## 2 Knowledge Representation for VSP Wavefield Separation
### Physical Knowledge Representation Based on Energy Flow  
The core principle for physically distinguishing upgoing and downgoing waves lies in their direction of energy propagation. This study leverages the energy flux density vector, commonly known as the Poynting vector \(\mathbf{e}\), to capture this directional information [23; 24; 25]. The extraction process involves three key steps: first, obtaining an approximate subsurface velocity model through Full Waveform Inversion (FWI); second, performing elastic wave forward modeling using this model; and third, calculating the Poynting vector from the simulated wavefield to derive the knowledge representations. To obtain accurate geological model parameters, we adopted a Physics-Informed Recurrent Neural Network (PIRNN)-based full-waveform-inversion method [26]. The geological model parameters mainly comprise the P- and S-wave velocities (\(c_{p},c_{s}\)) and density \(\rho\), which are key parameters describing the elastic properties of underground media. We can establish the mapping relationship between the geological model parameters and the resulting VSP wavefield. Specifically, we assume that the P- and S-wave velocities in the geological model are represented as \(c_{p}\in R^{N\times M}\) and \(c_{s}\in R^{N\times M}\) respectively, where N and M represent the number of vertical and horizontal sampling points. The forward modeling process can be expressed abstractly as  
\[\widetilde{X}=F\big{(}c_{p},c_{s}\big{)}, \tag{1}\]  
where \(\widetilde{X}\in R^{T\times N}\) represents the VSP wavefield obtained through forward modeling, \(T\) is the number of time sampling points, and \(F(\cdot)\) represents the forward process, which is specifically implemented using the PIRNN method. To invert the geological model parameters, \(c_{p}\) and \(c_{s}\), using the observed VSP wavefield, \(X\), as the label, the inversion problem of the geological model parameters can be converted into a constrained optimization problem:  
\[ \begin{array}{l}\min_{c_{p},c_{s}}||X-\widetilde{X}||\\ \text{subject to }\widetilde{X}=F(c_{p},c_{s})\end{array} , \tag{2}\]  
where \(\widetilde{X}\in R^{T\times N}\) is the observed VSP wavefield. By solving the above optimization problem, we can obtain the optimal solution of the geological model parameters, \(c_{p}^{*}\)_and_\(c_{s}^{*}\), and obtain the approximate solution of the seismic wavefield, \(\widetilde{X}^{*}\), through forward modeling. To achieve VSP wavefield simulation, the PIRNN incorporates the physical constraints of the wave equation into the neural network using the RNN's time-stepping mechanism for forward modeling. Through the backpropagation algorithm, the PIRNN can invert the geological model parameters from the observed VSP wavefield, thereby providing a foundation for the subsequent upgoing- and downgoing wave knowledge representation. After obtaining \(c_{p}^{\star}\) and \(c_{s}^{\star}\), the approximate solution of the seismic wavefield, \(\widetilde{X}^{\star}\), is obtained through a forward modeling process. During this process, the energy flux density vector [27; 28] is introduced to describe the propagation direction and distribution of energy within the wavefield. We simulated the wavefield propagation in the two-dimensional depth-offset space. The behavior of the displacement field \(u=(u_{x},u_{z})\) can be effectively described by decomposing it using the dilatation scalar \(\theta=\
abla\cdot\mathrm{u}\) and the rotation vector \(\omega=\
abla\times u\). The elastic wave equation, expressed in terms of these P-wave and S-wave related potentials, is given by [28; 29]  
\[\frac{\partial^{2}\mathrm{u}}{\partial\mathrm{t}^{2}}={c_{p}^{\star}}^{2} \
abla\theta-{c_{s}^{\star}}^{2}\
abla\times\omega, \tag{3}\]  
where the estimated velocities \(c_{p}^{\star}\) and \(c_{s}^{\star}\) are used. Solving Equation (3) numerically yields an approximate full wavefield solution \(X^{\star}\).