## 4 Numerical Examples
### Validation on Homogeneous Layered Velocity Model
#### 4.1.1 Ablation Experiments on the Homogeneous Layered Model  
In accordance with Equation (11), the loss function weights for the experiments in this section were configured as listed in Table 3. Each case runs for 10,000 iterations, with each epoch taking 100 ms.  
To validate the effectiveness of physical knowledge representation and gradient loss in constraining the upgoing and downgoing wavefields, three scenarios were examined:  
1. Application of reconstruction loss and physical knowledge representation loss; \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Forward Modeling** \\ \hline Central Frequency of Source & 50 Hz \\ Sampling Time & 4 ms \\ Location of launch point & 50 m below the ground \\ Offset & 12 m \\ Distance between adjacent receivers & 10 m \\ Receivers’ distribution range & Range: from 250 to 4250 m underground \\ Depth of model & 4500 m \\ Width of model & 50 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Homogeneous model forward parameters. Figure 8: Parameters of homogeneous-layer model and its beam diagram. (**a**) Generated raw VSP records. (**b**) P-wave velocity, S-wave velocity, and density parameters of the velocity model. (**c**) Beam diagram of the raw VSP records. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Case 1** & **Case 2** & **Case 3** \\ \hline \(\lambda_{1}\) & 0.2 & 0.2 & 0.2 \\ \(\lambda_{2}\) & 0 & \(1\times 10^{-4}\) & \(1\times 10^{-4}\) \\ \(\lambda_{3}\) & 0 & 0 & \(1\times 10^{-3}\) \\ \(\alpha\) & 0 & \((129^{\circ},139^{\circ})\) & \((129^{\circ},139^{\circ})\) \\ \(\beta\) & 0 & \((41^{\circ},52^{\circ})\) & \((41^{\circ},52^{\circ})\) \\ Epoch & 10,000 & \(10,000\) & \(10,000\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Loss function parameter settings for three cases. Application of reconstruction loss, Physical Knowledge Representation Loss, and Morphological Knowledge Representation Loss;
* 3. Application of reconstruction loss, Physical Knowledge Representation Loss, Morphological Knowledge Representation Loss, and Directional Difference Knowledge Representation Loss. Figure 9a-h shows the experimental results for the three cases. Specifically, (a) and (e) show the knowledge representation of the layered model; (b) and (f) illustrate the separation results from the first case; (c) and (g) present the outcomes from the second case; and (d) and (h) display the results from the third case. The first and second rows represent the separated upgoing and downgoing wavefields, respectively. Based on (a) and (e), the knowledge representation of the upgoing and downgoing wavefields was not entirely accurate. As the energy flux density vectors can only reflect the total energy information of the wavefield, the wave knowledge representation derived from their directions exhibits notable separation artifacts. Consequently, significant axial discontinuities appear at the convergence of upgoing and downgoing waves. Figure 9b,f demonstrates the network output when applying reconstruction and representation losses. Although the output should align with its knowledge representation when only representation loss is applied, it notably improved the axial discontinuity issue. This enhancement is attributed to parameter sharing in the CNNs and the multiscale invariance of the wave axial characteristics. However, the separation artifacts remained visible in the output. Figure 9c,g shows the network output under the second configuration. With the addition of gradient regularization, the separation artifacts in the wavefields were effectively reduced, although not completely eliminated, whereas the axial continuity of the wavefields was enhanced. However, due to the strong residual energy in the first arrival portion of the upgoing wave representation, some downgoing wave residuals persisted in this region. Figure 9d,h presents the network output under the third configuration.