# Signal Separation Based on Knowledge Representation
## 4 Numerical Simulation
### P- and S-Wave Separation via Physical Knowledge Representation
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Model** & **MSE** & **SNR (dB)** \\ \hline Initial Model (\(V_{p}\)) & 135,977.581 & 7.632 \\ Initial Model (\(V_{s}\)) & 79,916.215 & 7.727 \\ \hline Inversion-A (\(V_{p}\)) & 78,317.453 & 10.066 \\ Inversion-A (\(V_{s}\)) & 51,775.479 & 9.612 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: MSE and SNR for initial and inversion models. Figure 8: Model-A complex model with size 100 \(\times\) 100. **(a)** P-wave velocity model. **(b)** S-wave velocity model. **(c)** Density model. As shown in the table above, the inversion model exhibits a significant reduction in MSE compared to the initial model, indicating that the inversion process effectively optimizes the velocity model, making it closer to the true geological conditions. Meanwhile, the increase in SNR suggests that the inversion model has better signal quality and improved overall reliability. Specifically, the SNR of the P-wave velocity model increases from 7.632 dB to 10.066 dB, while the S-wave velocity model improves from 7.727 dB to 9.612 dB, further demonstrating the effectiveness of the inversion process. These results confirm the validity of the physical knowledge representation. The implementation of FWI through the RNN enables the acquisition of knowledge representations for P- and S-waves. The results of this process are shown in Figure 10.  
The inversion knowledge guidance results presented in Figure 11 highlight the effectiveness of the proposed method in reconstructing observation data. Notably, the separation of P- and S-waves improved the data reconstruction when compared to cases without knowledge guidance. In the absence of knowledge guidance, the network branches lack any constraints. As a result, the separated signals exhibit significant discrepancies from the true signals, rendering effective signal separation unachievable. Furthermore, noticeable low-frequency biases are observed among the separated target signals, indicating the network's inability to effectively capture essential features within the target signals. This is a common challenge encountered in ill-posed problems, where a mixed signal comprises multiple combinations of two signals. This improvement was evident in the reduced effective energy residuals between the predicted P- and S-wave results and actual results. Figure 9: RNN implements velocity inversion and obtains estimates of P- and S-wave velocity models. (**a**) P-wave initial velocity model. (**b**) S-wave initial velocity model. (**c**) Inversion-A: P-wave inversion velocity model. (**d**) Inversion-A: S-wave inversion velocity model. When there is no knowledge guidance, that is, when the loss function only contains the reconstruction loss in Equation (18), the separation of P- and S-waves cannot be achieved, as shown in Figure 12. In the absence of knowledge guidance, spurious strong energy axes were observed in the separated results, which were attenuated with the inclusion of knowledge guidance. Moreover, the initially discontinuous trends achieved continuity under knowledge guidance. Importantly, the separation results closely approximated the observed signal, thereby ensuring the reliability of the separation target.