## 4 Method Verification
### Incidence along the Longitudinal Direction  
For the accuracy of the longitudinal response of the method proposed, according to the results proposed by [PERSON] and [PERSON] (2018), the P-SV waves incident obliquely along the tunnel axis were analyzed. The substructure model is shown in Figure 10b; the oblique incidence angle \(\theta_{v}=30^{\circ}\) and the conversion angle \(\theta_{t}=0^{\circ}\) of the seismic wave are input. The corresponding dimensionless frequency is \(\eta=\omega r_{1}/\tau c_{s}\). Tunnel are buried with a depth of \(H=5.0r_{1}=4.545r_{0}\) and a lining thickness of \(d=0.1r_{1}=0.0909r_{0}\). The material parameters are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Verification**} & \multicolumn{4}{c}{\(\mathbf{U_{2,5D}}/\mathbf{U_{3D}}\)} \\ \cline{2-5}  & \(\mathbf{R^{2}}\) & **Maximum Error** & **Mean Error** & **Standard Deviation** \\ \hline P wave \(U_{r\text{ }-1}\) & 0.9998 & 0.0240 & 0.0057 & 0.0075 \\ \hline SV wave \(U_{r\text{ }-1}\) & 0.9999 & 0.0317 & 0.0049 & 0.0064 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: R\({}^{2}\), maximum error, mean error and standard deviation of the ratio between maximum seismic response calculated by 2.5D finite element method and 3D reference solution \(U_{2,5D}/U_{3D}\). Figure 8: Distribution of the radial displacement amplification along the tunnel external surface under oblique incidence of seismic waves in the plane (**a**) P wave incidence and (**b**) SV wave incidence. Figure 9: Normalized radial displacement amplitude \(U_{r\text{ }-1}\) (dimensionless frequency \(\eta=0.132\)) when seismic waves are obliquely incident along the tunnel plane (**a**) P wave incidence and (**b**) SV wave incidence. Figure 11: Distribution of the radial displacement amplification along the tunnel external surface under longitudinal oblique incidence of seismic waves (a) P wave incidence and (b) SV wave incidence. \begin{table}
\begin{tabular}{c c c c} \hline
**Soil** & **Elastic Modulus \(\mathbf{E_{1}}\) (GN/m\({}^{3}\))** & **Poisson’s Ratio \(\mathbf{v_{1}}\)** & **Density \(\mathbf{\rho}1\) (kg/m\({}^{3}\))** \\ \hline \hline \(\mathrm{Lining}\) & 16 & 0.2 & 2240 \\ \hline \(\mathrm{Rock}\) & 7.567 & 0.333 & 2664 \\ \hline \end{tabular}
\end{table} 
Table 3: Material constants of lining and rock. Figure 10: P-SV wave along longitudinal scattering by tunnel in a homogenous half space: (a) 3D finite element model and (b) 2.5D substructure model. Figures 13 and 14 show the comparison between the 2.5D calculated value and the 3D reference solution for each node when seismic waves are obliquely incident along the longitudinal direction. The closer the data points are to the 1:1 line, the more accurate the proposed method out of the plane. It can be seen from Table 4 that all \(R^{2}\) values are close to 1, indicating that the 2.5D substructure method proposed in the paper is consistent with the 3D reference solution. On the whole, the statistical indicators used for evaluation show that the 2.5D model proposed in this article has high accuracy. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Verification**} & \multicolumn{4}{c}{\(\mathbf{U_{2.5D}/U_{3D}}\)} \\ \cline{2-5}  & \(\mathbf{R^{2}}\) & **Maximum Error** & **Mean Error** & **Standard Deviation** \\ \hline P wave \(U_{t-2}\) & 0.9990 & 0.1255 & 0.0610 & 0.0386 \\ \hline SV wave \(U_{t-2}\) & 0.9986 & 0.1011 & 0.0347 & 0.0359 \\ \hline P wave \(U_{l}\) & 0.9954 & 0.1406 & 0.0440 & 0.0360 \\ \hline SV wave \(U_{l}\) & 0.9993 & 0.0730 & 0.0244 & 0.0197 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: R\({}^{2}\), maximum error, mean error and standard deviation of the ratio between maximum seismic response calculated by 2.5D finite element method and 3D reference solution \(U_{2.5D}/U_{3D}\). Figure 12: Distribution of the axial displacement amplification along the tunnel external surface under longitudinal oblique incidence of seismic waves (**a**) P wave incidence and (**b**) SV wave incidence.