# Frequency Domain Full Waveform Inversion Method of Acquiring Rock Wave Velocity in Front of Tunnels
## 3 Numerical Simulation
### Karst Geological Model  
The karst geological models with a thickness of 10 m and 30 m are established respectively (Figure 10a,b). The size of the forward model is 160 m \(\times\) 80 m, and the grid spacing is \(\Delta x=\Delta y=1m\). The CFS-PML absorption boundary around the model is set referring to the tunnel size in the actual project. The tunnel size in the forward model is set to length 50 m \(\times\) width 12 m and the tunnel cavity is filled with air. The cavern is elliptical with a  
Figure 8: Seismic records of CSP gathers of 30 m thick fault numerical model in underside wall.  
Figure 9: (**a**) FWI result of 30 m thick fault rock wave velocity numerical model. (**b**) FWI result of 110 m thick fault rock wave velocity numerical model.  
width of 20 m up and down, a thickness of 10 m left and right, and the nearest point of the front interface is 30 m away from the tunnel face. The wave velocity of the surrounding rock of the tunnel is set to 4000 m/s. The cave is filled with broken rock mass, and the wave velocity is 2000 m/s.  
Figure 11a,b show the CSP seismic records of the common shot gathers of all the lower wall geophones with the lower wall 5 m away from the tunnel face selected as the seismic source in the karst cave model with a diameter of 10 m and 30 m. The original seismic records of no anomalous body in front of the tunnel face are also subtracted. It can be seen that the three sets of strong event axes all show the characteristics of weaker energy as the distance between the geophone and the cavern increases. The first strong event axis corresponds to the front interface of the cave. The second group of events consists of two phase axes, corresponding to the back interface of the cave. The third phase axis and the following multiple weak phase axes are considered to be multiple reflection waves and diffracted waves after analysis.  
The frequency domain FWI is carried out via the observation mode shown in Figure 3, and the results are shown in Figure 12a,b. In addition, only the wave velocity of the front of tunnel face is updated. A white dotted line represents the position and shape of the real lithologic interface. It can be seen that the inversion results of the cave with a diameter of 20 m are ideal, the position and shape of the cave inversion are more accurate, and the wave velocity inside the cave is very close to the real wave velocity. In the inversion result, there is a wave velocity interface between the cave and the surrounding rock, which makes the cave slightly larger. Overall, the location of the cavern is accurate and easy to identify, which can provide good guidance for construction.  
Figure 10: (a) A 10 m karst rock wave velocity numerical model. (b) A 30 m karst rock wave velocity numerical model.  
Figure 11: (a) Seismic records of CSP gathers of 10 m karst cave numerical model in underside wall. (b) Seismic records of CSP gathers of 30 m karst cave numerical model in underside wall.  
Figure 12: (a) FWI result of 10 m karst cave numerical model. (b) FWI result of 30 m karst cave numerical model.