## 3 Proposed Ground Subsidence Prediction Method
### Selection of Extraction Elements within the Calculation Area  
By making use of DTM, an irregularly shaped mining area could be divided into a series of triangular extraction elements. Because of the cantilever effect of the coal wall, only the extraction elements within the calculation area would induce ground subsidence. Although the calculation area can be easily determined though deviations of the inflection point for the regular-shaped mining area (i.e., rectangle-shaped mining area), the determination of the calculation area for an irregularly shaped mining area is a challenging problem, due to the complicated mining area boundary. Due to the fact that the calculation area is related to the deviation of the inflection point, it is possible to determine whether an extraction element is within the calculation area by comparing the deviation of the inflection point and the distance from the mining area boundary to the incenter of the extraction element. If the latter is greater than the former, the extraction element is free of the effect of the cantilever, that is, the element is within the calculation, whilst this is not true otherwise. As shown in Figure 4, a quadrilateral mining area is constructed by four boundary lines (\(L_{1}\), \(L_{2}\), \(L_{3}\) and \(L_{4}\)); \(P_{1}\), \(P_{2}\), \(P_{3}\) and \(P_{4}\) are four vertexes of the area. \(C\) is the incenter of an extraction element. The coordinates of the incenter of the element can be calculated by:  
\[\left\{ \begin{array}{l}x_{0}=\frac{ax_{1}+bx_{2}+cx_{3}}{a+b+c}\\ y_{0}=\frac{ay_{1}+by_{2}+cy_{3}}{a+b+c}\end{array} \right. \tag{6}\]  
where (\(x_{0}\), \(y_{0}\)) is the coordinate of the extraction element incenter; \(a\), \(b\) and \(c\) are the lengths of three sides of the triangular extraction element; (\(x_{1}\), \(y_{1}\)), (\(x_{2}\), \(y_{2}\)) and (\(x_{3}\), \(y_{3}\)) are the coordinates of three vertexes of the extraction element. Boundary distance (BD), which is the distance from the extraction element to a mining boundary, is treated as the minimum distance from the incenter of the element to the boundary line. For example, the BD from the incenter of the element to a mining boundary line is \(d_{1}\) for the boundary line \(L_{1}\), and \(d_{2}\) for \(L_{2}\). Therefore, the amount of BDs is equals to the amount of mining area boundary lines for each extraction element. As shown in Figure 4, four BDs are termed as \(d_{1}\), \(d_{2}\), \(d_{3}\) and \(d_{4}\) respectively for the extraction element. For each BD, there is a corresponding point on the boundary line, which construct a BD with the incenter of the extraction element. These points (or boundary points, BPs) are termed as \(P_{1}\)', \(P_{2}\)', \(P_{3}\)'and \(P_{4}\)' respectively in Figure 4. Note that, because of the different relative positions between the incenter of the extraction element and the boundary line of the mining area, the determination of BPs is different. As shown in Figure 5a (i.e., Case A), the incenter of the extraction element is located between two endpoints of the mining boundary line \(L_{3}\)  
Figure 3: Example of segmentation result of an irregularly shaped mining area using DTM. the BP of the boundary line is the projection of the incenter on the boundary line (i.e., \(P_{3}\)?). As shown in Figure 5b (i.e., Case B), the incenter is out of two endpoints of the mining boundary line \(L_{2}\); the BP of the boundary line is the endpoint of the boundary line which is the closet to the incenter (i.e., \(P_{3}\) or \(P_{2}\)?). These two cases can be determined by:  
\[\left\{ \begin{array}{c}P_{i}\overrightarrow{P_{i+1}}\times \overrightarrow{P_{i}}C>0\\ P_{i+1}P_{i}\times P_{i+1}C>0\end{array} \right.\quad\text{(Case A)} \tag{7}\]  
\[\left\{ \begin{array}{c}P_{i}\overrightarrow{P_{i+1}}\times \overrightarrow{P_{i}}C>0\\ P_{i+1}P_{i}\times P_{i+1}C<0\end{array} \right.\quad\text{or}\quad\left\{ \begin{array}{c}P_{i}\overrightarrow{P_{i+1}}\times\overrightarrow{P_{i}}C <0\\ P_{i+1}P_{i}\times P_{i+1}C>0\end{array}