## 4 Experimental Results
### Validation Using Direct Leveling Based Subsidence Observations  
The proposed method was validated by direct leveling-based subsidence observations, which were collected from a coal mine in Shandong Province, China. An irregular mining area is located in the northern No.1 district of the coal mine; as shown in Figure 12, the irregular mining area is constructed by eight longwall working faces. The maximum and minimum underground elevations are about \(-\)375 m and \(-\)670 m respectively, and the ground of the mining area is quite flat, with an elevation of 42.1 m before the underground extraction. The average coal seam thickness is 6.65 m. The average dip angle of the coal seam is about 10\({}^{\circ}\) and the dip direction and strike direction of the coal seam are as shown in Figure 12. The maximum lengths of the mining area in these two directions are about 1560 m and 1010 m, respectively. The starting mining date of the first working face (No. 1301) of the mining area was 12 March 2006, and the last working face (No.1308) was terminated on 16 May 2015. The terminating date of each working face is also shown in Figure 12. In January 2006, three ground observation lines (observation line of A, B and C) were set up, with a total of 130 observation points; the relative position of the mining area and these three observation lines is as shown in Figure 12. The observation line of A, B and C includes 46, 59 and 25 observation points respectively. Subsidence observations of these ground points were measured 22 times with fourth-order direct leveling from 25 February 2006 to 9 March 2017. However, because many points were destroyed during the observation period by human or subsidence-induced water pooling, only 46 surface points were available when the last measurement was performed. The numbers of available ground observation points are 15, 19 and 12 for the observation line of A, B and C respectively. The last subsidence observations of these available ground points are given in Figure 13.  
Figure 11: Scatterplot of the RMS of the proposed method-based subsidence estimation errors versus the ratios of the amount of random points to 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
100 & 2.45 & 23.68 & 23.81 \\
200 & \(-\)1.76 & 10.17 & 10.32 \\
300 & \(-\)0.53 & 5.77 & 5.79 \\
400 & 0.32 & 2.13 & 2.15 \\
500 & 0.18 & 1.01 & 1.03 \\
600 & \(-\)0.15 & 0.75 & 0.76 \\
700 & 0.06 & 0.26 & 0.27 \\
800 & \(-\)0.01 & 0.07 & 0.07 \\
900 & 0.04 & 0.05 & 0.06 \\
1000 & 0.00 & 0.02 & 0.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean, STD and RMS of the proposed method-based subsidence estimations errors under different ratios of the amount of random points to the area of the corresponding extraction element. According to the previously observed data in the southern No.1 district of the coal mine, the subsidence prediction parameters could be obtained, including the subsidence factor (0.83), the tangent of the main effect angle (2.0), and the inflection point offset coefficient (0.05). A minimum deviation of the inflection point of the mining area of 0.5 times was selected as the mesh size for the proposed method, which is equal 10.4 m, and the amount of random points within an extraction element was set as 800 times the area of the element. By making use of these parameters, the proposed method-based subsidence predictions on the ground points and the predicted subsidence contours could be obtained. The predicted subsidence contours of the mining area are also shown in Figure 12. Taking the direct leveling-based subsidence observations as the in-situ ones, the proposed method-based subsidence prediction error could be calculated by deducting the direct leveling-based subsidence observation from the predicted one.