing the contributions of their respective features, which can be done thanks to SHAP values being an additive feature attribution method. To determine the SHAP percentage for a feature, we first calculate the mean of its absolute SHAP values over the entire time series. This time-averaged mean is then expressed as a percentage of the total mean SHAP value contributed by all features.

The atmospheric circulation has the largest importance (66 %–86 %) at all locations. Apart from being intuitive, this result is in line with for instance , who show that geopotential height at 500 hPa is the dominant driver of hot extremes worldwide, particularly in mid-latitudes, and that the relevance of atmospheric drivers increases with the duration of the hot event. The Land SHAP value is more important in northern regions. In Marrakech, the land SHAP percentage is very small (0.2 %), which is reasonable given the region is generally very dry during the summer, therefore the interannual variability of soil moisture is low and consequently coupling with atmosphere very little. Córdoba shows a higher percentage (9.4 %) explained by the land SHAP compared to Marrakech. The importance of swvl3 in Córdoba is slightly larger than swv1 and swvl2, consistent with top soil layers being fully dissicated in summer. The land SHAP value increases for higher-latitude locations, being specially important in Lyon. This result is in line with , who assessed how changes in the mean temperature and moisture influence the land–atmosphere coupling strength over Europe.

As for the large-scale fields, there is a predominance of the geopotential at 500 hPa, with only Marrakech having sea level pressure being more important than the atmospheric fields at higher altitude.

The atmospheric CO₂ feature has a higher importance in regions that exhibit a stronger trend in extreme temperature days (see Fig. ). Except for Hannover, the importance of the CO₂ feature is generally high (> 10 %). These differences among locations align with the work of in that different regions show different responses to climate change, with the forced fingerprint manifesting through varying signal-to-noise ratios across the globe.

Lastly, Fig.  illustrates how SHAP values vary with standardized anomalies of the soil-moisture predictors. For the upper soil layer (swvl1), negative anomalies are generally associated with positive SHAP contributions to the extreme class, consistent with the expected amplifying role of surface drying on hot extremes. In contrast, the relationship for the intermediate layer (swvl2) is less robust across locations, with weak mean contributions and substantial uncertainty. This behaviour likely reflects the strong correlation between neighbouring soil layers, which limits the model's ability to uniquely attribute predictive skill to swvl2 independently of swvl1. The deepest layer (swvl3) shows a more location-dependent behaviour, with a clearer contribution in Córdoba. The inverse SHAP–anomaly relationship identified for the dominant shallow soil-moisture contribution is also found in the 80th percentile model (not shown) and the model using SPEI (see Fig. ), further confirming the robustness of the land-drying feedback mechanism identified by the model across different methodological choices.

The discussed explainability results are robust to the definition of an extreme event. For the 80th percentile model, the SHAP values (see Fig. ) show the same main characteristics: the circulation component remains dominant (63.9 %–83.7 %), and the land component still exhibits a northward gradient in importance, though less pronounced. Except for Córdoba and Marrakech, the relative contribution of soil moisture is, however, generally smaller than for the 90th percentile model. This would suggest a more marginal role of soil moisture in less severe heat events for which the land–atmosphere feedbacks, a key amplifier of extreme temperatures, may not be fully activated. The fact that the land component contribution is similar for Marrakech and Córdoba in the two extreme definitions might be an indication of the land–atmosphere coupling being rarely activated there in summer due to the scarce coupling in very dry conditions, independently of the extreme definition. Land variables could play an even more important roles for more restrictive percentiles like the 95th percentile. However, as mentioned in Sect. , we do not show the explainability results for this case due to the lack of stability in the accuracy metrics when using this higher percentile as threshold to identify extremes.

To further constrain our analysis with observational records and evaluate the framework's performance beyond reanalysis data, we examine the SHAP results using observational data from E-OBS for TX and the SPEI30d drought index as a proxy for soil moisture (Fig. ).

The driver contributions are overall consistent when using observational data from E-OBS instead of ERA5-Land for TX and the SPEI drought index as a proxy for the soil moisture variables. In general, the XAI results reproduce the established north–south gradient in land driver importance and the dominant importance of the atmospheric circulation component (see Fig. ). In Fig.  we see that we also have a negative correlation between SHAP value and SPEI30d feature values.

When using the SPI index, the XAI results for the land component are less similar to the original ERA5-Land soil moisture results (see Fig. ). Although a correlation analysis between the indices and soil moisture shows that SPEI generally correlated slightly better with soil moisture than SPI (see Fig. ), the differences are relatively small. Nevertheless, the XAI outcomes reveal that SPI captures less of the original soil moisture signal. This aligns with physical understanding, as SPEI, by incorporating evapotranspiration, provides a more comprehensive representation of surface moisture balance and its coupling with the atmosphere than SPI, which is based solely on precipitation. This suggests that for the future application of this framework to climate models, SPEI is a suitable proxy that avoids inter-model differences in representation of soil layers, and therefore allows more consistent evaluation of the relationships across different models and observations/reanalysis.

is a suitable proxy that avoids inter-model differences in representation of soil layers in therefore allows more consistent evaluation of the relationships across different models and observations/reanalysis

Overall, the feature with the largest importance shown by the XAI results for the g500 field. As a result, we decided to test the robustness of the model when using only this field to represent the atmospheric circulation features, obtaining similar results to the ones discussed above (see Fig. ). The predominance of the g500 field is not surprising, as it is known to have a key role in determining the state and evolution of the troposphere, as well as a key indicator of the climate change response . and also show the important role of the atmospheric circulation component in triggering the heatwave event, and the land compound acting as an amplifier of extreme conditions. Moreover, highlight the importance of circulation in the 2018 summer heatwave over Stockholm, and its capability to force warm, long-lasting conditions without much contribution of surface feedbacks.

For the first case study, we test the CombinedModel during the period 10 to 20 August 2021 in the Córdoba region, when a record-shattering heatwave struck the Iberian Peninsula. During this period, temperatures up to 47.6 °C were registered in La Rambla (Córdoba) .

Figure a shows the temporal evolution of the extreme class SHAP values for the atmospheric circulation features in terms of mean absolute SHAP (spatially aggregated), and the three soil moisture levels. Additionally, it shows the standardized temperature anomalies and the true and predicted labels. By looking at the predicted labels, we can see that the model correctly predicts all the true positives of this event but yields false positives at the developing and decaying stages of the heatwave event. Figure b shows the SHAP values for the g500 feature at the temperature anomaly peak identified in Fig. a (14 August 2026). Grid points with |SHAP| < 0.008 are masked out for clarity. Lastly, Fig. c displays the standardized anomalies for the lag1 (13 August 2021) of the g500 field (colours) and the geopotential height at 500 hPa data (black contours). During the thirteenth of August, the positive anomalies and anticyclonic pattern are right above the Iberian Peninsula, and our model gives mainly positive SHAP values for that region, showing strong locality around the predicted location. This signal starts to be strong and positive when the positive anomalies reach the predicted location (Fig. a).

The soil moisture anomalies during this event are negative but not especially low (not shown), resulting in SHAP values for the soil moisture levels being small and remaining quite constant. A curious result is the CO₂ being generally the second most important feature. This suggests that even for a specific extreme event like this one, we can identify a clear anthropogenic contribution.

We select the Hannover region for the second case study to represent a different land–atmosphere coupling regime, characterized by higher soil water content, compared to the Córdoba region . We focus on the 2018 European heatwave, which was an exceptional climatic event that brought prolonged periods of extreme heat and drought to much of Northern and Central Europe, including Germany.

The period selected is 22 July 2018 to 12 August 2018. In Fig.  we show the same type of plots as in the former case study, with the same filtering for the SHAP values applied in Fig. b. Again we see a strong locality in the spatial SHAP values for g500, with positive values surrounding the predicted location, where the positive anomalies are located. The spatial distributions are noisy for the rest of circulation variables, but they all explain less than 10 % of the total mean absolute SHAP value. In Fig. a we can see that in this case the level one of soil moisture plays a more important role compared to the Córdoba event. The soil moisture at level one has positive SHAP values during the whole period, which is comparable to g500 in magnitude for some of the days. Regarding the CO₂ concentrations, in this case its relevance is much lower compared to the Córdoba case.