## 2 Knowledge Representation for VSP Wavefield Separation
### Morphological Knowledge Representation  
In near-VSP data, odd-order-reflected waves (downward waves) typically propagate downward, whereas even-order-reflected waves (upward waves) typically propagate upward. This difference in the direction of propagation confers distinct apparent velocity and slope characteristics for the upward and downward waves in the depth-time domain. Specifically, the upgoing wavefield forms a negative-slope inverse pattern in the depth-time domain, whereas the downgoing wavefield forms a positive-slope direct pattern. As shown in Figure 2, the near-offset VSP record of a simple layered model reveals that upgoing and downgoing waves have a set of axial dip angles with the horizontal interface, which is denoted as \(\alpha=\{\alpha_{i}|1\leq i\leq p,\alpha_{i}\in R\}\) and \(\beta=\{\beta_{i}|1\leq i\leq q,\beta_{i}\in R\}\). To utilize these morphological features to constrain the upgoing and downgoing wavefields, we introduced a knowledge representation based on morphological priors. This representa  
Figure 1: Physical knowledge representation extraction process. (**a**) Energy flux density vector with \(e_{pz}>0\). (**b**) Energy flux density vector with \(e_{pz}<0\). (**c**) Approximate solution, \(X^{*}\), of the seismic wavefield. (**d**) Upgoing wave knowledge representation, \(U_{rp}\). (**e**) Downgoing wave physical knowledge representation, \(D_{rp}\). tion aims to emphasize the axial continuity and slope differences between the upward and downward wavefields, thereby improving the accuracy of wavefield separation. To extract this knowledge representation, we used a modified Sobel operator to calculate the gradients in the axial direction. The Sobel operator is a commonly used discrete differential operator for calculating the approximate gradients of images and signals. It detects edges and directional changes in images using convolution operations. Figure 3a,b shows the basic Sobel operators, which represent the horizontal and vertical convolution kernels, respectively. To accommodate the slope characteristics of the upward and downward wavefields, the Sobel operator must be modified to capture the gradient information in different directions. Specifically, we rotated the Sobel operator as follows:  
\[\
abla_{\alpha_{1}}=\cos\alpha_{1}*A+\sin\alpha_{1}*B, \tag{7}\]  
where \(\alpha_{1}\) represents the rotation angle, \(A\) and \(B\) represent the horizontal and vertical Sobel operators shown in Figure 3a,b, and \(\
abla_{\alpha_{1}}\) represents the rotated gradient operator. Taking \(\alpha_{1}=45^{\circ}\) as an example, the rotated Sobel operator is shown in Figure 3c, which can effectively capture the axial gradient information of the downgoing waves that form a \(45^{\circ}\) angle with the horizontal plane. Figure 3: Basic Sobel operators and rotated Sobel operator. (**a**) Horizontal Sobel operator. (**b**) Vertical Sobel operator. (**c**) Sobel operator rotated by \(45^{\circ}\). Figure 2: Near-offset VSP record of a simple layered model. \(\alpha_{1}\) and \(\alpha_{2}\) represent the axial dip angles between upgoing waves and the horizontal interface, whereas \(\beta_{1}\) and \(\beta_{2}\) represent the axial dip angles between downgoing waves and the horizontal interface. [PERSON] et al. [21] proposed a gradient knowledge representation based on morphological priors. Let \(\widetilde{U}\) be the network's prediction for upgoing waves and \(\widetilde{D}\) be the prediction for downgoing waves.