## 4 Experimental Results
### Simulation Result
Figure 10: Scatterplot of RMS of the proposed method-based subsidence prediction errors versus ratio of mesh size to deviation of inflection point of the mining area. In addition, it can be observed in Figure 10 and Table 2 that the proposed method-based subsidence predictions are much closer to the in-situ ones (RMSE smaller than 0.2 cm) when the ratio is less than or equal to 0.5 (or the mesh size is less than or equal to 0.5 times the deviation of the inflection point). This observation is useful because we could select a reasonable mesh size to minimize the mining area segmentation-induced prediction error. If the mesh size is small enough, the mining area segmentation-induced subsidence prediction error could be ignored; in this circumstance, the accuracy of the proposed method-based subsidence prediction depends on the integration calculation accuracy. For a given extraction element, the integration calculation accuracy is proportional to the density of random points within the auxiliary domain, which is the ratio of the amount of generated random points within an auxiliary domain to the volume of the auxiliary domain; a larger ratio would produce a larger number of random points within the unit's volume, resulting in an accurate convergence of the probability integral function, and thus more accurate subsidence predictions. Because the volume of the auxiliary domain is equal to the area of the corresponding extraction element, the ratio could also be expressed as a ratio of the amount of generated random points within an auxiliary to the area of the corresponding extraction element. Taking the ratio of the amount of generated points to the area of the corresponding extraction element as the independent variable, and the RMS of the proposed method-based subsidence prediction errors as dependent variable, a scatterplot of the RMSE versus the ratio is shown in Figure 11. Note that the mesh size is set as 0.1 times the deviation of the inflection point (i.e., 2.25 m); because the mesh size is small enough, the mining area segmentation-induced subsidence prediction error could be ignored. It can be seen from Figure 11 that the RMS values of the proposed method-based subsidence prediction errors exponentially decrease with the increase of the ratio. When the ratio is greater than 800, the amount of generated random points within an extraction element is large enough, and the proposed method-based values of probability integral function are much closer to the in-situ ones for the proposed method. This could also be seen from Table 3, which shows the Mean, STD and RMS of the proposed method-based subsidence prediction errors under different ratios. This observation could be used to select a reasonable number of random points within the auxiliary domain for the purpose of minimizing Monte Carlo integration-induced prediction errors; specifically, the number of random points within an auxiliary domain should be greater than or equal to 800 times the area of the corresponding extraction element. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ratio** & **Mean (cm)** & **STD (cm)** & **RMSE (cm)** \\ \hline
0.1 & 0.00 & 0.02 & 0.02 \\
0.2 & \(-\)0.01 & 0.04 & 0.04 \\
0.3 & 0.02 & 0.07 & 0.07 \\
0.4 & \(-\)0.03 & 0.09 & 0.09 \\
0.5 & 0.03 & 0.15 & 0.15 \\
1.0 & 1.10 & 1.12 & 1.57 \\
2.0 & 4.03 & 3.48 & 5.32 \\
3.0 & 5.84 & 5.04 & 7.72 \\
4.0 & 5.86 & 5.11 & 7.78 \\
5.0 & 5.90 & 5.33 & 7.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean, STD and RMS of the proposed method-based subsidence estimation errors under different ratios of mesh size to deviation of inflection point.