# A Fast Ray-tracing Method for Locating Mining-induced Seismicity by Considering Underground Voids
## 3 Synthetic Tests
### Comparison and Analysis  
We use an example to compare the proposed method with the classic FMM and SPR. As shown in Figure 10, this void model consists of 8 vertices and 12 faces, with a range of \(x\in[40,70],\ y\in[40,70],\ z\in[40,70]\). The source coordinates are (0,50,50) and the coordinates of the 25 receivers R1-R25 are shown in Table 1. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & **R1** & **R2** & **R3** & **R4** & **R5** & **R6** & **R7** & **R8** & **R9** \\ \hline  & x (m) & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\  & y (m) & 0 & 0 & 0 & 0 & 0 & 21 & 21 & 21 \\  & z (m) & 0 & 21 & 42 & 63 & 84 & 0 & 21 & 42 & 63 \\ \hline \multirow{3}{*}{Travel time (ms)} & Proposed method & 24.50 & 23.10 & 22.42 & 22.51 & 23.37 & 23.10 & 21.62 & 20.89 & 20.99 \\  & FMM & 24.55 & 23.16 & 22.46 & 22.56 & 23.43 & 23.16 & 21.67 & 20.93 & 21.03 \\  & SPR & 24.49 & 23.60 & 22.70 & 22.92 & 23.81 & 23.60 & 22.61 & 21.71 & 21.92 \\ \hline  & & R10 & R11 & R12 & R13 & R14 & R15 & R16 & R17 & R18 \\ \hline  & x (m) & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\  & y (m) & 21 & 42 & 42 & 42 & 42 & 42 & 63 & 63 & 63 \\  & z (m) & 84 & 0 & 21 & 42 & 63 & 84 & 0 & 21 & 42 \\ \hline \multirow{3}{*}{Travel time (ms)} & Proposed method & 21.91 & 22.42 & 20.89 & 20.32 & 20.43 & 21.33 & 22.51 & 20.99 & 20.43 \\  & FMM & 21.97 & 22.46 & 20.93 & 20.71 & 20.61 & 21.49 & 22.56 & 21.03 & 20.61 \\  & SPR & 22.84 & 22.70 & 21.71 & 21.00 & 21.22 & 21.98 & 22.92 & 21.92 & 21.22 \\ \hline  & & R19 & R20 & R21 & R22 & R23 & R24 & R25 & & \\ \hline  & x (m) & 100 & 100 & 100 & 100 & 100 & 100 & 100 & & \\  & y (m) & 63 & 63 & 84 & 84 & 84 & 84 & 84 & & \\  & z (m) & 63 & 84 & 0 & 21 & 42 & 63 & 84 & \\ \hline \multirow{3}{*}{Travel time (ms)} & Proposed method & 21.27 & 21.43 & 23.37 & 21.91 & 21.33 & 21.43 & 22.33 & \\  & FMM & 22.83 & 21.59 & 23.43 & 21.97 & 21.49 & 21.59 & 22.68 & \\  & SPR & 21.86 & 22.19 & 23.81 & 22.84 & 21.98 & 22.19 & 23.09 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Receiver coordinates and travel time calculated using the proposed method, FMM, and SPR. Figure 10: The ray path of the receiver R20 is calculated by the proposed method, FMM, and SPR. (**a**) The view on XY plane, (**b**) the view on YZ plane, and (**c**) the view on XZ plane. We arranged the second category of nodes on the edges of the model with a step length of 1. According to Equation (4), the total number of nodes is 634. Then, we used the proposed method for ray tracing. The travel time of the wave from the source to the receivers is shown in Table 1. Similarly, we use classic FMM and SPR for ray tracing. The grid size is \(1\times 1\times 1\), and there is a total of \(100\times 100\times 100\) grid nodes. The results obtained with FMM and SPR are shown in Table 1. The difference between the arrival time calculated by the proposed method and the theoretical value is less than \(10^{-5}\) s, so the theoretical arrival time is not shown in Table 1.  
In Figure 10, we take the receiver R20 as an example to show the ray paths calculated by three methods. The ray path calculated by the proposed method coincides with the theoretical ray path. It also can be seen that the ray path calculated by the proposed method is similar to that calculated by FMM, but is quite different from that calculated by SPR. We also compared the error between the arrival time of R1-R20 and the corresponding theoretical value, as shown in Figure 11. The error of arrival time calculated by our proposed method is the smallest and the maximum error is less than 0.0093%. According to [PERSON]'s principle, the wave propagates along the minimum arrival path, so the ray path calculated by the proposed method is more accurate. SPR and FMM have the same grid size, but the error of the ray path calculated by SPR is greater. Above we conclude that the ray-tracing accuracy of the proposed method is higher than that of the classic FMM and SPR. Next, we compare the efficiency of the proposed method with the classic FMM and SPR.