2.1 Datasets 2.1 数据集

The first phase of NLDAS forcing data (NLDAS-1) was initiated to couple atmosphere-land-ocean models to improve weather and seasonal climate predictions (Mitchell et al., 2004). The NLDAS produces Land Surface Model (LSM) forcing data, including meteorology reanalysis, gauge-based precipitation, and shortwave radiation. The second phase of the NLDAS model (NLDAS-2) applies some corrections to the gauge precipitation data and contains several enhancements to the equations and calibration of LSM (Peters-Lidard et al., 2011). The NLDAS-2 forcing data provides hourly maps at 0.125-degree spatial resolution from 1979 to the present. This study uses daily Evapotranspiration (ET), Soil Moisture (SM), Temperature (T), and Leaf Area Index (LAI) products of NLDAS-2-Noah from 2016 to 2020 as inputs to the model. These inputs are selected after performing feature selection on all NLDAS-2-Noah components. The feature selection is performed using a subset selection method with a forward stepwise approach. In this method, the goal is to find the best input data that contains the least dimension that most contributes to the model accuracy (Hastie et al., 2020). Data is downloaded from NASA's website providing LDAS datasets (https://ldas.gsfc.nasa.gov/nldas/nldas-2-model-data).
NLDAS 强迫数据（NLDAS-1）的第一阶段旨在将大气-陆地-海洋模型耦合起来，以改善天气和季节性气候预测（Mitchell 等，2004）。 NLDAS 生成陆地表面模型 (LSM) 强迫数据，包括气象再分析、基于仪表的降水和短波辐射。 NLDAS 模型的第二阶段 (NLDAS-2) 对测量降水数据进行了一些修正，并对 LSM 的方程和校准进行了一些增强（Peters-Lidard 等，2011）。 NLDAS-2 强迫数据提供了从 1979 年到现在 0.125 度空间分辨率的每小时地图。本研究使用 2016 年至 2020 年 NLDAS-2-Noah 的每日蒸散量 (ET)、土壤湿度 (SM)、温度 (T) 和叶面积指数 (LAI) 产品作为模型的输入。这些输入是在对所有 NLDAS-2-Noah 组件执行功能选择后选择的。使用具有前向逐步方法的子集选择方法来执行特征选择。在此方法中，目标是找到包含对模型精度贡献最大的最小维度的最佳输入数据（Hastie 等人，2020）。数据是从 NASA 提供 LDAS 数据集的网站下载的（https://ldas.gsfc.nasa.gov/nldas/nldas-2-model-data）。

Recently, Gavahi et al. (2022) used the DA technique to produce daily SSI maps over the CONUS, which have been shown to have more reliable drought characterization compared to their counterparts. However, the product experiences a delay of 21 days, and the method requires significant computational resources. Consequently, this data collection is not appropriate for immediate decision-making or rapid analysis of flash drought occurrences. In this research, the daily SSI maps from 2016 to 2020 serve as both the targets and inputs. The selection of 2016–2020 as the timeframe for this study is due to the availability of the SSI maps exclusively for these years. Consequently, employing a GAN would be advantageous in managing the limited size of the data.
最近，Gavahi 等人。 (2022) 使用 DA 技术生成美国大陆上空的每日 SSI 地图，事实证明，与同类地图相比，该地图具有更可靠的干旱特征。然而，该产品出现了 21 天的延迟，并且该方法需要大量的计算资源。因此，这种数据收集不适合对突发干旱事件进行立即决策或快速分析。在本研究中，2016年至2020年的每日SSI地图既作为目标又作为输入。选择 2016 年至 2020 年作为本研究的时间范围是因为仅可获取这些年份的 SSI 地图。因此，采用 GAN 将有利于管理有限的数据量。

2.2 Data Preprocessing 2.2 数据预处理

The decision to use “21” in SSI_t-21 is based on the 21-day latency observed in the target data collection, as mentioned earlier. By incorporating this lag period into the network, the resulting outputs will account for and cover the delay in the target data collection.
如前所述，在 SSI _t-21 中使用“21”的决定是基于在目标数据收集中观察到的 21 天延迟。通过将此滞后期合并到网络中，所得输出将考虑并覆盖目标数据收集中的延迟。

In total, about 7,312 daily NLDAS-2 maps and 1,828 daily SSI maps from 2016 to 2020 are collected and masked over the CONUS. The maps are randomly selected for training, validation, and testing procedures. The maps are normalized by subtracting the mean and dividing by the standard deviation. The mean and standard deviation are scalars calculated for the training dataset and used to normalize training, validation, and testing data. Zero padding, the process of adding zeros to the rows and columns of maps to preserve the spatial dimensions of the maps before applying the following operations, is applied to the maps and the final map size for both inputs and targets is 512×512.
2016 年至 2020 年期间，总共收集了约 7,312 张每日 NLDAS-2 地图和 1,828 张每日 SSI 地图，并在 CONUS 上进行屏蔽。这些地图是随机选择的用于训练、验证和测试程序。通过减去平均值并除以标准差来对图进行归一化。平均值和标准差是为训练数据集计算的标量，用于标准化训练、验证和测试数据。零填充是在应用以下操作之前向地图的行和列添加零以保留地图的空间尺寸的过程，该过程应用于地图，输入和目标的最终地图大小均为 512×512。

3.1 GAN 3.1 生成式网络

The primary objective of this study is to develop a GAN for daily flash drought monitoring using the SSI index across the CONUS in real time. GANs work based on a game between two models, one called the “generator”, and the other player is the “discriminator”. The generator model is responsible for producing synthetic data, while the discriminator model assesses the accuracy of the generated outputs. By continually evaluating the discriminator's feedback, the generator model improves its ability to generate more realistic and accurate synthetic data. In the GAN framework, the discriminator model acts as a classifier with the objective of distinguishing between the synthetic data generated by the generator and real data. The outputs produced by the discriminator serve as a component of the loss function for the generator. This adversarial training process forms a minimax game, where both models strive to minimize their cost functions. Over time, the generator becomes better at producing realistic data, while the discriminator becomes better at identifying synthetic data.
本研究的主要目标是开发一个 GAN，使用整个 CONUS 地区的 SSI 指数进行每日突发干旱监测。 GAN 的工作原理基于两个模型之间的博弈，一个称为“生成器”，另一个称为“鉴别器”。生成器模型负责生成合成数据，而鉴别器模型则评估生成输出的准确性。通过不断评估鉴别器的反馈，生成器模型提高了生成更真实、更准确的合成数据的能力。在 GAN 框架中，判别器模型充当分类器，其目的是区分生成器生成的合成数据和真实数据。鉴别器产生的输出充当生成器损失函数的组成部分。这种对抗性训练过程形成了一个极小极大游戏，两个模型都努力最小化其成本函数。随着时间的推移，生成器变得更擅长生成真实数据，而鉴别器也变得更擅长识别合成数据。

Following the success of GANs in generating realistic data distributions, researchers aimed to have a more controlled generation process (Antipov et al., 2017). Conventional GANs lacked the ability to condition the generated output on certain desired attributes. To address this, Conditional GANs (CGANs) were proposed. The main idea behind CGANs is to provide both the generator and the discriminator with additional conditional information, usually in the form of a label or some other kind of auxiliary data (Elaraby et al., 2022). Figure 1 presents a general architecture for a GAN and a CGAN.
随着 GAN 在生成真实数据分布方面取得成功，研究人员的目标是建立一个更加受控的生成过程（Antipov 等人，2017）。传统的 GAN 缺乏根据某些所需属性调节生成输出的能力。为了解决这个问题，提出了条件 GAN（CGAN）。 CGAN 背后的主要思想是为生成器和鉴别器提供额外的条件信息，通常以标签或某种其他类型的辅助数据的形式（Elaraby 等人，2022）。图 1 展示了 GAN 和 CGAN 的一般架构。

In this study, a CGAN is developed for flash drought monitoring; hereafter, it will be referred to as Drought GAN (DroGAN). The discriminator and generator architectures of DroGAN are adapted from Isola et al. (2017) and Radford et al. (2015). DroGAN employs a modified version of the U-Net model as its generator. Additionally, a Markovian discriminator is used as the discriminator component.
在本研究中，开发了 CGAN 用于突发干旱监测；以下将其称为干旱 GAN（DroGAN）。 DroGAN 的判别器和生成器架构改编自 Isola 等人。 （2017）和雷德福德等人。 （2015）。 DroGAN 采用​​ U-Net 模型的修改版本作为其生成器。另外，马尔可夫判别器被用作判别器组件。

To generate flash drought maps using DroGAN, the outputs, which are SSI maps, are then categorized according to the specifications outlined in Table 1.
为了使用 DroGAN 生成突发干旱地图，输出（即 SSI 地图）将根据表 1 中列出的规格进行分类。

3.2 Generator (U-Net) 3.2 生成器（U-Net）

In this study, a modified version of the U-Net model is employed as the generator in the network. The purpose of this modified U-Net is to process maps as inputs and generate output maps as well. The utilized U-Net model in this study contains skip connections added between the nth layer and the (N-n)^th layer, where N is the total number of layers in the U-Net (Xu et al., 2021). Figure 2 presents the architecture of a U-Net model which allows low-level (highly local features) information to shortcut across the model.
在本研究中，采用 U-Net 模型的修改版本作为网络中的生成器。修改后的 U-Net 的目的是将地图作为输入进行处理并生成输出地图。本研究中使用的 U-Net 模型包含在第 n 层和 (N-n) ^th 层之间添加的跳跃连接，其中 N 是 U-Net 中的总层数（Xu 等人， 2021）。图 2 展示了 U-Net 模型的架构，它允许低级（高度局部特征）信息在模型中快捷传输。

U-Net is a neural network model consisting of two paths. The first path, known as the analysis path or the contracting path, is similar to a convolutional neural network (CNN) layer and extracts information from the input maps. The second path which is known as the synthesis path or the expansion path consists of up-convolutions, allowing the network to learn the localized information. The skip connections allow concatenation between the maps in a step of the contracting path and a similar step of the synthesis path, helping to avoid the gradient vanishing problem (Siddique et al., 2021). The modified U-Net model consists of standardized single 2D convolution and 2D transpose-convolution blocks with batch normalization, dropout, and activation functions installed on them. Srivastava et al. (2014) showed that the dropout method considerably improves the performance of the network compared to other regularization techniques. In addition, in the modified version of the U-Net model, the input image goes down to the bottleneck until it reaches a one-by-one feature map and then, the upsampling process starts. This modification helps the model fully extract the features in the input domain. The textual summary of the generator model is tabulated in Table S1 in Supporting Information S2.
U-Net 是由两条路径组成的神经网络模型。第一条路径称为分析路径或收缩路径，类似于卷积神经网络 (CNN) 层，从输入图中提取信息。第二条路径称为合成路径或扩展路径，由上卷积组成，允许网络学习局部信息。跳跃连接允许收缩路径的步骤和合成路径的类似步骤中的映射之间的串联，有助于避免梯度消失问题（Siddique et al., 2021）。修改后的 U-Net 模型由标准化的单 2D 卷积和 2D 转置卷积块组成，并安装了批量归一化、dropout 和激活函数。斯里瓦斯塔瓦等人。 （2014）表明，与其他正则化技术相比，dropout 方法显着提高了网络的性能。此外，在U-Net模型的修改版本中，输入图像下降到瓶颈，直到达到一对一的特征图，然后开始上采样过程。这一修改有助于模型充分提取输入域中的特征。生成器模型的文本摘要列于支持信息 S2 的表 S1 中。

3.3 Markovian Discriminator (PatchGAN)
3.3 马尔可夫判别器（PatchGAN）

The Markovian discriminator (PatchGAN) is used as the discriminator model to penalize at the scale of patches. The PatchGAN operates convolutions across the images, aggregating the patch responses to produce the discriminator's final output, which is commonly referred to as adversarial loss. The PatchGAN discriminator categorizes sections of maps as "real" or "synthetic". Such a discriminator is often referred to as a Markovian discriminator, given its approach of representing the image as a Markov random field (MRF), where there is an assumption of independence for pixels distanced more than a patch's diameter from one another (Li & Wand, 2016). The PatchGAN discriminator is essentially a convolutional neural network designed to receive an image segment and produce a singular value signifying if the segment is real or synthetic. This approach places additional constraints that promote clear, high-frequency details.
使用马尔可夫判别器（PatchGAN）作为判别器模型，在补丁规模上进行惩罚。 PatchGAN 对图像进行卷积操作，聚合补丁响应以产生鉴别器的最终输出，这通常称为对抗性损失。 PatchGAN 判别器将地图部分分类为“真实”或“合成”。这种判别器通常被称为马尔可夫判别器，因为其将图像表示为马尔可夫随机场 (MRF) 的方法，其中假设距离大于块直径的像素彼此独立（Li & Wand ，2016）。 PatchGAN 判别器本质上是一个卷积神经网络，旨在接收图像片段并产生表示该片段是真实的还是合成的奇异值。这种方法施加了额外的限制，可以促进清晰的高频细节。

In the architecture of the DroGAN network, a PatchGAN is utilized as the discriminator. The discriminator contains modules in the form of Convolution-BatchNorm-LeakyReLu. All LeakyReLu functions have a slope of 0.2. A convolutional layer is applied after the last layer, followed by a sigmoid activation function to output a one-dimensional image. Figure 3 presents the general architecture of the Markovian discriminator used in DroGAN.
在 DroGAN 网络的架构中，PatchGAN 被用作鉴别器。判别器包含 Convolution-BatchNorm-LeakyReLu 形式的模块。所有 LeakyReLu 函数的斜率为 0.2。在最后一层之后应用卷积层，然后是 sigmoid 激活函数以输出一维图像。图 3 展示了 DroGAN 中使用的马尔可夫判别器的总体架构。

By incorporating PatchGAN, DroGAN is able to generate output maps with improved accuracy in terms of capturing the spatial structure of drought within specific regions. The textual summary of the discriminator model is tabulated in Table S2 in Supporting Information S2.
通过整合 PatchGAN，DroGAN 能够生成输出地图，在捕获特定区域内干旱的空间结构方面具有更高的准确性。鉴别器模型的文本摘要列于支持信息 S2 的表 S2 中。

3.4 Batch Normalization 3.4 批量归一化

One of the complications of training DL models is the dynamic distribution of input to each layer, which changes during the training phase. Modifying the parameters of the preceding layer in this process can hinder the training speed since it necessitates using a lower learning rate. To address the issue of internal covariance shift and mitigate the problems related to vanishing gradient descent, batch normalization is employed in both the generator and discriminator blocks of DroGAN (Ioffe & Szegedy, 2015). Internal covariance shift refers to the changing distribution of input to layers in deep networks during the training phase, which can lead to the vanishing gradient descent problem and hinder the training process. The vanishing gradient descent problem becomes significant in deep neural networks as adding more layers can cause the weights to diminish, resulting in computations that yield nearly constant values.
训练深度学习模型的复杂性之一是每层输入的动态分布，该分布在训练阶段会发生变化。在此过程中修改前一层的参数可能会阻碍训练速度，因为它需要使用较低的学习率。为了解决内部协方差偏移问题并减轻与消失梯度下降相关的问题，DroGAN 的生成器和判别器模块都采用了批量归一化（Ioffe & Szegedy，2015）。内部协方差偏移是指在训练阶段深层网络各层的输入分布发生变化，这可能导致梯度下降问题消失并阻碍训练过程。梯度下降消失问题在深度神经网络中变得很重要，因为添加更多层会导致权重减小，从而导致计算产生几乎恒定的值。

3.5 Activation Function 3.5 激活函数

The activation function plays a crucial role in a neural network by transforming the summed inputs of a neuron into its output. Typically, this function is nonlinear and applies mathematical equations to determine whether the neuron should be activated or not. Additionally, it introduces nonlinearity to the neuron's outputs, enabling more complex representations and behaviors in the network (Vijayakumar et al., 2021). In this study, the LeakyReLU activation function is used in the contracting path of the generator, and ReLU activation function is used in the expansion path, and at the end, the hyperbolic tangent (Tanh) function is used(Varshney & Singh, 2021).
激活函数通过将神经元的输入总和转换为其输出，在神经网络中发挥着至关重要的作用。通常，该函数是非线性的，并应用数学方程来确定是否应激活神经元。此外，它还为神经元的输出引入了非线性，从而在网络中实现更复杂的表示和行为（Vijayakumar 等人，2021）。本研究中，生成器的收缩路径使用LeakyReLU激活函数，扩张路径使用ReLU激活函数，最后使用双曲正切（Tanh）函数（Varshney & Singh，2021） 。

3.6 Loss Function 3.6 损失函数

In DroGAN, we need to define two distinct loss functions for the discriminator and the generator. The objective of the discriminator's loss function is to minimize the negative log-likelihood of correctly identifying observed and simulated SSI maps while considering the SSI observed maps as conditioning information. To achieve this, the Binary Cross Entropy (BCE) loss function is employed in the discriminator (Ho & Wookey, 2020). To identify the optimal loss function for the generator, four different models are constructed, each employing a distinct loss function. In the first model, the loss function solely consists of Mean Absolute Error (MAE). The second model only uses adversarial loss as the loss function. The third model combines the adversarial loss, as provided by the discriminator, with MAE (L1). Lastly, the fourth model incorporates a combination of the adversarial loss and Mean Square Error (MSE) (L2) in its loss function.
在 DroGAN 中，我们需要为判别器和生成器定义两个不同的损失函数。判别器损失函数的目标是最小化正确识别观察到的和模拟的 SSI 图的负对数似然，同时将 SSI 观察到的图视为条件信息。为了实现这一目标，鉴别器中采用了二元交叉熵 (BCE) 损失函数 (Ho & Wookey, 2020)。为了确定生成器的最佳损失函数，构建了四个不同的模型，每个模型都采用不同的损失函数。在第一个模型中，损失函数仅由平均绝对误差（MAE）组成。第二个模型仅使用对抗性损失作为损失函数。第三个模型将判别器提供的对抗性损失与 MAE (L1) 相结合。最后，第四个模型在其损失函数中结合了对抗性损失和均方误差（MSE）（L2）。

3.7 DroGAN 3.7 卓甘

As mentioned previously, DroGAN, same as other GANs, contains a generator and a discriminator model. The modified version of the U-Net model is employed to serve as the generator and PatchGAN is used as the discriminator of DroGAN. In each block of the generator and discriminator, the 2D convolutional layers are used to extract the information embedded in the input data to develop the network.
如前所述，DroGAN 与其他 GAN 一样，包含生成器和判别器模型。采用 U-Net 模型的修改版本作为生成器，PatchGAN 用作 DroGAN 的判别器。在生成器和鉴别器的每个块中，2D 卷积层用于提取嵌入在输入数据中的信息以开发网络。

During the network training process, an Adam solver with a learning rate of 10⁻⁴ was set, and the padding mode was configured as “reflect”, a method that adjusts the image size by creating a reflected boundary around the image, aiming to reduce edge effects (Liu et al., 2018). To mitigate the risk of overfitting, three dropout regularization techniques are implemented in the first three layers of the generator's expansion path, employing a dropout rate of 50%.
网络训练过程中，设置学习率为10 ⁻⁴ 的Adam求解器，填充模式配置为“reflect”，通过在图像周围创建反射边界来调整图像大小的方法。图像，旨在减少边缘效应（Liu et al., 2018）。为了减轻过度拟合的风险，在生成器扩展路径的前三层中实施了三种 dropout 正则化技术，采用 50% 的 dropout 率。

The U-Net architecture is similar to Figure 2, except that it contains more layers and in the expansion path, the skip connections double the number of channels. The batch size is set to 1, 16, 32, 64, and 128 to study the impacts of batch size on the runtime.
U-Net 架构与图 2 类似，不同之处在于它包含更多层，并且在扩展路径中，跳过连接使通道数量增加了一倍。批大小设置为 1、16、32、64 和 128，以研究批大小对运行时的影响。

3.8 Accelerating With GPU
3.8 使用GPU加速

DroGAN is coded in a way that takes advantage of both CPU and GPU simultaneously. The datasets are iteratively processed in parallel by multiple CPUs and then transferred to the GPU to train the network. Additionally, Compute Unified Device Architecture (CUDA) is used to optimize the calculations over GPU, which helps improve the training process. CUDA is a novel programming architecture for running computations on GPU, released by NVIDIA (Luebke (2008)). The CUDA toolkit helps to develop and deploy methods on GPU-accelerated systems. In this study, Pytorch in conjunction with CUDA is used to run DroGAN on GPU.
DroGAN 的编码方式同时利用 CPU 和 GPU。数据集由多个 CPU 并行迭代处理，然后传输到 GPU 来训练网络。此外，统一计算设备架构 (CUDA) 用于优化 GPU 上的计算，这有助于改进训练过程。 CUDA 是一种用于在 GPU 上运行计算的新颖编程架构，由 NVIDIA 发布（Luebke (2008)）。 CUDA 工具包有助于在 GPU 加速系统上开发和部署方法。在本研究中，Pytorch 结合 CUDA 用于在 GPU 上运行 DroGAN。

To run DroGAN, we used GPU: NVIDIA A100 80 GB PCIe, installed on the High-Performance Computing (HPC) system in the Center for Complex Hydrosystems Research (CCHR) at The University of Alabama.
为了运行 DroGAN，我们使用了 GPU：NVIDIA A100 80 GB PCIe，安装在阿拉巴马大学复杂水系统研究中心 (CCHR) 的高性能计算 (HPC) 系统上。

3.9 Model Comparison 3.9 模型比较

Inspired by the CNNs, the well-known U-Net, initially conceptualized by Ronneberger et al. (2015) for biomedical imagery segmentation, has shown promising results in climatic studies. These include refining weather forecasts as observed in works by Gronquist et al. (2021) and Hess and Boers (2022), downscaling application as noted in Adewoyin et al. (2021), and precipitation forecasting, as mentioned in Trebing et al. (2021). Additionally, Larraondo et al. (2019) explored multiple encoder-decoder designs and identified U-Net-based structures as optimal for predicting overall rainfall using geopotential height. Meanwhile, Weyn et al. (2020) employed a U-Net design, transforming input grid values into a cubed-sphere. Based on the success of U-Net across varied applications, we trained a U-Net with the same architecture as the generator to compare the results with DroGAN. The inputs to the U-Net model are the same as the inputs to DroGAN. These inputs are divided into training, validation, and testing datasets in the same manner as for DroGAN. The U-Net model is trained on the same GPU, and an early stopping strategy is utilized to mitigate overfitting issues.
受 CNN 的启发，著名的 U-Net 最初由 Ronneberger 等人提出概念。 （2015）用于生物医学图像分割，在气候研究中显示出了有希望的结果。其中包括根据 Gronquist 等人的作品中观察到的改进天气预报。 (2021) 以及 Hess 和 Boers (2022)，缩小应用，如 Adewoin 等人所述。 （2021）和降水预报，如 Trebing 等人所述。 （2021）。此外，Larraondo 等人。 （2019）探索了多种编码器-解码器设计，并确定基于 U-Net 的结构是使用位势高度预测总体降雨量的最佳结构。与此同时，Weyn 等人。 （2020）采用了 U-Net 设计，将输入网格值转换为立方球体。基于 U-Net 在各种应用中的成功，我们训练了与生成器具有相同架构的 U-Net，以将结果与 DroGAN 进行比较。 U-Net 模型的输入与 DroGAN 的输入相同。这些输入按照与 DroGAN 相同的方式分为训练、验证和测试数据集。 U-Net 模型在同一 GPU 上进行训练，并利用提前停止策略来缓解过度拟合问题。

As another model, a Naïve approach is implemented, taking the SSI value at time t-21 to directly represent the prediction. Hence, SSI at time t is equated to SSI at time t-21.
作为另一个模型，实现了 Naïve 方法，采用时间 t-21 时的 SSI 值来直接表示预测。因此，时间 t 处的 SSI 等于时间 t-21 处的 SSI。

3.10 Evaluation Metrics 3.10 评估指标

4.1 Loss Function 4.1 损失函数

Choosing an appropriate loss function is an essential aspect when creating a GAN model consisting of two deep learning models operating together in a competitive manner (Elaraby et al., 2022). We compare the performance of the networks with different loss functions to identify the most effective loss function. The evaluation of the models is conducted using R², employing a batch size of 64 and 600 epochs. We have set the lambda value for the composite loss functions at 100. The outcomes of this evaluation for the testing phase are presented in Figure 4.
创建由两个以竞争方式一起运行的深度学习模型组成的 GAN 模型时，选择合适的损失函数是一个重要方面（Elaraby 等人，2022）。我们比较具有不同损失函数的网络的性能，以确定最有效的损失函数。模型的评估是使用 R ² 进行的，批量大小为 64 和 600 epoch。我们将复合损失函数的 lambda 值设置为 100。测试阶段的评估结果如图 4 所示。

In the comparative analysis of models, we opted to use R² as our evaluative metric. This decision is guided by the interpretability of R² in quantifying the proportion of the variance in the dependent variable that is predictable from the independent variables.
在模型的比较分析中，我们选择使用R ² 作为我们的评价指标。该决策由 R ² 在量化可从自变量预测的因变量方差比例方面的可解释性指导。

In our study, we employ both MSE and MAE as the most widely used loss functions in training DL-based models to train DroGAN. This approach is chosen to explore and capture diverse error characteristics: MSE, known for its sensitivity to larger errors, emphasizes error variance, while MAE provides a more balanced focus on the average magnitude of errors. In this study, R² is not used as a loss function since it does not provide a directly actionable gradient for optimization in the same way as MSE or MAE. Particularly in scenarios with non-linear or complex relationships in data, R² can be less effective as a training criterion (Nie et al., 2018).
在我们的研究中，我们采用 MSE 和 MAE 作为训练基于 DL 的模型中最广泛使用的损失函数来训练 DroGAN。选择这种方法是为了探索和捕获不同的误差特征：MSE 以其对较大误差的敏感性而闻名，强调误差方差，而 MAE 则更加平衡地关注误差的平均幅度。在本研究中，R ² 不用作损失函数，因为它不像 MSE 或 MAE 那样提供直接可操作的优化梯度。特别是在数据具有非线性或复杂关系的场景中，R ² 作为训练标准可能不太有效（Nie et al., 2018）。

The results of this study (Figure 4) indicate that using the combination of adversarial loss and MAE in DroGAN leads to better performance than the combination of adversarial loss and MSE. In the combination that includes MAE, as the absolute value of an error is calculated, all the errors will be considered on one linear scale. Therefore, the loss function focuses more on how well the model is generally working. Both combined loss functions resulted in better R² than utilizing only MAE as the loss function of the generator. These results are in alignment with previous studies that reported using only one of the common loss functions for the generator model leads to blurry output images and using the adversarial loss alone (λ = 0 in Equation 1) outputs sharper images (Isola et al., 2017). Integrating MAE with adversarial loss (Figure 4c) offers enhanced performance in comparison to employing adversarial loss in isolation (Figure 4a). While the primary objective of the adversarial loss is to calculate the probability that the generated image is a real one, it falls short of guaranteeing pixel-to-pixel similarity with the desired target. Incorporating the L1 loss (MAE) ensures a pixel-level structural resemblance between the generated and target images.
这项研究的结果（图 4）表明，在 DroGAN 中使用对抗性损失和 MAE 的组合比对抗性损失和 MSE 的组合具有更好的性能。在包含 MAE 的组合中，当计算误差的绝对值时，所有误差都将在一个线性尺度上考虑。因此，损失函数更关注模型的总体运行情况。与仅使用 MAE 作为生成器的损失函数相比，两种组合损失函数产生的 R ² 更好。这些结果与之前的研究一致，即仅使用生成器模型的一种常见损失函数会导致输出图像模糊，而单独使用对抗性损失（等式 1 中的 λ = 0）会输出更清晰的图像（Isola 等人， 2017）。与单独使用对抗性损失（图 4a）相比，将 MAE 与对抗性损失集成（图 4c）可提供增强的性能。虽然对抗性损失的主要目标是计算生成的图像是真实图像的概率，但它无法保证与所需目标的像素到像素的相似性。结合 L1 损失 (MAE) 确保生成图像和目标图像之间的像素级结构相似性。

The above-mentioned loss function is designed that produces a structured loss function and penalizes the difference between the spatial dependencies of output and target maps. In DroGAN, there is no need for hand-engineering the loss function because it is performed inherently in the network as it not only learns to simulate the SSI maps but also learns the loss function in the discriminator section. Combining the L error with adversarial loss also promotes the adherence of the GAN model to the input maps; otherwise, the model only aims to synthesize the maps that look realistic, without considering the input maps. Since the loss functions provided by the discriminator model dynamically change with respect to the growth of the generator model, the performance of the generator model improves during the training process. This finding is in agreement with a study stating that training a machine learning model with a dynamic loss function leads to more accurate results (Wu et al., 2018).
上述损失函数被设计为产生结构化损失函数并惩罚输出和目标图的空间依赖性之间的差异。在 DroGAN 中，不需要手工设计损失函数，因为它是在网络中固有执行的，因为它不仅学习模拟 SSI 图，而且还学习鉴别器部分中的损失函数。将 L 误差与对抗性损失相结合也可以促进 GAN 模型对输入图的依从性；否则，该模型仅旨在合成看起来真实的地图，而不考虑输入地图。由于判别器模型提供的损失函数随着生成器模型的增长而动态变化，因此生成器模型的性能在训练过程中得到改善。这一发现与一项研究一致，该研究表明使用动态损失函数训练机器学习模型可以得到更准确的结果（Wu et al., 2018）。

4.2 Optimum Learning Rate
4.2 最佳学习率

The learning rate determines the magnitude of steps the optimizer takes while adjusting the weights of both the generator and discriminator networks throughout the training process. A higher learning rate might expedite the learning process, but there is a risk that it might lead to a sub-optimal final weight configuration (Zeiler, 2012). In this section, we delve into the effects of varying learning rates on the efficacy of DroGAN in the testing phase, as depicted in Figure 5.
学习率决定了优化器在整个训练过程中调整生成器和鉴别器网络权重时所采取的步骤的大小。较高的学习率可能会加快学习过程，但存在可能导致最终权重配置次优的风险（Zeiler，2012）。在本节中，我们将深入研究测试阶段不同学习率对 DroGAN 功效的影响，如图 5 所示。

The results (Figure 5) indicate that selecting the learning rate is crucial in training the model as it has a significant impact on the performance of the model. Upon training the model with various learning rates including 10⁻², 10⁻³, 10⁻⁴, and 10⁻⁵, the distinct patterns in convergence are observed. As shown in Figure 5, training with a high learning rate (e.g., 10⁻²) leads to erratic R² maps, suggesting that the model might be overshooting the minima in the loss landscape. In contrast, an extremely low learning rate (e.g., 10⁻⁵) results in a very slow convergence, affecting the computational efficiency. Here, we trained the model with a 10⁻⁵ learning rate for 2,000 epochs; however, the performance of the model is not as efficient as when the model is trained with a 10⁻⁴ learning rate for 600 epochs.
结果（图 5）表明，选择学习率对于训练模型至关重要，因为它对模型的性能有重大影响。使用各种学习率（包括 10 ⁻² 、 10 ⁻³ 、 10 ⁻⁴ 和 10 ⁻⁵ ）训练模型后，观察到收敛。如图 5 所示，使用高学习率（例如 10 ⁻² ）进行训练会导致不稳定的 R ² 图，这表明模型可能会超出损失情况中的最小值。相反，极低的学习率（例如 10 ⁻⁵ ）会导致收敛速度非常慢，影响计算效率。在这里，我们以 10 ⁻⁵ 学习率训练模型 2,000 个 epoch；然而，模型的性能不如以 10 ⁻⁴ 学习率训练 600 个 epoch 时的效率高。

4.3 Determining the Optimum Epoch Number
4.3 确定最佳纪元数

The inherent complexity of the dataset is the main factor in determining the optimum number of epochs, which is the number of times the learning algorithm performs on the training dataset. The overfitting problem is a significant issue in training a DL model. One of the causes for this problem is setting the wrong number of epochs in the training phase. To determine the optimum epoch number, the performance of the network is evaluated using the R² metric in different epochs from 10 to 1,000 over the validation set. Figure 6 presents the R² results for different epoch numbers in the validation phase.
数据集的固有复杂性是确定最佳时期数的主要因素，最佳时期数是学习算法在训练数据集上执行的次数。过拟合问题是训练深度学习模型的一个重要问题。造成这个问题的原因之一是在训练阶段设置了错误的epoch数。为了确定最佳历元数，使用 R ² 指标在验证集上的 10 到 1,000 个不同历元中评估网络的性能。图 6 显示了验证阶段不同历元数的 R ² 结果。

The results indicate that 600 is the optimum epoch number to train the DroGAN model. According to the results (Figure 6), starting from 400 epochs, the performance of the model becomes stable and some improvements in R² are seen up to 600. Training the model with less than 400 epochs leads to an abundant amount of noise in the simulated SSI maps. When the epoch number is increased from 600 to 1,000, it potentially results in a decline in the network's performance. This decrease in performance may indicate the presence of an overtraining problem. In most of the studies, GANs need a large epoch number (>100) to reach the best performance (Bird et al., 2022; Robic-Butez & Win, 2019; Sarp, Kuzlu, Pipattanasomporn, & Guler, 2021; Sarp, Kuzlu, Wilson, & Guler, 2021); while there are a few studies that reported lower epoch numbers to train their proposed GAN or use an already existing GAN for an application (Laloy et al., 2018; Li et al., 2020). Generally, there is no universal agreement on the optimal epoch number for training a GAN network. The number depends on the type, size, and application.
结果表明 600 是训练 DroGAN 模型的最佳历元数。根据结果​​（图 6），从 400 epoch 开始，模型的性能变得稳定，并且 R ² 的一些改进在达到 600 时可见。用少于 400 epoch 训练模型会导致模拟 SSI 图中存在大量噪声。当纪元数从 600 增加到 1,000 时，可能会导致网络性能下降。这种表现下降可能表明存在过度训练问题。在大多数研究中，GAN 需要较大的纪元数 (>100) 才能达到最佳性能 (Bird et al., 2022; Robic-Butez & Win, 2019; Sarp, Kuzlu, Pipattanasomporn, & Guler, 2021; Sarp,库兹鲁、威尔逊和古勒，2021）；虽然有一些研究报告了较低的纪元数来训练他们提出的 GAN 或使用现有的 GAN 进行应用（Laloy 等人，2018 年；Li 等人，2020 年）。一般来说，对于训练 GAN 网络的最佳历元数没有普遍的共识。数量取决于类型、尺寸和应用。

4.4 Optimum Batch Size on GPU
4.4 GPU 上的最佳批量大小

To determine the optimum batch size and compare the execution times, multiple iterations of DroGAN are conducted using different batch sizes and a maximum of 600 epochs. The batch size refers to the number of training samples processed before modifying the model's internal parameters. After each epoch in each setup, the timing data is recorded, and the cumulative results are displayed in Figure S1 in Supporting Information S1.
为了确定最佳批量大小并比较执行时间，使用不同批量大小和最多 600 个时期对 DroGAN 进行多次迭代。批量大小是指修改模型内部参数之前处理的训练样本数量。在每个设置中的每个纪元之后，都会记录计时数据，并且累积结果显示在支持信息 S1 中的图 S1 中。

The initial epoch in each configuration takes longer to run compared to subsequent epochs. The variation in timing between the first epoch and subsequent epochs can be attributed to the generation and loading of data into memory, which occurs at the beginning of the first epoch. Furthermore, during the coding process, the CUDA auto-tuner is activated to evaluate different algorithm variations in the first epoch. This selection process identifies the fastest algorithm for the specific configuration and utilizes it to train the model in the subsequent epochs. While this initialization process takes more time in the first epoch, it optimizes the training procedure in the following epochs and reduces the overall cumulative training time consumption.
与后续纪元相比，每个配置中的初始纪元需要更长的时间来运行。第一个纪元和后续纪元之间的时序变化可归因于数据的生成和加载到内存中，这发生在第一个纪元的开始。此外，在编码过程中，CUDA 自动调节器被激活以评估第一个时期的不同算法变化。此选择过程确定特定配置的最快算法，并利用它来训练后续时期的模型。虽然此初始化过程在第一个 epoch 中需要更多时间，但它优化了后续 epoch 中的训练过程并减少了总体累积训练时间消耗。

The results (Figure S1 in Supporting Information S1) indicate that increasing the batch size to a larger number (up to 64) helps to decrease the run time due to the multi-threading and matrix multiplication abilities of GPUs. The optimum batch size in this study for training DroGAN on the GPU is 64. Increasing the batch size from 64 to 128 increases the run time which is due to the finite capabilities of the GPU.
结果（支持信息 S1 中的图 S1）表明，由于 GPU 的多线程和矩阵乘法能力，将批量大小增加到更大的数量（最多 64）有助于减少运行时间。本研究中在 GPU 上训练 DroGAN 的最佳批量大小为 64。由于 GPU 的能力有限，将批量大小从 64 增加到 128 会增加运行时间。

4.5 Model Performance 4.5 模型性能

The performance of DroGAN is evaluated using the performance measures discussed in Section 14, including R², NSE, and MSE, the results of which are presented in Figure 7 for the testing phase. The results suggest that the model effectively estimates SSI in various regions of CONUS. However, the degraded model performance is seen in the upper Midwest, Mideastern, Northeastern, and Western states of CONUS (shown by gray rectangles in Figure 7a). These states, characterized by mountainous terrain and high snowfall, exhibit lower performance for DroGAN. This observation aligns with previous research findings, where several studies have also reported decreased model performance in snow-dominant and mountainous regions (e.g., Cai et al., 2014; Markstrom et al., 2016; Mazrooei & Sankarasubramanian, 2019; Wrzesien et al., 2019).
DroGAN 的性能使用第 14 节中讨论的性能测量进行评估，包括 R ² 、NSE 和 MSE，其测试阶段的结果如图 7 所示。结果表明，该模型有效地估计了美国大陆各个地区的 SSI。然而，模型性能下降的情况出现在美国本土的中西部、中东、东北部和西部各州（如图 7a 中的灰色矩形所示）。这些州的特点是山区地形和高降雪量，DroGAN 的性能较低。这一观察结果与之前的研究结果一致，其中一些研究也报告了在积雪为主的地区和山区模型性能下降（例如，Cai 等人，2014 年；Markstrom 等人，2016 年；Mazrooei 和 Sankarasubramanian，2019 年；Wrzesien 等人） .，2019）。

The diminished performance of DroGAN in mountainous and snow-dominant regions may stem from a scarcity of data representative of these areas. Given the limited geographical spread of such climates compared to others, the available datasets may not sufficiently capture the unique environmental dynamics present. In mountainous regions, the interaction between hydrological processes and snow cover bears significant importance, similar to other snow-dominant areas. Temperature plays a crucial role in influencing streamflow in these snow-rich environments. Furthermore, the interaction of temperature with elevation can lead to varying effects on drought onset, particularly due to the presence of snow. Soil characteristics are another vital aspect, influencing infiltration, water capacity, and rooting depth, which in turn, affect subsurface water flow, discharge rates, and soil moisture in mountain regions (Bennie et al., 2008; Moeslund et al., 2013; Strachan & Daly, 2017). Baseflow, shaped by region-specific natural factors, also exerts a direct and indirect influence on drought dynamics in these areas (Konapala & Mishra, 2020). The slow movement of groundwater in mountainous areas results in extended baseflow periods, subsequently impacting soil moisture. Moreover, certain climatic variables that significantly influence soil moisture and agricultural drought in other regions may manifest differently in mountainous landscapes. For instance, the limited time for rainfall storage at higher altitudes impacts the primary sources of soil moisture in these regions. There are notable disparities in NSE values across different snow-dominant regions. For example, DroGAN exhibits lower performance in Northern Michigan compared to areas like California's Sierra Nevada. This variability is likely due to differences in snow density and water content, which are shaped by regional temperature, precipitation types, and other climatic factors, thus influencing soil moisture dynamics. Consequently, the model struggles to accurately capture these dynamics in certain regions compared to others. The primary reason for this reduced efficacy may be the limitations within the training dataset, particularly in representing conditions prevalent in snow-dominated regions. Inadequate representation of data from similar climatic areas in the training set can hinder the model's ability to generalize effectively to such environments.
DroGAN 在山区和多雪地区的性能下降可能是由于缺乏代表这些地区的数据。鉴于与其他气候相比，此类气候的地理分布有限，可用的数据集可能无法充分捕捉当前独特的环境动态。在山区，水文过程与积雪之间的相互作用具有重要意义，与其他积雪为主的地区类似。在这些积雪丰富的环境中，温度对影响水流起着至关重要的作用。此外，温度与海拔的相互作用可能会对干旱的发生产生不同的影响，特别是由于降雪的存在。土壤特性是另一个重要方面，影响入渗、持水量和根系深度，进而影响山区的地下水流、排放率和土壤湿度（Bennie 等，2008；Moeslund 等，2013；斯特拉坎和戴利，2017）。由区域特定自然因素形成的基流也对这些地区的干旱动态产生直接和间接的影响（Konapala & Mishra，2020）。山区地下水运动缓慢，导致基流周期延长，从而影响土壤湿度。此外，某些显着影响其他地区土壤湿度和农业干旱的气候变量在山区景观中可能表现不同。例如，高海拔地区降雨储存时间有限会影响这些地区土壤水分的主要来源。不同降雪地区的 NSE 值存在显着差异。例如，与加利福尼亚州内华达山脉等地区相比，DroGAN 在密歇根州北部的表现较低。 这种变化可能是由于雪密度和含水量的差异造成的，而雪密度和含水量是由区域温度、降水类型和其他气候因素决定的，从而影响土壤湿度动态。因此，与其他区域相比，该模型难以准确捕捉某些区域的这些动态。效率降低的主要原因可能是训练数据集的局限性，特别是在代表雪占主导地位的地区普遍存在的条件方面。训练集中类似气候地区的数据表示不充分可能会阻碍模型有效推广到此类环境的能力。

NLDAS-2 could also contribute to the model's suboptimal performance in certain areas. Data provided by NLDAS-1 in mountainous regions tends to have a significant negative bias (Pan et al., 2003). To mitigate this bias in NLDAS-2, parameter-elevation regressions are incorporated into the Independent Slopes Model climatology, which helps reduce the negative bias. However, the precipitation data has not yet been bias-corrected, leading to biases in the evapotranspiration (ET) simulations (Xia et al., 2015). Furthermore, studies have shown that the ET product of NLDAS exhibits poor accuracy when compared to observation data in forested regions (Xia et al., 2012, 2014). The presence of biased input data in mountainous and forested areas could be one of the reasons for the lower performance of DroGAN in these regions. It is crucial to address the bias in the input data because it is likely that the model retains this biased information during the training process, as highlighted by Kim et al., in 2019.
NLDAS-2 还可能导致模型在某些领域的性能不佳。 NLDAS-1 在山区提供的数据往往存在显着的负偏差（Pan 等，2003）。为了减轻 NLDAS-2 中的这种偏差，参数海拔回归被纳入独立坡度模型气候学中，这有助于减少负偏差。然而，降水数据尚未经过偏差校正，导致蒸散量（ET）模拟存在偏差（Xia et al., 2015）。此外，研究表明，与森林地区的观测数据相比，NLDAS 的蒸散结果的准确性较差（Xia 等，2012，2014）。山区和森林地区存在有偏差的输入数据可能是 DroGAN 在这些地区表现较低的原因之一。解决输入数据中的偏差至关重要，因为正如 Kim 等人在 2019 年强调的那样，模型很可能在训练过程中保留了这些有偏差的信息。

This reduced performance could also be attributed to the use of MAE in the loss function, as it introduces blurriness to the synthesized image when the model is uncertain about the exact location of an edge.
这种性能下降也可能归因于在损失函数中使用 MAE，因为当模型不确定边缘的确切位置时，它会给合成图像带来模糊。

4.6 Comparative Performance Analysis
4.6 性能对比分析

In this section, we compare the performance of our proposed model, DroGAN, against the U-Net architecture and a Naïve model for predicting SSI. The hyperparameter tuning results for the U-Net model is provided in Figures S2–S3 in Supporting Information S1. The U-Net model is trained with the MSE loss function and learning rate 10⁻⁴. The comparison of the models is conducted using R² and the outcomes of this evaluation are presented in Figure 8 for the testing phase.
在本节中，我们将我们提出的模型 DroGAN 与 U-Net 架构和用于预测 SSI 的 Naïve 模型的性能进行比较。支持信息 S1 中的图 S2-S3 提供了 U-Net 模型的超参数调整结果。 U-Net 模型使用 MSE 损失函数和学习率 10 ⁻⁴ 进行训练。使用 R ² 进行模型比较，测试阶段的评估结果如图 8 所示。

In our comparative analysis, we observe that the DroGAN model outperformed the U-Net model in predicting SSI. While both models have performed considerably better than the Naïve model, the specific nuances of SSI prediction seem to be more amenable to the structure and mechanics of the DroGAN model. Table S3 in Supporting Information S2 presents the training phase run times for both DroGAN and U-Net using the same computational resources. Although U-Net's training is faster than DroGAN, it results in lower accuracy. Our comparative study between the DroGAN and U-Net models (Figure 8) demonstrates that while DroGAN requires a longer training duration, this investment is justified by its superior accuracy and robustness. The DroGAN model excels in capturing complex patterns in data, which results in enhanced performance metrics. Furthermore, the adversarial training mechanism of DroGAN contributes to higher accuracy and its generalization capabilities. One possible explanation for DroGAN's enhanced performance is its adaptive loss function, which adjusts according to the network's performance. This function optimizes the network to have a better performance during both training and testing phases. While U-Net is undoubtedly a powerful architecture, DroGAN, by integrating the U-Net model with a discriminator, shows to be more adept at capturing both local and global features. Furthermore, the adversarial loss encourages the network to account for spatial dependencies between neighboring regions during the modeling stage. Additionally, the adversarial training approach used by DroGAN ensures the model is optimized such that its generated maps (predictions) closely mirror observed SSI maps and increases the generalizability of the model.
在我们的比较分析中，我们观察到 DroGAN 模型在预测 SSI 方面优于 U-Net 模型。虽然这两个模型的表现都比 Naïve 模型好得多，但 SSI 预测的具体细微差别似乎更适合 DroGAN 模型的结构和机制。支持信息 S2 中的表 S3 列出了使用相同计算资源的 DroGAN 和 U-Net 的训练阶段运行时间。虽然 U-Net 的训练速度比 DroGAN 更快，但导致准确率较低。我们对 DroGAN 和 U-Net 模型的比较研究（图 8）表明，虽然 DroGAN 需要更长的训练时间，但这项投资因其卓越的准确性和鲁棒性而得到了证明。 DroGAN 模型擅长捕获数据中的复杂模式，从而提高性能指标。此外，DroGAN 的对抗性训练机制有助于提高准确性和泛化能力。 DroGAN 性能增强的一种可能解释是其自适应损失函数，该函数根据网络性能进行调整。此功能优化网络，使其在训练和测试阶段都有更好的性能。虽然 U-Net 无疑是一个强大的架构，但 DroGAN 通过将 U-Net 模型与判别器集成，显示出更擅长捕获局部和全局特征。此外，对抗性损失鼓励网络在建模阶段考虑相邻区域之间的空间依赖性。此外，DroGAN 使用的对抗性训练方法可确保模型得到优化，使其生成的地图（预测）密切反映观察到的 SSI 地图，并提高模型的通用性。

4.7 Examples of Spatial and Temporal Distribution
4.7 时空分布示例

The simulated SSI maps for random days are compared to the observed SSI maps across the CONUS in Figure 9. In addition, the outputs of the network are categorized into drought intensity maps using Table 1 and compared with the relevant drought intensity maps provided by USDM over the CONUS in Figure 10.
将随机日期的模拟 SSI 地图与图 9 中观察到的美国大陆 SSI 地图进行比较。此外，使用表 1 将网络的输出分类为干旱强度图，并与 USDM 提供的相关干旱强度图进行比较。图 10 中的 CONUS。

The results presented in Figure 9 demonstrate that the simulations align with the observed maps, indicating that the network is capable of capturing the patterns and values of SSI. Overall, the network successfully replicates the large-scale patterns observed in SSI. However, Figure 9 exemplifies that the SSI values in the upper Midwest, Midwestern, Northeastern, and Western states of CONUS exhibit somewhat erratic spatial patterns, which can explain the lower accuracy of DroGAN in these regions. On the other hand, the network effectively captures the detailed patterns of SSI spatial distribution in other areas. Nevertheless, in a few regions such as the Northeastern, Southeastern, and east-central regions, the network tends to display a white color (SSI = 0), indicating non-drought conditions. This behavior may be related to the MAE component of the loss function, which encourages an average value (in this case, 0) when the model is unsure about the appropriate SSI value for a specific pixel (Cheng et al., 2022).
图 9 中的结果表明，模拟与观察到的地图一致，表明网络能够捕获 SSI 的模式和值。总体而言，该网络成功复制了 SSI 中观察到的大规模模式。然而，图 9 举例说明了 CONUS 上部中西部、中西部、东北部和西部各州的 SSI 值表现出有些不稳定的空间模式，这可以解释 DroGAN 在这些地区的准确性较低。另一方面，该网络有效地捕获了其他区域的 SSI 空间分布的详细模式。然而，在东北部、东南部和中东部等少数地区，网络倾向于显示白色（SSI = 0），表明非干旱条件。这种行为可能与损失函数的 MAE 组件有关，当模型不确定特定像素的适当 SSI 值时，它会鼓励使用平均值（在本例中为 0）（Cheng 等人，2022）。

Investigating Figure 10, the random drought maps from the testing dataset illustrate the success of DroGAN in flash drought monitoring. In general, the network effectively captured both general drought patterns across almost all regions. Although both the USDM and DroGAN drought maps are categorized based on Table 1, there are some differences between the provided maps. For instance, according to the USDM drought map on 14 January 2020, the D0, and D1 drought conditions were observed in the northwestern, west-central, and south-central regions, and only a small area in Texas experienced drought with category D3. However, the DroGAN map indicates that the majority of these regions experienced D2 and D3 drought conditions, and some regions were in category D4. Similarly, on 25 February 2020, the DroGAN drought map displayed D0, D1, and D2 drought conditions in the northwestern areas, whereas the USDM map indicates D0 and D1 conditions in these regions. Another example can be explained for 15 December 2020, where the USDM characterized D0 and D1 drought conditions in the northeastern CONUS, while DroGAN shows a wider range of drought conditions, including D0–D4, in these areas. The primary reason for these discrepancies between the maps may be that the USDM maps consider various variables, including groundwater data, for mapping drought. In contrast, DroGAN's simulations are solely based on SSI maps, and the model generates drought maps exclusively using soil moisture information.
研究图 10，测试数据集中的随机干旱地图说明了 DroGAN 在突发干旱监测方面的成功。总的来说，该网络有效地捕捉了几乎所有地区的一般干旱模式。尽管 USDM 和 DroGAN 干旱地图都是根据表 1 进行分类的，但提供的地图之间存在一些差异。例如，根据2020年1月14日的USDM干旱地图，西北、中西部和中南部地区出现D0和D1级干旱，德克萨斯州只有小部分地区出现D3级干旱。然而，DroGAN 地图表明，这些地区大多数经历了 D2 和 D3 干旱条件，一些地区属于 D4 类。同样，2020 年 2 月 25 日，DroGAN 干旱地图显示了西北地区的 D0、D1 和 D2 干旱状况，而 USDM 地图则显示了这些地区的 D0 和 D1 状况。另一个例子可以解释 2020 年 12 月 15 日，其中 USDM 描述了美国本土东北部的 D0 和 D1 干旱条件，而 DroGAN 显示了这些地区更广泛的干旱条件，包括 D0-D4。地图之间存在这些差异的主要原因可能是 USDM 地图在绘制干旱地图时考虑了各种变量，包括地下水数据。相比之下，DroGAN 的模拟仅基于 SSI 地图，并且该模型仅使用土壤湿度信息生成干旱地图。

One of the main proposed approaches to investigate flash drought is studying soil moisture variations because they are important drivers of vegetation stress during drought situations particularly when soil moisture reaches the wilting point. Plants respond to soil moisture conditions by regulating their water consumption and balancing evaporative demand with the availability of moisture. Therefore, soil moisture is an important indicator of early vegetation drought stress. It has been shown that soil moisture rapidly decreases during the early stages of a flash drought due to increasing ET (Ford & Labosier, 2017).
研究突发干旱的主要方法之一是研究土壤湿度变化，因为它们是干旱期间植被胁迫的重要驱动因素，特别是当土壤湿度达到枯萎点时。植物通过调节耗水量并平衡蒸发需求与可用水分来应对土壤湿度条件。因此，土壤水分是早期植被干旱胁迫的重要指标。研究表明，由于蒸散量的增加，在突发干旱的早期阶段，土壤湿度迅速下降（Ford & Labosier，2017）。

Due to the escalating effects of climate change and human activities, the need for drought risk management policies has increased, necessitating the examination of spatial and temporal drought patterns in various regions (J. Li et al., 2021; P. Li et al., 2021). Moreover, it is essential to analyze the patterns of SSI time series to assess the efficacy of using DroGAN for studying flash droughts, which are characterized by rapid intensification within a short period. To address this, DroGAN was employed to estimate daily SSI maps across the CONUS in 2020. Subsequently, specific locations were randomly selected in different regions, and the corresponding time series were plotted to evaluate the model's performance. The outcomes of this analysis are depicted in Figure 11.
由于气候变化和人类活动的影响不断升级，干旱风险管理政策的需求增加，需要研究不同地区干旱的时空格局（J. Li等，2021；P. Li等，2021）。 ，2021）。此外，有必要分析 SSI 时间序列的模式，以评估使用 DroGAN 研究突发干旱的有效性，突发干旱的特点是在短时间内迅速加剧。为了解决这个问题，采用 DroGAN 来估计 2020 年美国大陆的每日 SSI 地图。随后，在不同地区随机选择特定位置，并绘制相应的时间序列来评估模型的性能。该分析的结果如图 11 所示。

The results (Figure 11) indicate that the model effectively captured the temporal patterns of SSI in various locations. Overall, the model demonstrates excellent performance, with coefficient of determination (R² ≥ 0.85) in the time series, reproducing the temporal patterns of SSI. However, there are instances of lower model performance observed in New York and Montana states, corresponding to points located in the northeastern and western regions of the CONUS. The reasons for this reduced performance in these regions were previously discussed in the preceding sections. According to Figure 11, the model proves to be reliable for studying flash droughts, as it effectively captures sudden changes in SSI temporal patterns in different locations.
结果（图 11）表明该模型有效地捕获了不同位置的 SSI 时间模式。总体而言，该模型表现出了优异的性能，时间序列中的决定系数（R ² ≥ 0.85）再现了 SSI 的时间模式。然而，在纽约州和蒙大拿州观察到模型性能较低的情况，对应于位于美国大陆东北部和西部地区的点。这些区域性能下降的原因已在前面的章节中讨论过。根据图 11，该模型被证明对于研究突发干旱是可靠的，因为它有效地捕获了不同地点 SSI 时间模式的突然变化。

4.8 Evolution of 2020 Flash Drought in the CONUS
4.8 2020 年美国大陆骤发干旱的演变

Since the early 2000s, extensive portions of the western U.S. have suffered a prolonged “megadrought”, influenced by both natural climate conditions and human activity (Williams et al., 2020). A temporary improvement occurred in 2019, but the drought worsened again in the mid-2020s due to a La Niña event, which led to reduced rainfall and extended heatwaves (Sehgal et al., 2021). To assess the effectiveness of DroGAN in generating the drought maps, we compared DroGAN-generated drought maps with that of the USDM and the flash drought detection by ESI. Previously, the ESI maps were demonstrated to provide reliable assessments of flash drought evolution (Anderson et al., 2016; Otkin et al., 2013). This section demonstrates how well DroGAN-generated drought maps perform in an actual flash drought situation.
自 2000 年代初以来，受自然气候条件和人类活动的影响，美国西部大部分地区遭受了长期的“特大干旱”（Williams 等，2020）。 2019年出现了暂时的好转，但由于拉尼娜事件，干旱在2020年代中期再次恶化，导致降雨量减少和热浪持续时间延长（Sehgal等，2021）。为了评估 DroGAN 在生成干旱地图方面的有效性，我们将 DroGAN 生成的干旱地图与 USDM 的干旱地图以及 ESI 的突发干旱检测进行了比较。此前，ESI 地图已被证明可以对突发干旱演变提供可靠的评估（Anderson 等人，2016 年；Otkin 等人，2013 年）。本部分演示了 DroGAN 生成的干旱地图在实际的突发干旱情况下的表现如何。

The temporal evolution of drought severity in the western U.S. during July–September 2020 (Figure 12) reveals a strong spatial correlation between ESI and SSI (DroGAN outputs) indices. Drought maps based on ESI and SSI methods reveal an escalation of drought conditions in August 2020 across several states including Utah, western Colorado, southern Nevada, western Texas, Arizona, and New Mexico. These indices indicate a temporary decline in drought levels by late September 2020, which can be attributed to winter rainfall in the central plains (NOAA/NIDIS, 2021). While the USDM drought maps are generally in line with other drought maps, they seem to be less effective in identifying the onset of flash droughts in August. Moreover, because USDM maps consider a variety of factors, the winter rainfall does not result in a reduction of drought levels according to these maps.
2020 年 7 月至 9 月美国西部干旱严重程度的时间演变（图 12）揭示了 ESI 和 SSI（DroGAN 输出）指数之间存在很强的空间相关性。基于 ESI 和 SSI 方法的干旱地图显示，2020 年 8 月，犹他州、科罗拉多州西部、内华达州南部、德克萨斯州西部、亚利桑那州和新墨西哥州等几个州的干旱状况有所升级。这些指数表明，到 2020 年 9 月下旬，干旱水平暂时下降，这可归因于中部平原的冬季降雨（NOAA/NIDIS，2021）。虽然 USDM 干旱地图总体上与其他干旱地图一致，但它们在识别 8 月份突发干旱的发生方面似乎不太有效。此外，由于 USDM 地图考虑了多种因素，根据这些地图，冬季降雨并不会导致干旱程度降低。

The USDM drought maps are provided based on multivariable inputs which is a robust aspect of these maps for prolonged drought monitoring. However, the validation of the drought maps provided by USDM for flash drought identification has shown that the maps can successfully determine the spatial extent of a developed flash drought event, while it contains lags in showing the rapid onset. The delay in capturing the flash drought features and intensities is related to the multivariable characteristics of the USDM drought maps (Otkin et al., 2013, 2018). While the ESI maps are suitable for flash drought monitoring, they are provided on a weekly basis. However, DroGAN is able to provide daily SSI maps for flash drought monitoring.
USDM 干旱地图是根据多变量输入提供的，这是这些地图用于长期干旱监测的强大方面。然而，对 USDM 提供的用于突发干旱识别的干旱地图的验证表明，这些地图可以成功确定已发生的突发干旱事件的空间范围，但在显示快速干旱方面存在滞后。捕捉突发干旱特征和强度的延迟与 USDM 干旱地图的多变量特征有关（Otkin 等，2013，2018）。虽然 ESI 地图适用于突发干旱监测，但它们每周提供一次。然而，DroGAN 能够提供每日 SSI 地图以进行突发干旱监测。