# Signal Separation Based on Knowledge Representation
## 4 Numerical Simulation
### P- and S-Wave Separation via Physical Knowledge Representation  
Marmousi-II is an updated and extended geophysical model based on the original Marmousi, designed with a fully elastic structure and high-quality synthetic seismic data [40]. It is widely used for studying complex geological structures and supporting various geophysical research applications. In this study, a subset of the Marmousi-II complex model, labeled Model-A, was used for simulation purposes. The acquisition system, P-wave velocity, S-wave velocity, and density models are shown in Figure 8. Model-A's grid dimensions are 100 \(\times\) 100, with a spatial sampling interval of 10 m. The RNN effectively implements FWI and enables  
Figure 6: Up\(-\) and downgoing wavefield separation without observed knowledge representation. (**a**) Mixed signal of observed up- and downgoing wavefield. (**b**) Mixed wavefield signal reconstructed by the autoencoder network. (**c**) Upgoing wavefield signal predicted by the U-Net network branch. (**d**) Downgoing wavefield signal predicted by the U-Net network branch. Figure 7: Up\(-\) and downgoing wavefield separation via observed knowledge representation. (**a**) Mixed signal of observed up- and downgoing wavefield. (**b**) Mixed wavefield signal reconstructed by the autoencoder network. (**c**) Upgoing wavefield signal predicted by the U-Net network branch. (**d**) Downgoing wavefield signal predicted by the U-Net network branch. efficient knowledge representation of P-waves and S-waves by incorporating elastic wave propagation and utilizing tensors to capture wavefield gradients across time steps, with this information leveraged within the backpropagation computational graph to facilitate the training of a nonlinear velocity model for nonlinear inversion. During the FWI process, the forward modeling of the elastic wave equation using an RNN network involved a 20 Hz Ricker wavelet excitation as the shot point, spaced at horizontal intervals of 10 m (totaling 10 shots, marked as the top red explosion icons in Figure 8). To obtain the knowledge representation results of P-waves and S-waves from the RNN, a single shot was placed at a horizontal distance of 0.25 km (marked as the top white explosion icon in Figure 8). The geophone array was positioned at a horizontal distance of 0.5 km (marked as vertical golden dotted lines in Figure 8). The simulation included 1280 time sampling points, with a time sampling interval of 0.5 ms, covering the time period from 0 to 0.64 s.  
The effective realization of P- and S-wave velocity inversion is facilitated by the design of a suitable RNN architecture, as documented in previous studies [32; 41; 42; 43]. In the context of Model-A, the initial P- and S-wave velocity models and inversion velocity model (Inversion-A) obtained via RNN are presented in Figure 9. The initial velocity model in Figure 9,b results from Gaussian blur applied to Model-A, resulting in the loss of fine details. Employing RNN for the velocity model estimation enables a more accurate representation of knowledge. It is apparent from the inversion velocity model (Figure 9,c) using the RNN that numerous geological structural details were recovered, and the results closely resemble that of the actual velocity model, Model-A. The velocity model obtained through FWI exhibited diminished quality on both sides, primarily because of the absence of recorded reflected wavefield signals near the boundaries in the observation data. The central segment of the estimated model parameters closely resembles the structure of Model-A, with the primary distinctions arising from variations in numerical values and the blurring of detailed information at the model interface. This paper evaluates the difference between the applied model and the true model (Model-A) using mean squared error (MSE) and signal-to-noise ratio (SNR). Table 2 presents the MSE and SNR metrics for the initial and inversion models in terms of the P-wave velocity model and the S-wave velocity model.