## 2 Establishment of Prediction Model  
### TLM-PML Method  
The Green's function of the layered soil is derived using the TLM in the cylindrical coordinate system. The main idea is as follows: Firstly, the fundamental equation of motion in an elastic half-space expressed in displacement form in the time domain is transformed into the frequency domain by Fourier transform, and the equation of motion in the frequency-space domain is obtained. Subsequently, the ordinary differential equation of displacement in the wavenumber domain with respect to depth direction z is obtained by introducing the Fourier-Bessel integral transform, that is, the fundamental differential equation of the TLM. Subsequently, for each thin-layer element, the layer element equation (the relationship between displacement and stress in the thin-layer element) is derived by using the Galerkin weighted residual method for a discrete solution. Finally, the displacement fundamental solution is obtained by using the stiffness matrix spectral decomposition method. (1) Fundamental differential equation of TLM  
First, for the elastic half-space depicted in Figure 1, the fundamental equation of motion expressed in the form of stress is established under the cylindrical coordinate system as follows:  
\[ \begin{array}{l}\frac{\partial\sigma_{r}}{\partial r_{r}}+\frac{1}{r}\frac{ \partial\sigma_{r}}{\partial\theta}+\frac{\partial\sigma_{rz}}{\partial z}+ \frac{\sigma_{rz}-\sigma_{\theta\theta}}{r}+F_{r}=0\\ \frac{\partial\sigma_{rz}}{\partial r_{r}}+\frac{1}{r}\frac{\partial\sigma_{ \theta\theta}}{\partial\theta}+\frac{\partial\sigma_{rz}}{\partial z}+\frac{2 }{r}\sigma_{\theta r}+F_{\theta}=0\\ \frac{\partial\sigma_{rz}}{\sigma_{r}}+\frac{1}{r}\frac{\partial\sigma_{ \theta\theta}}{\partial\theta}+\frac{\partial\sigma_{rz}}{\partial z}+\frac{1 }{r}\sigma_{zr}+F_{z}=0\end{array}  \tag{1}\]  
where \(F_{r}\), \(F_{\theta}\), and \(F_{z}\) are the body force per unit volume; \(\sigma_{rr}\), \(\sigma_{r\theta}\), \(\sigma_{rz}\), \(\sigma_{\theta r}\), \(\sigma_{\theta\theta}\), \(\sigma_{rz}\), \(\sigma_{z\theta}\), and \(\sigma_{zz}\) are the stress components in the cylindrical coordinate system. \[ \begin{array}{l}(\lambda+\mu)\frac{\partial\Lambda}{\partial r}+\mu\left[ \
abla^{2}u_{r}-\frac{1}{r}\left(\frac{2}{r}\frac{\partial u_{r}}{\partial \theta}+\frac{u_{r}}{r}\right)\right]=\rho\frac{\partial^{2}u_{r}}{\partial t ^{2}}\\ (\lambda+\mu)\frac{\partial\Lambda}{\partial r\partial}+\mu\left[\
abla^{2}u _{\theta}-\frac{1}{r}\left(\frac{u_{\theta}}{r}-\frac{2}{r}\frac{\partial u_ {r}}{\partial\theta}\right)\right]=\rho\frac{\partial^{2}u_{\theta}}{\partial t ^{2}}\\ (\lambda+\mu)\frac{\partial\Lambda}{\partial z}+\mu\
abla^{2}u_{z}=\rho\frac{ \partial^{2}u_{z}}{\partial t^{2}}\end{array}  \tag{2}\]  
where \(u_{r}\), \(u_{\theta}\) and \(u_{z}\) represent the displacement components in the radial, circumferential and vertical directions in the column coordinate system, respectively. \(\rho\) is the density of foundation soil, and \(\
abla^{2}=\frac{\partial^{2}}{\partial t^{2}}+\frac{\partial}{\partial r }+\frac{\partial^{2}}{\partial t^{2}\partial\theta^{2}}+\frac{\partial^{2}}{ \partial z^{2}}\) is a Laplace operator. The Fourier transform is applied to Equation (2) to obtain the equation of motion in the frequency-space domain:  
\[ \begin{array}{l}(\lambda+\mu)\frac{\partial\alpha}{\partial r}+\mu\left[\
abla^ {2}\overline{u}_{r}-\frac{1}{r}\left(\frac{\overline{u}_{r}}{\overline{r}}\frac{ \overline{u}_{\theta}}{\overline{u}}+\frac{\overline{u}_{r}}{\overline{r}} \right)\right]=\rho\omega^{2}\overline{u}_{r}\\ (\lambda+\mu)\frac{\partial\alpha}{\partial\overline{u}}+\mu\left[\
abla^{2} \overline{u}_{\theta}-\frac{1}{r}\left(\frac{\overline{u}_{\theta}}{r}-\frac{2 }{r}\frac{\overline{u}_{\theta}}{\overline{u}\overline{u}}\right)\right]= \rho\omega^{2}\overline{u}_{\theta}\\ (\lambda+\mu)\frac{\partial\alpha}{\partial z}+\mu\
abla^{2}\overline{u}_{z}= \rho\omega^{2}\overline{u}_{z}\end{array}