## 3 Methodology
### Network Structure of CSNGAN-ASPP
This design enables the discriminator to focus on the correlations between patches, capturing fine-grained features of the geological structure with fewer parameters [34]. The discriminator function \(f(x)\) that satisfies the Lipschitz continuity condition can ensure that the local variation in the input space is controlled and that the gradient remains relatively stable. This property helps prevent issues such as gradient explosion and vanishing, thereby enhancing the training process stability [35]. Specifically, any two points \(x\) and \(x^{\prime}\) in the input space must satisfy the condition  
\[\big{|}f(x)-f(x^{\prime})\big{|}\leq K\big{|}\big{|}x-x^{\prime}\big{|}\big{|} \tag{3}\]  
That is, the maximum value of the derivative is less than the constant \(K\), and the derivative function is bounded. The spectral normalization strategy effectively controls the spectral norm of the weight matrix, limiting the gradient variation of the discriminator function and making the model more stable [36]. The spectral norm represents the maximum singular value of the weight matrix \(W\), which is equivalent to the square root of the maximum eigenvalue of \(W^{T}W\), which can be written as  
\[\sigma(W)=\sqrt{\lambda_{1}} \tag{4}\]  
to satisfy the Lipschitz constraint; let \(\sigma(W)=1\). For any input weight matrix \(x\), the matrix multiplication can satisfy  
\[|Wx|<\sigma(W)|x| \tag{5}\]  
Therefore, it is only necessary to normalize the weight matrix during the training process. The new weight matrix \(W_{SN}=W/\sigma(W)\) not only satisfies the Lipschitz continuity constraint but also avoids sensitivity to specific directions in the hidden layers [37]. In summary, the spectral normalization strategy is implemented in the CSNGAN-ASPP discriminator. In order to further improve the ability of CSNGAN-ASPP to consider feature information during the reconstruction process, we introduce the channel attention mechanism and convolutional block attention module in the generator and discriminator, respectively [38]. The convolutional block attention module uses an end-to-end training approach, as illustrated in Figure 5.  
In particular, the channel attention mechanism, which is used in the ASPP module, compresses the spatial dimension of the feature map through pooling and fully connected layers to obtain a one-dimensional feature map tensor. After feature structure extraction by multiple parallel dilation convolutions, the channel attention mechanism can efficiently extract significant information from different channels. Under the adaptive combination of low-level and high-level features, the generator can effectively evaluate the significance  
Figure 5: The structure of the attention mechanism modules. of different feature channels in the input data, thereby the network feature extraction capability is improved. Additionally, spatial attention compresses the feature map along the channel dimension to learn dependencies between spatial features. By using maximum pooling and average pooling, it can aggregate channel information and extract the most spatially crucial features. This process enables the network to better comprehend the spatial structure of feature map across the entire channel dimension. The convolutional block attention module incorporates both channel and spatial attention mechanisms, and the relationship between the input feature \(f\) and the output result \(f^{\prime}\) can be expressed as follows:  
\[f^{\prime}=M_{S}\Big{[}M_{c}(f)\bigotimes f\Big{]}\bigotimes[M_{c}(f)\bigotimes f \Big{]} \tag{6}\]  
Among them, \(M_{S}\) and \(M_{c}\) are denoted as spatial and channel attention mechanism modules, respectively, and \(\bigotimes\) represents element-level multiplication. In summary, the overall network model effectively addresses the challenges posed by varying geological feature scales in the reconstruction task and better perceives and integrates the features of nonstationary geological bodies.