## 4 Numerical Examples
### Validation on Homogeneous Layered Velocity Model
#### 4.1.2 Comparative Experiments on the Homogeneous Layered Model  
To further investigate the effectiveness of the proposed KGCAE methodology, we conducted wavefield separation on the aforementioned simple-layered model and VSP data, comparing it with the conventional f-k filtering method and the morphological feature-guided (dualCAE) approach [21]. Table 4 lists the hyperparameter configurations for KGCAE and dualCAE methods. The proposed KGCAE method runs for 10,000 iterations, with each epoch consuming 100 ms. For the f-k filtering, a rectangular window was applied in the f-k domain. The separation was performed based on the sign of the apparent velocity associated with the wave energy: energy corresponding to negative apparent velocities was attributed to the upgoing wavefield, while energy with positive apparent velocities was assigned to the downgoing wavefield. Figure 10 presents a comparison of the separation results from the three methods. The f-k filtering results are shown in (a) and (d), (b) and (e) display the separation results using the dualCAE method, (c) and (f) show the results obtained using the proposed KGCAE methodology. The corresponding frequency-beam diagrams are illustrated in (g-l). From a comprehensive perspective, three methodologies successfully achieved distinct wavefield separation, attaining axial continuity and zero-mean characteristics in both the upward and downward waves. However, as observed in Figure 8c, spatial aliasing exists in the raw VSP data. When the energy distributions of upgoing and downgoing waves overlap near zero apparent velocity, energy leakage occurs in the separation results (Figure 10a,d). Additionally, ringing artifacts may emerge during the conversion from the frequency-wavenumber (f-k) domain to the time-depth domain, as highlighted within the dashed box in Figure 10a. In contrast, the proposed KGCAE method demonstrated superior performance in terms of energy intensity and separation clarity. A notable advantage of our methodology lies in its ability to process raw VSP data directly without domain transformation. It automatically allocates frequency-domain aliasing to its respective upgoing and downgoing components without requiring explicit frequency-domain filtering, thereby effectively mitigating spectral aliasing challenges. The energy is correctly partitioned into upgoing and downgoing wavefields, reducing leakage and yielding clearer f-k spectra in the separated wavefields. Compared to conventional f-k filtering and dualCAE methods, this approach achieves higher fidelity. Comparing the convergence processes of KGCAE and dualCAE algorithms, we focused on the reconstruction loss evolution rather than the total loss, because the loss function compositions and weights differ. As illustrated in Figure 11, the blue curve (dualCAE) exhibits significant fluctuations during convergence, whereas the red curve (KGCAE) demonstrates notably smaller oscillations. The reconstruction loss convergence curves indicate that the proposed method surpasses dualCAE in terms of convergence speed and stability. The f-k method, being a direct filtering process based on the described criterion, does not have an analogous iterative convergence behavior. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **KGCAE** & **dualCAE** \\ \hline \(\lambda_{1}\) & 0.2 & 0 \\ \(\lambda_{2}\) & \(1\times 10^{-3}\) & \(1\times 10^{-3}\) \\ \(\lambda_{3}\) & \(1\times 10^{-4}\) & \(1\times 10^{-4}\) \\ \(\alpha\) & \((129^{\circ},139^{\circ})\) & \((129^{\circ},139^{\circ})\) \\ \(\beta\) & \((41^{\circ},52^{\circ})\) & \((41^{\circ},52^{\circ})\) \\ Epoch & \(10,000\) & \(10,000\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulation parameters of KGCAE and dualCAE. Figure 10: Comparison of wavefield-separation results on homogeneous-layer VSP data using three methods. (**a**,**d**) Predicted upgoing and downgoing wavefields using the F-K filtering method. (**b**,**e**) Predicted upgoing and downgoing wavefields using the dualCAE method.