## 3 Field Measurements and Model Verification
### Model Results and Validation  
Based on the tunnel excitation test, thin-layer modeling was adopted. The soil layer parameters were based on the soil layer parameters near the laboratory. The specific soil layer parameters are shown in Table 1. For the tunnel model, the horseshoe tunnel with a net height and width of 4 m was simplified into a circular shield tunnel with an outer diameter of 4 m. The wall thickness \(\delta\) is 0.5 m, the buried depth of the tunnel rail surface \(H\) is 10 m, the dynamic elastic modulus \(E_{\text{d}}\) is 29 GPa, the dynamic Poisson ratio \(\
u\) is 0.2, and the bulk weight \(\gamma\) is 25 kN/m\({}^{3}\).  
The displacement frequency response functions at different observation points in the tunnel-soil system under vertical unit harmonic load were calculated. The frequency calculation range is between 0 and 100 Hz, and the frequency resolution is 0.5 Hz. A total of 17 isometric surface observation points (Q0-Q16) perpendicular to the driving direction were selected from directly above the excitation point \(x=0\) m to \(x=80\) m. The  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Soil Layer**} & **Thickness** & **Volume Weight** & **Dynamic Elastic** & **Shear Wave** & **Dynamic** & **Damping Ratio** \\  & _d/m_ & \(\gamma\)/(kN/m\({}^{3}\)) & **Modulus** & **Velocity** & **Poisson Ratio** & **\$** \\  & & & & & & \\ \hline
1 & 3 & 16.5 & \(1.4\times 10^{5}\) & 176 & 0.35 & 0.02 \\
2 & 30 & 20.1 & \(3.1\times 10^{5}\) & 238 & 0.33 & 0.02 \\
3 & \(\infty\) & 20.5 & \(6.7\times 10^{5}\) & 353 & 0.29 & 0.02 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Soil parameters of laboratory model.  
Figure 14: Time histories and Fourier spectra of ground accelerations at different positions.  
distance between the two adjacent observation points is 5 m. The schematic diagram of each observation point on the surface is shown in Figure 15.  
According to the derivation method of displacement frequency response function in Section 2.2, the displacement frequency response functions at each observation point on the surface under vertical unit harmonic load were calculated, as shown in Figure 16.  
Figure 16: Displacement frequency response function of vertical ground vibrations at different locations.  
Figure 15: Schematic diagram of surface observation points.  
The Figure 17 calculated and experimental results are illustrated. It can be observed that the calculated results are close to the experimental ones, both exhibiting the fluctuation with frequency. The calculated and experimental ground vibrations are in good agreement in most frequency bands at 5, 30, and 80 m and tend to be consistent in most frequency bands but with a slightly different magnitude at 20, 25, and 30 m. Between 25 and 60 m, the experimental results are significantly larger than the calculated ones in the frequency band between 40 and 80 Hz. In general, considering that there are actually two layers of tunnels in the laboratory, the calculated model does not take into account the influence of the tunnel structure of the underground two-layer rail surface at a buried depth of 18 m. In addition, the actual soil layer parameters and tunnel structure were greatly simplified in the calculation process. In addition, the difference between the prediction and measurement results were measured by calculating the Root Mean Square Error (RMSE) of each position, as shown in Figure 18. It can be observed that the smaller the value of the RMSE, the closer the predicted result of the model is to the measured result. The RMSE gradually decreases with the increase in distance, and all magnitude values are one magnitude smaller than the curve. Combined with the comparison results, it is considered that the TLM-VM-PML can be used to study the analysis of the tunnel-foundation dynamic interaction.  
Figure 17: Comparison of the vertical ground vibrations at different locations.