## 3 Proposed Ground Subsidence Prediction Method
### Irregularly Shaped Mining Area Segmentation Using DTM  
The Delaunay triangulation method, which was invented by [PERSON] in 1934, is an important preprocessing technique for computational geometry and mathematics [33]. The method could divide arbitrarily shaped polygons (including convex polygons and concave polygons) into multiple triangular sub-regions. Because DTM has many advantages (e.g., optimality, uniqueness) compared with other triangular mesh methods, and can be easily implemented by programing languages, it is more applicable in the segmentation of irregularly shaped mining areas. Based on DTM and the Monte Carlo integration method, computer software (Subsidence Observation Data processing Program, SODP) has been developed by the Lunan Geo-Engineering Exploration Institute of Shandong Province and Wuhan University. The software was used to perform the following numerical calculation. Figure 2a shows the flowchart of the DTM-based segmentation algorithm of an irregularly shaped mining area. The inputs of the algorithm are the vertex coordinates of the mining area and mesh size. Based on the inputting parameters, mesh points in the mining area could be generated. There are two kinds of generated mesh points; one is the points at the boundary of the mining area and the other is the points in the interior of the mining area. The former could be generated by inserting points along with the boundary lines of the mining area (parallel to the direction of the \(x\) axis and \(y\) axis, inserting mesh points respectively), and the latter could be generated. The distance between two adjacent mesh points would be equal to the input mesh size in general. After the generation, all the mesh points are marked as \"unused\". Two adjacent mesh points (e.g., \(p_{1}\) and \(p_{2}\)) are selected to construct a vector, \(\overrightarrow{p_{1}p_{2}}\) (or starting edge), as shown in Figure 2b, and at the same time these two points are marked as \"used\"; then, the third point, \(p_{3}\), is selected as follows:
Once the third point is selected, these three points (\(p_{1}\), \(p_{2}\) and \(p_{3}\)) could be used to construct a Delaunay triangle. Using \(p\overset{\rightarrow}{1}p_{3}\) and \(p\overset{\rightarrow}{2}p_{3}\) as the starting edge respectively and repeating the process above produces the other Delaunay triangles. When all the points are marked as \"used\", the segmentation algorithm is finished, and the irregular mining area has been divided into multiple triangular extraction elements. Note that, because of the uniqueness property of the Delaunay triangle, constructing the starting edge with any two adjacent points would produce the same result. Figure 3 shows an example of the segmentation result of an irregularly shaped mining area by making use of DTM. The minimum and maximum lengths of the mining area boundary line are 26.8 m and 180.0 m, respectively, and the mesh size is set as 10.0 m. It can be seen that all the generated mesh points are evenly distributed in the mining area. Due to the optimality property of the Delaunay triangle, the lengths of the three sides are almost the same for each extraction element. This characteristic is important, because it is helpful in determining the calculation area accurately.  
Figure 2: (a) Flowchart of DTM-based segmentation algorithm of irregularly shaped mining area. (b) Example of Delaunay triangulations generated by DTM.