## 2 Knowledge Representation for VSP Wavefield Separation
### Physical Knowledge Representation Based on Energy Flow
From this solution, we can derive the necessary quantities for energy analysis, including the particle velocity \(\mathrm{v}=\frac{\partial u}{\partial t}\) the dilatation scalar \(\theta\), and the rotation vector \(\omega\). Lastly, based on the simulated wavefield \(X^{\star}\), we calculate the energy flux density vector, \(e\), to understand the direction and distribution of energy propagation [27; 28]. The vector \(\mathrm{e}\) is composed of P-wave and S-wave components, \(e_{p}\) and \(e_{s}\), respectively:  
\[\left\{ \begin{array}{l}e_{p}=-\theta v_{p}\\ e_{\mathrm{s}}=v_{s}\times\omega\end{array} \right., \tag{4}\]  
where \(e_{p}=\left(e_{px},e_{pz}\right)\) is the energy flux density vector of the P waves and \(e_{s}=\left(e_{sx},e_{sz}\right)\) is the energy flux density vector of the S waves. \(v_{p}\) is the P-wave particle vibration velocity and \(v_{s}\) is the S-wave particle vibration velocity. Equation (4) clarifies the interrelationship between the P- and S-waves during time propagation and elucidates the mechanism whereby they couple with the dilatation scalar and rotation vectors. The total energy flux density vector is expressed as follows:  
\[e=e_{p}+e_{s}. \tag{5}\]  
The energy flux density vector, \(e\), not only describes the direction and intensity of energy propagation but also provides a basis for distinguishing between upgoing and downgoing waves. From Equation (5), the P-wave component, \(e_{p}\), is related to the dilatation scalar and the P-wave particle vibration velocity field, \(v_{p}\), whereas the S-wave component, \(e_{s}\), is related to the rotation vector, \(\omega\). We utilize the sign of the vertical component of the P-wave energy flux vector, \(e_{pz}\), to determine the primary wave propagation direction. A positive \(e_{pz}\) generally corresponds to upward energy flow, and a negative value to downward flow. This directional information allows us to mask the seismic wavefield approximate solution, \(X^{\star}\), to extract the knowledge representation of the upward and downward wavefields:  
\[U_{rp}(t,z_{r})=\left\{ \begin{array}{ll}\widetilde{X}^{\star}(t,z_{r}),& \text{if }e_{pz}\geq 0\\ 0,&\text{otherwise}\\ \widetilde{X}^{\star}(t,z_{r}),&\text{if }e_{pz}<0\\ 0,&\text{otherwise}\end{array} \right., \tag{6}\]  
where \(U_{rp}\) represents the knowledge representation of the upward P-waves, \(D_{rp}\) represents the knowledge representation of the downward P-waves, and \(e_{pz}\) is the vertical component of the P-wave energy flux density vector, \(e_{p}\). Figure 1 shows the extraction process of the physical knowledge representation, where (a) and (e) represent the parts of the P-wave energy flux density vector, with \(e_{pz}>0\) and \(e_{pz}<0\) obtained through the above forward process, showing the distribution of the upward and downward waves. By applying this equation to the seismic wavefield approximate solution, \(X^{*}\), in (c), we can obtain the physical knowledge representation of the upgoing and downgoing waves in (d) and (e), respectively. This knowledge representation, which is derived directly from physical models, has a clear physical meaning and can reflect the direction of wavefield propagation to provide important information on physical constraints for subsequent upgoing and downgoing wave separation.