## 4 Discussion  
To evaluate the performance of our developed method based on SADCGANs, we use two popular performance metrics to effectively quantify the neural network output. The first performance metric is the structural similarity index (SSIM) [38,45] which provides the perceptual difference between two samples and has proven to be highly effective for measuring the signal fidelity. The expression of SSIM can be written as  
\[SSIM=\frac{\left(2\mu_{x}\mu_{y}+C_{1}\right)\left(2\sigma_{xy}+C_{2}\right)}{ \left(\mu_{x}^{2}+\mu_{y}^{2}+C_{1}\right)\left(\mu_{x}^{2}+\mu_{y}^{2}+C_{2}\right)} \tag{9}\]  
and  
\[C_{1}=(K_{1}L)^{2},\ C_{2}=(K_{2}L)^{2} \tag{10}\]  
where \(\mu_{x}\) is the mean value of sample \(x\), \(\mu_{y}\) is the mean value of sample \(y\), \(\sigma_{xy}\) is the covariance of samples \(x\) and \(y\), \(\mu_{x}^{2}\) is the variance of sample \(x\), \(\mu_{y}^{2}\) is the variance of sample \(y\), \(C_{1}\) and \(C_{2}\) are the stability factors, \(L\) is the dynamic range of the pixel values, and \(K_{1}<1\) and \(K_{2}<1\) are small constants. The second performance metric is the coefficient of determination (\(R^{2}\)) [38] which is a statistical measure of closeness between two samples, and its expression can be described as  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{N}\left(z^{i}-\bar{z}^{i}\right)^{2}}{\sum \limits_{i=1}^{N}\left(z^{i}-\bar{z}^{i}\right)^{2}} \tag{11}\]  
where \(z^{i}\) and \(\bar{z}^{i}\) are the original samples and the generated samples using the developed SADCGANs, \(\bar{z}^{i}\) is the mean value of the original samples, and \(N\) is the number of samples. Table 1 lists the SSIM and \(R^{2}\) values for different models with different source positions mentioned in the previous section. According to Table 1, we can find that the decomposed P- and S-wave modes using the developed algorithm based on SADCGANs and the wavenumber domain decomposition algorithm are close to each other even if the source position changes, which effectively proves that our developed vector decomposition method based on SADCGANs has good application performance. Deep learning methods often require a large number of sample labels to train the neural network. For our vector decomposition method of the P- and S-wave modes, it requires a large number of decomposed vector P-wave modes as sample labels to train the network structure of SADCGANs. In our work, we use the vector decomposition algorithm in the wavenumber domain to effectively separate the decomposed vector P-wave modes from the time slices of simulated elastic seismic wavefields at different times as training sample labels. After training, the decomposed vector P-wave modes can be separated from the coupled elastic seismic wavefields which are not included in training sample \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Models**} & **Simple Model** & **Simple Model** & **Hess Model** & **Hess Model** \\  & **(2.56 km, 0.02 km)** & **(2.56 km, 1.28 km)** & **(3.5 km, 2.2 km)** & **(3.0 km, 0.2 km)** \\ \hline \(u_{xP}\) (SSIM) & 0.999 & 0.985 & 0.997 & 0.978 \\ \(u_{xP}\) (\(R^{2}\)) & 0.993 & 0.964 & 0.990 & 0.955 \\ \(u_{xP}\) (SSIM) & 0.998 & 0.981 & 0.992 & 0.976 \\ \(u_{xP}\) (\(R^{2}\)) & 0.990 & 0.963 & 0.984 & 0.953 \\ \(u_{xS}\) (SSIM) & 0.989 & 0.978 & 0.983 & 0.959 \\ \(u_{xS}\) (\(R^{2}\)) & 0.976 & 0.953 & 0.964 & 0.940 \\ \(u_{zS}\) (SSIM) & 0.983 & 0.970 & 0.976 & 0.954 \\ \(u_{zS}\) (\(R^{2}\)) & 0.965 & 0.949 & 0.952 & 0.933 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structural similarity index (SSIM) and determination coefficient (\(R^{2}\)) for different models with different source positions. labels using the trained neural network, and the decomposed vector S-wave modes can be obtained using the strategy of wavefield subtraction shown in Equation (3).