# Strengths and limitations of statistical and dynamical downscaling for the representation of compound dry and hot events over Spain

M. N. Legasa, A. Casanueva, R. Manzanas

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M. N. Legasa, A. Casanueva, R. Manzanas. Strengths and limitations of statistical and dynamical downscaling for the representation of compound dry and hot events over Spain. International Journal of Climatology, 2025, 2025, pp.e70183. 10.1002/joc.70183. hal-05369016

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RESEARCH ARTICLE

OPEN ACCESS

# Strengths and Limitations of Statistical and Dynamical Downscaling for the Representation of Compound Dry and Hot Events Over Spain

M. N. Legasa<fnref n="1" />,² ID | A. Casanueva<fnref n="1" /> ID | R. Manzanas<fnref n="1" /> ID

**Correspondence:** M. N. Legasa (mikel.legasa@lsce.ipsl.fr)

**Received:** 4 March 2025 | **Revised:** 17 October 2025 | **Accepted:** 30 October 2025

**Funding:** This research is part of the project COMPOUND (TED2021-131334A-I00), co-funded by MCIU/AEI/10.13039/501100011033 and by the European Union NextGenerationEU/PRTR. A. Casanueva acknowledges support from Grant PID2023-149997OA-I00 funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU. M. N. Legasa benefited from funding managed by the Agence Nationale de la Recherche under France 2030 bearing the references ANR-22-EXTR-0005 (TRACCS-PC4-EXTENDING project) and ANR-22-EXTR-0011 (TRACCS-PC10-LOCALISING project).

**Keywords:** climate change | compound events | deep learning | dynamical downscaling | machine learning | regional climate model | statistical downscaling

## ABSTRACT

Compound events pose significant threats to society and ecosystems, making their analysis crucial under climate change. Global climate models, the primary tools for studying future climates, require downscaling to bridge their coarse resolution to local scales. This study evaluates the performance of the two main downscaling approaches—statistical and dynamical—in reproducing compound dry-hot events (co-occurring high temperatures and low precipitation), as represented by the standardised dry and hot index (SDHI). We compare three statistical downscaling (SD) methods—generalised linear models, a posteriori random forests, and convolutional neural networks—against three EURO-CORDEX regional climate models (RCMs), over mainland Spain and the Balearic Islands. Although all the models considered in this work (both statistical and dynamical) provide good results for downscaling precipitation and temperature and are capable of capturing standard multivariate metrics (such as the Spearman correlation between both variables), their performance declines when it comes to the reproduction of compound extremes like dry-hot events. For this particular aspect, neither of the two approaches (statistical and dynamical) consistently outperforms the other. In particular, while SD methods outperform RCMs in reproducing the observed temporal variability of compound dry-hot events, RCMs are better at simulating these events' intensity, likely due to their foundation in physical processes, which enhances inter-variable consistency. Based on the different limitations of both statistical and dynamical models found for properly capturing the tails (dry and hot) of the multivariate distribution, we conclude that more advanced model development is needed for accurate analysis of compound events at the local scales needed for most practical applications.

## 1 | Introduction

In recent years, the study of compound events has attracted increasing attention. Compound events are defined as either combinations of extreme conditions occurring simultaneously,

or as combinations of events that are not themselves extreme but lead to an extreme event or impact when combined (Zscheischler et al. 2022). Among them, compound dry-hot extremes (episodes with co-occurring high temperature and low precipitation, see e.g., Hao et al. 2022) have

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2025 The Author(s). *International Journal of Climatology* published by John Wiley & Sons Ltd on behalf of Royal Meteorological Society.

<footnote n="1">¹Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Santander, Spain | ²Laboratoire des Sciences du Climat et de l'Environnement (LSCE-IPSL), CEA/CNRS/UVSQ, Université Paris Saclay, Centre d'Etudes de Saclay, Gif-sur-Yvette, France</footnote>
been shown to be among the most detrimental to society and ecosystems (Gouveia et al. 2016; Turco et al. 2019; Ruffault et al. 2020), impacting energy production (Giannakopoulos et al. 2009; Rübbelke and Vögele 2011; Ahmad et al. 2023; Zeighami et al. 2023), human health (World Meteorological Organization 2021; Tabari and Willems 2023) as well as agricultural systems (Zampieri et al. 2017; Beillouin et al. 2020; Christian et al. 2020; Ciais et al. 2005; Ribeiro et al. 2020). An increase in climate extremes, including compound events, has been associated with climate change (De Luca and Donat 2023; Hosseinzadehtalaei et al. 2024; Shi et al. 2024). Thus, it is essential to have a spatially detailed and reliable projection of such events, in order to inform mitigation and adaptation policies.

Global climate models (GCMs) constitute the main and most comprehensive tool for studying current climate processes and investigating potential future climates. However, the existing computational constraints, together with the need for large ensembles of long simulations, continue to limit nowadays the temporal and spatial resolution GCMs can achieve. Partly because of this reason, they are known to exhibit systematic biases (i.e., deviations from the observed statistics when simulating historical climates). Their low resolution and biases hinder the use of GCMs for most practical applications (see, e.g., Doblas-Reyes et al. 2013; and references therein), including the investigation of compound events.

Statistical and dynamical downscaling (SD and DD, respectively) are the main two approaches used to bridge the gap between the coarse-resolution simulations provided by GCMs and the local-to-regional information required by most practical applications. On the one hand, DD is based on the use of regional climate models (RCMs; Giorgi 2019). RCMs are numerical models nested into GCMs that solve the governing equations of the atmosphere over limited spatial domains at higher spatial and temporal resolutions. On the other hand, SD relies on statistical and machine learning techniques to learn empirical relationships between observed records of large-scale variables and local, high-resolution target variables at the surface, like temperature or precipitation (Maraun and Widmann 2018; Maraun et al. 2019). Previous studies suggest that SD methods and RCMs exhibit comparable skill in representing regional climate features, albeit with distinct strengths and limitations. Their performance varies across regions and is influenced by local conditions and the specific climate variable under consideration, with no single approach (statistical or dynamical) performing best in all evaluated aspects (Murphy 1999; Haylock et al. 2006; Menéndez et al. 2010; Huth et al. 2015; Vaittinada Ayar et al. 2016; Bettolli et al. 2021; Byun et al. 2022). Moreover, the relative performance of SD methods and RCMs may depend on the assessed metric (Gutiérrez et al. 2019) and on the observational dataset considered for evaluation (Casanueva et al. 2013; Casanueva, Herrera, et al. 2016). For instance, Hertig et al. (2012) indicates that SD methods are better suited to represent the frequency of extreme precipitation events, whereas RCMs better represent event intensity. One aspect that is worth highlighting is that, as opposed to RCMs, SD methods do not explicitly solve any physical process, and thus, inter-variable consistency is not granted. However, RCMs are well known to have biases both in the marginal and multivariable distributions, inheriting them from their driving GCMs

and/or generating their own (Christensen et al. 2008; Sørland et al. 2018).

Taking all this into consideration, this work presents a comparison of SD and DD approaches targeting, specifically, the reliable reproduction of compound dry and hot events. We do so by using the standardised dry and hot index (SDHI; Hao et al. 2018), which is based on the bivariate distribution of temperature and precipitation and provides a severity measure of a compound dry-hot event. A key property of this index is that it is not sensitive to the potential biases in the marginal distributions of precipitation and temperature, allowing for a fair intercomparison between SD methods and RCMs. To our knowledge, no study has analysed this timely issue to date.

## 2 | Data and Methodology

### 2.1 | Observational Reference: ROCIO_IBEB Gridded Dataset

ROCIO_IBEB (Peral et al. 2017; Amblar-Francés et al. 2020) is used as observational reference in this work. Among other variables, this gridded (5 km spatial resolution) dataset provides daily maximum temperature and daily accumulated precipitation over mainland Spain and the Balearic islands (9.3° West–4.3° East, 36° North–43.7° North) for the period of study, 1979–2008 (note this is the longest available common overlapping period for all the datasets considered, see next subsections). In order to be closer to the resolution of the available RCM simulations (0.11° ~ 12.5 km; see next subsection), we have upscaled it by a factor of 2 to a 10-km resolution grid, using interpolation to the nearest neighbour.

### 2.2 | Dynamical Downscaling: EURO-CORDEX RCM Simulations

The Coordinated Regional Climate Downscaling Experiment (CORDEX) is an international initiative which provides an appropriate framework for examining RCM uncertainties affecting regional climate projections (Gutowski et al. 2016). As of October 2025, the RCMs with the highest resolution available over this spatial domain are the EURO-CORDEX (the European branch of CORDEX) RCMs, spanning 1979–2008 and driven by the ERA-Interim reanalysis (Dee et al. 2011) for the evaluation simulations (Vaittinada Ayar et al. 2016; Kotlarski et al. 2014; Vautard et al. 2013), which we use in this work. These simulations have a spatial resolution of 0.11° (~ 12.5 km) for the EUR-11 domain, which includes our region of interest. Specifically, we use daily maximum temperature and daily accumulated precipitation values from 3 EURO-CORDEX RCMs, namely the RCA4, RACMO22E and REMO2009 (r1i1p1 run in the three cases). For evaluation purposes, the simulations are spatially interpolated to the upscaled observational reference grid (10 km, see previous section) using nearest neighbours.

Previous assessments of the EURO-CORDEX evaluation simulations show that they are able to capture the basic features of the European climate (Kotlarski et al. 2014), and particularly they reproduce the spatial pattern and variability of mean
temperature and precipitation in the Iberian Peninsula (Bedia
et al. 2020). More precisely, RACMO shows negligible biases
of summer mean temperature and REMO and RCA simulate
warmer temperatures, while they show contrasting (RACMO
and REMO positive, RCA negative) biases in mean summer
precipitation (Kotlarski et al. 2014). A well-known problem of
RCMs is the excess of days with low precipitation (drizzle effect;
Vaittinada Ayar et al. 2016; Dai 2006), which is often associated
with the misrepresentation of the lower tail of the precipitation
distribution (e.g., an overestimation of the wet-day frequency).
For the simulations considered, negligible biases in the summer
wet-day frequency are found for RACMO (except in the north-
ernmost part of the peninsula) and REMO, while RCA overesti-
mates it in the eastern part (Casanueva, Kotlarski, et al. 2016).
Most EURO-CORDEX RCMs exhibit an overestimation of sum-
mertime temperature extremes and heatwave duration; how-
ever, RACMO and RCA represent rather well the frequency of
high-amplitude heatwaves (Vautard et al. 2013). Overall, within
the full EURO-CORDEX ensemble, RACMO is one of the best-
performing RCMs considering a comprehensive set of metrics
(Herrera et al. 2020). Note that the short period of these evalu-
ation simulations hampers the comparison of long-term trends
with observed ones. Previous works using historical runs show
that some models underestimate surface solar downward radi-
ation, due to the use of constant aerosol concentrations (Bartók
et al. 2017; Boé et al. 2020; Nabat et al. 2014), which together
with natural variability challenges trend evaluations.

## 2.3 | Statistical Downscaling Methods

Three statistical downscaling methods, representative of differ-
ent classes of algorithms and recently shown to be among the
best performing methodologies, have been selected for assess-
ment in this work. These methods are generalised linear models
(GLMs, see e.g., Chandler and Wheater 2002), a posteriori ran-
dom forests (APRFs, see Legasa et al. 2022) and dense convo-
lutional neural networks (CNNs, see Baño-Medina et al. 2021;
Soares et al. 2024). We refer the reader to (Legasa et al. 2023)
for an in-depth assessment of these three methodologies in the
context of downscaling climate change precipitation scenarios.

In this work we build on the perfect prognosis downscaling
framework (Gutiérrez et al. 2019; Von Storch et al. 1993), that is,
the SD methods are trained using observed predictands and pre-
dictors (reanalysis is used for large-scale predictor variables). In
order to be consistent with the RCM evaluation simulations, we
employ the ERA-Interim reanalysis (Dee et al. 2011), covering
the geographical domain 10.5° West–5.25° East, 35.25° North–45°
North, at 0.75° spatial resolution for all the SD methods. Note
that this domain has been shown optimal for the target variables
of interest in previous studies: (Gutiérrez et al. 2013) for tem-
perature and (San-Martín et al. 2017) for precipitation. The three
methods are trained to downscale daily maximum temperature
and daily accumulated precipitation from the ROCIO_IBEB
data over the period 1979–2008, using the same set of predic-
tors: geopotential, specific humidity, air temperature, eastward
wind and northward wind at 500, 700 and 850 hPa vertical lev-
els. This selection of predictors is also based on previous stud-
ies on statistical downscaling dealing with temperature and
precipitation (Gutiérrez et al. 2019; Baño-Medina et al. 2021;

Soares et al. 2024; Legasa et al. 2023; Doury et al. 2023; Doury
et al. 2024), and adheres to the so-called perfect prognosis con-
ditions (Maraun and Widmann 2018): essentially, the predictors
used to build the SD models must be realistically simulated (ide-
ally, bias-free) in the present climate and credibly projected into
the future by GCMs.

As typically done in the literature (see, e.g., Gutiérrez et al. 2019;
Baño-Medina et al. 2021; Soares et al. 2024; Hernanz et al. 2023),
the three SD methods considered build separate models to
downscale temperature and precipitation, and thus they are not
specifically trained to capture the relationship between the two
variables. Despite this, standard dependence measures are al-
ready well captured by using separate models. Specifically, these
methods capture the daily correlation between downscaled pre-
cipitation and maximum temperatures (see Figure S2). This
correlation is overall low in magnitude: there is a negative cor-
relation pattern with a spatial average of – 0.27, and the maxi-
mum observed correlation (in magnitude) is – 0.516.

In order to fairly assess the SD methods and intercompare them
against RCMs, a cross-validation scheme is carried out for all
the results shown in this article. That is, for all the SD meth-
ods, we split the calibration period, 1979–2008, into 5 sets of 6
consecutive years. To predict each 6-year period, each statisti-
cal model is trained with the remaining 24 years. This process
is repeated 5 times using each of the 5 sets as predicting set.
This way, we assess the performance of the SD methods when
applied on unseen data. Moreover, the considered SD methods
perform probabilistic downscaling and are used in a genera-
tive way: for each gridpoint and day, each SD method produces
a probability distribution for each variable, Gaussian for tem-
perature and Bernoulli-Gamma for precipitation (i.e., they fit
5 parameters in total, hereby called *μ* and *σ* for the mean and
standard deviation of predicted temperature, and *p*, *α* and *β* for
the probability of rain, shape and rate parameters of precipita-
tion intensity, respectively). These distributions are used to sto-
chastically produce 250 samples, representing the variability of
possible expected outcomes for each large-scale predictors' state
(i.e., for each day). All the validation measures shown in this
article are computed from these simulations, as the average of
the corresponding 250 values. We devote the rest of this section
to briefly describe the methodologies and the differences among
them that are relevant for this work.

With respect to GLMs, we build three separate models. As
GLMs for modelling normally-distributed variables are just lin-
ear models, the GLM for temperature models it linearly from the
predictors. Precipitation is modelled using two separate mod-
els, as typically done: a logistic GLM using the *logit* link for oc-
currence and a Gamma GLM for modelling the expected value
of precipitation intensity using the logarithm link (Gutiérrez
et al. 2019; Abaurrea and Asín 2005; Manzanas et al. 2015). Note
that the latter GLM, in contrast to APRFs and CNNs, assumes *β*
to be constant conditional on the predictors' state and it is thus
estimated from the residuals (see, e.g., Manzanas et al. 2015 for
the details). For each target site, then, we have 3 GLMs, which
use as predictors the principal components explaining 95% of
the variance over the whole domain. This configuration corre-
sponds exactly to the GLMs used in Gutiérrez et al. (2019), and
are assessed in this work since they show a good compromise
between simplicity (in terms of a low number of parameters), and predictive and distributional performance. For simplicity, we refer to this methodology as a whole (i.e., the three GLMs with the configuration explained above) as the GLM in this article.

A posteriori random forests (APRFs) were introduced and thoroughly assessed by Legasa et al. (2022) for downscaling precipitation intensity under the perfect prognosis paradigm. APRFs are a modification of traditional random forests (Breiman 2001) that allows for accurately estimating the parametric distribution of precipitation and temperature by employing a Bernoulli-Gamma and a Gaussian model for each variable, respectively. The Bernoulli-Gamma model uses a combination of the Gamma deviance and Bernoulli entropy as split function, tasked with splitting the predictors' space for each tree in the forest to provide predictive samples of each target variable. The specific configuration for the precipitation model uses 200 trees and estimators defined in Louzada et al. (2019) for the Gamma distribution (α, β), whereas the temperature model uses 100 trees and maximum likelihood estimation for μ and σ. Both models are configured to have at least 5 observations in each terminal leaf. The reader is referred to Legasa et al. (2023) for an in-depth description and assessment of their performance. As for the predictors, APRFs use the 4 nearest large-scale gridpoints to each target location in the observational grid, since this yields good performance while keeping computational cost moderate. As with GLMs, these two models are simply referred to as the APRF in this article.

The third SD method belongs to the deep learning category. CNNs (Lecun et al. 1998) are a type of neural network originally developed for applications in computer vision, where data has a known, grid-like spatial structure, as is the case with climate models. The CNN configuration used here stems from that introduced by Baño-Medina et al. (2020) and later refined by Soares et al. (2024), who perform downscaling on a domain that overlaps the one used in this work. Specifically, we use in the present work the configuration called BMdense by Soares et al. (2024), which is shown to be the best performing both for temperature and precipitation. In BMdense, hereby simply called CNN, the input layer is convolutionally and sequentially connected to 3 hidden layers with 50, 25 and 1 feature maps, with a standard kernel size (3 × 3) in each convolutional layer. After the convolutional layers, the CNN has 2 additional dense layers, each with 50 units. Rectified linear unit (ReLU) activation functions are used in all the layers. We train the CNNs with Adam optimizer (adaptive moment estimation; Kingma and Ba 2015), using early stopping with 10% of the dataset as validation set. The convolutional layers automatically handle the spatial predictors, so the whole domain is used for the CNN. As with APRFs, two separate models are used, to downscale temperature and precipitation for each day. The loss functions for each are the negative log-likelihood of the Gaussian and Bernoulli-Gamma distributions, respectively. Note that this is a minor difference with respect to the implementation assessed by Soares et al. (2024), who used the mean squared error loss for the temperature CNN. Minimising the mean squared error is equivalent to maximising the likelihood if constant downscaled standard deviation σ is assumed, and, in this work, we want to explicitly allow σ to vary with the large-scale predictors, hence the use

of the log-likelihood. Finally, note also that APRFs and GLMs build a separate statistical model for each location, whereas CNNs downscale all the gridpoints simultaneously with a single model.

From now on, we use the term *model* to refer to both statistical methods and RCMs.

## 2.4 | Standardised Dry and Hot Index (SDHI)

The comparison of downscaling approaches is performed based on the standardised dry and hot index (SDHI; Hao et al. 2018). It can be used to measure the severity of a compound dry and hot condition. As we will later see, its standardised nature grants the characterisation of events in a relative manner, thus allowing for spatial and inter-model comparisons.

The SDHI is derived from the yearly averages of daily accumulated precipitation *P* and daily maximum temperature *T*, as follows:

$$SDHI = \Phi^{-1}(F_X(X)), \text{ with } X = \frac{F_P(P)}{F_T(T)}$$

Here *F* denotes the cumulative distribution function (CDF) of each respective variable and *Φ* is the CDF of a standard normal distribution. Therefore, each *P* and *T* are first transformed to a standard uniform distribution, their ratio is computed (*X*, see Figure 1a, bottom right), and the resulting distribution is inverse transformed to a standard normal distribution. The computation of the CDF for each variable is performed using the Gringorten plotting position formula (Gringorten 1963), as in the original definition of the index (Hao et al. 2018).

We show, in Figure 1b, the SDHI for three illustrative gridpoints of the domain, together with the standardised precipitation index (SPI) and the standardised temperature index (STI) (Mckee et al. 1993). Since the index entails dividing precipitation by temperature, the lower the value the more intense the dry-hot compound event. While lower values of the SDHI typically correspond to high STI and low SPI values, the SDHI can also detect events where neither the STI or SPI are particularly high/low, such as the year 2001 for the first gridpoint shown in Figure 1b. Similarly to the SPI, the authors consider thresholds for the SDHI below −0.6, −0.8, −1.3, −1.6 and −2 standard deviations indicating an abnormal, moderate, severe, extreme and exceptional dry-hot event, respectively. Due to the limitations on the available data (30 years), in this work we do not distinguish between extreme and exceptional events, since it is expected to obtain approximately 1 extreme event at each gridpoint over the whole period (~5.5%).

By construction, a key property of the SDHI is that, due to the transformations performed, it does not account for the potential biases produced by each model. In this regard, there are two points to consider: On the one hand, the transformation of *P* and *T* to standard uniform distributions makes *X* only consider the consistency of the quantiles of precipitation and temperature, beyond specific model biases in the marginal distributions. This property allows us to perform a direct and fair comparison of
<figure><img src="image_4.png" /><figcaption>**FIGURE 1** | Legend on next page.</figcaption></figure>
**FIGURE 1** | (a) Observed climatologies for daily accumulated precipitation (top left), daily maximum temperature (top right), ratio of climatologies (daily accumulated precipitation divided by daily maximum temperature, bottom left); and mean $X$ (bottom right), defined as the ratio of cumulative distribution functions of precipitation and temperature (see Section 2.4). (b) For 3 illustrative gridpoints, the observed annual standardised dry-hot index SDHI (black), standardised temperature index (STI, in red) and standardised precipitation index (SPI, in blue). Dashed lines mark the thresholds for abnormal (SDHI below – 0.6), moderate (below – 0.8), severe (below – 1.3) and extreme events (below – 1.6), defined in Section 2.4. [Colour figure can be viewed at wileyonlinelibrary.com]

statistical and dynamical methods, since RCMs are well known to have important biases, especially when compared to SD methods (Vaittinada Ayar et al. 2016; Bettolli et al. 2021; Casanueva et al. 2013). This is desirable for our comparison since it avoids the need to use bias correction methods, which would introduce an additional layer of complexity to the analysis. On the other hand, the index also lacks sensitivity to the magnitude of $X$ due to the final inverse transformation to the standard normal distribution. This transformation is useful to assess how the different models reproduce the temporal component of the SDHI and their capability to detect specific compound dry-hot events driven by the large-scale within each model simulations' domain, but ignores the distribution of $X$ itself. This distribution is also relevant, as it speaks of the models' skill for reproducing paired values of temperature and precipitation coherent with observations.

Therefore, and following from these considerations, we consider as well the magnitude of $X$, in order to address the distributional consistency (with observations) of the simulations provided by all models. To do so, we analyse how the models capture the average magnitude of $X$ for the thresholds defined before, as follows:

$$ \bar{X}_{abnormal} = \{x \in X \mid x \le 28^{th} \text{ percentile}\}. $$

Similarly, we define $\bar{X}_{moderate}$ for values below the 21st percentile, $\bar{X}_{severe}$ for values below the 9.7th percentile and $\bar{X}_{extreme}$ for values below the 5.4th percentile. In all cases, we obtain the bias, computed as the difference between the simulated values and the observed ones. Note that, since the marginal distributions are still transformed to be uniform, we are specifically targeting consistency of temperature and precipitation on a percentile-to-percentile basis, ignoring the marginal distribution shapes themselves. Together, all these measures provide a detailed analysis of the consistency of maximum daily temperature and precipitation, targeting compound dry-hot events.

## 3 | Results

### 3.1 | Reproduction of the Interannual Series of the SDHI

We first look at the detection of compound dry-hot events, by measuring the Spearman correlation between the interannual time-series of the SDHI simulated by each model and its observed counterpart (Figure 2). With respect to RCMs, the results vary considerably both among models and spatially. RACMO is the best performing one, by a clear margin, with a mean spatial correlation of 0.68. RCA and REMO yield an average spatial correlation of 0.55 and 0.48, respectively, finding geographical

areas where they fail to reproduce the temporal structure of the index. In particular, the correlation with the observed SDHI for REMO is systematically lower, with areas exhibiting no correlation with the observed SDHI.

Differently, the SD methods exhibit similar performance among them. Also, the geographical distribution is clearly alike for the three SD methods. While SD methods exhibit an overall good performance (the correlation is higher than 0.5 for most of the domain), they still show systematic, rather than model-dependent, regions where the SDHI is not reproduced properly. Among the SD methods, GLMs fall slightly behind, with an average spatial correlation of 0.63, compared to that of the CNN (0.67), and the APRF (0.7), which is the best performing model of the six analysed in this work. Overall, SD methods outperform RCMs (specially considering their respective multi-model mean; see bottom panels in Figure 2a), but the best performing RCM (RACMO) is comparable to the SD counterpart (APRF).

Generally, the index is particularly well reproduced by the SD methods in the eastern part of the peninsula, as well as the northwesternmost area. Such spatial differences cannot be attributed to deficiencies in the models' ability to reproduce temperature and precipitation (Figure S1). Dynamical methods show comparatively good performance in the northwest and southwest of the peninsula as well, but not in the southeast, where they work worse, excluding RACMO. The performance in the Balearic islands is particularly good for the SD methods, whereas RCMs fail to produce correlations with observations higher than 0.4.

Looking at the violin plots (Figure 2b), results for the CNN and GLM are more stretched downwards but reach the same predictive performance as the APRF, showing how the former particularly underperforms in some geographical areas. Only the RACMO RCM exhibits a performance in line with SD methods, with its average performance only beaten by the APRF by a small margin. The overall good performance of RACMO in reproducing compound events is consistent with previous studies which have evaluated its suitability to properly simulate temperature and precipitation (Kotlarski et al. 2014; Herrera et al. 2020; Hernanz et al. 2023). Finally, note that we have also analysed other metrics beyond the Spearman correlation, such as the root mean squared error between the simulated and observed indices (see Figure S3), leading to similar results.

### 3.2 | Evaluation of the Compound Dry-Hot Event Intensity Distribution

Having assessed the temporal representation of the index in the previous section, we now address the multivariate performance
<figure><img src="image_5.png" /><figcaption>**FIGURE 2** | Legend on next page.</figcaption></figure>
**FIGURE 2** | (a) Spearman correlation between the observed SDHI and those obtained from the statistical (left) and dynamical (right) downscaling models. (b) Violin plots summarising the results from (a). Each violin plot encompasses all the gridpoints of the studied region. The colour displays the spatial mean of maps, and the horizontal dashed lines represent the medians for each group of models (i.e., statistical and dynamical). [Colour figure can be viewed at wileyonlinelibrary.com]

of the models in a distributional sense. That is, whether the models are appropriately capturing the bivariate interaction between precipitation and temperature on the lower tail of X (the most intense dry-hot events). As discussed in Section 2.4, the fact that $F_T(T)$ and $F_P(P)$ have uniform distributions isolates the multivariate aspect, and assesses whether high temperature and low precipitation percentiles are reproduced with a dependence pattern coherent with the observed one. In order to do so, we consider the thresholds for compound dry-hot events defined in Section 2.4 ($\bar{X}_{abnormal}$, $\bar{X}_{moderate}$, $\bar{X}_{severe}$ and $\bar{X}_{extreme}$) and assess the models' bias with respect to the observational reference (Figure 3). There is, thus, no temporal aspect being taken into account here. Instead, we test the models' ability to reproduce the intensity of dry-hot events exceeding certain thresholds, regardless of when they occur.

Unlike the evaluation results of temporal aspects, SD methods tend to exhibit higher biases than RCMs. Indeed, except in some small regions for the extreme events in the case of the APRF, the biases in SD methods are always positive, indicative of underestimation of the intensity of dry-hot events. These biases are distributed quite uniformly over the entire spatial domain for abnormal and moderate events and are especially intense in the northeastern part of the domain for severe and extreme events, particularly for the GLM and APRF. Among the SD methods, CNNs perform better for this particular aspect, with an overall slightly lower bias for all the thresholds considered except for extreme events, for which CNN and APRF perform similarly. However, no SD method is able to beat any of the RCMs regarding this aspect.

Indeed, RCMs are much less biased reproducing the tail of X. RCM bias is much closer to 0 for all the intensity thresholds and models, with no clear pattern of over- or underestimation. RCA is the worst performing RCM, with a slightly positive bias (i.e., underestimation of compound dry-hot event intensity), especially for abnormal and moderate events. Yet, it exhibits comparatively lower biases than all SD methods. RACMO and REMO perform similarly.

## 3.3 | Disentangling the Effect of Precipitation and Temperature on Intensity Biases of Compound Dry-Hot Events

Finally, in this section we analyse the separate contribution of each variable (precipitation and temperature) to the bias on the compound dry-hot distribution. That is, we address, for each gridpoint, the bias exhibited by each model in $\bar{X}$, decomposed into the relative bias in $F_T(T)$ and $F_P(P)$ (Figure 4). For brevity, the results shown here correspond exclusively to abnormal episodes (similar conclusions are obtained for the remaining intensity thresholds). This analysis is useful to address the influence of each individual variable in the bivariate distribution in terms of percentiles, to avoid the impact of potential biases in

the marginal distributions. It also allows us to check whether any model may be compensating biases, such as compensating wetter with hotter years.

Overall, positive biases of abnormal compound events in the SD methods (Figure 4, left column) mainly come from positively biased precipitation percentiles compared to fairly accurate temperature percentiles. Indeed, for more than half the gridpoints, biases in $F_P(P)$ are larger than 50% and a substantial fraction surpasses 100% (i.e., the simulated percentiles double the observed ones). Conversely, for most of the gridpoints, temperature bias is contained within ± 25%. As concluded in the previous section, bias of abnormal compound events for the CNN is the lowest among the SD methods, which comes from a slightly better simulation of low percentiles of precipitation. In addition, we observe in some cases that, even for low biases in both temperature and precipitation (≤ ± 25%), SD methods still exhibit comparatively higher bias in $\bar{X}$ than RCMs (see the corresponding quadrant in the right panels), indicating that the distributions of precipitation and temperature are not being coupled properly by the SD methods.

For the RCMs (Figure 4, right column) all the points are more centred around the origin. Precipitation biases still have a higher influence than temperature counterparts, but this effect is smaller than for the SD methods. Indeed, a clear difference with respect to SD methods is that RCMs are also producing negative (dry) biases for precipitation, which somewhat counteracts negative (cold) biases in temperature (bottom left quadrant), leading to small biases in compound events. For RACMO and REMO, most biases (for both variables) are contained within ± 50%.

## 4 | Conclusions and Discussion

In this work, we address the following question: How do statistical and dynamical downscaling reproduce compound dry-hot events? To answer it, we present a comprehensive evaluation of three established and well-performing statistical downscaling (SD) methods against three state-of-the-art regional climate models (RCMs) based on the standardised dry and hot index (SDHI). This index allows us to perform a direct and fair comparison of statistical and dynamical methods, with a focus on the multivariate aspect. We assess the models by examining both their predictive performance, as measured by how well they reproduce the observed temporal series of the SDHI, and their distributional performance, as indicated by how well they simulate the tail of the compound dry-hot event distribution.

Our results show that SD methods outperform RCMs in reproducing the temporal variability of the observed annual series of the SDHI from large-scale conditions, as measured by the Spearman correlation of the simulated index with the observed
<figure><img src="image_6.png" /><figcaption>FIGURE 3 | Legend on next page.</figcaption></figure>
**FIGURE 3** | Bias in *X* (defined in Section 2.4) for each of the models considered in this work, for different intensity thresholds (columns): X̄<sub>abnormal</sub>, X̄<sub>moderate</sub>, X̄<sub>severe</sub> and X̄<sub>extreme</sub>. Observations are shown in the top row (recall that lower values, indicate higher intensity of the compound dry-hot event). The remaining rows show the bias for each of the models considered in this work, with red/blue colours indicating more/less intense than observed dry-hot events. For each model (in rows) and intensity threshold (in columns), the averaged bias, in absolute value, is given (numbers close to each map). For each intensity threshold, the best performing model (in terms of absolute biases) is highlighted in bold. [Colour figure can be viewed at wileyonlinelibrary.com]

<figure><img src="image_7.png" /></figure>

**FIGURE 4** | Analysis of the biases produced by SD methods and RCMs (left and right column, respectively) for the X̄<sub>abnormal</sub> intensity threshold (see Section 2.4). Each dot represents a gridpoint, with red/blue colours indicating more/less intense than observed compound events (see the colour bar). Additionally, the position of the dot in the graph allows for assessing the relative bias for F<sub>T</sub>(T) (in the x-axis) and F<sub>P</sub>(P) (in the y-axis), the two individual components contributing to X̄<sub>abnormal</sub>. [Colour figure can be viewed at wileyonlinelibrary.com]
counterpart. Moreover, despite their methodological differences, we find a general agreement among all the SD methods analysed in terms of performance, suggesting the presence of a systematic, rather than method-dependent, bias. However, among the RCMs, there are notable differences, with RACMO performing similarly to the SD methods, while REMO and RCA fall behind by a considerable margin.

Conversely, RCMs better simulate the severity of compound dry-hot events in distributional terms (over the entire period of study considered), being able to reproduce these events with an intensity closer to the observed one. We attribute this to their foundation in physical principles, which allows for a better simulation of inter-variable relationships (precipitation and temperature in this case). Instead, SD methods exhibit consistently positive biases at the tail of the distribution, indicating a general inability to simulate sufficiently intense compound dry-hot events.

Overall, we conclude that neither approach (statistical or dynamical) performs best in all aspects analysed. It is important to highlight that, although all the models considered have been shown to perform well when downscaling precipitation and temperature separately, our study evaluates them in a much more challenging context. As a result, the comparatively worse performance with respect to other, easier to capture, metrics (see Figure S1 for the univariate performance for the SD methods) is expected, given the difficulty of accurately reproducing compound extremes. This is also true for traditional metrics used to assess multivariate performance such as inter-variable correlations: the models capture them well (Figure S2), yet they fail to reliably reproduce more complex interactions, in this case those occurring in the tails (dry and hot) of the multivariate distribution.

Early intercomparison works (for univariate extremes, i.e., precipitation and temperature assessed separately) (Murphy 1999) already highlighted the complementarity of statistical and dynamical downscaling approaches, showing that SD methods tended to perform better for summer temperature anomalies while RCMs provided advantages for winter precipitation and for reproducing mean climatological features, although both struggled with extremes. Building on this, Vaittinada Ayar et al. (2016) compared a wide range of SD methods and RCMs over Europe and showed that SD methods often outperform RCMs for precipitation occurrence and intensity, including extreme quantiles, while RCMs remain superior in terms of spatial coherence, which may follow from their foundation in physical principles. Hertig et al. (2012) further reported that dynamical downscaling provides a better representation of the intensity of extremes than SD methods, findings that are broadly consistent with our compound dry-hot analysis.

In this work, which focuses on compound dry-hot events, this division of strengths becomes sharper: SD methods excel at reproducing the temporal variability and detection of compound events, whereas RCMs are clearly more skillful at capturing their intensity, owing to their physically consistent treatment of temperature-precipitation interactions. Therefore, to properly study compound dry-hot events at the scales required for most practical applications, further research and development of models are needed. In this regard, future lines of work include

improving SD methods to better capture realistic inter-variable relationships. We also plan to extend the analysis to other compound events, including those involving the combination of more than two variables, such as hot-dry-windy events (Feng and Sun 2024). Finally, analysing future climate change projections in the context of downscaling compound events is also an important research perspective.

## Author Contributions

**M. N. Legasa:** conceptualization, methodology, software, validation, formal analysis, data curation, visualization, writing – original draft, writing – review and editing, investigation. **A. Casanueva:** conceptualization, methodology, formal analysis, supervision, funding acquisition, writing – review and editing, project administration, resources, investigation. **R. Manzanas:** methodology, conceptualization, writing – review and editing, funding acquisition, formal analysis, project administration, resources, supervision, investigation.

## Acknowledgements

The authors are grateful to the World Climate Research Programme's Working Group on Regional Climate, and the Working Group on Coupled Modelling, the former coordinating body of CORDEX and responsible panel for CMIP5. We also thank the climate modelling groups for producing and making available their model output: the Earth System Grid Federation infrastructure, an international effort led by the U.S. Department of Energy's Program for Climate Model Diagnosis and Intercomparison; the European Network for Earth System Modelling, and other partners in the Global Organisation for Earth System Science Portals (GO-ESSP). The authors would like to express their gratitude to the Santander Climate Data Service (https://scds.es) for hosting and maintaining an ESGF (https://esgf-metagrid.cloud.dkrz.de) node from which RCM data can be accessed through OPeNDAP protocols (http://data.meteo.unican.es) using the facilities provided by climate4R (https://github.com/SantanderMetGroup/climate4R). Finally, the authors thank the anonymous reviewers for their valuable comments and suggestions, which helped improve the quality of this manuscript.

## Conflicts of Interest

The authors declare no conflicts of interest.

## Data Availability Statement

EURO-CORDEX simulations can be downloaded from the Copernicus Climate Data Store (CDS, https://cds.climate.copernicus.eu/datasets/projections-cordex-domains-single-levels?tab=download), by selecting the Europe and Evaluation simulations for the three RCMs: **KNMI-RACMO22E** (Netherlands), **SMHI-RCA4** (Sweden) and **MPI-CSC-REMO2009** (Germany). ERA-Interim data can be downloaded from https://cds.climate.copernicus.eu/datasets/reanalysis-era-interim?tab=d_download. The observational reference dataset (ROCIO Interim) can be downloaded from the official AEMET website (https://www.aemet.es/en/serviciosclimaticos/cambio_climat/datos_diarios?w=2). The R packages *downscaleR* (Bedia et al. 2020; https://github.com/SantaiderMetGroup/downscaleR), *RandomForestDist* (Legasa et al. 2022; https://github.com/MNLR/RandomForestDist) and the Python library *PyTorch* (Paszke et al. 2019) were used, respectively, to train the generalised linear models, a posteriori random forests and convolutional neural networks used in this work.

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**Supporting Information**

Additional supporting information can be found online in the Supporting Information section. **Data S1:** Supplementary Information.