# A Fast Ray-tracing Method for Locating Mining-induced Seismicity by Considering Underground Voids  
[PERSON]  
1 School of Resources and Safety Engineering, Central South University, Changsha 410083, China; [EMAIL_ADDRESS] (P.P. ); [EMAIL_ADDRESS] (L.W. ); [EMAIL_ADDRESS] (Z.H. ); [EMAIL_ADDRESS] (S.T)  
[PERSON]  
1 School of Resources and Safety Engineering, Central South University, Changsha 410083, China; [EMAIL_ADDRESS] (P.P. ); [EMAIL_ADDRESS] (L.W. ); [EMAIL_ADDRESS] (Z.H. ); [EMAIL_ADDRESS] (S.T)  
[PERSON]  
1 School of Resources and Safety Engineering, Central South University, Changsha 410083, China; [EMAIL_ADDRESS] (P.P. ); [EMAIL_ADDRESS] (L.W. ); [EMAIL_ADDRESS] (Z.H. ); [EMAIL_ADDRESS] (S.T)  
[PERSON]  
1 School of Resources and Safety Engineering, Central South University, Changsha 410083, China; [EMAIL_ADDRESS] (P.P. voids (also known as \"goaves\" or \"stopes\"). A constant velocity model without considering the great difference between rock mass and underground voids will result in large location error. To solve this problem, [PERSON] [7] proposed a sectional velocity model in the tunnel, but the voids of the mine are many and complicated, and the velocity from each sensor to the source is different. Furthermore, [PERSON] [8] proposed an anisotropic velocity model for location. This method requires association with rockburst events to solve the underdetermined equation, and it is not suitable for situations where there is no rockburst information. From another perspective, [PERSON] and [PERSON] [9,10] use a ray-tracing method to greatly improve the accuracy of earthquake location. For small-scale underground mines with a large number of known voids, it is appropriate to use ray tracing to improve the accuracy of the microseismic location. Ray tracing is a method to find the propagation path of the wave from the source to the receiver. Traditional ray-tracing methods include shooting methods [11, 12, 13] and bending methods [14,15]. However, the accuracy and efficiency of such methods in complicated models are notably low, and they cannot satisfy the computational requirements of complicated rock masses in engineering. Newer methods include wavefront reconstruction methods [16], slowness matching methods [17], shortest path ray-tracing (SPR) methods [18,19], simulated annealing methods [20], etc. Among them, the finite-difference (FD) approximation to the eikonal equation is very popular to calculate the ray path. In the field of earthquakes, the eikonal solver is used for many seismological problems with very large data sets. The eikonal solvers based on the fast-marching method (FMM) [21] are also common methods in the field of microseismic. [PERSON] developed multi-stencils fast marching methods (MSFMM) [22] based on FMM to improve accuracy. Moreover, the FMM is also applied to the microseismic location by many scholars [23, 24, 25]. The FMM used to solve the eikonal equation implies a plane wave approximation, so it causes larger errors in places close to the source [26]. To overcome this singularity at the source, [PERSON] [27] and [PERSON] [28] use a simpler algorithm based on the factored eikonal equation, but this algorithm causes the error far away from the source to become larger. To fulfill all these objectives, [PERSON] [29] develops a hybrid algorithm that combines a spherical approximation by solving the factored eikonal equation when close to the source and a plane wave approximation by solving the standard eikonal equation when far away from the source. However, one disadvantage of the above method is that it needs to be meshed to calculate the ray path, and a regular cuboid grid is generally used. The complicated geometry of the geological media requires fine-scale grids that may make computationally intensive, particularly for 3D models [30]. The more grid nodes, the lower the computational efficiency. Therefore, the above methods are not suitable for underground mines with many voids.