# InSAR-CTPIM-Based 3D Deformation Prediction in Coal Mining Areas of the Baisha Reservoir, China
## 2 Methods
### InSAR-3D-CTPIM  
Based on Section 2.2, the time series function model between the InSAR LOS-oriented deformation and the 3D coordinate-time function 3D-CTPIM is constructed, and the InSAR LOS-oriented deformation can be expressed as follows [29]:  
\[\Delta d(t_{i})=d(t_{B})-d(t_{A}) \tag{11}\]  
\[d(t_{i})=W\cos\theta-\sin\theta[U_{N}\cos(\alpha_{h}-3\pi/2)+U_{E}\sin(\alpha_ {h}-3\pi/2)] \tag{12}\]  
In Equation (11), \(d(t_{B})\) and \(d(t_{A})\) are the LOS-direction deformation components at moments \(t_{B}\) and \(t_{A}\), respectively, and the LOS-direction deformation component at \(t_{i}\) moments can be expressed as Equation (12), where \(\theta\) is the radar incidence angle, and \(\alpha_{h}\) is the satellite heading angle, and it is known from Equations (4) and (5) that \(W\), \(U_{E}\) and  
Figure 1: SBAS-InSAR technique to acquire one-dimensional time series deformation principle in coal mine area.  
Figure 2: (a) Spatial coordinate system of 3D-CTPIM model; (b) mine strike limited mining; (c) mine inclination limited mining.  
denote the vertical, east-west and north-south direction deformation components. At this time, the surface three-dimensional displacement can be expressed as follows:  
\[\left\{\begin{aligned} & W(x,y,ti)=f_{W}\\ & U(x,y,\gamma N,ti)=f_{Un}\\ & U(x,y,\gamma E,ti)=f_{Ue}\end{aligned}\right. \tag{13}\]  
where \(f_{W}\), \(f_{Un}\), and \(f_{Ue}\) denote the mapping functions of vertical and horizontal displacements containing unknown parameters, respectively. According to the geometric relationship of the mining area, \(\gamma_{N}\) and \(\gamma_{E}\), respectively, are the counterclockwise angles between the underground mining direction and the north direction and the east direction, which can be obtained by bringing into Equations (14) and (15):  
\[f_{w}=\frac{W_{0}(x,ti)W_{0}(y,t_{i})}{w_{0}} \tag{14}\]  
\[f_{Un}=\frac{U_{0}(x,t_{i})W_{0}(y,t_{i})\cos\gamma_{N}}{w_{0}}+\frac{U_{0}(y,t_ {i})W_{0}(x,t_{i})\sin\gamma_{N}}{w_{0}} \tag{15}\]  
\[f_{Ue}=\frac{U_{0}(x,t_{i})W_{0}(y,t_{i})\cos\gamma_{E}}{w_{0}}+\frac{U_{0}(y,t_ {i})W_{0}(x,t_{i})\sin\gamma_{E}}{w_{0}} \tag{16}\]  
Substituting Equation (13) into Equation (12) to construct the functional relationship between the InSAR LOS deformation and the parameters of the probabilistic integral model, the LOS deformation component at the moment \(t_{i}\) can be expressed as follows:  
\[d(t_{i})=\begin{bmatrix}\cos\theta\\ -\sin\theta\cos(\alpha_{h}-3\pi/2)\\ -\sin\theta\sin(\alpha_{h}-3\pi/2)\end{bmatrix}^{T}\begin{bmatrix}f_{W}\\ f_{Ue}\\ f_{Ue}\end{bmatrix} \tag{17}\]  
Substituting Equation (17) into Equation (1), the function between the InSAR differential interferometric phase and the PIM parameters can be expressed as below:  
\[\delta\varphi^{m}(x,r)=\delta\varphi_{B}(x,r)-\delta\varphi_{A}(x,r)\approx \frac{4\pi}{\lambda}[d(t_{B})-d(t_{A})]+\frac{4\pi B}{\lambda R\sin\theta} \Delta H(x,r)+\delta\varphi^{m}_{res}(x,r) \tag{18}\]  
The above equation describes the time series functional relationship between the InSAR temporal phase and the 3D-CTPIM predicted parameters, incorporating the temporal parameters, which can be used directly to estimate the model predicted parameters using the InSAR phase values. Among them, the mine geological parameters \(G_{1}=[m,H_{1},H_{2},\alpha,\omega,\)\(D_{1},D_{3},\upsilon_{m}]\) can be determined based on the information of the mine geological conditions, which are considered as known parameters. The PIM parameters \(G_{2}=[q,\) tan\(\beta,\)\(d,\theta_{0},\)\(S_{1},\)\(S_{2},\)\(n]\) and the time parameter \(c\). The dynamic prediction of the three-dimensional movement of the mine surface can be realized by substituting the obtained unknown parameters into Equations (4) and (5).