Elastic full-wave field simulation in 3D tunnel space using a variable staggered-grid finite-difference method in cylindrical coordinates

Tunnel advance detection technology is an important method for determining the structure of a complex geological body in front of the tunnel face. Among the tunnel advance detection technologies, the seismic method is one of the most accurate methods with long detection distances. In seismic tunnel advance detection, the cylindrical configuration aggravates the complexity of the wave field in the tunnel space and significantly influences the accuracy of the detection results. Thus, it is crucial to simulate an accurate seismic full-wave field of the tunnel space and to understand the propagation and wave-field characteristics of individual seismic waves for seismic tunnel advance detection. Usually, in 3D Cartesian coordinates, the tunnel wall is approximated with a staircase boundary, but it is not sufficiently accurate in shape and generates numerical dispersion in the simulation, especially in the presence of surface waves. Therefore, we developed a variable staggered-grid finite-difference method in cylindrical coordinates to simulate the elastic full-wave field in a 3D tunnel space. Setting free-surface boundary conditions solves the propagation of surface waves on the tunnel wall and face. The free-surface boundary condition was validated by comparing the simulated seismic records with the finite element method. The interference of the instability and discontinuity of the pole axis in the seismic wave field simulation was eliminated using our method. Using this scheme, we simulated the elastic full-wave field of three geological bodies in front of the tunnel face, including the vertical interface, inclined interface, and karst cave. The results of the three models show that the excitation near the tunnel face is more conducive to the detection of geological bodies. Compared with the simulation results in Cartesian coordinates, the results in cylindrical coordinates show that numerical dispersion is negligible and conclude that a higher signal-to-noise ratio and more accurate seismic wave field can be simulated with cylindrical coordinates in the tunnel space. The new method can also be used as an accurate elastic wave propagator for reverse-time migration and full-waveform inversion under tunnel-observing geometries. Our simulation method provides theoretical and practical guidance for analyzing and interpreting seismic wave fields in tunnel advance detection.