To handle seasonal effects in weather data, which can bias results and complicate fitting statistical models, deseasonalisation and trend removal is carried out as a pre-processing step. We use the notation s|t:RT→RT to denote a generic deseasonalisation function. Many methods of varying complexity are available, ranging from simple (subtracting climatology or filtering by season) to complex (fitting generalized linear models with seasonal covariates). The choice of method will depend on the data being modelled and the context (see Sect. ).

The deseasonalised weather data is used to identify hazard events. To do this, we use a severity function r|ijk:RH×W×T→RT to construct a time series of hazard intensities. The definition of this severity function determines the characteristics of the hazards that will be selected, e.g., calculating the mean or sum of a variable across a region will prioritize widespread events while selecting the maximum over the region will identify events that reach higher peak intensities and may be more spatially localised. More specialised severity functions could also be used, such as for example hazard indices like the storm severity index for windstorms or the fire weather index for fire potential .

To identify independent hazard events, a declustering algorithm d|t:RT→RT is applied to the time series r|ijk(x). In this framework, hazard days are defined as consecutive days in which r|ijk(x) exceeds some specified threshold vr, separated by a specified minimum number of non-exceedences days ℓr. The choice of vr and ℓr will depend on the data being modelled and the context. Since we need to fit statistical distributions to the margins, the data should be independent along the sample dimension, which favours using high thresholds vr and extracting fewer storms. However, small sample sizes in the tails will lead to high variance in the parametric fits, so this trade-off must be handled. The simplest solution is to perform a standard grid search over the space of (vr,ℓr) to select the largest number of events while maintaining independence between the extracted r|ijk(x) values. Independence can be verified using a standard Ljung–Box test .

Finally, a hazard event set is created by extracting the deseasonalised data corresponding to the declustered hazard days. This creates an event set of smaller spatiotemporal data cubes corresponding to each hazard event and variable. Each data cube will have dimensions H×W×T, re-purposing the T notation to represent the duration of an arbitrary event. A feature of this approach to event identification is that the extracted variables are sampled conditionally on the occurrence of events, as defined by the severity function. This is intentional: we seek to model the joint behaviour of all variables during hazard events rather than the independent natural extremes of each variable. However, the implications of this for fitting statistical models should be considered and will be discussed later. Broadly, this approach to storm detection bears similarities to the Method of Independent Storms for univariate wind speeds and to the methods of in the spatial setting.

Each spatiotemporal hazard event can be projected into a 2-D footprint using a temporal aggregation function hk|t:RH×W×T→RH×W. The specific definition of hk|t will depend on the hazard of interest and how its impact materialises over the event. A measure of cumulative precipitation, for example, may be more relevant for assessing flood risk, while the maximum wind speed may be more relevant for assessing storm damages. Applying a temporal aggregation function to hazard variable creates a set of 2-D event footprints, which can be stacked together to create multi-hazard event footprints with dimensions H×W×K.

To generate realistic hazard events, we need to learn the multivariate distribution of the multi-hazard footprints. Critically, we need to learn the marginal and joint distributions of the most extreme values. As showed, the ability of a GAN to learn the extremal dependence structure of a dataset can be improved by training it on the empirical distribution functions of its margins. This approach is analogous to classical methods in multivariate extremes, which often disentangle the margins of a multivariate distribution from its dependence structure . However, 's approach relies on fitting generalised extreme value (GEV) distributions to the margins, requiring data in the form of annual block maxima. The use of block maxima creates spatial incoherence, entails significant loss of information, and is unsuitable for modelling the cumulative impact of hazards, such as storms, which materialise over the duration of the event, not just at the peak.

To overcome the limitations of the block maxima approach, we opt for a peaks-over-threshold (POT) approach, allowing us to make more efficient use of data. Eq. 1.3 used a semiparametric function, which allowed them to model the entire distribution of the data, using a parametric distribution for the extreme values to guide extrapolation to new extremes, and an empirical distribution for the non-extremes where there is already sufficient data to provide a good approximation. Considering a random variable X and suitably extreme threshold vX, the semiparametric distribution function can be written in its most general form as: 1F̃(x)=1-(1-F^(vX))(1-FD(x))for x>vXF^(x)for x≤vX where F^ is the empirical distribution function (ECDF) and D is an extreme value distribution used to model the tails. For most applications, D will be a generalised Pareto distribution (GPD), as this is the only nondegenerate limiting distribution for exceedances over a high threshold . In this case, the shape, location, and scale parameters of D are denoted by ξ, μ, and σ, respectively. However, we have kept the formulation general to allow for alternative parametric distributions where appropriate see, for example. A semiparametric distribution as in Eq. () is fitted to all the margins of the multi-hazard event footprints x, transforming them to have standard uniform margins u∼U(0,1).

The event identification approach means that, technically, the distribution of all but one of the margins will be a conditional distribution, conditional on r|ijk(x) having exceeding the threshold vr. While in theory conditioning does not violate the assumptions of a GPD fit, the question of whether the conditioned data will be independent and identically distributed is more challenging to address. Although the deseasonalisation and declustering should ensure stationarity and independence of samples, it is still possible that the conditioning will select events arising from different meteorological mechanisms. The POT approach should mitigate this somewhat by isolating the most extreme events, which are more likely to arise from a single dominant mechanism. In Sect.  we will use an automated threshold selection method as a further safeguard, which will fail and revert to empirical distributions for any margins where no suitable GPD threshold can be identified. The validity of this approach will depend on the weather variables being modelled and need to be assessed on a case-by-case basis.

From a machine learning perspective, transforming the margins to a uniform distribution is a natural standardisation step. However, the uniform probability transform means that the extremes occupy a small region at the edge of the domain. Furthermore, work by and demonstrated that a GAN with a light-tailed latent space struggles to learn the tail behaviour of a heavy-tailed distribution, leading to underestimation of the tails. developed a GAN with a heavy-tailed latent space to address this issue, but this required replacing the adversarial loss with a custom loss function.

Drawing on these results, we hypothesise that what matters is not the specific tail behaviour of the latent space, but rather that it matches that of the data. A sufficient strategy would therefore be to transform heavy-tailed data to a light-tailed distribution before training, and invert the transformation afterwards. This mirrors common practice in the multivariate extremes literature, where margins are transformed to standard distributions such as the Laplace, Gumbel, Fréchet, or unit Pareto to exploit certain desirable properties . We verify this hypothesis by repeating the experiments of in the Supplement and find that we can match the results of using a standard GAN on light tail-transformed data. In Sect. , we will explore results of transforming the training data to a standardised distribution G and compare the model performance for when G is a uniform distribution or another light-tailed distribution.

Depending on the distribution chosen for training G, the transformed multi-hazard footprints (with marginals y∼G) may require an additional rescaling step to ensure they lie in the range (0,1). A standard rescaling approach is min–max scaling, which maps the largest value of a dataset to 1 and the smallest value to 0. But this would prevent the GAN from generating values outside the range of the training data, which is our goal.

An alternative approach that allows us to specify a sensible maximum range for the generated data, using only the training data, is to set the maximum range using quantiles of the training distribution G, which we can specify to correspond specific return periods. For example, we can set it so that the maximum possible value our method can produce, at any point on the domain, corresponds to an intensity that is on average exceeded once every R events at that point (or similarly, once every R′=λR years, with λ the number of events per year). For the distribution G, the minimum and maximum values associated with R are FG-1(1/R) and FG-1(1-1/R), respectively. Following this approach, the data can be rescaled according to 2y′=y-FG-1(1/R)FG-1(1-1/R)-FG-1(1/R), assuming that R is sufficiently large that FG-1(1/R) is smaller than the smallest value in the training data and FG-1(1-1/R) is larger than the largest value in the training data. While this will result in the same maximum and minimum values across the G-distributed training data, it will map back to different values in physical space.

After rescaling, the weather fields are stacked such that each multivariate footprint is now a three-channel tensor. Finally, it is converted to an RGB image, ready to feed into a generative model.

In theory, this framework is agnostic to the choice of deep generative model, meaning common models like variational auto-encoders , diffusion models , or flow-based models could be used instead of GANs. In practice, however, historical weather datasets such as ERA5 generally contain fewer than 100 years of data, and for rare events, this does not provide enough extreme events to train standard deep generative models.

Some GANs have been developed to work well with small datasets, such as FastGAN and StyleGAN2-ADA , which use a self-supervised discriminator and adaptive discriminator augmentation (ADA), respectively, to prevent overfitting. Augmentation regularises the discriminator by adding semantics-preserving augmentations (e.g. additive noise, rotations, isometric scaling, saturation changes) to all samples. When combined with additional differentiable augmentation (DA), StyleGAN2-ADA successfully learned the distribution of a dataset of only 100 images from scratch . In Sect. , we will use StyleGAN2-ADA with differentiable augmentations (StyleGAN2-ADA+DA) to train a generative model on a set of multi-hazard footprints.

To measure the similarity between any two distributions, we will use the Wasserstein distance. The Wasserstein distance, also known as the Earth Mover's Distance, measures the minimum cost of transporting mass to transform one distribution into another. For one-dimensional distributions, the Wasserstein distance can be computed from distribution P to distribution Q as 3W(P,Q)=∫R|P(x)-Q(x)|dx. The Wasserstein distance is non-negative and takes a value of zero if and only if P and Q are the same distribution.

To assess whether the dependence structures are being learned correctly, we will first calculate dependence metrics between pairs of marginal distributions for each dataset. We will then compare the distributions of the generated dependence metrics to those of the training data. We will use Pearson correlation and mean-squared-error to assess the similarity between the calculated dependence metrics.

To measure the dependence between non-extreme values, we will use the Pearson correlation coefficient to measure linear agreement between the variables. To measure the level of extremal dependence between two variables X1∼F1 and X2∼F2, the extremal correlation between them, χ, above a fixed high threshold u can be written as p. 346, 4χ(u)=Pr⁡(F1(X1)>u,F2(X2)>u)Pr⁡(F1(X1)>u)=Pr⁡(F1(X1)>u,F2(X2)>u)1-u. We will use the simple empirical estimator for χ(u), which can be calculated from a sample of size n as 5χ^(u)=∑i=1n1{F^1(X1i)>u,F^2(X2i)>u}∑i=1n1{F^1(X1i)>u}, where F^i indicates the empirical distribution function of variable X. The true extremal correlation χ is defined as the asymptotic limit of χ(u) as u→1. The extremal correlation takes values in [0,1], where χ=0 indicates asymptotic independence and χ>0 indicates asymptotic dependence. The higher the value of χ, the stronger the extremal dependence between the two variables. In practice, however, a finite, high threshold u can be used to approximate χ.

The χ metric tells us the strength of extremal dependence, and it is often supplemented by the χ‾ metric, which measures the strength of asymptotic independence – where the variables maintain some dependence at finite, high levels but are ultimately independent. To assess the strength of asymptotic independence, we can use the χ‾ metric, which is defined as p. 348, 6χ‾(u)=2log⁡(1-u)log⁡Pr⁡(F1(X1)>u,F2(X2)>u)-1.

To benchmark the performance of the method, we compare it to a widely used model for multivariate extremes: the model. The model learns the conditional distribution of a set of variables, given that one of them exceeds a high threshold. For two variables standardised to Gumbel-distributions Yi and Yj, the probability that Yj exceeds a high threshold u given that Yi exceeds u is given by Eq. 4.1: 7Yj=a(Yi)+b(Yi)Z,Z∼G|i where G|i is the residual distribution after normalising Yj with the scalars a(Yi) and b(Yi). The form of G|i may vary, and depends on whether the margins of the residual distribution are asymptotically dependent. This naturally scales up to the multivariate case, where we can model the distribution of n other variables j=1,…,n, modelling Yj conditional on Yi exceeding u.

Ranking scores according to the severity function, as defined by the domain-maximum wind speed r|ijk(x)=max⁡k=0|i,j(x), Fig.  compares the 16 most severe storm footprints derived from the ERA5 dataset (top rows) with the most extreme footprints generated using the GAN (bottom rows) for each of the four training configurations: rescaling-only, uniform margins, Gaussian margins, and Gumbel margins. Corresponding figures for sea-level pressure and precipitation fields are also provided in the Supplement.

Qualitatively, the rescaled and Gumbel footprints look most similar to the ERA5 training data. Uniform-trained events are more extreme, overly-widespread, and do not exhibit the gradual decay in intensity with distance from the storm centre that would normally be expected. The footprints generated by standard rescaling also look reasonable, although the track shapes appear more simple and ellipsoidal than those produced by the Gumbel or Gaussian-trained models. The latter exhibit longer tracks and more pronounced changes of direction, better capturing the curved trajectories seen in the ERA5 reference footprints.

To assess the GAN's ability to capture marginal distributions, we calculate the Wasserstein distance between the generated and training data for each margin using Eq. (), scaling by the standard deviation of the training data distribution. The average rescaled Wasserstein distance across all margins is 0.57 for the rescaled model, 0.21 for the uniform model, 0.13 for the Gaussian model, and 0.14 for the Gumbel model, indicating that the Gaussian and Gumbel models are overall better capturing the marginal distributions of the training data.

To assess how well each model captures the overall distribution of pixel values, Fig.  shows the flattened distributions of all pixels, transformed to Gumbel scale to enable cross-comparison. The uniform-trained model shows a pronounced spike at the maximum value, suggesting the GAN attempted to extrapolate beyond its allowed range, saturating at the 10 000-year return period. The Gaussian-trained model exhibits the same behaviour, though far less severely. The Gumbel and rescaled models avoid this saturation: the Gumbel model produces the smoothest tail decay, while the rescaled model shows a more stepped distribution.

To assess how well the GAN captures the storm severity distribution, we again calculate the severity r|ijk(x) for each storm in the training and generated data and compare their distributions in Fig. . The rescaled and uniform-trained models perform poorly here with high Wasserstein distances (3.7 and 16.8, respectively) compared to the Gaussian and Gumbel models (0.71 and 1.0, respectively).

To assess how the model is learning the spatial dependence structures, we estimate the tail dependence coefficients χ^(u) (choosing u=0.8 based on initial data exploration) between all pairs of pixels across the domain, generating for each variable a 4096×4096 matrix for each of the four generated datasets (Fig. ). We quantify the level of agreement between the ERA5 and GAN-generated correlation structures by calculating the Pearson correlation and mean absolute error (MAE) between the two extremal correlation matrices. The rescaled model performs worst, with a correlation of 0.380 (MAE = 0.309) between the ERA5 and GAN-generated correlation fields for wind speed, while the uniform, Gaussian, and Gumbel models perform much better, with correlations of 0.839 (MAE = 0.088), 0.837 (MAE = 0.089), and 0.857 (MAE = 0.083), respectively.

For a more detailed view of bivariate spatial relationships, Fig.  shows scatter plots of 10 m wind speed anomalies between Chittagong and Dhaka, two cities in Bangladesh. The rescaling-only model produces a noticeably simpler, more ellipsoidal dependence structure than is seen in the ERA5 data, while the uniform-trained model struggles to resolve marginal extremes, producing clustering visible at the boundaries. Both the Gaussian and Gumbel-based models show considerably better agreement with the observed dependence structure, with the Gaussian model appearing to provide the closest match. Similar results were observed for all other variables and pairs of locations tested.

Figure  maps the extremal correlation estimates χ^(0.8) between 10 m wind speed and total precipitation across the Bay of Bengal for training data and the four GAN-generated datasets. The GANs trained on uniform, Gaussian and Gumbel margins show the best agreement with the ERA5 data, with Pearson correlations of 0.664 (MAE = 0.104), 0.666 (MAE = 0.089) and 0.705 (MAE = 0.08), respectively. The rescaling-only and uniform-trained model performs much worse, with correlation of 0.193 (MAE = 0.117). In all cases, the generated data had a tendency to overestimate the strength of the inter-variable dependence, with the mean overestimation (across all pairs of variables and locations) of 0.292 (uniform), 0.256 (Gaussian), 0.205 (Gumbel), and 0.241 (rescaling-only). The Gumbel-trained model performs best here, with the lowest mean error and highest correlation with the training data, while the uniform-trained model performs worst. We observe that these scores are slightly lower than the spatial dependence scores, which may be due to the architecture of StyleGAN prioritising spatial relationships or because the inter-variable dependence structure is more complex and harder to learn than the spatial dependence structure.

We choose the Gaussian model as the best-performing model and compare its performance to the conditional exceedance model, which is provided in R's texmex package . In this package, the mexDependence function integrates the GPD fitting of Eq. (3.2) and the fitting of the dependence model Eq. 4.1 into a single function, but our data has already been transformed to a uniform scale using Eq. (), so we customise the mexDependence function to skip the GPD-fitting step and directly accept data on the uniform scale.

Using the dataset of the top-150 storms, we randomly sample 1000 points across the domain and fit a conditional exceedance model between (i) different weather variables at that point, and (ii) for the same weather variable between that location and a second, randomly sampled, location. We then use the fitted model to generate 500 years of synthetic data, and calculate χ^(0.8) between the generated and training data. Figure a plots the distribution of spatial extremal correlations for each weather variable and Fig. b plots the inter-variable extremal correlations for all combinations of wind speed, precipitation, and sea-level pressure.

Both models struggle more to model the inter-variable dependence structure than the spatial structure, suggesting that the inter-variable dependence structure is indeed more challenging to learn. Nonetheless, the GAN achieved higher correlations and lower MAE compared to the training data than the model for both spatial and inter-variable dependence. Overall, this result is encouraging, particularly because the hazGAN model learns the full joint distribution of all variables and locations simultaneously, whereas the model is learning pairwise relationships separately so does not capture higher-order dependencies. As a result, the GAN may be able to borrow strength across the full dataset, enabling it to learn a more accurate dependence structure, while the model fits many separate models, each with less data. In the future, it would be interesting to compare the GAN to spatial models for extremes, such as r-Pareto processes , which are designed to model extremal spatial fields. However, we consider this beyond the scope of the current work, which aims only to demonstrate the effectiveness of the hazGAN approach.