## 2 Establishment of Prediction Model
### TLM-PML Method
\begin{array}{l} \widetilde{u}_{r}\\ \widetilde{u}_{\theta}\\ \widetilde{u}_{z}\end{array} \right\}_{L+1} \tag{8}\]  
Substituting Equation (8) of any plane in the thin-layer element into Equation (7) in the wavenumber domain, the displacement can be written as follows:  
\[\left\{\begin{array}{l}k_{r}^{2}\left[ \begin{array}{cc}\mathbf{A}_{r}^{e} &\mathbf{O}\\ \mathbf{B}_{rz}^{e}{}^{T}&\mathbf{A}_{z}^{e}\end{array} \right]+\left[ \begin{array} []{cc}\mathbf{C}_{r}^{e}&\mathbf{B}_{rz}^{e}\\ \mathbf{O}&\mathbf{C}_{z}^{e}\end{array} \right]\right\}\left[ \begin{array}{c} \widetilde{\mathbf{u}}_{r}^{e}\\ \widetilde{\mathbf{u}}_{z}^{e}\end{array} \right]=\left[ \begin{array}{c} \widetilde{\mathbf{e}}_{zr}^{e}\\ k_{r}\widetilde{\mathbf{e}}_{zz}^{e}\end{array} \right] \tag{9}\]  
Figure 3: Sections and coordinates of a thin-layer unit with quadratic function. Figure 2: Cylindrical coordinate system of subsoil model. where \(\widetilde{\mathbf{u}}_{r}^{\mathbf{e}}=[\widetilde{u}_{r,L}\ \ \widetilde{u}_{r,L+1/2}\ \ \widetilde{u}_{r,L+1}], \widetilde{\mathbf{u}}_{z}^{\mathbf{e}}=[\widetilde{u}_{z,L}\ \ \widetilde{u}_{z,L+1/2}\ \ \widetilde{u}_{z,L+1}], \widetilde{\mathbf{u}}_{\theta}^{\mathbf{e}}=[\widetilde{u}_{\theta,L}\ \ \ \widetilde{u}_{\theta,L+1/2}\ \ \widetilde{u}_{\theta,L+1}]\), \(\widetilde{\mathbf{u}}_{z\prime}^{\mathbf{e}}=[\widetilde{v}_{z\prime,n}\ \ 0\ \ \widetilde{v}_{z\prime,n+1}]\), \(\widetilde{\mathbf{u}}_{z\prime}^{\mathbf{e}}=[\widetilde{v}_{z\prime,n}\ \ 0\ \ \widetilde{v}_{z\prime,n+1}]\) and \(\widetilde{\mathbf{u}}_{z\theta}^{\mathbf{e}}=[\widetilde{v}_{z\theta,n}\ \ 0\ \ \widetilde{v}_{z\theta,n+1}]\) represent the displacement and stress in the \(r\), \(z\), and \(\theta\) directions of thin layer element sections in wave-number domain, respectively. Thin-layer element submatrices \(\mathbf{A}_{r}^{\mathbf{e}}\), \(\mathbf{A}_{z}^{\mathbf{e}}\), \(\mathbf{A}_{\theta}^{\mathbf{e}}\), \(\mathbf{B}_{rz}^{\mathbf{e}}\), \(\mathbf{C}_{r}^{\mathbf{e}}\), \(\mathbf{C}_{z}^{\mathbf{e}}\), and \(\mathbf{C}_{\theta}^{\mathbf{e}}\) are matrices represented by the parameters of layered soil materials:  
\[\mathbf{A}_{r}^{\mathbf{e}}=\frac{(\lambda+2G)h}{30} \begin{bmatrix}4&2&-1\\ 2&16&2\\ -1&2&4\end{bmatrix} \qquad\mathbf{A}_{z}^{\mathbf{e}}=\mathbf{A}_{\theta}^{ \mathbf{e}}=\frac{Gh}{30} \begin{bmatrix}4&2&-1\\ 2&16&2\\ -1&2&4\end{bmatrix}  \tag{10}\]  
\[\mathbf{B}_{rz}^{\mathbf{e}}= \begin{bmatrix}\frac{-\lambda+2G}{2}&\frac{2( \lambda+G)}{3}&-\frac{\lambda+G}{6}\\ -\frac{2(\lambda+G)}{3}&0&\frac{2(\lambda+G)}{3}\\ \frac{\lambda+G}{6}&-\frac{2(\lambda+G)}{3}&\frac{\lambda-G}{2}\end{bmatrix}  \tag{11}\]  
\[\mathbf{C}_{r}^{\mathbf{e}}=\mathbf{C}_{\theta}^{\mathbf{e}}=\frac{G}{3h} \begin{bmatrix}7&-8&1\\ -8&16&-8\\ 1&-8&7\end{bmatrix} -\omega^{2}\frac{\rho h}{30} \begin{bmatrix}4&2&-1\\ 2&16&2\\ -1&2&4\end{bmatrix}  \tag{12}\]  
After lumping each layer, the fundamental equation of the thin-layer method in the frequency-wavenumber domain can be expressed as follows:  
\[\widetilde{\mathbf{P}}=\mathbf{K}\widetilde{\mathbf{U}}=\left(\widetilde{ \mathbf{A}}k_{r}^{2}+\widetilde{\mathbf{C}}\right)\widetilde{\mathbf{U}} \tag{13}\]  
where \(\widetilde{\mathbf{P}}=\left\{\widetilde{\mathbf{P}}_{r}\ k_{r}\widetilde{ \mathbf{P}}_{z}\ \widetilde{\mathbf{P}}_{\theta}\right\}^{\mathrm{T}}\) is the load vector. \(\widetilde{\mathbf{U}}=\left\{\widetilde{\mathbf{U}},k_{r}\widetilde{\mathbf{ U}}_{z}\ \widetilde{\mathbf{U}}_{b}\right\}^{\mathrm{T}}\) is the displacement vector. \(\mathbf{K}=\widetilde{\mathbf{A}}k_{r}^{2}+\widetilde{\mathbf{C}}\) is the total stiffness matrix; matrices A and C can be written as:  
\[\widetilde{\mathbf{A}}= \begin{bmatrix}\mathbf{A}_{r}&\mathbf{0}&\mathbf{0}\\ \mathbf{B}_{rz}^{\mathrm{T}}&\mathbf{A}_{z}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{A}_{\theta}\end{bmatrix} \quad\widetilde{ \mathbf{C}}= \begin{bmatrix}\mathbf{C}_{r}&\mathbf{B}_{rz}^{\mathrm{T}}& \mathbf{0}\\ \mathbf{0}&\mathbf{C}_{z}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{C}_{\theta}\end{bmatrix}