# Frequency Domain Full Waveform Inversion Method of Acquiring Rock Wave Velocity in Front of Tunnels
## 4 Discussion  
This paper mainly discusses the forward and inverse problems in the frequency domain FWI. First, the nine-point difference scheme is used in frequency domain wave equation forward modeling. Compared to the large-grid forward simulation on the ground, the nine-point difference scheme not only meets the accuracy requirements, but also greatly reduces the calculation amount and improves calculation efficiency. The tunnel ahead prospecting has a smaller scale than the seismic exploration on the surface, so the boundary condition needs to be more detailed. The CFS-PML boundary condition is adopted to effectively suppress the boundary false reflection and ensures the accuracy of the forward modeling. In the inversion process, the objective function is initially established. The FWI gradient formula is obtained using the adjoint state method, which greatly reduced the calculation amount compared with the Jacobian matrix derivation. The parabola method is used to obtain the optimized step length, thereby speeding up the convergence speed. In the optimization algorithm, the LBFGS algorithm is used to perform local optimization solutions, which greatly reduces the amount of calculation and storage while ensuring high accuracy. Finally, the theoretical framework of FWI is established.  
In the research of several typical undesirable geological numerical examples, we have explored some rules. For the lithological interface, if the dip angle is large (90\({}^{\circ}\)), the FWI method can give the accurate position and shape of the lithological interface. The inversion value of the rock and wave velocity behind the interface is also accurate. For a small inclination interface (45\({}^{\circ}\)), the FWI method cannot give the accurate position and shape of the interface. The rock wave velocity inversion behind the interface also has a certain error. However, it can still give an accurate wave velocity change trend, which can provide guidance for construction. For faults, the position of the interface before and after the fault and the approximate wave velocity of the internal rock mass can be accurately given. When the thickness of the fault is small, the upper part of the fault has a certain distortion and the inversion result will be further deteriorated when the thickness of the fault is further reduced. For karst caves, the location, shape, and approximate wave speed of the internal rock mass can be inverted accurately. However, when the cave is small, the FWI method can only invert the presence of low-velocity anomalies in front of the cave, and the specific position and shape cannot be obtained. As the size of the cave is further reduced, the effect will be further deteriorated.  
This paper only studies the two-dimensional FWI method in tunnels. Actually, ahead prospecting of tunnel engineering is a three-dimensional full-space problem. Thus, the three-dimensional FWI method in tunnels needs to be studied further. However, the FWI method has a large amount of calculation and long calculation time, which does not meet the requirement of rapid construction in the project. Therefore, more effective parallel methods should be studied continuously to improve the computing efficiency and increase the practicality of the three-dimensional FWI method in tunnels in future work.