## 3 Proposed Ground Subsidence Prediction Method
### Extraction Element-Induced Subsidence Prediction Based on the Monte Carlo Method  
In Section 3.2, the extraction elements within the calculation area have been selected. The mining of extraction element-induced ground subsidence could be obtained by making use of Equation (2). However, given the complicated calculation of the double integral function and integral domain, it is hard to calculate the extraction element induced surface subsidence using Equation (2) directly. Figure 6: Flowchart of determination of an extraction element within the calculation area. Referring to Equation (2), the coal mining-induced surface subsidence for a given extraction element within the mining area can be written as:  
\[W_{E}(\varepsilon,\eta)=\frac{W_{cm}}{r^{2}}\cdot\iint\limits_{E}e^{-\pi\frac{(x -r)^{2}+(y-y)^{2}}{r^{2}}}dxdy=\frac{W_{cm}}{r^{2}}\cdot I_{E} \tag{10}\]  
where \(E\) is the domain of extraction element, or integral domain of the probability integration function, and \(I_{E}\) is a definite integral:  
\[I_{E}=\iint\limits_{E}f(x,y)dxdy=\iint\limits_{E}e^{-\pi\frac{(x-r)^{2}+(y-y)^ {2}}{r^{2}}}dxdy \tag{11}\]  
Clearly, the integrand \(f(x,y)\) satisfy the following condition over the domain \(E\):  
\[0\leq f(x,y)\leq 1 \tag{12}\]  
Letting \(D_{E}\) represent a three-dimensional domain \(\{x\in E,y\in E,0\leq z\leq f(x,y)\}\) and \(V_{E}\) represent the volume of the domain, it can be determined from the geometric interpretation of double the integral that the value of the integral \(I_{G}\) is equal to the volume of the domain \(D_{E}\). Letting \(D_{A}\) (auxiliary domain) represent a three-dimensional domain \(\{x\in E,y\in E,0\leq z\leq 1\}\), it can be easily derived that the volume of the auxiliary domain is equal to the area of the extraction element, which can be obtained by the vertex coordinates of the element:  
\[V_{A}=S_{E}=\frac{1}{2}[(x_{1}y_{2}-x_{2}y_{1})+(x_{2}y_{3}-x_{3}y_{2})+(x_{3} y_{1}-x_{1}y_{3})] \tag{13}\]  
where \(V_{A}\) is the volume of the auxiliary domain, and \(S_{E}\) is the area of the extraction element. The volume ratio of the domain \(D_{E}\) to the domain \(D_{A}\) can be written as:  
\[\mu=\frac{V_{E}}{V_{A}} \tag{14}\]  
where \(V_{E}\) is the volume of the domain \(D_{E}\). If the volume ratio can be estimated, the volume of the domain \(D_{E}\) or the integral \(I_{E}\) can be obtained by:  
\[V_{E}=I_{E}=\mu\cdot V_{A}=\mu\cdot S_{E} \tag{15}\]  
Then, substituting (15) into (10), the extraction element-induced ground subsidence can be rewritten as:  
\[W_{E}(\varepsilon,\eta)=\frac{W_{cm}}{r^{2}}\cdot\mu\cdot S_{E} \tag{16}\]  
As shown in Figure 7, by randomly rolling a larger number of three-dimensional points over the domain \(D_{A}\), it can be derived via the Monte Carlo integration method [34] that the volume ratio in Equation (14) is approximately equal to the ratio of the number of points within the domain \(D_{E}\) to the number of points within the domain \(D_{A}\). \[\mu\approx\frac{n_{E}}{n_{A}} \tag{17}\]  
where \(n_{E}\) is the number of points within the domain \(D_{E}\), and \(n_{A}\) is the number of points within the domain \(D_{A}\) (or the number of rolling points). Substituting Equation (17) into Equation (16), the extraction element-induced ground subsidence can be estimated by:  
\[W_{E}(\varepsilon,\eta)\approx\frac{W_{cm}}{r^{2}}\cdot\frac{n_{E}}{n_{A}} \cdot S_{E} \tag{18}\]Note that the Monte Carlo integration method-based ratio estimation of \(n_{E}\) to \(n_{A}\) is guaranteed to converge with the in-situ ratio of \(V_{E}\) to \(V_{A}\) when the number of rolling points increases to infinity [35]. In addition, the accuracy of the Monte Caro method-based subsidence prediction is proportional to the number of rolling points (i.e., \(n_{A}\)). The data processing results show that the difference between the Monte Caro method-based subsidence prediction and the in-situ one is marginal for a given extraction element if the number of rolling points is greater than or equal to 800 times the area of the extraction element.