## 3 Numerical Experiment
### Single Vertical Geological Interface Model
Figure 4: Kars cave model filled with air. Figure 3: Seismic profile of the model with a single vertical interface of (**a**) \(v_{r}\), component, (**b**) \(v_{\theta}\) component and (**c**) \(v_{z}\) component. The four columns, from left to right, denote the seismic data obtained from the upper sidewall, down sidewall, left sidewall, and right wall. We select the snapshot of the 3D wavefield of the karst cave model filled with soft soil and its slice diagram for demonstration. It can be seen from Figure 5 that the seismic wave reflects for the first time when it passes through the first interface of the karst cave and then spreads in the spherical form with the cave as the center. The other part passes through the cave space in the form of a transmission wave, which produces diffraction inside the cavity, and finally, a second reflected wave is transmitted back to the receivers. As shown in Figure 6b, the first \(P\)-wave reflection \(PP1\) reaches the tunnel face at 32 ms, and the second reflection \(P\)-wave \(PP2\) formed after diffraction is formed at 40 ms and is received by the receivers at 52 ms. Similarly, the two reflected waves \(SS1\) and \(SS2\) of \(S\)-wave arrive at the survey line at 52 ms and 76 ms, respectively. The diffracted waves on the tunnel face can be observed in the transverse wave field, which shows that although the method can avoid the interference of the diffracted waves at the edges of the tunnel face emerging in the Cartesian coordinate system, it cannot suppress the diffracted waves generated at the junction of the surface edge and the sidewall. Figure 5: \(Cont.\)  
When the cave is filled with other media, the basic characteristics of the seismic wavefield are similar to those of soft soil. The main difference lies in the energy intensity of the reflected waves and whether the \(P/S\)-waves can pass through the cave space. In the following, we will further compare the three-component seismic profiles of survey line 1 when the cave is filled with different media. Figure 6a shows the three-component seismic profile when the karst cave is filled with soft soil. Similar to the model with a vertical single interface, the energy of the \(S\)-wave is still stronger than that of the \(P\)-wave. In the seismic profile of the \(v_{z}\) component, the energy of \(PP1\) is relatively weak and almost unrecognizable, whereas in the seismic profile of the \(v_{\theta}\) component, only the reflected \(S\)-wave can still be observed. The energy of the reflected \(P\)-wave is the strongest in the seismic profile of the \(v_{r}\) component, and the energy of the reflected \(S\)-wave is the strongest in the seismic profile of the \(v_{z}\) component. Comparing the energy of the two kinds of reflected waves, it can be found that the energy of the second reflected waves is stronger than that of the first. This is mainly because the first interface of the cave has a \"divergence\" effect for the seismic wave. This interface and the inside of the cavernous cavity play the role of \"gathering\" seismic waves. Figure 6b is the seismic profile obtained under a water-filled karst cave. Same as the aforementioned result, the energy of the reflected \(P\)-wave is still the strongest in the seismic profile of the \(v_{r}\) component and the energy of the reflected \(S\)-wave is strongest in the seismic profile of the \(v_{z}\) component. However, in the \(v_{\theta}\) component, the energy of the reflected waves is weakest. This is different from the observation of that under a soft-filled karst cave, mainly in the value of maximum energy of the reflected \(P\)-wave and \(S\)-wave and the waveform of the reflected \(S\)-wave. From the perspective of the energy of the reflected wave, the energy of the reflected wave is different for the difference of the reflection coefficient of media.