## 3 Numerical Experiment
### Single Vertical Geological Interface Model  
As shown in Figure 1, we set up a model with a single vertical geological interface and its radius and width are 35 m and 120 m, respectively. The tunnel is 40 m long and has a radius of 6 m. The center of the interface is 45 m away from the tunnel face in the \(z\)-direction. The \(P\)-velocity and \(S\)-velocity of the surrounding rock are \(V_{P}=4000\) m/s and \(V_{S}=2300\) m/s, respectively; those of the medium behind the interface are \(V_{P}=3400\) m/s and \(V_{S}=2000\) m/s; the grid spacing is \(\Delta r=\Delta z=0.5\) m. The time sampling interval is 0.2 ms, and the total time window is 0.08 s.  
In Figure 2a, it can be seen that this method can simulate the propagation process of the tunnel full-space seismic wavefield well and separate the \(P\)-wave and \(S\)-wave fields. Since the source is put on the sidewall, the energy of \(S\)-waves is stronger than that of \(P\)-waves. It can be seen from Figure 2b,c that the direct \(P\)-wave reaches the reflector at about 24 ms, the reflected \(PP\)-wave was collected by receivers at 40 ms, and the direct \(S\)-wave reaches the reflector at 32 ms. The reflected \(SS\)-wave is received at 56 ms.  
Figure 1: The model with a single vertical geological interface. Figure 2: _Cont._  
As for the seismic profiles, the direct wave has been removed from all of the seismic data shown in the figure to prevent it from affecting reflected waves. Figure 3a is a three-component seismic profile of a single-layer vertical interface on each survey line. First, the shear waves mainly propagate along the \(z\) and \(r\) directions, so the energy of the reflected waves in the \(v_{\theta}\) component is weak, and \(S\)-waves are the main ones. The \(P\)-wave mainly propagates along the \(r\) direction in the \(roz\) plane. The energy of the reflected \(P\)-wave in the \(v_{r}\) component is the strongest, followed by that in the \(v_{z}\) component, and the energy of the reflected \(P\)-wave in the \(v_{\theta}\) component seismic profile is the weakest. Then, let us compare the difference between the reflected waves of each survey line, analyze the amplitude energy characteristics of the event axis of the reflected wave, and remove the \(v_{\theta}\) component. Two laws can be summarized: 1 In the seismic profile from the left and right sidewall, the energy of the \(S\)-wave is stronger than the \(P\)-wave. However, in that of upper and lower sidewalls, that is opposite. 2 The energy of the reflected \(P\)-wave from the right wall is stronger than that from the left side wall, and the same observation can be made for the reflected \(S\)-wave energy. In summary, in the three-dimensional tunnel space, different components and different survey lines can receive different reflected wave information. Therefore, in tunnel forward seismic prospecting, the seismic data from multicomponent and multisurvey lines should be used comprehensively. As shown in Figure 4, the karst cave model is a cylindrical space with a radius of 35 m and a height of 140 m. The karst cave has a diameter of 20 m and a distance of 45 m in front of the tunnel face. It is filled with three media: water, soft soil, and air, respectively. The sampling interval is 0.2 ms, and the total calculation time is 0.1 s.  
The specific physical parameters of the filling medium in the karst cave are shown in Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Medium** & \(\rho\) **(kg/m3)** & \(V_{P}\) **(m/s)** & \(V_{S}\) **(m/s)** \\ \hline Surrounding rock & 2000 & 4000 & 2300 \\ Soft soil & 1800 & 2500 & 1700 \\ Water & 1000 & 1500 & 0 \\ Air & 1.0 & 340 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of the different mediums in the karst cave model. Figure 2: Wavefield in \(v_{r}\) component of the model with a single vertical interface at different times. (**a**) 3D wavefield snapshot at different time. Snapshot of (**b**) \(P\)-wavefield and (**c**) \(S\)-wavefield in \(ro\theta\) plane at different time.