## 2 Establishment of Prediction Model
### TLM-PML Method
\begin{array}{l}U_{xx}^{mm}=\sum\limits_{j}^{NR}I_{3j}^{\prime}\phi_{xj}^{m} \phi_{xj}^{n}+\sum\limits_{j}^{NR}I_{4j}^{\prime}\phi_{yj}^{m}\phi_{yj}^{n}\\ U_{yy}^{mm}=\sum\limits_{j}^{NR}I_{4j}^{\prime}\phi_{xj}^{m}\phi_{xj}^{n}+\sum \limits_{j}^{NR}I_{3j}^{\prime}\phi_{yj}^{m}\phi_{yj}^{n}\\ U_{zz}^{mm}=\sum\limits_{j}^{NR}I_{1j}^{\prime}\phi_{zj}^{m}\phi_{zj}^{n}\\ I_{1j}^{\prime}=r_{0}\Bigg{\{}\frac{\pi}{2 ik_{j}}J_{0}\left(k_{j}r\right)H_{1} ^{(2)}\left(Z_{j}\right)-\frac{1}{r_{0}k_{j}^{2}}\Bigg{\}}\\ I_{3j}^{\prime}=\frac{\pi r_{0}}{2 ik_{j}}H_{1}^{(2)}\left([PERSON]_{j}\right)\left\{ \cos^{2}\theta J_{0}\left(k_{j}r\right)-\frac{\cos^{2}\theta}{k_{j}r}J_{1} \left(k_{j}r\right)+\frac{\sin^{2}\theta}{k_{j}r}J_{1}\left(k_{j}r\right) \right\}-\frac{1}{2k_{j}^{2}}\\ I_{4j}^{\prime}=\frac{\pi r_{0}}{2 ik_{j}}H_{1}^{(2)}\left(Z_{j}\right)\left\{ \sin^{2}\theta J_{0}\left(k_{j}r\right)-\frac{\sin^{2}\theta}{k_{j}r}J_{1} \left(k_{j}r\right)+\frac{\cos^{2}\theta}{k_{j}r}J_{1}\left(k_{j}r\right) \right\}-\frac{1}{2k_{j}^{2}}\end{array}  \tag{22}\]  
(3)  
Perfect match layer (PML) is applied to TLM  
In order to consider the influence of the lowest elastic half-space of layered soil, the PML is used to simulate the lowest elastic half-space. The PML is a numerical technique, similar to the absorption or transmission boundary [20] based on a discrete finite element model to suppress the reflection of waves in an infinite medium. The PML stretches the space through a position-dependent function. Starting from the interface that defines the elastic layer, the complex values of this function become larger to exponentially attenuate the waves within the PML as the distance from this interface increases. In addition, the  
Figure 4: Meaning of the fundamental solution of displacement. impedance difference at the PML boundary is consistent, which means that no reflection occurs regardless of the propagation angle of the wave entering the PML region. Considering that the SH wave propagates as \(C_{\mathrm{s}}\) in a uniform elastic layer with a depth of \(H\), the elastic layer is usually converted to the PML by converting the vertical coordinate \(z\) to the complex form \(\overline{z}\) defined as follows:  
\[\overline{z}=z-\mathrm{i}\Psi(z) \tag{23}\]  
where the real part of \(\overline{z}\) is the actual boundary layer depth; the imaginary part is the function that changes with depth. The imaginary part \(\Psi(z)\), the undetermined function, changes with depth, as expressed in Equation (24). \[\Psi(z)=\int\limits_{0}^{z}\psi(s)ds(0\leq z\leq H) \tag{24}\]  
where \(\Psi(s)\) is an extension function that is always positive. The commonly used extension function [26] is as follows:  
\[\psi(z)=\frac{\omega_{0}}{\omega}\Big{(}\frac{z}{H}\Big{)}^{m} \tag{25}\]  
where \(\omega_{0}\) represents the absorption level of the wave at the boundary. Parameter \(m\) is the elongation which should meet the requirement that \(m>0\). Consider a plane wave propagating into the PML domain along the \(z\)-axis at a \(\theta\) angle (Figure 5); this wave can be expressed as follows:  
\[u(x,\overline{z},t)=Ae^{\mathrm{i}(\omega t-x\Psi\sin\theta-2\overline{\Psi} \cos\theta)}=Ae^{\mathrm{i}(\omega t-x\Psi\sin\theta-2\overline{\Psi}\cos \theta)}e^{-\Psi(z)\Psi\cos\theta} \tag{26}\]  
The wave can be reflected at the bottom of the PML domain and on the free plane, and its total energy attenuation \(\Delta\) is  
\[\Delta=e^{-2\omega\Omega}\frac{\mu}{\cos\theta} \tag{27}\]  
where \(\Omega=\omega_{0}/\omega(m+1)\). According to \(\lambda=2\pi C/\omega\), and defining \(\eta=H/\lambda\), Equation (24) can be expressed as follows:  
\[\Delta=e^{-4\pi\Omega\eta\cos\theta} \tag{28}\]  
For the given incidence angle, the effectiveness of the PML layer depends only on the parameter \(\Omega\).