## 2 Numerical Simulation
### Calculating the Critical Angle and Petrophysical Parameters  
We can determine from [PERSON]'s quantification in Equation (2) that the incident angle must be greater than or equal to the critical angle for total reflection to occur. There is a weak dispersion region in which the incident angle is equal to the critical angle; thus, total reflection occurs throughout the entire propagation process with a uniform amplitude. Combining the Zoeppritz Equations (3) and (4) with the parameters of the strata in Table 1 can reveal the relationship between the energy distribution of seismic waves on the elastic interface, and the relationship between the reflection coefficient and the incident angle can be calculated:  
\[\frac{1}{V_{x}}=\frac{\sin\alpha_{1}}{V_{P1}}=\frac{\sin\beta_{1}}{V_{S1}}= \frac{\sin\alpha_{2}}{V_{P2}}=\frac{\sin\beta_{2}}{V_{S2}}, \tag{2}\]  
\[\left[ \begin{array}{cccc}-\sin\alpha_{1}&-\cos\beta_{1}&\sin\alpha_{2}&-\cos \beta_{2}\\ \cos\alpha_{1}&-\sin\beta_{1}&\cos\alpha_{2}&\sin\beta_{2}\\ \sin 2\alpha_{1}&\frac{V_{P1}}{V_{S1}}\cos 2\beta_{1}&\frac{\rho_{2}V_{S2}^{2}V_{P1} }{\rho_{1}V_{S1}^{2}V_{P2}}\sin 2\alpha_{2}&-\frac{\rho_{2}V_{P1}V_{S2}}{\rho_{1}V_{S 1}^{2}}\cos 2\beta_{2}\\ -\cos 2\beta_{1}&\frac{V_{S1}}{V_{P1}}\sin 2\beta_{1}&\frac{\rho_{2}V_{P2}}{ \rho_{1}V_{P1}}\cos 2\beta_{2}&\frac{\rho_{2}V_{S2}}{\rho_{1}V_{P1}}\sin 2\beta_{2} \end{array} \right]\left[ \begin{array}{c}R_{P}\\ R_{SV}\\ T_{P}\\ T_{SV}\end{array} \right]=\left[ \begin{array}{c}\sin a_{1}\\ \cos a_{1}\\ \sin 2a_{1}\\ \cos 2\beta_{1}\end{array} \right], \tag{3}\]  
where the P-wave incident angle and reflection angle are both \(\alpha_{1}\); the P-wave refraction angle is \(\beta_{1}\); the P-wave transmission angle is \(\alpha_{2}\); the P-wave critical refraction angle is \(\beta_{2}\); the P-wave velocity in the coal seam is \(V_{P1}\); the P-wave velocity in the surrounding rock is \(V_{P2}\); the S-wave velocity in the coal seam is \(V_{S1}\); the S-wave velocity in the surrounding rock is rock is \(V_{S2}\); \(T_{P}\) and \(R_{P}\) are the P-wave transmission coefficient and reflection coefficient, respectively, indicating the amplitude ratios of the transmitted P-wave and the reflected P-wave to the incident P-wave, respectively; and \(T_{SV}\) and \(R_{SV}\) are the S-wave transmission coefficient and reflection coefficient, respectively, indicating the amplitude ratios of the transmitted S-wave and reflected S-wave to the incident S-wave. \[\left[ \begin{array}{cccc}-1&1\\ V_{S1}\cos\alpha&\frac{\rho_{2}}{\rho_{1}}V_{S2}\cos\beta\end{array} \right] \left[ \begin{array}{c}R_{SH}\\ T_{SH}\end{array} \right]=\left[ \begin{array}{c}1\\ V_{S1}\cos\alpha\end{array} \right] \tag{4}\]  
In Equation (4), \(R_{SH}\) and \(T_{SH}\) are the SH-wave reflection and transmission coefficients, respectively; \(\alpha\) and \(\beta\) are the SH-wave reflection and transmission angles, respectively; \(\rho_{1}\) and \(\rho_{2}\) are the densities of the coal seam and surrounding rock, respectively; and \(V_{S1}\) and \(V_{S2}\) are the S-wave velocities of the coal seam and surrounding rock, respectively. Combining Equations (3) and (4), we can obtain the following:  
\[R_{SH}=\frac{\rho_{1}V_{S1}\cos\alpha-\rho_{2}V_{S2}\cos\beta}{\rho_{1}V_{S1} \cos\alpha+\rho_{2}V_{S2}\cos\beta}. \tag{5}\]  
The relationships of the P- and S-wave reflection coefficients with the incident angle are shown in Figure 6. When it does not reach the critical angle, most of the reflected energy is released. When the incident SH-wave reaches the critical angle, the energy increases to the maximum and remains constant with an increase in the angle. This corresponds to the leakage mode and normal mode of channel wave propagation in the process of wave field propagation. In contrast, the P-wave energy is also highest when the incident angle is the critical angle of refraction, but the energy weakens as the angle decreases below or increases above the critical angle, causing the intensities of transmission and reflection to decay rapidly.