MSG-GAN-SD: A Multi-Scale Gradients GAN for Statistical Downscaling of 2-Meter Temperature over the EURO-CORDEX Domain

Abstract

One of the most important open challenges in climate science is downscaling. It is a procedure that allows making predictions at local scales, starting from climatic field information available at large scale. Recent advances in deep learning provide new insights and modeling solutions to tackle downscaling-related tasks by automatically learning the coarse-to-fine grained resolution mapping. In particular, deep learning models designed for super-resolution problems in computer vision can be exploited because of the similarity between images and climatic fields maps. For this reason, a new architecture tailored for statistical downscaling (SD), named MSG-GAN-SD, has been developed, allowing interpretability and good stability during training, due to multi-scale gradient information. The proposed architecture, based on a Generative Adversarial Network (GAN), was applied to downscale ERA-Interim 2-m temperature fields, from 83.25 to 13.87 km resolution, covering the EURO-CORDEX domain within the 1979–2018 period. The training process involves seasonal and monthly dataset arrangements, in addition to different training strategies, leading to several models. Furthermore, a model selection framework is introduced in order to mathematically select the best models during the training. The selected models were then tested on the 2015–2018 period using several metrics to identify the best training strategy and dataset arrangement, which finally produced several evaluation maps. This work is the first attempt to use the MSG-GAN architecture for statistical downscaling. The achieved results demonstrate that the models trained on seasonal datasets performed better than those trained on monthly datasets. This study presents an accurate and cost-effective solution that is able to perform downscaling of 2 m temperature climatic maps.

1. Introduction

Downscaling is a procedure that allows making predictions at local scales, starting from climatic field information available at large scale. In climate science, a well-established representation of climatic fields involves the use of multi-dimensional structures (i.e., 2D, 3D or even 4D if the time dimension is considered) that can be treated as single images or image sequences [1,2,3,4,5]. Therefore, the grid points in a climatic map can be represented as image pixels. This is the reason why the downscaling problem is closely related to the super-resolution (SR) problem in computer vision and image processing, corresponding to the enhancement of the spatial resolution of an image beyond its original resolution [6,7,8,9,10,11]. Thus, a prior extensive knowledge about the structure of the images at the target finer resolution is critical for SR models, and enables them to generate upsampled images that are coherent with the input data [10]. The task of producing a super-resolution image (from now on called an HR image), starting from its lower resolution counterpart (from now on called an LR image), is recognized in the literature as single-image super resolution (SISR) [10]. This problem is generally ill-posed because it does not have a unique solution, as many different HR images can be generated starting from the same LR image [11]. In fact, the upsampling procedure involves the synthesis of artificial information which serves to scale-up the image towards the target resolution. (In this scenario, the terminology may lead to confusion as the terms “upsampling” and “upscaling” are both used in computer vision to indicate the process of increasing the number of pixels in an image, whereas the term “downsampling” indicates the inverse process. In climate science, the term “downscaling” refers to the generation of maps with a finer resolution (i.e., with a higher number of grid points), starting from their coarser resolution. This is due to the fact that the finer resolution maps, generated through a downscaling process, will consist of grid points with a smaller horizontal resolution.). The need for downscaling is generally motivated by the typically unsatisfactory coarse resolution of global climate models (GCMs). Although these models are used for a better understanding of climate change at global and up to continental scales, and provide information for a large number of climatic fields, they are not able to resolve processes that manifest at regional and local scales, whose dynamics are often critical for assessing the impacts of a changing climate on society [3,12]. Downscaling can be carried out through two classes of techniques: dynamical and statistical. The former is performed through a physics-based model, namely a regional climate model (RCM), that involves a set of physical equations for modeling different components of the climatic system and their interactions. The physical laws are numerically solved in order to simulate the outcomes for a series of different climatic fields at a finer resolution. The statistical techniques, which are typically more accurate than the model’s raw output [13], involve the learning of empirical statistical relationships between coarse GCM outputs and HR products [3,4,12,14]. Inspired by the work presented in [15], a multi-scale deep architecture was developed for downscaling the 2-m temperature (T2M), from 0.75° × 0.75° up to 0.125° × 0.125° of spatial resolution, in the past 40 years (1979–2018), over the European domain by exploiting Era-Interim analysis data [16]. The aim was to provide a novel deep learning-based solution to the downscaling task, as an alternative to traditional dynamical and statistical techniques. This was also motivated by the cost-effectiveness of deep learning models that, once trained, provide outcomes through a limited amount of computing resources and execution time. The proposed architecture, named Multi-Scale Gradients GAN for Statistical Downscaling (from now on called MSG-GAN-SD), features a Generative Adversarial Neural Network (GAN) composed of Generator (from now on called G) and Discriminator (from now on called D) networks, that are both convolutional and set to work at multiple scales. Each of these networks is made up of several blocks exploiting multiple versions of the same image at different resolutions, depending on the specific scale. This allows the propagation of gradients coming from multiple scales during the training phase. As opposed to other similar works based on statistical downscaling of climatic fields, low-resolution T2M heatmaps were not artificially downsampled from the high-resolution counterpart, because images were directly gathered at low and high resolutions. Furthermore, the present study introduced an experimental multi-stage framework for evaluation purposes.

Related Work

According to [17], statistical downscaling is classified into three sub-categories: regression-based, weather classification-based, and weather generators-based approaches. Concerning regression-based approaches, several attempts at downscaling a variety of climatic fields—mainly temperature, precipitation and wind fields—have been proposed in recent years. Machine learning (ML) techniques have been widely adopted for downscaling. In particular, the LASSO regression was used in [18] for downscaling precipitation, whereas genetic programming (GP) was exploited in [19] and [20] for downscaling precipitation and temperature, and daily minimum and maximum temperature, respectively. Moreover, the random forest (RF) was used in [21] for land surface temperature, whereas a novel hybrid dynamical-statistical approach was presented in [22] focusing on the resolution of fine-scale rainfalls with lower computational costs, through a combination of dynamical and statistical downscaling. Artificial Neural Networks (ANNs) have been used in [23] to perform precipitation downscaling, applying a further residual correction method based on Particle Swarm Optimization (PSO), an Imperialist Competitive Algorithm (ICA), and a Genetic Algorithm (GA). Furthermore, a Back-Propagation Neural Network (BPNN) and Support Vector Machine (SVM) fusion approach was adopted in [24] to downscale precipitation. Several works moved towards deep architectures in the context of Deep Learning (DL), especially concerning Long-Short Term Memory (LSTM) networks, Convolutional Neural Networks (CNNs) and Generative Adversarial Networks (GANs). Since their introduction presented in [25], LSTMs have been proven to be suitable for recovering and bridging information arbitrarily far in time, while avoiding the vanishing gradient problem. LSTMs have also been widely adopted for time-series related problems and, in the context of climate downscaling, for statistical downscaling of precipitation [26] and rainfall forecasting [27] in combination with Feed Forward Neural Networks (FFNNs). Convolutional Neural Networks (CNNs), due to their ability to deal with spatio-temporal multi-dimensional structures, have been demonstrated to be particularly suitable for accomplishing SR tasks. Several attempts to use deeper architectures have been proposed [28,29,30,31,32,33] for the extraction of high-level image characteristics and the downscaling of climatic fields. A deep neural network based on a CNN and a LSTM recurrent module was proposed in [34] to estimate precipitation based on well-resolved atmospheric dynamical fields. A novel architecture, named DeepSD, based on the super resolution framework, was presented in [1,2] for downscaling precipitation fields, and a CNN-based approach was proposed in [13] as an alternative solution to the existing precipitation-related parameterization schemes for the numerical precipitation estimation. A CNN model for downscaling the occurrence of precipitation was also proposed in [3], whereas different configurations of CNN were adopted in [35] to downscale daily temperature and precipitation over China. A competitive DL framework based on a CNN was presented in [36] for downscaling temperature and precipitation, and it performed particularly well in generating spatio-temporal details at very fine-grid scales. A U-Net-type CNN was also used in [37] to learn a one-to-one mapping of low-resolution (input) to high-resolution (output) wind fields simulations data, and a conditional variational autoencoder (based on CNN) was exploited for learning the conditional distributions from data, assessing multimodalities and uncertainties. A CNN was adopted in [38] to perform smart dynamical downscaling (SDD) for extreme precipitation events, whereas a surrogate model, based on a Deep CNN (DCNN), was evaluated in [39] for surface temperature, and was found to estimate image details that were not retained in the inputs. Recently, remarkable results were reported in several studies exploiting GANs for SR tasks in climate science. An Enhanced Super-Resolution GAN (ESRGAN) [40] was adopted and presented in [41] to downscale wind speeds from a coarse grid, capturing high frequency power spectra and high order statistics in the dataset, thus generating images of superior visual quality compared to the SR-CNN. A novel method (ClimAlign) was introduced in [42] for unsupervised, generative downscaling of temperature and precipitation based on normalizing flows for variational inference. Further works [4,43,44,45,46,47,48,49,50] opted for downscaling based on ML and DL, and they helped assess both strengths and weaknesses of such methods. The results reported in these studies show that downscaling models based on ML allow better performance with respect to the other statistical approaches presented in [51,52,53].

2. Materials and Methods

2.1. Data

The proposed work aimed at downscaling 2-meter temperature over the EURO-CORDEX domain [54], by learning a statistical relationship between HR images and their LR counterparts, each one representing a temperature map. Analysis data was gathered in NetCDF4 format [55] from the COPERNICUS ERA-Interim global atmospheric reanalysis dataset [16], and treated as images throughout the various experiments, according to the SISR framework. All the background information can be found in [56]. From the global domain, the EURO-CORDEX subdomain was selected for both resolutions, from −48.5° E to 69.75° E of longitude and from 73.9° N to 20.15° N of latitude. The horizontal resolutions of HR and LR images are about 13.87 km (0.125° × 0.125°) and 83.25 km (0.75° × 0.75°) respectively, whereas the related images sizes are 431 × 947 and 72 × 158 pixels. The data covers a temporal range from January 1979 to December 2018 (40 years) and is made up of 6-hourly samples. Thus, each day of a year consists of four samples, at 00:00, 06:00, 12:00 and 18:00, respectively.

2.2. The Architecture: Multi-Scale Gradients GAN for Statistical Downscaling

The proposed Multi-Scale Gradients GAN architecture for statistical downscaling (MSG-GAN-SD, reported in Figure 1) is based on the fusion of two frameworks, that is, the super resolution GAN defined in [57] and the Multi-Scale Gradients GAN proposed in [15]. The multi-scale framework in [15] was considered in order to speed up both the development and the tuning of the model, overcoming the difficulties of a conventional GAN caused by its training instabilities. The nature of the statistical downscaling task required the employment of a SR architecture [57], in which a LR version of the HR image (target) is used to feed G, instead of the random latent vector proposed in [58]. In [15], two implementations of the MSG framework were described: the MSG-ProGAN and the MSG-StyleGAN. They share the same architectural structure (layers) for G, but differ for D layers and the loss functions used. The present study selected the MSG-ProGAN [15] as the baseline architecture and used the Wasserstein GAN with Gradient Penalty (WGAN-GP) loss function [59]. Both G and D retain the ascending and descending complexity pattern of the MSG-ProGAN layers. At the early stages of development, the architectures reported in

Table A1. Generator architecture for the MSG-GAN-SD.

Block	Layer	Activation	Output Shape
0.	Input Conv 1 × 1 Conv 2 × 2	– LReLU LReLU	2 × 4 × 1 2 × 4 × 480 2 × 4 × 480
$G_0$	Conv 1 × 1	Tanh	2 × 4 × 1
1.	Input Upsampling (2, 2) Conv 3 × 3 Conv 3 × 3	– – LReLU LReLU	2 × 4 × 480 4 × 8 × 480 4 × 8 × 480 4 × 8 × 480
$G_1$	Conv 1 × 1	Tanh	4 × 8 × 1
2.	Input Upsampling (2, 2) Conv 3 × 3 Conv 3 × 3	– – LReLU LReLU	4 × 8 × 480 8 × 16 × 480 8 × 16 × 240 8 × 16 × 240
$G_2$	Conv 1 × 1	Tanh	8 × 16 × 1
3.	Input Upsampling (2, 2) Conv 3 × 3 Conv 3 × 3	– – LReLU LReLU	8 × 16 × 240 16 × 32 × 240 16 × 32 × 120 16 × 32 × 120
$G_3$	Conv 1 × 1	Tanh	16 × 32 × 1
4.	Input Upsampling (2, 2) Conv 3 × 3 Conv 3 × 3	– – LReLU LReLU	16 × 32 × 120 32 × 64 × 120 32 × 64 × 60 32 × 64 × 60
$G_4$	Conv 1 × 1	Tanh	32 × 64 × 1
5.	Input Upsampling (3, 3) Conv 3 × 3 Conv 3 × 3	– – LReLU LReLU	32 × 64 × 60 96 × 192 × 60 96 × 192 × 20 96 × 192 × 20
$G_5$	Conv 1 × 1	Tanh	96 × 192 × 1
6.	Input Upsampling (5, 5) Conv 3 × 3 Conv 3 × 3 Conv 3 × 3	– – LReLU LReLU Tanh	96 × 192 × 20 480 × 960 × 20 480 × 960 × 4 480 × 960 × 4 480 × 960 × 1

and

Table A2. Discriminator architecture for the MSG-GAN-SD.

Block	Layer	Activation	Output Shape
0.	Input Conv 3 × 3 MiniBatchStd Conv 3 × 3 Conv 3 × 3 AvgPool (5, 5)	– LReLU – LReLU LReLU –	480 × 960 × 1 480 × 960 × 4 480 × 960 × 5 480 × 960 × 4 480 × 960 × 20 96 × 192 × 20
$Au{x}_0$	Auxiliary Image	–	96 × 192 × 1
1.	Input Concat MiniBatchStd Conv 3 × 3 Conv 3 × 3 AvgPool (3, 3)	– – – LReLU LReLU –	96 × 192 × 20 96 × 192 × 21 96 × 192 × 22 96 × 192 × 20 96 × 192 × 60 32 × 64 × 60
$Au{x}_1$	Auxiliary Image	–	32 × 64 × 1
2.	Input Concat MiniBatchStd Conv 3 × 3 Conv 3 × 3 AvgPool (2, 2)	– – – LReLU LReLU –	32 × 64 × 60 32 × 64 × 61 32 × 64 × 62 32 × 64 × 60 32 × 64 × 120 16 × 32 × 120
$Au{x}_2$	Auxiliary Image	–	16 × 32 × 1
3.	Input Concat MiniBatchStd Conv 3 × 3 Conv 3 × 3 AvgPool (2, 2)	– – – LReLU LReLU –	16 × 32 × 120 16 × 32 × 121 16 × 32 × 122 16 × 32 × 120 16 × 32 × 240 8 × 16 × 240
$Au{x}_3$	Auxiliary Image	–	8 × 16 × 1
4.	Input Concat MiniBatchStd Conv 3 × 3 Conv 3 × 3 AvgPool (2, 2)	– – – LReLU LReLU –	8 × 16 × 240 8 × 16 × 241 8 × 16 × 242 8 × 16 × 240 8 × 16 × 480 4 × 8 × 480
$Au{x}_4$	Auxiliary Image	–	4 × 8 × 1
5.	Input Concat MiniBatchStd Conv 3 × 3 Conv 3 × 3 AvgPool (2, 2)	– – – LReLU LReLU –	4 × 8 × 480 4 × 8 × 481 4 × 8 × 482 4 × 8 × 480 4 × 8 × 480 2 × 4 × 480
$Au{x}_5$	Auxiliary Image	–	2 × 4 × 1
6.	Input Concat MiniBatchStd Conv 2 × 2 Conv 2 × 4 Fully Connected	– – – LReLU LReLU Linear	2 × 4 × 480 2 × 4 × 481 2 × 4 × 482 2 × 4 × 480 1 × 1 × 480 1 × 1 × 1

of Appendix A were adopted for G and D, respectively, in order to match the 2 × 4 low-resolution and 480 × 960 high-resolution padded images. Starting from a 2 × 4 input image, obtained by progressively undersampling the original LR image (see Section 2.3), G produces upsampled images at finer scales. Specifically, at the end of each block of G, an image is generated at the associated scale (see $G_0$–$G_5$ blocks in Table A1). The generated images have the property of being intelligible (once denormalized), thus representing the downscaled temperature map at a particular resolution. When discriminating the generated (fake) samples, images produced by $G_0$–$G_5$ are to be considered as auxiliary images that are further fed to the blocks of D matching the corresponding resolutions. By comparison, when D discriminates real samples, D blocks are additionally fed with the correspondingly downsampled version of higher resolution real images. In both cases, the auxiliary images in a certain block of D are concatenated with the activation volume coming from the preceding block, by means of a simple combine function. After each convolution, a Leaky ReLU (LReLU) with $\alpha =0.2$ was used in order to keep positive values unchanged and lower the negative values. In addition, only for G, the PixelNormalization scheme [60] was used after each activated convolution, in order to make convergence faster and prevent signals escalation. A MinBatchStdDev [61,62] layer was used at the beginning of each block of D to obtain statistics about the batch formed by the previous activation volume and the auxiliary image along with the different scales, thus improving sample diversity [15]. Concerning the WGAN-GP loss, because D is a function of multi-scale input images produced by G or the corresponding ground truth HR images at different resolutions, the gradient penalty was modified to be the average of the penalties over each input. This dependence makes the multi-scale gradients able to flow between intermediate layers of both D and G [15].

2.3. Data Preprocessing

In order to match architectural needs, HR images were artificially padded through edge-padding before training and test phases, going from 431 × 947 to 480 × 960 pixels. Edge-padding consists of adding a certain number of rows and columns at the image edges, by replicating the pixel values at the edges. Subsequently, starting from the 480 × 960 resolution, different scale versions of the padded images were progressively generated by means of the bilinear interpolation undersampling technique: 96 × 192, 32 × 64, 16 × 32, 8 × 16, 4 × 8, and 2 × 4. The edge-padding was also used for LR images, going from 72 × 158 to 80 × 160 pixels. Subsequently, the bilinear interpolation technique was applied to undersample padded LR images from 80 × 160 to 2 × 4 grid points, passing through the following intermediate resolutions: 40 × 80, 20 × 40, 10 × 20. The progressive undersampling procedure allows losing as little information as possible. Additionally, each resolution was chosen to maintain the same proportionality between rows (latitude) and columns (longitude), in order to satisfy the multi-scale framework requirements. Data was partitioned into three non-overlapping subsets, namely training, validation and test sets, that correspond to 87.5% (1979–2013, 35 years), 2.5% (2014, 1 year) and 10% (2015–2018, 4 years) of the whole dataset, respectively. Both HR and 2 × 4 images were normalized in the range [−1, 1] by computing their own maximum and minimum on the training set, and then by scaling all images in the training, validation and test sets, accordingly. Furthermore, because D takes multiple images at different resolutions as input, and these images were constructed from the HR image by applying progressive downsampling, the intermediate scales were also normalized by using HR image extrema. The same procedure was also applied for normalizing 2 × 4 images to be fed to G. Once the output is produced by G, a denormalization of the HR images is required to get back to the original temperature values. Afterwards, both generated and real HR images in full resolution were processed in order to remove the edge-padding. As explained in Section 2.4, two dataset arrangements were derived: the monthly set-up, where data is organized in twelve subsets, each one referring to a particular month, across different years (e.g., January 1979, January 1980, …, January 2018); and the season-based set-up, where data is organized in four subsets, each one referring to a particular season across different years (e.g., DJF 1979, DJF 1980, …, DJF 2018). In the remainder of the manuscript, seasons are intended to be DJF (December–January–February), MAM (March–April–May), JJA (June–July–August) and SON (September–October–November), according to the climate science scientific literature.

2.4. Experimental Setup

Experiments were carried out exploiting the Marconi100 GPU cluster hosted by CINECA [63]. The HPC system is based on the IBM Power9 architecture with NVIDIA Volta GPUs. Specifically, each node hosts 2 × 16 cores IBM POWER9 AC922 at 3.1 GHz with 256 GB/node of RAM memory and 4 × NVIDIA Volta V100 GPUs per node, Nvlink 2.0, 16 GB [64]. Regarding the software adopted for our implementation, both architecture and training/test control flows were written in Python v3.8.2 based on the Keras API v2.4.3 [65] and relying on the TensorFlow v2.2.0 [66] backend. Training was performed in a distributed fashion by means of the TensorFlow Distributed Training [67] and MirroredStrategy. The results of the present study were achieved by running the model on just one node of the Marconi100 cluster exploiting all 4 GPUs available.

2.4.1. Training Set Arrangements

Because the dataset consists of 40 years of data, two dataset arrangements were derived: (i) monthly, where the dataset was organized in twelve subsets, each having data referring to a particular month across different years (e.g., January 1979, January 1980, …, January 2018); and (ii) season-based, where the dataset was organized in four subsets, each having data referring to a particular season across different years (e.g., DJF 1979, DJF 1980, …, DJF 2018). Clearly, the seasonal arrangement dataset is three times bigger than the monthly dataset because the same number of years is considered. Training the model on month- and season-based arrangements of the same dataset allows gaining valuable insights into the ability of the MSG-GAN-SD to capture intra-monthly, intra-seasonal and/or inter-annual climatic dynamics [14]. In this manner, the models’ capability of capturing daily anomalies and extreme events is ensured, as reported in [12]. Consequently, a total of 12 monthly and 4 seasonal generator models were collected after the training phase. Therefore, during the inference phase, each monthly model was able to generate a downscaled T2M map for the corresponding month, whereas each seasonal model was able to provide maps for each month belonging to the corresponding season.

2.4.2. Training Configurations

In the literature, a well-known strategy for WGAN-GP framework is to make D able to learn more quickly, leading it to be more powerful than G at a particular training epoch. This is based on the intuition that, during the training phase, if D is sufficiently accurate in the discrimination task, then its gradients flowing back and the subsequent update of G weights allow an improvement of the generation task [59,68]. This strategy can be carried out in at least three ways: (i) raising D learning rates with respect to G; (ii) making D deeper, thus increasing its number of weights and enhancing its capacity; and (iii) using an imbalanced training, particularly suited for deep GANs. In this case, for each epoch, the Discriminator weights are updated more times with respect to those used for the Generator. The number of the discriminator updates is, from now on, referred to as DtrainUpdates. In the present study, a batch size of 64 and the RMSProp optimizer [69] with learning rates of 0.0003 and 0.0001 were used for G and D, respectively. However, because of memory issues, it was not possible to deepen D. Despite this, the discriminator was trained using three different update configurations (DtrainUpdates = 1, 2, 3 respectively). Therefore, a total of 48 models (12 monthly models × 3 DtrainUpdates + 4 seasonal models × 3 DtrainUpdates) were trained for 1000 epochs, saving them every 50 epochs.

2.4.3. The Validation Framework

The experimental workflow consisted of training the proposed architecture on the four climatological seasons (i.e., DJF, MMA, JJA, SON) and all twelve months, for all DtrainUpdates configurations (i.e., 1, 2, 3). The validation framework was based on three phases: (1) extraction of the best models through the minimization of a mathematical expression aiming to select the epoch in which the architecture performed better, by looking at training and cross-validation errors; (2) an evaluation procedure, which assessed the quality of the previously selected models by computing several metrics on the test set; and (3) a final test procedure, in which insightful climatological aggregated maps were created, using the previously selected best models.

(1)

Best Models Selection

For each seasons/DtrainUpdates and months/DtrainUpdates combination, it was necessary to select a model at the training epoch in which the network was sufficiently mature. This means that it well approximated the mapping between the ground truth and the generated distributions, without incurring a lack of generalization. Therefore, this epoch should be the point where the trade-off between underfitting and overfitting is reached. At the same time, accuracy and variety (or sample diversity) capabilities also need to be considered. In order to consider these factors during the training phase, different random batches of samples, coming from both training and validation sets, were monitored by computing the batch-averaged MSE between generated and real HR images, across scales. By using these variable batches, it was possible to check both the accuracy and the variety of the ongoing trained models at the same time. Ideally, the epoch at which the training and validation sets MSEs are sufficiently close was supposed to be the best epoch. In fact, there is usually a sort of convergence of the two aforementioned errors beyond this point. However, as a consequence of the GAN’s training instabilities, although both MSEs could be close in multiple epochs, they can report high values in these epochs. Even if MSG-GAN mitigates these issues by increasing the stability during training through the use of additional multi-scale gradient information, the choice of the best epoch in which the model performed better is not a trivial task. In order to tackle this problem, the following expression was proposed:

$$e_{best}=\underset{e}{argmin}\left({\lambda }_{tr} MS{E}_{tr,e}+{\lambda }_{tr-cv} \left|MS{E}_{tr,e}-MS{E}_{cv,e}\right|\right)$$

(1)

where $MS{E}_{tr,e}$ and $MS{E}_{cv,e}$ are the MSEs computed at each epoch e on a random batch of 64 training and validation samples, respectively; ${\lambda }_{tr}$ and ${\lambda }_{tr-cv}$(both set to 1) weight the $MS{E}_{tr,e}$ and the difference term $\left|MS{E}_{tr,e}-MS{E}_{cv,e}\right|$, respectively; and $e_{best}$ represents the point where the trade-off between underfitting and overfitting is reached. Further details on Equation (1) are reported in Appendix B. Once the selection of the best-epoch model was completed, a total of 48 models for the various seasons/DtrainUpdates and months/DtrainUpdates combinations were analyzed.

(2)

Evaluation Procedure

The previously described selection procedure leads to a pool of model candidates, which are further tested by making predictions on the test set and computing evaluation metrics such as:

• Mean Squared Error (MSE):

$$MSE=\frac{1}{N} {\displaystyle \sum }_{i=1}^N{\left(T_{real,i}-T_{gen,i}\right)}^2$$

• Peak Signal-to-Noise Ratio (PSNR):

$$PSNR=10lo{g}_{10}\left(\frac{T_{gen}}{MSE\left(T_{real , }{T}_{gen}\right)}\right)\left[\mathrm{dB}\right]$$

• Log Spectral Distance (LSD):

$$LSD=\sqrt{\frac{1}{N}{\displaystyle \sum }_{i=1}^N{\left(10lo{g}_{10}\frac{T_{real,i}}{T_{gen,i}}\right)}^2}$$

• Structural Similarity Index Measure (SSIM):

$$SSIM\left(x,y\right)=\frac{\left(2{\mu }_x{\mu }_y+c_1\right)\left(2{\sigma }_{xy}+c_2\right)}{\left({\mu }_x^2+{\mu }_y^2+c_1\right)\left({\sigma }_x^2+{\sigma }_y^2+c_2\right)}$$

• Fréchet Inception Distance (FID):

$$FID=d^2\left(\left(m,C\right),\left(m_w,C_w\right)\right)=\Vert m-m_w\Vert {}_2^2+tr\left(\mathrm{C}+{\mathrm{C}}_w-2{\left({\mathrm{CC}}_w\right)}^{\frac{1}{2}}\right)$$

where:

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$T_{real}$ and $T_{gen}$ are real and generated T2M maps, respectively;

-

$x$ and $y$ refer to square windows of fixed size;

-

${\mu }_x$ and ${\sigma }_x$ are mean intensity and standard deviation of the $x$ window (similarly for $y$);

-

${\sigma }_{xy}$ is the covariance between $x$ and $y$;

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$c_1$ and $c_2$ are non-negative constants used to stabilize the division with weak denominator;

-

$d^2\left(\left(m,C\right),\left(m_w,C_w\right)\right)$ represents the Fréchet distance between the Gaussian with mean ($m,C)$ obtained from the probability of generating model data and the Gaussian ($m_w,C_w)$ obtained from the probability of observing real-world data.

Further details about SSIM and FID metrics are reported in [70,71], respectively.

Moreover, several temporal metrics are recorded, such as the total predictions’ elapsed time and mean prediction elapsed time for the sample. When testing the models on the test set, the model that reaches the best trade-off on the quality metrics for each season/month is referred to as the best model. All these evaluation metrics were used to define a novel comprehensive metric called 5-fold Accuracy Perceptivity Product ($5f_{APP}$) reported in Equation (2).

$$5f_{APP}{}_{ }:=Accuracy\times Perceptivity=\frac{\left(\frac{{\lambda }_{MSE}}{MSE}\right) \left({\lambda }_{PSNR }PSNR\right) \left({\lambda }_{SSIM }SSIM\right)}{({\lambda }_{FID }FID) \left({\lambda }_{LSD }LSD\right)}$$

(2)

All the $\lambda$ values in Expression (2) can be arbitrarily selected. In the present setup these parameters were set to 1 in order to equally weight all the metrics involved in the computation. Specifically, the higher the $\lambda$ value, the more the corresponding metric affects the $5f_{APP}$ result. More details on Equation (2) are reported in Appendix B. At this point, it is possible to claim whether seasonal or monthly training performed better. Furthermore, the best models were used to produce T2M maps that were compared with the corresponding real high-resolution maps. Pixel-wise Mean Absolute Error (MAE), and Pearson (${\rho }_{X,Y})$ and Spearman $\left(r_s\right)$ correlations between generated and real images, were also assessed and reported, according to the following formulations:

$$MAE=\frac{1}{d} {\displaystyle \sum }_{i=1}^d\left|X_i-Y_i\right|$$

$${\rho }_{X,Y}=\frac{cov\left(X,Y\right)}{{\sigma }_X{\sigma }_Y}$$

$$r_s=\frac{cov\left(R\left(X\right),R\left(Y\right)\right)}{{\sigma }_{R\left(X\right)}{\sigma }_{R\left(Y\right)}}$$

where:

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$X$ and $Y$ represent vectors of T2M values for the same pixel location in the generated and real images, respectively. The dimension of these vectors is

○

d = # daily samples x # days in a month (for the monthly arrangement)

○

d = # daily samples x # days in a season (for the seasonal arrangement);

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$cov\left(X,Y\right)$ is the covariance between $X$ and $Y$;

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${\sigma }_X, {\sigma }_Y$ are the standard deviations of $X$ and $Y$, respectively;

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$R\left(X\right), R\left(Y\right)$ are ranks of $X$ and $Y$, respectively;

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${\sigma }_{R\left(X\right)}, {\sigma }_{R\left(Y\right)}$ are the standard deviations of $R\left(X\right)$ and $R\left(Y\right)$, respectively.

Additionally, the statistical significance of the Spearman correlation was reported using the Spearman’s associated p-value (CI 95%).

(3)

Final Test Procedure

In this phase, monthly samples were temporally averaged. In order to identify the interannual trends, this was done for each month of every test set year. The monthly maps were further averaged along the test set years for obtaining interannual monthly maps, which allowed the assessment of the overall T2M averaged value in the considered period. This kind of aggregation was useful to capture large scale climate trends for different temporal scales [12]. The same procedure was also applied to correlation maps, such as Pearson, Spearman, and Spearman’s associated p-value between generated and ground truth samples.

3. Results and Discussion

This section presents the results of the training phase and evaluation procedure. For the best model selection among the various training epochs, both ${\lambda }_{tr}$ and ${\lambda }_{tr-cv}$ in Equation (1) have been set to 1. After the validation procedure, the best model was found to be the one trained on the JJA season (from now on, it will be indicated as the JJA model) with DtrainUpdates = 1.

3.1. Training Results

Figure 2 reports the MSE calculated on random training batches of the aforementioned model. Both errors exhibit an early decreasing behavior that leads to a substantial stable convergence in the last epochs. This indicates that the mapping was learnt very quickly, starting around the 50th epoch. The remaining training time was used to improve the model perception capabilities, learning complex details, geometric structures, and high-frequency details, and to enhance sample diversity. Despite this, in the last epochs, the errors did not change significantly. The simultaneous training of all MSG-GAN-SD multi-scale layers makes each epoch slower (especially for models with DtrainUpdates equal to 2 or 3), but fewer epochs were needed for reaching convergence. This means that the overall time required to train the model was reduced compared to that of traditional GANs [15].

Figure 3 reports the resulting generated images of a single fixed sample along the training epochs and resolutions. Throughout the training, all the scales synchronize with each other and mature together during the epochs in terms of accuracy and perceptivity. Concerning the training time performance, the average execution time of each experiment is reported in

Table 1. Training execution time performance.

Training Set Arrangements	Month-Based	Season-Based
DtrainUpdates = 1	~1 day	~2 days
DtrainUpdates = 2	~1 day	~3 days
DtrainUpdates = 3	~2 days	~4 days

. It must be remarked that the experiments were run under the operative conditions explained in Section 2.

3.2. Evaluation Procedure

Table A3. Best models’ properties.

Training Set Arrangements	Monthly		Seasonal
Months	# D updates	Epoch	# D updates	Epoch
January	2	950	1	850
February	1	500	1	850
March	1	850	1	1000
April	2	600	1	1000
May	3	750	3	350
June	2	850	1	800
July	1	800	1	800
August	1	850	1	800
September	2	750	3	750
October	1	950	3	750
November	3	600	3	750
December	1	950	1	850

reported in Appendix B shows the number of discriminator updates in the training phase (DtrainUpdates) and the selected epoch of the best models, for both month- and season-based arrangements. For each of the three DtrainUpdates models, the best epoch was obtained by applying Equation (1), then the best model was the one that produced the highest $5f_{APP}$ score after a test cycle. As shown in Table A3, there was no evidence of the improvement caused by training D more times than G in a single epoch.

Table A4. Evaluation results based on monthly models’ outcomes.

Monthly-Based Training	MSE (↓)	PSNR (↑)	SSIM (↑)	FID (↓)	LSD (↓)	Accuracy (↑)	Perceptivity (↑)	$5f_{\mathit{A}\mathit{P}\mathit{P}}(\uparrow )$
January	0.012	17.478	0.811	0.099	8.294	1173.585	1.213	1423.007
February	0.010	18.052	0.834	0.062	8.086	1526.165	1.981	3023.297
March	0.009	18.562	0.842	0.051	8.918	1746.389	2.208	3856.081
April	0.009	18.507	0.839	0.047	9.761	1697.090	2.180	3699.364
May	0.010	18.137	0.799	0.066	9.505	1455.684	1.585	2307.581
June	0.006	20.284	0.831	0.067	9.192	2844.603	1.631	4639.858
July	0.006	20.009	0.812	0.064	9.245	2548.737	1.697	4325.116
August	0.005	20.692	0.836	0.045	7.960	3502.819	2.822	9885.056
September	0.007	19.840	0.894	0.063	7.767	2580.708	2.053	5298.983
October	0.007	20.040	0.905	0.046	8.378	2602.556	2.573	6697.147
November	0.009	19.367	0.877	0.091	7.893	1940.007	1.396	2708.218
December	0.010	18.699	0.853	0.082	8.416	1545.117	1.457	2251.966

and

Table A5. Evaluation results based on seasonal models’ outcomes.

Season-Based Training	MSE (↓)	PSNR (↑)	SSIM (↑)	FID (↓)	LSD (↓)	Accuracy (↑)	Perceptivity (↑)	$5f_{APP}$ (↑)
January	0.011	17.584	0.817	0.079	8.760	1324.248	1.438	1904.269
February	0.009	18.414	0.843	0.061	8.388	1645.494	1.968	3239.104
March	0.008	17.586	0.797	0.041	9.871	1651.174	2.478	4091.508
April	0.007	18.837	0.873	0.042	10.120	2233.076	2.351	5249.693
May	0.007	20.051	0.924	0.053	9.689	2739.662	1.949	5340.450
June	0.005	20.450	0.829	0.046	9.170	3110.453	2.382	7407.763
July	0.005	21.084	0.873	0.037	8.699	3733.963	3.109	11,607.422
August	0.004	21.395	0.877	0.039	7.316	4217.381	3.540	14,929.312
September	0.005	21.618	0.958	0.048	7.047	3970.463	2.966	11,777.042
October	0.005	20.731	0.908	0.043	7.538	3748.529	3.085	11,565.111
November	0.007	18.326	0.825	0.066	7.740	2138.194	1.954	4178.579
December	0.009	18.677	0.852	0.072	8.973	1794.051	1.542	2767.413

in Appendix B report the outputs of the evaluation procedure on the test set, for both monthly and seasonal models’ outcomes, respectively. For each considered month, it is evident from the $5f_{APP}$ metric (higher is better) that the seasonal models perform better against the respective monthly models during the test set years (2015–2018). Evidently, still looking at the $5f_{APP}$ metric, they reach a sort of best trade-off among all the metrics involved in the Accuracy and Perceptivity terms. Thus, the higher number of samples in seasonal dataset arrangements was of greater benefit than the more specific climate dynamics exhibited by the monthly arrangements. Moreover, from a practical standpoint, it is more convenient to manage four seasonal models rather than twelve monthly ones. According to the $5f_{APP}$, it is be easily noted that the JJA model was found to be the best among the seasonal models, because the metric scores on the average are higher than those reported for the months in the other seasons (see Table A5 in Appendix B). Additionally, the JJA model reached the highest $5f_{APP}$ score in predicting the August month.

3.3. Test Results

Figure 4 shows the results generated by the JJA model in August (left panel), in addition to the ground truth (center panel) and the MAE (right panel). In all cases, the generated and real maps appear to be nearly indistinguishable. In fact, the MAE maps in Figure 4 are mainly blue or dark blue, representing a low error. However, there are recurrent regions in which the behavior is not as good with errors greater than 1 degree, as shown by critical yellow and light blue zones. Additionally, the highest errors (red hotspots) are mainly located in the North-West zone, where very low T2M values are recorded in the ground truth images. Consequently, the weight of the unavoidable error in the network forecasting phase is greater at very low real values. These errors may also depend on the use of a large amount of data during the network training phase. High temporal resolution data (6-hourly) has actually been used, which allowed for training the deep architecture introduced in the present work. By doing this, overfitting could be avoided but, at the same time, it inevitably led to fitting the noise due to day and night cycles that characterize the daily temperature trend. An alternative solution may be considering data at a coarser time resolution (e.g., daily or monthly) with an inevitable reduction in the number of samples available for the training phase. It would therefore be necessary to extend the time interval of the analysis (longer than 40 years) or look for simpler architectural solutions requiring fewer data. In addition, as a pre-processing step, each data sample may be expressed as an anomaly with respect to the daily or monthly average, in order to potentially fasten convergence. Nonetheless, the critical hotspots are mitigated by the interannual mean shown in Figure 5, at the expense of an increased error in the remaining zones. Figure 6 and Figure 7 present, respectively, the monthly and interannual monthly means of Pearson (left panel) and Spearman (center panel) correlation metrics, along with the Spearman associated p-value with a confidence interval (CI) of 95% (right panel). The latter values were computed between the ground truth samples and those generated by the JJA model in August. Looking at these maps, it can be observed that there exists a positive moderate-to-strong correlation—of both Pearson and Spearman types—between generated and real samples, mostly in land areas. This correlation is higher than in sea areas, where there still exists a negative moderate-to-weak correlation. However, there are also some infrequent sea area hotspots in which a modest correlation exists. Considering the Spearman associated p-value, there exist some border-like zones, mostly located in the Western and Northern regions, where the non-linear correlation is not statistically significant (p-value > 0.05).

The complete collection of the rest of the monthly and interannual monthly means comparisons is available at: https://drive.google.com/drive/folders/1GenkyhZHDGxfTF2K12lw2-5dxbkjFLpW?usp=sharing, (accessed on 14 November 2021). The considered architecture required about 43.942 s (on average) to predict a particular month for all test set years. Each atomic sample was processed in about 0.090 s on average. The aforementioned times were computed by averaging the execution times of all test runs. All these tests have been carried out by exploiting only one GPU on a single node of the cluster mentioned in Section 2.

4. Conclusions

This work is the first attempt to use the MSG-GAN architecture for statistical downscaling. The proposed architecture was used for downscaling 2 m temperature from the resolution of 83.25 to 13.87 km over the EURO-CORDEX domain. The season-based training set arrangement was found to overcome the monthly one. Formally, this was confirmed by means of a novel metric—the $5f_{APP}$—introduced in the present study to simultaneously take both Accuracy and Perceptivity issues into account. In fact, the quality of the images generated by MSG-GAN-SD is very high as they appear nearly indistinguishable from the ground truth samples. However, some critical hotspots were highlighted in the North-West area of the EURO-CORDEX domain, and these are worth of further investigations in order to improve the overall MSG-GAN-SD accuracy. From a computational standpoint, the training of a seasonal model on 36 years of the corresponding season required three days in the worst case (DtrainUpdates = 3), exploiting all the four GPUs of a single node. Clearly, the use of more compute nodes would have led to a significant reduction in the training time. Moreover, the inference phase for each month required less than a minute, considering the four years of the test set. The present work is not aimed at providing a comparison between MSG-GAN-SD output and other downscaling approaches, and the authors leave this kind of investigation for future work. Obviously, errors should be further reduced for this approach to be operationally adopted in climate science. Additional architectural variants shall therefore be taken into consideration as future work. Moreover, a new dataset with either a daily or a monthly temporal resolution will also be considered to avoid the noise coming from daily temperature cycles. Over all, the solution presented in this work paves the way for possible scenarios regarding the climate science context and the use of DL techniques coming from the SISR image processing domain, which offer flexible, powerful and computationally convenient solutions. The authors also intend to experiment with the introduction of additional climatic fields providing more information as inputs, in addition to the explicit embedding of the temporal dimension.