## 2 Establishment of Prediction Model
### TLM-PML Method
 \tag{14}\]  
(2) Fundamental solution of 3D foundation soil vibration by TLM  
The stiffness matrix spectral decomposition method is used to solve Equation (13) for the fundamental solution of 3D soil vibration in TLM:  
The right eigenvalue analysis of \(\mathbf{K}\) is carried out. According to the structure of matrix \(\widetilde{\mathbf{A}}\) and \(\widetilde{\mathbf{C}}\), it can be decomposed into two right eigenvalue problems: the problem of the Rayleigh wave (\(r\) and \(z\) direction) and the problem of the Love wave (\(\theta\) direction). Similarly, there is the left eigenvalue problem. The first \(j\) is a set of corresponding left \(k_{\mathrm{j}}\) eigenvalues \(\mathbf{\Omega}_{\mathrm{L,j}}=[k_{j}\phi_{rj}\ \phi_{zj}\ \phi_{\theta j}]^{ \mathrm{T}}\) and right eigenvalues \(\mathbf{\Omega}_{\mathrm{R,j}}=[\phi_{rj}\ \dot{k}_{j}\phi_{zj}\ \phi_{\theta j}]^{ \mathrm{T}}\). By substituting the eigenvalues into the total stiffness matrix of the fundamental equation of the TLM and combining them, the left and right eigenvectors have the following relationship:  
\[\mathbf{\Omega}_{\mathrm{L}}^{\mathrm{T}}\widetilde{\mathbf{A}} \mathbf{\Omega}_{\mathrm{R}} = \begin{bmatrix}\overline{\mathbf{K}_{rz}}\mathbf{\Phi}_{r}\mathbf{A}_{r} \mathbf{\Phi}_{r}+\mathbf{\Phi}_{z}\mathbf{B}_{rz}^{\mathrm{T}}\mathbf{\Phi}_{r}+ \overline{\mathbf{K}_{rz}}\mathbf{\Phi}_{z}\mathbf{A}_{z}\mathbf{\Phi}_{z}& \mathbf{0}\\ \mathbf{0}&\mathbf{\Phi}_{\theta}\mathbf{A}_{\theta}\mathbf{\Phi}_{\theta} \end{bmatrix}  \tag{15}\] \[= \begin{bmatrix}\overline{\mathbf{K}_{rz}}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}\end{bmatrix} =\mathbf{X}\]  
\[\mathbf{\Phi}_{\mathrm{L}}^{\mathrm{T}}\widetilde{\mathbf{C}}\mathbf{\Phi}_{ \mathrm{R}}=-\mathbf{\Phi}_{\mathrm{L}}^{\mathrm{T}}\widetilde{\mathbf{A}} \mathbf{\Phi}_{\mathrm{R}}\overline{\mathbf{K}}_{\mathrm{R}}^{2}=-\mathbf{X} \overline{\mathbf{K}}^{2}= \begin{bmatrix}\overline{\mathbf{K}_{rz}^{2}}&\mathbf{0 }\\ \mathbf{0}&\overline{\mathbf{K}_{\theta}^{2}}\end{bmatrix}  \tag{16}\]where \(j=1,2,\,\cdots,\,3N;\mathbf{\Phi}_{r}\) and \(\mathbf{\Phi}_{z}\) are \(N\times 2N\) matrices; \(\mathbf{\Phi}_{\theta}\) is an \(N\times\) N matrix; \(\overline{\mathbf{K}}_{rz}\) are the first 2N elements of the diagonal matrix \(\overline{\mathbf{K}}\), and \(\overline{\mathbf{K}}_{\Theta}\) are the last \(N\) elements of the diagonal matrix \(\overline{\mathbf{K}}\). Introducing \(\mathbf{\Omega}_{\text{R}}\Omega_{\text{R}}^{-1}=\mathbf{I}\), and combining Equations (13), (15) and (16), the displacement equation in the frequency and wavenumber domain can be expressed as follows:  
\[ \begin{bmatrix}\widetilde{\mathbf{u}}_{r}\\ \widetilde{\mathbf{u}}_{z}\\ \widetilde{\mathbf{u}}_{0}\end{bmatrix} = \begin{bmatrix}\mathbf{\Phi}_{r}\mathbf{ E}_{rz}\mathbf{\Phi}_{r}^{T}&k_{r}\mathbf{\Phi}_{r}\overline{\mathbf{K}}_{ \mathbf{K}}^{-1}\mathbf{E}_{rz}\mathbf{\Phi}_{z}^{T}&\mathbf{0}\\ \mathbf{\Phi}_{z}\mathbf{E}_{rz}\mathbf{E}_{rz}\mathbf{\Phi}_{r}^{T}/k_{r}& \mathbf{\Phi}_{z}\mathbf{E}_{rz}\mathbf{\Phi}_{z}^{T}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{\Phi}_{\theta}\mathbf{E}_{\theta}\mathbf{\Phi}_ {\theta}^{T}\end{bmatrix}\begin{bmatrix}\widetilde{\mathbf{P}}_{r}\\ \widetilde{\mathbf{P}}_{z}\\ \widetilde{\mathbf{P}}_{\theta}\end{bmatrix}  \tag{17}\]  
where \(\mathbf{E}_{rz}=(k_{r}^{2}\mathbf{I}-K_{rz}^{2})^{-1}\) and \(\mathbf{E}_{\theta}=(k_{r}^{2}\mathbf{I}-K_{\theta}^{2})^{-1}\). Based on the inverse Fourier-Bessel integral transform method, the displacement can be transformed into the spatial domain:  
\[ \begin{cases}\overline{\mathbf{u}}_{r}\\ \overline{\mathbf{u}}_{z}\\ \overline{\mathbf{u}}_{\theta}\end{cases} =\sum\limits_{n=0}^{\infty}T_{n}( \theta)\int_{0}^{\infty}k_{r}C_{n}(k_{r}r) \begin{cases}\widetilde{\mathbf{u}}_ {r}\\ \widetilde{\mathbf{u}}_{z}\\ \widetilde{\mathbf{u}}_{\theta}\end{cases} \text{d}k_{r} \tag{18}\]  
All derivations in this paper are carried out in column coordinate system. The fundamental solution of displacement in the rectangular coordinate system can be transformed by Equation (19). \[