# Frequency Domain Full Waveform Inversion Method of Acquiring Rock Wave Velocity in Front of Tunnels
## 3 Numerical Simulation
### Geological Interface Model  
The lithologic interface models with a 90\({}^{\circ}\) and 45\({}^{\circ}\) dip angle are established respectively (Figure 4a,b). The size of the forward model is 160 m \(\times\) 80 m and 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 rock wave velocity on one side of the tunnel is set to 4000 m/s and the rock wave velocity in front of the lithological interface is set to 2000 m/s. The seismic source adopts Ricker wavelet, and the main frequency is 200 Hz. The sampling interval of the detector is 0.005 ms. Figure 5a,b shows the Common Source Point (CSP) gather seismic records of the common shot gather of all the lower side wall geophones selected as the seismic source at a distance of 5 m from the tunnel face of the lower side wall. In order to make the reflected waves in the seismic records clearer, the interference of the tunnel cavity and the tunnel surface scattered wave are removed. In other words, here the original seismic records of no anomalous bodies in front of the tunnel are subtracted from the 90\({}^{\circ}\) and 45\({}^{\circ}\) dip angle lithology interface model CSP gather seismic records. In Figure 5a, it can be seen that the seismic wave detection response characteristic of 90\({}^{\circ}\) lithologic interface is a continuous homogeneous axis of inclined straight line. The weaker homogeneous axes at the back is caused by multiple reflections of the reflected wave between the 90\({}^{\circ}\) lithologic interface and the tunnel face. \begin{table}
\begin{tabular}{c c} \hline
**Geological Structures** & **Further Explanations** \\ \hline Lithologic interface & The rock interface of two different lithology. \\ \hline Fault fracture zone & In the process of fault formation, fracture of surrounding rock is often caused by the dislocation of upper and lower plates, forming a fault fracture zone. \\ \hline \multirow{2}{*}{Karst geological body} & Karst systems such as caverns and crevices often exist in karst areas. \\  & Cavities created by the long-term dissolution of groundwater are called caverns. \\ \hline Composite strata & For instance, Wedge bad geological bodies, a group of caverns consisting of multiple caverns, etc. \\ \hline \end{tabular}
\end{table} 
Table 1: The explanations of undesirable geological structures in tunnels. Figure 3: Schematic diagram of the observation system. Figure 4: (**a**) A 90\({}^{\circ}\) lithologic interface rock wave velocity numerical model, where the red dotted line indicates the tunnel environment. Its specific observation method is shown as Figure 3. (**b**) A 45\({}^{\circ}\) lithologic interface rock wave velocity numerical model. Figure 5: (**a**) Seismic records of CSP gathers of 90\({}^{\circ}\) lithologic interface numerical model in underside wall. (**b**) Seismic records of CSP gathers of 45\({}^{\circ}\) lithologic interface numerical model in underside wall. The frequency domain FWI is carried out via the observation mode shown in Figure 3 and the results are shown in Figure 6a,b. The actual engineering pays more attention to the geological conditions of the unexcavated section; consequently, only the wave velocity in front of the tunnel face is updated here. There is an obvious wave velocity interface in the figure, and its position corresponds well to the position of the real lithological interface (white dotted line). It can be seen that the inversion result can basically reflect the position and shape of the lithological interface. In terms of the accuracy of the inverted wave velocity value, there is a wave velocity transition zone near the interface, and the inverted wave velocity at other locations is slightly higher than the actual situation.