## 4 Discussion  
In order to verify the effectiveness of the proposed method in this paper, we compared it with the 3D staggered-grid finite-difference scheme in the Cartesian coordinate system. In order to show the advantages of this method, we use the 3D elastic wave equation to simulate the wave field propagation ahead of the tunnel.  
Take the model setting in the Cartesian coordinate system as an example (the model size and observation setup setting in the cylindrical coordinate system are exactly the same); the model size is \(40\times 40\times 140\) m, the size of grid is 0.5 m. The tunnel is 30 m long and has a width (diameter) of 12 m. In terms of observation methods, a source is set at the center point of the top of the tunnel, 5 m from the tunnel face, and the main frequency is 120 Hz. There are two survey lines. Survey line 1 is set on the face of the tunnel, around the tunnel; survey line 2 is set 20 m to 50 m in front of the tunnel, arranged along the \(z\) axis, with an interval of 1 m, totally 31. In addition to the air medium in the tunnel area, the surrounding rock area is set as a homogeneous medium, with \(V_{p}=4000\) m/s and \(V_{s}=2300\) m/s.  
Comparing the wavefield results in \(r\theta\) plane in Cartesian coordinate system and cylindrical coordinate system, it can be seen that the interference of diffracted waves is mainly caused by \(SH\)-waves, which detour along the tunnel boundary during the  
Figure 9: The seismic profile of the upper and lower wall lines of the wedge fault. (**a**) The seismic profile of the lower wall lines. (**b**) The seismic profile of the upper wall lines.  
propagation process. When the seismic wave meets the mutation point (indicated by the red dots in Figure 10), a strong diffraction wave will be generated. In the cylindrical coordinate system, the wave fields is clean. However, because the \(S\)-wave can propagating along the two sides of the tunnel meet each other at the bottom of the tunnel, there will be a slight dispersion phenomenon (shown by the arrow in Figure 10).  
Further comparing the seismic profiles of survey line 1 and survey line 2, as shown in Figure 11, it can also be seen that the forward simulation method in cylindrical coordinates is significantly better than that in the Cartesian coordinate system in suppressing the interference of diffracted waves at the mutation point. Because survey line 1 is arranged around the tunnel, the seismic profileing event axis presents a typical \"X\" shape. The intersection of the event axis corresponds to the receiver at the bottom of the tunnel. Therefore, the dispersion caused by the interference of seismic waves is only generated in the lower half of the \"X\" shape. The frequency is higher than the direct wave, and the energy is weaker than the direct wave, and weakens as the seismic wave propagates toward the front of the tunnel. The diffracted wave interference caused by the mutation point is different. It can be seen from Figure 11a,b that the wavelength of the diffracted wave interference is close to the direct wave, and will propagate toward the front of the tunnel. When the direct wave energy weakens with its propagation distance increases, the diffracted waves even conceal the in-phase axis of the shear wave.  
Figure 10: Wavefield results in \(r\theta\) plane. Wavefield results at different time (**a**) in the cylindrical coordinate system and (**b**) the Cartesian coordinate system.