# Shallow Subsurface Wavefield Data Interpolation Method Based on Transfer Learning
## 2 Method
### Deep Network Model for Interpolation
#### 2.1.1 Down-Sampling Block  
As shown in Figure 3, we designed the down-sampling module with the following structure: The input first passes through a wavelet down-sampling layer, which can compress the data without loss. To avoid degradation issues caused by the obstruction of  
Figure 1: Overall schematic of transfer learning. Figure 2: Overall schematic of the depth interpolation model. forward and backward information flow during training, a residual block was added. This block consisted of \"convolution, LeakyReLU layer, convolution, LeakyReLU layer\". Finally, to enable the model to capture features at different scales, a multi-scale convolution module was incorporated after the residual block. This module included multiple parallel convolution paths with kernels of varying sizes. Below is a brief explanation of each component:  
Wavelet down-sampling layer  
Wavelet transform can effectively mitigate such information loss [26] compared to traditional down-sampling methods (such as max pooling and average pooling), which often lose critical information, like boundaries and textures, when reducing resolution. Therefore, we used the Haar wavelet transform as the down-sampling layer. By decomposing the feature map into four components (one low-frequency component and three high-frequency components), the spatial resolution was reduced to half its original size, but the information was encoded into the channel dimension without any loss. This preserved more raw information for subsequent feature learning, enabling the model to better capture representative features, thereby improving the reconstruction performance. As shown in Figure 4, in the two-dimensional case, when applying the Haar wavelet transform to an image (which can be regarded as a two-dimensional signal), four components are obtained: the approximation (low-low) component, the horizontal detail (low-high) component, the vertical detail (high-low) component, and the diagonal detail (high-high) component [26]. The transformation formulas are as follows:  
\[ \begin{cases}x_{LL}(i,j)=x(2i-1,2j-1)+x(2i-1,2j)+x(2i,2j-1)+x(2i,2j)\\ x_{LH}(i,j)=-x(2i-1,2j-1)-x(2i-1,2j)+x(2i,2j-1)+x(2i,2j)\\ x_{HL}(i,j)=-x(2i-1,2j-1)+x(2i-1,2j)-x(2i,2j-1)+x(2i,2j)\\ x_{HH}(i,j)=x(2i-1,2j-1)-x(2i-1,2j)-x(2i,2j-1)+x(2i,2j)\end{cases}  \tag{1}\]  
where \(x_{LL}(i,j)\), \(x_{LH}(i,j)\), \(x_{HL}(i,j)\), and \(x_{HH}(i,j)\) denote the pixel values at the i-th row and j-th column of \(x_{LL}\), \(x_{LH}\), \(x_{HL}\), and \(x_{HH}\), respectively. Based on the orthogonality of the wavelet transform, the inverse transform can accurately reconstruct the input image, as shown in Equation (2). Clearly, the discrete wavelet transform has adaptive spatial-frequency resolution, achieving better spatial resolution at high frequencies and better frequency resolution at low frequencies. Therefore, we introduced the Haar wavelet transform and its inverse transform to retain the feature maps of the convolutional layer, thereby enhancing the proposed seismic data reconstruction model. After performing the wavelet transform, the number of channels expands fourfold, necessitating a convolutional layer to restore the number of channels to what it was before the wavelet transform. Figure 3: Schematic diagram of the down-sampling block. After performing the wavelet transform, the number of channels expands to four times the original, and the size of the feature map is reduced to one-quarter of its original size. At this point, a convolutional layer is needed to restore the number of channels to what it was before the wavelet transform. Similarly, in the up-sampling module, following the inverse wavelet transform, the number of channels becomes one-quarter of the original, thus it is also necessary to set up a convolutional layer to restore the number of channels to their pre-transform state:  
\[\left\{