# Frequency Domain Full Waveform Inversion Method of Acquiring Rock Wave Velocity in Front of Tunnels
## 1 Introduction  
With the expanding Chinese economy, many tunnels are being designed and constructed. However, when tunneling in hazardous geologic terrain, with faults, fractures, water-bearing openings, and other adverse geological conditions, construction safety is seriously endangered [1]. The ahead geophysical prospecting method is the main good method to ensure the safety of tunnel construction. The commonly used methods include the seismic ahead prospecting method [2, 3], electromagnetic ahead prospecting method, etc. The main electromagnetic ahead prospecting methods include GPR (Ground-penetrating radar) [4, 5], transient electromagnetic method [6, 7], NMR (nuclear magnetic resonance) [8], etc. Among the above methods, the seismic ahead prospecting method has the advantages of long prospecting distance, good structure recognition effect, and sensitivity to the interface of undesirable geological bodies [9]. Therefore, it is often used for medium- and long-distance prospecting in tunnel ahead prospecting. Since the 1990s, various seismic methods have been popularized and applied in engineering. Initially, the tunnel seismic wave method was mainly used and referred to surface seismic methods such as VerticalSeismic Profiling (VSP) and the multi-wave train exploration method. Based on VSP, some scholars have proposed and developed Horizon Sound Profiling (HSP) [10]. In this method, the reflected wave signal is analyzed in the time domain and frequency domain, and the geological conditions in front of the tunnel are inferred by combining with geological survey. The Seismic Ahead Prospecting (SAP) system, which has the advantages of flexible layout and convenient operation, is suitable for deployment in the narrow space of a tunnel. At present, the most commonly used seismic wave detection method in tunnel construction is the Tunnel Seismic Prediction (TSP) method developed by Swiss Amberg Measurement Technology Co. Ltd (Switzerland, Regensdorf-Watt). By disposing the collected seismic data with the methods of band-pass filtering, first arrival picking, energy balance, reflection wave extraction, wavefield separation, velocity analysis, and depth migration, the distribution of the front adverse geological body and the mechanical parameters of the rock can be obtained. Although the above methods can detect the undesirable geological body in front of the tunnel more accurately, the wave velocity distribution of the geological body cannot be obtained. The Full Wave Inversion (FWI) method, which has the characteristics of high precision and high resolution, makes full use of the dynamic and kinematic information of seismic waves. It can accurately characterize the complex underground structures and invert the physical parameters of the earth's medium. In the 1980s, [PERSON] and [PERSON] proposed generalized least squares FWI in the time domain [11; 12; 13; 14; 15; 16; 17], drawing on [PERSON]'s idea of reverse-time offset [18]. In this paper, the gradient of the objective function is obtained via cross-correlation of the forward wavefield and the back-propagated wavefield with the data residual as the seismic source, which avoids the direct calculation of the Frechet derivative. In the 1990s, [PERSON] et al. [19; 20; 21; 22; 23] developed FWI from the time domain to the frequency domain and realized multi-scale inversion from low frequency to high frequency, which lays the foundation for FWI in the frequency domain. [PERSON] [24] combined time domain forward simulation and frequency domain inversion reconstruction algorithms. [PERSON] and [PERSON] [25] applied the Limited memory Broyden Fletcher Goldfarb Shanno (LBFGS) algorithm to FWI and proved that its convergence speed is fast and easy to achieve. [PERSON] and [PERSON] [26] studied a method of using elastic FWI to locate and characterize the disturbance zone in front of the tunnel. However, there is a lack of frequency domain research.