## 2 Methods
### Network Structure of SADCGANs  
In addition to the input and output channels, the network structure of DCGANs has two different models (Figure 1): a generator (G) and a discriminator (D). The generator is mainly used to generate the approximately real samples to deceive the discriminator as much as possible, and the discriminator is mainly used to accurately classify whether the input samples come from the real samples or the generated samples [41]. In other words, there is a competitive relationship between the generator and the discriminator, and their competition until the latter cannot distinguish between the generated samples and the corresponding real samples [42]. In our work, the generator is used to generate the horizontal and vertical components of the decoupled vector P-wave mode \((u_{xP},u_{zP})\) by inputting the coupled elastic seismic wavefields \((u_{x},u_{z})\), and the discriminator is used to determine whether two generated vector P-wave modes \((u_{xP},u_{zP})\) are real. Due to the limitation of the size of the receptive field of a fixed-size convolution kernel, the convolution kernel, which can be regarded as a small patch, usually only acquires the local features of input samples; in other words, the global features of input samples will be lost for the conventional DCGAN-based methods. To effectively solve this problem, self-attention [43] is incorporated into the conventional DCGANs in our work and therefore our developed method can be called a self-attention deep convolutional generative adversarial network (SADCGAN). By introducing the self-attention modules into the generator and the discriminator, SADCGANs can simultaneously acquire the local and global features of input samples, which can greatly improve the training process of the neural network and further generate more accurate wavefield decomposition results. The working principle of the self-attention module is shown in Figure 2. Firstly, the feature maps \(x\) are transformed into the feature spaces \(f(x)\), \(g(x)\) and \(h(x)\) using three \(1\times 1\) convolutions, and they can be written as  
\[f(x)=W_{f}x,g(x)=W_{g}x,h(x)=W_{h}x \tag{4}\]  
where \(W_{f}\), \(W_{g}\) and \(W_{h}\) are the weight matrix of the convolutional layer. Figure 2: The working principle of self-attention module. Secondly, the attention feature map \(\beta_{j,i}\) can be obtained by multiplying the transpose of \(f(x)\) by \(g(x)\) and then normalizing the corresponding result using the softmax function, which can be expressed as  
\[\beta_{j,i}=\frac{\exp\big{(}s_{ij}\big{)}}{\sum\limits_{i=1}^{N}\exp\big{(}s_{ ij}\big{)}},s_{ij}=f(x_{i})^{T}g\big{(}x_{j}\big{)} \tag{5}\]  
Thirdly, the global feature map \(o_{j}\) can be obtained by multiplying the attention feature map \(\beta_{j,i}\) by \(h(x)\) and then dealing with the corresponding result using a \(1\times 1\) convolution, which can be expressed as  
\[o_{j}=v\left(\sum\limits_{i=1}^{N}\beta_{j,j}h(x_{i})\right),v(x_{i})=W_{v}x_ {i} \tag{6}\]  
where \(W_{v}\) is also the weight matrix of the convolutional layer. Finally, the final output \(y_{i}\) of the self-attention module can be obtained, and the corresponding expression can be described as  
\[y_{i}=\gamma o_{i}+x_{i} \tag{7}\]  
where \(\gamma\) is a transition parameter with an initial value of 0, and it can effectively achieve the feature acquisition of local and global space positions. The network structures of the generator and discriminator for SADCGANs, which are used in our work, are shown in Figure 3. According to Figure 3, the generator has 24 transposed convolutional layers to generate two samples of the decoupled vector P-wave mode (\(u_{xP},u_{zP}\)) from two samples of coupled elastic seismic wavefields (\(u_{x},u_{z}\)), and uses the ReLU activation function and the spectral normalization for all layers except its output layer which uses the Tanh activation function to effectively improve the training dynamics of its neural network.