## 2 Establishment of Prediction Model
### TLM-PML Method
 \tag{3}\]  
where \(\omega\) represents the excited circular frequency and the symbol \"-\" represents a physical quantity in the frequency-space domain. Then, the Fourier-Bessel integral transform is introduced:  
\[\widetilde{U}(k_{r},z)=\alpha_{n}\int_{0}^{\infty}rC_{n}(k_{r}r)\int_{0}^{2 \pi}T_{n}(\theta)\overline{U}(r,\theta,z)\mathrm{d}\theta\mathrm{d}r \tag{4}\]  
where \(k_{r}\) is the wavenumber in the direction of \(r\). \(T_{n}(\theta)\) is an expansion of the Fourier series in the \(\theta\) direction, and \(n\) is the number of Fourier series expansions. The coefficient term \(\alpha_{n}\) is defined as follows: when \(n=0\), \(\alpha_{n}=\)1/2\(\pi\); when \(n\
eq 0\), \(\alpha_{n}=\)1/\(\pi\). The transformation matrix \(C_{n}(k_{r}r)\) can be expressed by  
\[C_{n}(k_{r}r)=\left[ \begin{array}{ccc}\frac{\mathrm{d}}{d(k_{r}r)}J_{n}(k_{ r}r)&\frac{n}{k_{r}r}J_{n}(k_{r}r)&0\\ \frac{n}{k_{r}r}J_{n}(k_{r}r)&\frac{\mathrm{d}}{d(k_{r}r)}J_{n}(k_{r}r)&0\\ 0&0&J_{n}(k_{r}r)\end{array} \right] \tag{5}\]  
where \(J_{n}(k_{r}r)\) is the Bessel functions of the first kind. In the case of vertical vibration, with \(n=0\), the expression of the displacement component in the equation of motion in the frequency domain is as follows:  
\[\left\{ \begin{array}{c}\overline{u}_{r}\\ \overline{u}_{z}\end{array} \right\}=\int_{0}^{\infty}\left[ \begin{array}{cc }\frac{J_{1}(k_{r}r)}{2}&0\\ 0&J_{0}(k_{r}r)\end{array} \right]\left\{ \begin{array}{c}\widetilde{u}_{r}\\ \widetilde{u}_{z}\end{array} \right\}\mathrm{d}k_{r} \tag{6}\]  
Substituting Equation (6) into the motion Equation (3) in the frequency domain, the fundamental differential equation of the TLM can be written as follows after sorting:  
\[ \begin{array}{l}(\lambda+2\mu)k_{r}^{2}\widetilde{u}_{r}-\mu\frac{\mathrm{d }^{2}\widetilde{u}_{r}}{\mathrm{d}z^{2}}+k_{r}(\lambda+\mu)\frac{\mathrm{d} \widetilde{u}_{z}}{\mathrm{d}z}-\rho\omega^{2}\widetilde{u}_{r}=0\\ -k_{r}(\lambda+\mu)\frac{\mathrm{d}\widetilde{u}_{r}}{\mathrm{d}z}+k_{r}^{2} \widetilde{u}_{z}-(\lambda+2\mu)\frac{\mathrm{d}^{2}\widetilde{u}_{z}}{ \mathrm{d}z^{2}}-\rho\omega^{2}\widetilde{u}_{z}=0\end{array}  \tag{7}\]  
where the symbol \"-\" represents a physical quantity in the frequency-wavenumber domain. For the discrete equation of the TLM in the cylindrical coordinate system, the Galerkin method of weighted residuals is used to solve the fundamental differential Equation (7) of  
Figure 1: Elastic half-space cylindrical coordinate system. the TLM. As shown in Figure 2, the foundation soil model consists of a series of thin layers of layered soil and the lowest half-space. The soil is divided into \(N\) horizontal thin-layer elements along the depth direction of the subsoil model. In the same thin-layer element, the soil parameters are equal. It is assumed that the displacement inside any thin-layer element is linearly or conically distributed along the vertical axis, corresponding to the primary and quadratic functions respectively. Take the quadratic function as an example, an intermediate section (\(L+1/2\)) to the upper and lower sections of a thin-layer element \(L\) with thickness \(h\) is added as shown in Figure 3. The displacements of the three sections are \(\widetilde{U}_{L}=\{\widetilde{u}_{r},\widetilde{u}_{\theta},\widetilde{u}_{ z}\}_{L}^{T}\), \(\widetilde{U}_{L+1/2}=\{\widetilde{u}_{r},\widetilde{u}_{\theta},\widetilde{u}_ {z}\}_{L+1/2}^{T}\) and \(\widetilde{U}_{L+1}=\{\widetilde{u}_{r},\widetilde{u}_{\theta},\widetilde{u}_ {z}\}_{L+1}^{T}\), respectively. The displacement of any plane in the thin-layer element can be expressed by the following formula:  
\[\left\{ \begin{array}{l}\widetilde{u}_{r}\\ \widetilde{u}_{\theta}\\ \widetilde{u}_{z}\end{array} \right\}=N_{L}\left\{ \begin{array}{l}\widetilde {u}_{r}\\ \widetilde{u}_{\theta}\\ \widetilde{u}_{z}\end{array} \right\}_{L}+N_{L+\frac{1}{2}}\left\{ \begin{array} []{l}\widetilde{u}_{r}\\ \widetilde{u}_{\theta}\\ \widetilde{u}_{z}\end{array} \right\}_{L+1}+N_{L+1}\left\{