1. Introduction

As society tries to adapt to Earth’s changing climate, access to accurate, local-scale climate information is essential. Earth system models (ESMs) provide state-of-the-art projections on a global scale but provide insufficient spatial resolution for regional analyses. Thus, downscaling—creating high-resolution climate information from low-resolution, large-scale data—is an important practical tool. Spatially and temporally high-resolution fields of climate data over large scales are important for many applications, including simulation of extremes at local scales (e.g., fires, floods, storms; Fischer et al. 2021), local planning, and making climate-informed ecological decisions such as tree species selection (MacKenzie and Mahony 2021). Hence, accurate and computationally efficient downscaling methods are crucial for local climate adaptation.

Strategies for downscaling coarse-resolution climate model simulations to regional scales can be broadly classified into dynamic downscaling and statistical downscaling. Dynamic downscaling employs a limited-area numerical climate model to resolve fine-scale features, driven by large-scale weather patterns from the low-resolution ESM (Skamarock et al. 2001). Dynamic downscaling can capture complex spatial patterns at smaller scales and can provide high-accuracy downscaling. However, it is highly computationally intensive and can only be used for relatively short time periods. Statistical downscaling develops statistical relationships between low-resolution (LR) and high-resolution (HR) climate variables. Statistical downscaling can be applied either at individual points, often using a combination of bias correction and transformation of statistical moments (Maraun 2013), or can be used to downscale entire fields. Common techniques for the latter include parametric approaches (e.g., Gaussian process/kriging), where covariances are specified to allow analytic solutions. While these approaches have shown some success, such climate-field downscaling methods make strong assumptions about the distribution and homogeneity of statistics, which are often not satisfied (Chen et al. 2014). Also, many current methodologies struggle to accurately downscale spatially complex variables (i.e., variables with nonlinear dependence on elevation or spatially heterogeneous dependence structures) and to capture extremes (Harris et al. 2022). Recently, a new strategy for statistical downscaling of climate fields has been developed that uses deep learning algorithms to learn a mapping from LR to HR paired climate fields (Annau et al. 2023). It has been found that this strategy can produce downscaled fields with much higher accuracy than traditional statistical downscaling and does not require the prohibitive computation required for dynamic downscaling.

Downscaling is intrinsically an underdetermined problem with a distribution of possible HR realizations physically consistent with any given LR input (Afshari et al. 2023). This is especially true since weather is sensitive to initial conditions, small differences of which can result in drastically different outcomes through the development of internal variability (Lucas-Picher et al. 2021). Stochastic weather generators, which attempt to sample from the distribution of weather states, have been used to try and account for this variability (Wilks 2010). Being able to capture the full variability of a downscaling problem is crucial for quantifying uncertainty and for characterizing extremes. Ideally, downscaling techniques would allow sampling from the HR distribution, conditioned on the LR input.

Deep learning methods are a promising approach for such distributional downscaling problems. Generative adversarial networks (GANs) have been successful in various generative artificial intelligence (AI) fields, especially computer vision applications (Goodfellow et al. 2014). GANs generally use two separate deep neural networks during training: a generator network which is given input and attempts to create plausible counterfeits of the training data, and a discriminator or critic network which is provided training data mixed with generator output and attempts to distinguish between the counterfeits and the real data. During training, the two networks play a minimax game: The generator tries to improve its output to “fool” the discriminator, and the discriminator tries to improve its ability at distinguishing between real and generated samples. In the last few years, GANs have been introduced to deep learning–based downscaling and have shown success in drawing realizations from high-dimensional non-Gaussian distributions with complicated dependence structures. Conditional GANs, developed by Mirza and Osindero (2014), allow the GAN to draw realizations from distributions conditioned on covariates.

Much of the development of GANs for climate downscaling builds on work from the computer vision field of super resolution. Most studies in computer vision use conditional GANs, where the networks are provided LR information and learn to sample from the HR conditional distribution (Ledig et al. 2017). With the introduction of GANs to climate downscaling, difficulties with instability during training (Wang et al. 2020) were addressed by the introduction of the Wasserstein GAN (WGAN; Arjovsky et al. 2017). In the WGAN, instead of using a discriminator network to estimate the probability of individual realizations being real, a critic network estimates the Wasserstein distance between the true HR distribution and the generated distribution. Intuitively, the Wasserstein distance is a critic score, specifying the quality of the downscaled fields compared to the training data. Not only does this approach substantially improve stability during training, but for downscaling, it conceptually makes sense to focus on convergence in distribution of generated and truth fields.

The initial formulation of (unconditional) GANs used a stochastic approach where the only input to the generator was Gaussian noise, generating different realizations for each different noise input. With the development of conditional GANs (Mirza and Osindero 2014; Ledig et al. 2017) and the subsequent super-resolution GAN (SRGAN) and enhanced super-resolution GAN (ESRGAN) frameworks from the field of super resolution, the noise input was replaced by the conditioning fields, leading to a semideterministic network. In this setting, each trained generator would still draw a realization from the conditional distribution, but it would always draw the same realization for each set of conditioning fields (in principle, one could draw a different realization by training a new model). A few studies successfully used variations of this architecture for downscaling climate fields; for example, Stengel et al. (2020) adapted the SRGAN model to create very high-resolution downscaled wind fields by first downscaling to a moderate resolution and then further downscaling to the final HR.

Recently, studies have investigated methods for allowing explicit stochasticity in conditional GANs, usually using variations of adding noise covariates stacked with the LR conditioning fields (e.g., Miralles et al. 2022). Price and Rasp (2022) concatenated a noise layer partway through their generator network, while Harris et al. (2022) concatenated multiple noise inputs with the conditioning information at the beginning of the network. Both studies found that the stochastic results were underdispersive: Trained models were unable to capture the full range of variability, often only sampling from the center of the conditional distribution. Recent advancements in super resolution (e.g., nESRGAN+; Rakotonirina and Rasoanaivo 2020) have improved stochastic calibration but have not yet adapted it to climate downscaling. Furthermore, many downscaling studies using stochastic GANs have focused their analyses on image quality. While Harris et al. (2022) performed a comprehensive evaluation of stochastic calibration of downscaled precipitation using continuous ranked probability score (CRPS) and rank histograms, the general issue of correcting underdispersion in generated ensembles remains an open question.

We aim to fill this gap by improving stochastic GAN frameworks for climate downscaling. Most of this work builds on Annau et al. (2023). While their model showed success for downscaling wind fields, it was not stochastic; for each set of conditioning fields, only a single realization of the HR field was generated. We use a similar model architecture as Annau et al. (2023) adapted for full stochasticity. An obvious challenge with testing distributional quality in a downscaling setting is the lack of a truth conditional distribution, as in most applications we only have access to a single truth realization for each time step. To address this challenge, we first consider an idealized approach based on synthetic data with known distributional properties. Based on these experiments, we test a “noise injection” method, where hundreds of noise fields, at different spatial resolutions, are injected into the latent layers of the network. This approach provides excellent stochastic calibration on the synthetic data. We then test our modification on a real-world downscaling problem, predicting HR wind components from LR conditioning data. Challenges with underdispersion on the wind data lead to the development of an updated loss function using a probabilistic error function and modification of the training method to fully utilize the stochasticity. Our final model is successful at capturing variability and improves estimates of moderate extremes.

2. Methods

All models in this paper use the same basic super-resolution structure. We train the models on paired sets of LR conditioning fields (covariates) and HR truth fields. The GAN then learns a mapping to the HR fields from the input covariates. For consistency, we keep the same resolution and size of fields across all models: HR fields are 128 × 128 pixels, and LR fields are 16 × 16 pixels, resulting in a downscaling factor of 8.

b. Model

This work utilizes conditional GANs (Mirza and Osindero 2014), which have shown success at learning the mapping between low-resolution variables and the desired output variables. Specifically, we use the Wasserstein conditional GAN formulation (Arjovsky et al. 2017), where the critic network learns to estimate the Wasserstein distance between the high-dimensional distributions of the generated and true fields (${\mathbb{P}}_g$ and ${\mathbb{P}}_r$, respectively).

1) Architecture

Most GAN network architectures employ dense convolutional blocks, which have been shown to be effective at extracting representative features from images. Our network architecture is based on the ESRGAN (Wang et al. 2018) setup, using Residual-in-Residual Dense Blocks (RRDBs) as the main convolutional blocks in the generator. RRDBs, introduced by Zhang et al. (2018), employ densely connected convolutional layers to extract important features while also maintaining direct connections from the previous layers to all convolutional layers to create a contiguous memory. Such an architecture provides freedom for the network to extract complex features while also ensuring consistency with previous layers. Specifically, we adapt the architecture employed in Annau et al. (2023) to allow noise input and multiple covariate streams.

Unlike other applications of super resolution, climate downscaling often has access to pertinent HR information during training. A common example of such HR information is topography, which influences local climate strongly. While many previous studies have included topography as a covariate, it has often been input at low resolution with climate covariates, discarding potentially useful information. Harris et al. (2022) included HR covariates by first convolving them down to the shape of the LR covariates. We developed a different strategy, based on depthwise separable networks (Jiang et al. 2020) where the generator contains two input streams, one stream for each resolution. Each stream has equivalent convolutional blocks applied in parallel, and after the LR stream has passed through the upsampling blocks to increase the resolution, the two streams are concatenated and passed through a final convolutional block (Fig. 1).

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Architecture of GAN networks showing RRDB with noise injection. Green denotes locations where noise is added into the network. Rectified linear units (ReLUs) are used to introduce nonlinearity and show the negative slope parameter in parentheses. Labels on the convolutional blocks show parameter values, with k, n, and s denoting the size of the kernel, the number of filters, and the stride, respectively. Note that in the final convolutional block of the generator, npred corresponds to the number of predictands.

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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In the standard super-resolution formulation, the critic network is only given samples of the predictands (either generated or from the training data). However, Harris et al. (2022) found that passing all available covariates to the critic network can improve its predictions, and in the Wasserstein GAN formulation, it is important that the critic network is able to make relevant estimates of the Wasserstein distance. Given that we want the generator to sample from a conditional distribution, we thus want the critic to make use of conditioning information when quantifying this distribution. Similarly to the generator, we adapted the critic network to include a separate input stream for the LR covariates, which is concatenated after the HR stream has been downsampling via strided convolution (Fig. 1).

2) Noise injection

To improve the model’s ability to sample across the entire range of the conditional distribution, we adjusted the generator architecture to inject noise directly into the latent representations produced by the convolutional layers. Specifically, we based our approach on nESRGAN+ (Rakotonirina and Rasoanaivo 2020) and concatenate uncorrelated, mean zero, unit variance Gaussian noise fields with the latent layers inside each Dense Block (Fig. 1). With our architecture, this leads to six noise injection instances in each Residual Dense Block and 18 noise injections in each RRDB. Our full noise generator comprises 14 RRDBs and two RRDBs in the LR and HR input streams, respectively, followed by a single HR RRDB after concatenation. In total, this results in 252 LR and 54 HR noise layers.

To test the effect of the number of noise injection layers, we replaced some of the stochastic RRDBs with deterministic RRDBs (i.e., RRDBs without noise injection). Altogether, we tested three different levels of noise injection: low (two stochastic LR RRDBs and no stochastic HR RRDBs), moderate (seven stochastic LR RRDBs and no stochastic HR RRDBs), and full noise (14 stochastic LR RRDBs and two HR RRDBs). As a baseline model, we also considered the more standard noise-covariate approach (similar to that used in Leinonen et al. 2021), for which a Gaussian noise field is concatenated with the LR input covariates before passing through the generator network.

3) Losses

In Wasserstein GAN training, the critic tries to maximize the difference in the distributional distance between true fields and generated fields, while the generator attempts to minimize this distance (i.e., it attempts to make it difficult for the critic to distinguish the generated fields from the ground truth data). Previous studies have found that solely relying on the critic loss (adversarial loss) for the generator training can lead to instability, such that the training process does not converge (Wang et al. 2018). Here, we use the common approach of adding an extra content loss term to the generator—a pixelwise error metric between the training data and the generated fields, intended to aid convergence in realization.

We experimented with two different content loss functions: mean absolute error (MAE) and CRPS. MAE is commonly used as a content loss in deep learning, but it is a deterministic measure. CRPS is defined as

$$\text{CRPS}(F,x)={\displaystyle {\int }_{-\infty }^{\infty }}{[F(y)-H(y-x)]}^{2}dy,$$(13)

where F represents the CDF of the predicted distribution, H is the Heaviside function, and x is the ground truth value. The CRPS metric returns lower values to distributions whose mass is centered around the ground truth value. In a deterministic setting, the PDF of y is a delta function, and the CRPS reduces to MAE, so it is appropriate to use as a content loss. The pixelwise CRPS metric aims to encourage convergence in distribution at small scales, rather than convergence in realization at larger scales. To apply the CRPS, we calculated an empirical CRPS metric for each pixel, using the ground truth value and values from the stochastic sampling realizations (described below), and then took the mean across pixels.

4) Training

Since our goal is to sample realizations across the conditional distribution of HR fields, we do not want the loss function to force the generator to create copies of the training data—we consider the ground truth as one realization of the conditional distribution. Ideally, we would expect generated realizations to have similar statistics and large-scale features as the training data, but not be identical. Overreliance on content loss (particularly deterministic measures) can degrade model performance, as it overly penalizes small deviations in feature location/presence. In situations where features are spatially shifted, pixelwise error metrics such as the content loss will penalize the model twice: once for the feature not occurring where it does in the ground truth, and once for the feature occurring where it is not in the ground truth. This double-penalty problem is a well-known issue with pixel-based losses in generative networks (Rossa et al. 2008) and results in overly blurry output, where the model converges on the conditional median (or mean, depending on the form of the content loss), thus suppressing small-scale features and extremes (Annau et al. 2023). If there is a strong association between LR and HR scales, then there is less freedom for variability in the conditional HR distribution. Where there is broad separation, the conditional variability should be greater and could be more heavily penalized by the double-penalty problem. While the adversarial loss in GANs (in our case, the Wasserstein distance) is not a pixelwise metric and does not constrain the network in the same way, the use of content losses can suppress variability.

We considered two training techniques in our models to address the double-penalty problem while rewarding convergence at large scales common to both resolutions: frequency separation and stochastic sampling. Frequency separation, introduced by Annau et al. (2023), coarsens HR fields to only provide low frequencies to the content loss, whereas the full HR fields are used for the adversarial loss. The motivation of this approach is to allow the model to more freely develop high-frequency patterns by removing some of the content loss constraints. Stochastic sampling is an approach modified from Harris et al. (2022), where multiple stochastic realizations are passed to the content loss, which uses an ensemble metric to assess calibration (algorithm 1). In each generator training step, we generate six stochastic realizations of each field in the batch and pass these to the content loss. For the MAE loss, we take the pixelwise average across the stochastic realizations before computing the loss. Both frequency separation and stochastic sampling allow generated variability at small scales but encourage convergence at large scales.

With the stochastic sampling method, our generator loss function is

$$L_G=\underset{\text{Adversarial}\text{loss}}{\underbrace{-{\mathbb{E}}_{G(x)\sim {\mathbb{P}}_g}C[G(x)]}}+\underset{\text{Content}\text{loss}}{\underbrace{\alpha {\mathbb{E}}_{y\sim {\mathbb{P}}_r}{l}_c[y,G{(x)}_1,G{(x)}_2,\dots ,G{(x)}_6]}},$$(14)

where x represents LR conditioning fields; y is the HR ground truth; α is the content loss weighting; l_c is the content loss function; and G and C represent the generator and critic network, respectively. Note that an ensemble of six stochastic realizations is passed to the content loss.

Algorithm 1

Pseudocode for stochastic sampling algorithm using the CRPS metric. Note that we chose six stochastic realizations as the maximum number that did not exceed our graphics processing unit (GPU) memory.

 nRealisation ← 6

 for i ∈ 1: nBatch do

  LRBatch = LRBatches_i

  HRBatch = HRBatches_i

  adversarialLoss ← −mean[Critic(LRBatch)]

  for j ∈ 1: nSample do

   subBatch ← repeat(LRBatch_j, nRealisation)

   subRealisations ← Generator(subBatch, invariant)

   CRPSBatch_j ← mean[pixelwiseCRPS(subRealisations, HRBatch_j)]

  end for

  contentLoss ← mean(CRPSBatch)

  loss ← adversarialLoss + contentLoss

 end for

Throughout this paper, we will use the following naming conventions to specify models:

$${\text{Training}}_{\text{Noise}\text{level}}^{\text{Content}\text{loss}},$$

where training is represented, respectively, by F or S for frequency separation and stochastic sampling; noise level is represented by NC for noise covariate; and low, medium, and full are represented for noise injection (cf. section 2). For example, $S_{\text{full}}^{\text{CRPS}}$ specifies a model using stochastic sampling, the CRPS content loss, and full noise injection. Note that we have not investigated all combinations of generator parameters, as some only make sense in combination or in specific settings. Model training parameters are presented in the online supplemental material (Tables 1 and 2).

3. Results

a. Synthetic data

Noise injection

Models with full noise injection performed substantially better overall than the baseline models at matching the true marginal distribution of the synthetic data. A representative example of pixelwise marginal distributions for the truth and generated fields is shown in Fig. 2a. The baseline $F_{\text{NC}}^{\text{MAE}}$ model produced highly underdispersed distributions, while noise injection models were able to match the true distributions well. This result held across all pixels: KS statistics comparing the true marginal distributions with those from noise injection models were significantly smaller than with the baseline model (Fig. 2b). Both S class models had slightly larger median KS statistics than $F_{\text{full}}^{\text{MAE}}$ but still showed good distributional matching.

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(a) KDEs of marginal distributions of p(HR|LR) for the unimodal synthetic dataset for one example pixel (i = 5, j = 5) for the true distribution and generated distributions. KDEs are based on 500 realizations for a single conditioning field for each distribution. Dashed line shows the true marginal distribution. The inset figure shows the full y-axis range. (b) Violin plot showing KS statistic values comparing generated marginal conditional distributions to ground truth distributions for all pixels. Statistics are calculated for each pixel individually, using 500 realizations of a single conditioning field. Lines show 0.25, 0.5, and 0.75 quantiles, respectively. (c) CDF of the rank histogram on unimodal synthetic data, with four models, showing calibration of conditional distributions. The dashed line shows the reference uniform distribution. Rank histograms were calculated across 100 randomly selected conditioning fields, with 96 HR realizations generated for each. (d) KDEs of marginal conditional distributions for one example pixel of a bimodal dataset, comparing true (dashed line) and generated distributions. Distributions were estimated using the same approach as in (a).

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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Decreasing the number of noise injection layers in the generator decreased the performance of conditional distributions (Fig. 2b). Low noise injection ($F_{\text{low}}^{\text{MAE}}$) showed slight improvement over the baseline model but still produced underdispersed results. Medium noise injection ($F_{\text{med}}^{\text{MAE}}$) had improved statistics but was still generally underdispersive, and full noise injection ($F_{\text{full}}^{\text{MAE}}$) created results with the best agreement of pixelwise marginal distributions.

Rank histograms provide another tool for investigating the calibration of conditional distributions, averaged across multiple conditioning fields. Rank histograms showed good calibration for all models with full noise injection and severe underdispersion for the baseline model (Fig. 2c). The $S_{\text{full}}^{\text{MAE}}$ model showed almost perfect calibration, but all noise injection models performed well.

Learning multimodal distributions is challenging for GANs; they tend to show mode collapse, in which distributions are collapsed to the conditional mean (Saatci and Wilson 2017). We found that using the bimodal dataset [Eqs. (6)–(10)], the $F_{\text{full}}^{\text{MAE}}$ model could learn both modes of the marginal distributions. The baseline models usually showed mode collapse and could not meaningfully recreate any of the marginal distributions (Fig. 2d).

Investigating results of the full distribution p(HR), statistics of generated fields had fewer artifacts and were closer to the ground truth statistics using the $F_{\text{full}}^{\text{MAE}}$ model than the baseline $F_{\text{NC}}^{\text{MAE}}$ model (Fig. 3). Even after training metrics had converged, the baseline model showed noticeable traces of the convolutional filters as checkerboard artifacts, which were not apparent in the $F_{\text{full}}^{\text{MAE}}$ model. The baseline model also substantially underestimated the 99.9th percentiles, especially at the highest values. These were better captured (although not perfectly) by the $F_{\text{full}}^{\text{MAE}}$ model.

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Spatial fields of median and 99.9th percentiles of the full distributions across samples for ground truth and generated data from two models, using the unimodal synthetic dataset [Eqs. (1)–(3)].

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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As well as better capturing pixelwise marginal variability, all noise injection models performed better at representing covariance patterns. The RASP metrics (Fig. 4) demonstrate that the baseline model showed too little power for a range of low wavenumbers and then spurious spikes at other wavenumbers. The $F_{\text{full}}^{\text{MAE}}$ did not show the spikes but still had a lower power bias at low wavenumbers. Both S class models substantially improved the representation of power at low wavenumbers and in general were the closest to the true spectral power across all wavenumbers. There was no obvious difference in spectral power between the two S class models using this metric.

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RASP for four models using the unimodal synthetic dataset [Eqs. (1)–(3)]. Values are standardized to amplitudes of ground truth wavenumbers, so perfectly matched spectral power occurs at a value of one. Solid lines and shaded regions, respectively, show the mean and ± one standard deviation across 1200 randomly selected samples. The dashed line indicates the wavenumber corresponding to LR pixel size.

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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Overall, synthetic data experiments showed that models using full noise injection were substantially better at capturing conditional distributions than the baseline model. In general, all models with noise injection performed comparably well. There was also noticeable improvement in the quality of the full distributions using noise injection, and the S class models showed improved ability to represent spatial dependence.

b. Wind downscaling case study

As above, we first present results for distributions conditioned on a single set of LR fields before moving to the full distributions across the test set. Note that while all models predicted both zonal (eastward) and meridional (northward) wind components, the results were generally similar, and unless otherwise stated, we only show results for meridional components.

Even with noise injection, the F class models applied to a realistic downscaling problem produced underdispersed results (Fig. 5). While the $F_{\text{full}}^{\text{MAE}}$ model showed substantial improvement over the baseline $F_{\text{NC}}^{\text{MAE}}$ model, it did not fully capture the conditional variability. Both $S_{\text{full}}^{\text{MAE}}$ and $S_{\text{full}}^{\text{CRPS}}$ models had better calibration than the F class models; the $S_{\text{full}}^{\text{CRPS}}$ model, while still showing some underdispersion, performed best among the models considered (Fig. 5).

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CDFs of rank histograms for meridional wind components, using four different models. Rank histograms were calculated across 100 randomly selected conditioning fields, with 96 HR realizations generated for each. The dashed line shows the reference uniform CDF.

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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With the $S_{\text{full}}^{\text{CRPS}}$ model, individual realizations were visually realistic and showed noticeable differences in spatial patterns given a single set of conditioning fields (Fig. 6). Examples of standard deviation fields showed that the conditional distribution varies substantially with the state of the conditioning fields. Realizations of moderate wind component extremes (0.01 and 0.99 quantiles of spatial means) did not show evident bias and were similar in textures and range of values to the ground truth WRF fields. None of the other models performed as well (see supplemental material); all models, and especially the $F_{\text{NC}}^{\text{MAE}}$, had substantially lower conditional standard deviations. The $S_{\text{full}}^{\text{MAE}}$ model generated realizations that were blurry, and the $F_{\text{full}}^{\text{MAE}}$ model generated realizations that did not match the WRF fields as well as the S class models.

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Example meridional and zonal wind fields for coastal British Columbia using the $S_{\text{full}}^{\text{CRPS}}$ model. First two rows show hours representative of spatial-mean 0.01 quantiles, the middle two rows show randomly selected hours from the full test set, and the bottom rows show hours representative of spatial-mean 0.99 quantiles. Columns show (from left to right) WRF (i.e., ground truth), ERA5 (input conditioning field), three generated realizations, and the conditional standard deviations across 500 realizations.

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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Looking at the full distribution, spectral power was also better calibrated with the S class models (Fig. 7). The baseline model produced low spectral power at both low and high wavenumbers; the performance of the $F_{\text{full}}^{\text{MAE}}$ model was better but followed similar patterns and showed a substantial high bias at intermediate wavenumbers. The $S_{\text{full}}^{\text{CRPS}}$ model generally had the most similar distribution of spectral power to the truth fields, although it showed a modest high power bias at low and high wavenumbers, especially for meridional wind components. All models using the MAE loss function showed low-power bias at high wavenumbers, consistent with the suppression of fine-scale features by this deterministic metric.

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RASP metric (median and IQR) standardized to ground truth values for zonal and meridional wind fields. Spectral powers are calculated across 1200 randomly selected fields. The dashed line shows the wavenumber corresponding to LR grid size.

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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Pixelwise marginal statistics of full distributions differed slightly between models (Fig. 8). Differences in median and interquartile range were spatially smoother with the S class models, suggesting these models were able to capture fine spatial patterns better. The baseline model substantially underestimated interquartile range (IQR) in many locations. This bias was improved with the $F_{\text{full}}^{\text{MAE}}$ model, although it overestimated IQR in certain areas, and the S class models further improved the results.

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Pixelwise median and IQR of the full distribution of the test dataset for meridional wind fields. The first column shows truth statistics, followed by difference fields for each of the four models (truth − model).

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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Investigating the tail behavior of the full distributions, we found that models that were better at learning conditional distributions performed better at predicting accurate extremes (Fig. 9). Comparing moderately large marginal extremes (99.99th and 0.01th percentiles) across time steps, the $S_{\text{full}}^{\text{CRPS}}$ model had the least biased estimate of extremes, while biases from the $F_{\text{NC}}^{\text{MAE}}$ were largest, underestimating 99.99th percentiles and overestimating 0.01th percentiles (Fig. 9a). This pattern is also apparent in difference maps of pixelwise extremes (cf. 0.01th percentile in Fig. 9b). The map for $F_{\text{NC}}^{\text{MAE}}$ has many strongly negative areas, whereas the map for $S_{\text{full}}^{\text{CRPS}}$ shows reduced systematic bias and is fairly well centered around zero for the 0.01th percentiles. We also investigated less extreme 0.01th and 99.9th percentiles and more extreme 0.001th and 99.999th percentiles; all extreme percentiles showed the same pattern.

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Calibration of moderate extremes for meridional wind fields over full distributions. (a) Boxplots of distributions of difference in the 99.99th and 0.01th percentiles of ground truth and generated realizations for four models, based on 500 realizations for each of 350 randomly selected conditioning fields. Values below zero represent model overestimation; values above zero represent underestimation. (b) Difference maps of pixelwise 0.01th percentiles of ground truth and generated realizations for four models.

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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Calibration of spatial extremes was also improved in the S class models (Fig. 10). Here, rank histograms represent stochastic calibration of models in regards to spatial extreme values (i.e., large or small quantiles of values across the domain). Rank histograms of 0.01th and 99.99th percentiles across wind fields showed that the $S_{\text{full}}^{\text{CRPS}}$ model was the least underdispersive and had less bias than the other models (Fig. 10). By contrast, the rank histogram of the baseline $F_{\text{NC}}^{\text{MAE}}$ model showed the model systematically overpredicted 0.01th percentiles and underpredicted 99.99th percentiles.

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CDFs of rank histograms based on the 0.01th and 99.99th percentiles of meridional wind fields over 400 conditioning fields, with 96 realizations of each field. Dashed lines represent the CDF of a uniform distribution.

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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To investigate why the $F_{\text{full}}^{\text{MAE}}$ model showed good calibration on the synthetic dataset but was underdispersive when used to generate high-resolution wind fields, we tested the effect of increased spatial heterogeneity on synthetic data models (cf. section 2a). Models trained on datasets with high heterogeneity showed a greater degree of underdispersion than those with low or moderate heterogeneity (Fig. 11). Investigating the KS statistics of the pixelwise marginal distributions showed that distributions from datasets with high heterogeneity had very large ranges in quality, whereas those from low heterogeneity data were more consistent and better matches on average (Fig. 11b). The rank histograms of the results from these datasets showed similar patterns—the low heterogeneity model was relatively well calibrated, and the high heterogeneity model was underdispersive. Interestingly, while the moderate-level model did not show as much underdispersion, it showed the most bias, more often than not overestimating values. These results suggest that the F class models struggle to produce good stochastic calibrations when there is a high degree of spatial heterogeneity in the fields—as is often the case in a realistic setting.

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Comparison of stochastic calibration of the $F_{\text{full}}^{\text{MAE}}$ model based on synthetic data with low, moderate, and high spatial heterogeneity. (a) CDFs of rank histograms based on all pixels of 50 random conditioning fields with 96 realizations of each. (b) Distribution of pixelwise KS statistics between generated and true marginal distributions, using 500 stochastic realizations for a single conditioning field.

Citation: Artificial Intelligence for the Earth Systems 4, 2; 10.1175/AIES-D-24-0044.1

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4. Discussion

This paper discusses three main classes of GAN-based downscaling models distinguished by noise type (noise covariate vs noise injection), training method (frequency separation vs stochastic sampling), and content loss type (deterministic MAE vs probabilistic CRPS). We aim to improve stochastic calibration, creating models that can sample successfully from the full range of the conditional distribution. We first present a novel network architecture, where many layers of noise at different resolutions are injected into the generator. Compared to the baseline $F_{\text{NC}}^{\text{MAE}}$ model, our architecture performs better at capturing the conditional variability of the data and achieves good calibration on synthetic data. We then introduce a stochastic training method which greatly improves stochastic calibration and spatial structure, especially when combined with the probabilistic CRPS metric in the $S_{\text{full}}^{\text{CRPS}}$ model. The $S_{\text{full}}^{\text{CRPS}}$ model shows improved skill at estimating marginal conditional distributions, as well as marginal and spatial statistics of the full distribution. Using this model, we successfully downscale wind fields and show that ensembles of generated realizations are well calibrated. All models we investigate here use a GAN architecture with a separate stream for HR topography, as initial tests showed substantially improved results. Future work will fully investigate the advantages of this approach.

Most conditional GANs in super resolution are deterministic, and recent attempts at reintroducing stochasticity have added noise fields as additional covariates (e.g., Leinonen et al. 2021). Our approach of injecting noise directly into the convolutional layers fundamentally differs in that it adds noise to latent representations deep inside the network, instead of to the input. When noise is introduced as a covariate at the beginning of the network, we hypothesize that the network will learn low weights for the noise layers to optimize the loss function. By adding noise to the latent representations, we slightly alter features inside the network, leading to better representation of conditional and full distributions. Our approach is similar to the nESRGAN+ architecture (Rakotonirina and Rasoanaivo 2020) which injects noise inside the Residual-in-Residual Dense Blocks, but we inject noise one level deeper, inside the Dense Blocks, to alter the output of the basic convolutional layers. In contrast to the nESRGAN+, we also use noise injection at both the low and high resolution, allowing for more scales of stochasticity. Interestingly, we never experienced problems with overdispersion as we increased the number of noise injection layers; marginal distributions were closest with the maximum possible number of noise injection layers for a given architecture.

Noise injection is certainly not the only method of creating stochastic output using deep learning methods. Getter et al. (2024) investigated using convolutional neural networks to predict parameters of Gumbel distributions from which wind speed values could be sampled. This is an appealing approach, in that it allows relatively simple distributional analysis, especially for extremes. However, it requires the data to match the selected distributions and thus is less flexible than our nonparametric approach. Also, it is unclear how spatial dependence of realizations would be accounted for in this setup. Another theoretically interesting setup, developed by Saatci and Wilson (2017), involves a Bayesian training framework in which model weights are sampled from learned distributions. However, this method is computationally expensive and likely not applicable for realistic applications. Future work could investigate and compare these approaches.

Stochastic sampling is an approach adapted from Harris et al. (2022), in which the content loss function is calculated on a set of stochastic realizations. While the $F_{\text{full}}^{\text{MAE}}$ model showed excellent calibration on the synthetic data and much improved performance than the $F_{\text{NC}}^{\text{MAE}}$ model, it still failed to fully capture the variability in the wind downscaling application. The S class models, especially in combination with the CRPS loss function, resulted in substantially improved performance, although the $S_{\text{full}}^{\text{CRPS}}$ model sometimes produced too much variability at very fine spatial scales. This improvement could be the result of a few different factors. First, by using multiple realizations of each field in the loss function, the network has more information to use for backpropagation. Second, CRPS is a stochastic metric which aims to quantify distributional matching; it seems reasonable that it thus improves the stochastic calibration. In contrast to our study, Harris et al. (2022) did not find an improvement with using CRPS in the loss function. This difference in results could be due to differences in generator architecture. As Harris et al. (2022) did not use noise injection, perhaps the generated stochasticity was less suitable for optimization with the CRPS metric. Future research could also consider multivariate extensions of CRPS in the content loss.

All models considered in this study use a separation of spatial scales in the content loss in an effort to address the double-penalty problem inherent in the ill-conditioned nature of climate downscaling. In our synthetic data experiments, we found that partial frequency separation (PFS), as described in Annau et al. (2023), resulted in well-calibrated output. However, this method did not perform as well in the realistic setting of wind component downscaling, motivating consideration of the stochastic sampling approach. Fundamentally, PFS and stochastic sampling have similar goals: allowing the adversarial loss freedom to create small-scale features while rewarding consistency between generated and conditioning fields at large scales. While PFS achieves this by only providing low-frequency information to the content loss, the stochastic sampling approach applies the content loss function across an ensemble of stochastic realizations, thus “smoothing out” the smaller-scale features of the generated fields. The stochastic sampling approach is likely more accurate than PFS—instead of arbitrarily choosing a frequency for separation, the sample conditional means define the transition from conditioning scales to sampling scales. Indeed, we found that for downscaling wind components, stochastic sampling always outperformed models using PFS. A practical challenge with stochastic sampling is that it uses more computational resources during training than PFS models, as each of the stochastic realizations has to be used during backpropagation. Most notably, stochastic sampling nearly doubled the amount of memory required during training, and it increased training time by about 50% (using a stochastic batch size of six). In practice, the choice of training approach will likely depend on the desired outcome and the computational resources available. While the stochastic sampling models performed substantially better at capturing conditional and full distributions, they only performed slightly better at capturing the spatial patterns of single fields. Thus, if the goal is to produce downscaling without needing to capture conditional variability, it may be prudent to use PFS and reduce training requirements. It is, of course, possible that the improvement gained from using stochastic sampling on capturing the conditional distributions will depend strongly on the fields being considered.

Wind fields show a high level of spatial heterogeneity, which we expect is responsible for the difficulties experienced by the $F_{\text{full}}^{\text{MAE}}$ model in capturing the conditional variability. Our experiment with spatial heterogeneity showed that even with synthetic data, increasing heterogeneity lead to increased underdispersion and bias in conditional means. High heterogeneity will generally lead pixelwise metrics to be more sensitive—slight shifts in features from the truth fields will tend to result in poor pixelwise metrics compared to similar shifts in fields with low heterogeneity.

Data availability statement.

Our fully stochastic GAN with CRPS loss is implemented with PyTorch at https://github.com/nannau/ClimatExML/tree/stochastic. LR conditioning fields are available from ERA5 (https://cds.climate.copernicus.eu/) and HR WRF training data are available at https://www.gwfnet.net/Metadata/Record/T-2020-05-28-Q1KtfEjVdz0aBmauuvG9r9w.