# Signal Separation Based on Knowledge Representation
## 2 Theory
### Knowledge Representation
#### 2.2.2 Physical Knowledge Representation  
Taking the separation of P-waves and S-waves in the field of geoscience as an example, according to the direction of wave propagation, seismic waves can be classified into P-waves and S-waves. The observed signal can thus be represented as the sum of P-waves and S-waves, expressed as follows:  
\[\mathbf{X}=\mathbf{S}_{p}+\mathbf{S}_{s} \tag{5}\]  
In the task of separating P-waves and S-waves, it is not feasible to directly represent the separation targets based on human cognition from images. In isotropic media, the P-wave manifests as a curl-free wavefield and the S-wave manifests as a divergence-free wavefield. According to the Helmholtz theory [38], [PERSON] et al. [39] formulated first-order velocity-stress equations for the equivalent separation of P- and S-waves in elastic waves:  
\[\left\{ \begin{array}{c}v_{x}=v_{xp}+v_{xs}\\ v_{z}=v_{zp}+v_{z{\rm s}}\\ \frac{\partial v_{rp}}{\partial t}=\frac{V_{p}^{2}}{2\lambda+2\mu^{2}}\Big{(} \frac{\partial v_{xx}}{\partial x}+\frac{\partial v_{z{\rm s}}}{\partial x} \Big{)}\\ \frac{\partial v_{zp}}{\partial t}=\frac{V_{p}^{2}}{2\lambda+2\mu}\Big{(} \frac{\partial v_{xx}}{\partial x}+\frac{\partial v_{zz}}{\partial x}\Big{)} \\ \frac{\partial v_{z{\rm s}}}{\partial t}=V_{s}^{2}\Big{[}\frac{1}{\mu}\frac{ \partial v_{zz}}{\partial z}-\frac{1}{2\lambda\mu+2\mu^{2}}\Big{(}\big{(} \lambda+2\mu\big{)}\frac{\partial v_{zz}}{\partial x}-\lambda\frac{\partial v_ {xx}}{\partial x}\Big{)}\Big{]}\\ \frac{\partial v_{zz}}{\partial t}=V_{s}^{2}\Big{[}\frac{1}{\mu}\frac{ \partial v_{zz}}{\partial x}-\frac{1}{2\lambda\mu+2\mu^{2}}\Big{(}\big{(} \lambda+2\mu\big{)}\frac{\partial v_{zz}}{\partial x}-\lambda\frac{\partial v_ {xx}}{\partial x}\Big{)}\Big{]}\\ \frac{\partial v_{xx}}{\partial t}=(\lambda+2\mu)\frac{\partial v_{zz}}{ \partial z}+\lambda\frac{\partial v_{zz}}{\partial z}\\ \frac{\partial v_{zz}}{\partial t}=\mu\Big{(}\frac{\partial v_{zz}}{\partial z }+\frac{\partial v_{zz}}{\partial x}\Big{)}\\ V_{p}^{2}=\frac{(\mu+2\lambda)}{\rho}V_{s}^{2}=\frac{\mu}{\rho}\end{array} \right. \tag{6}\]  
Here, \(v_{x}\) and \(v_{z}\) are the particle velocities in the x direction and z direction, respectively. \(v_{xp}\) and \(v_{z{\rm s}}\) are the P-wave velocity component and S-wave velocity component in the x direction, respectively. \(v_{zp}\) and \(v_{z{\rm s}}\) are the P-wave velocity component and S-wave velocity component in the z direction, respectively. \(\tau_{xx}\) and \(\tau_{zz}\) are the normal stresses in the x direction and z direction, respectively. \(\tau_{xx}\) is the shear stress; \(\lambda\) and \(\mu\) are Lame coefficients; \(\rho\) is the density; \(V_{p}\) and \(V_{s}\) are the P- and S-wave velocity models, respectively. \(v_{p}=v_{xp}+v_{zp}\) and \(v_{s}=v_{xs}+v_{z{\rm s}}\) represent the nondivergent pure P-wave and noncurl pure S-wave, respectively. At a given time \(t\), let us assume the current state of each variable in the elastic wave equation is denoted by \(h_{t}\), including particle velocity, velocity model, density model, as well as normal and shear stresses. \[h_{t}=[v_{x}^{t},v_{xp}^{t},v_{z{\rm s}}^{t},v_{zp}^{t},v_{z{\rm s}}^{t},\tau_ {xx}^{t},\tau_{zz}^{t},\tau_{xz}^{t},V_{p},V_{s},\rho]^{\mathsf{T}} \tag{7}\]  
Once the partial differential equations governing the behaviors of P-waves and S-waves are derived, certain coefficients or terms in these equations are typically unknown. Existing inversion techniques can be applied to estimate these coefficients and terms. For instance, full waveform inversion (FWI) enables the accurate estimation of P-wave velocity, S-wave velocity, and density models within the elastic wave equation. When accurate P- and S-wave velocity models, along with a density model, are available within the time range \([0,T]\), which one can operate on the current seismic source\(s_{t}\) and the previous state \(h_{t-1}\) to produce the current time step's knowledge representation of P- and S-waves, denoted as \(K_{pt}\) and \(K_{st}\), respectively.