## 4 Numerical Examples
### Validation on Marmousi Model  
To evaluate the performance and stability of our proposed KGCAE method against the dualCAE method [21] and F-K filtering method for complex structures, this study employed a widely recognized Marmousi model. The Marmousi model used in this experiment is shown in Figure 12. Specifically, Figure 12a-c represents the P-wave velocity, S-wave velocity, and density distributions of the Marmousi model, respectively. The observation system configuration, as indicated in Figure 12a, consists of a source positioned at the surface with a 40 m offset while receivers are distributed underground at intervals of 5 m, spanning depths from 0 to 500 m. A Ricker wavelet with a dominant frequency of 30 Hz was employed as the source signal. The VSP data were generated through elastic wave forward modeling, with the specific simulation parameters detailed in Table 5. The resulting VSP forward modeling record is presented in Figure 12d, and the corresponding f-k spectra are shown in Figure 12e. The f-k spectrum clearly demonstrates the presence of distinct upgoing and downgoing wave energies in the original VSP record, with energy concentrated around the 30 Hz frequency band. Figure 11: Reconstruction loss of dualCAE and KGCAE method on the homogeneous model. Figure 12: Marmousi model and its forward modeling VSP data. (**a**) P-wave velocity; (**b**) S-wave velocity; (**c**) density; (**d**) VSP data obtained from forward modeling of the Marmousi geological model; and (**e**) frequency beam of (**d**). The spectral observation system has been marked in (**a**). Figure 13a,b illustrates the physical knowledge representation of the upgoing and downgoing waves, respectively, demonstrating the wavefield distribution extracted based on the energy flux density vectors. Figure 13c,d shows the frequency-beam spectra of (a,b). Figure 14 shows the separation results and their frequency-wavenumber spectra for three methods applied to the Marmousi model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **KGCAE** & **dualCAE** \\ \hline \(\lambda_{1}\) & 0.2 & 0 \\ \(\lambda_{2}\) & \(1\times 10^{-3}\) & \(1\times 10^{-3}\) \\ \(\lambda_{3}\) & \(1\times 10^{-4}\) & \(1\times 10^{-4}\) \\ \(\alpha\) & \((96^{\circ},102^{\circ})\) & \((96^{\circ},102^{\circ})\) \\ \(\beta\) & \((78^{\circ},81^{\circ})\) & \((78^{\circ},81^{\circ})\) \\ \(Epoch\) & \(10,000\) & \(10,000\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Simulation parameters of KGCAE and dualCAE. Figure 13: The physical knowledge representation of the Marmousi model. (**a,b**) Physical Knowledge representation of upgoing and downgoing wavefields; (**c,d**) frequency–beam spectra of (**a,d**). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Forward Modeling** \\ \hline Central Frequency of Source & 30 Hz \\ Sampling Time & 5 ms \\ Offset & 40 m \\ Distance between adjacent receivers & 5 m \\ Receivers’ distribution range & 0 to 500 m underground \\ Depth of model & 500 m \\ Width of model & 500 m \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Marmousi forward parameters. Figure 14a,d,g,j displays the separation results and their f-k spectra using the F-K filtering method. As evident from the f-k spectrum of the original VSP data in Figure 12c, severe spatial aliasing exists between upgoing and downgoing waves. Since F-K filtering separates wavefields by distinguishing opposite apparent velocity directions, noticeable aliasing artifacts remain in the separation results, as indicated by the black arrows in Figure 14d where upgoing wave components contaminate the downgoing wavefield. Moreover, similar to the comparative experiments on the homogeneous layered model, the F-K filtering results still exhibit leakage and ringing effects.