# Shallow Subsurface Wavefield Data Interpolation Method Based on Transfer Learning
## 2 Method
### Deep Network Model for Interpolation
#### 2.1.1 Down-Sampling Block
\begin{matrix}x(2i-1,2j-1)=(x_{LL}(i,j)-x_{LH}(i,j)-x_{HL}(i,j)+x_{HH}(i,j ))/4\\ x(2i-1,2j)=(x_{LL}(i,j)-x_{LH}(i,j)+x_{HL}(i,j)-x_{HH}(i,j))/4\\ x(2i,2j-1)=(x_{LL}(i,j)+x_{LH}(i,j)-x_{HL}(i,j)-x_{HH}(i,j))/4\\ x(2i,2j)=(x_{LL}(i,j)+x_{LH}(i,j)+x_{HL}(i,j)+x_{HH}(i,j))/4\end{matrix} \right. \tag{2}\]  
2. Residual block  
When training convolutional neural network (CNN) models, increasing the number of convolutional layers is a common strategy to enhance their training capacity. This is because additional layers enable the extraction of richer and more complex features, thereby endowing the model with greater expressiveness in handling various tasks. However, as the network depth continues to increase, a challenging issue arises: the training error may cease to decrease or even potentially increase. This phenomenon is known as the vanishing or exploding gradient problem in deep networks. When gradients vanish, parameter updates become exceedingly slow or even stall, rendering the model ineffective in learning. Conversely, exploding gradients cause excessively large updates, leading to instability and convergence difficulties. To mitigate this problem, researchers have developed several tools, among which the residual network is notably effective. Studies indicate that introducing residual layers facilitates network optimization. These modules, through a special type of connectivity, allow information to flow more smoothly through  
Figure 4: Haar wavelet experimental results: (**a**) original data (\(1024\times 64\)); (**b**) the approximation (low-low) component (\(512\times 32\)); (**c**) the horizontal detail (low-high) component (\(512\times 32\)); (**d**) the vertical detail (high-low) component (\(512\times 32\)); (**e**) the diagonal detail (high-high) component (\(512\times 32\)). the network, thereby alleviating the vanishing and exploding gradient issues. Moreover, by significantly increasing depth, residual blocks can markedly enhance accuracy, as deeper networks can extract more advanced and abstract features, thereby improving adaptability to complex tasks. We replaced the ReLU function with the LeakyReLU function to address the \"neuron death\" phenomenon that occurs with ReLU on the negative half-axis, where the derivative is zero for negative inputs, hindering backpropagation. The mathematical expression for the LeakyReLU activation function is shown in Equation (3). Compared to ReLU, LeakyReLU avoids the neuron death issue, enabling more effective training of deep neural networks, and has demonstrated superior performance in experiments:  
\[LeakyReLU(x)=\left\{ \begin{aligned} & x,& x>0\\ & ax,& x\leq 0\end{aligned} \right. \tag{3}\]  
3. Multi-scale convolution block  
We introduced a multi-scale convolution block. Due to the locality of convolution operations, the DWT has a relatively limited receptive field, making it difficult to capture long-range information. The introduction of multi-scale convolutions compensates for this limitation by processing images in parallel using convolutional kernels of different sizes. This allows the model to perceive image information at multiple scales, thereby effectively expanding the overall receptive field. The specific structure is shown in Figure 5, where three parallel convolutional layers were added. After channel concatenation, a \(1\times 1\) convolutional layer was used to balance the number of channels.