# InSAR-CTPIM-Based 3D Deformation Prediction in Coal Mining Areas of the Baisha Reservoir, China
## 2 Methods  
### SBAS-Based Estimation of Line-of-Sight Deformation  
Firstly, the small baseline subset (SBAS) technique was used to obtain the LOS deformation of the mine surface. This technique is used to invert the LOS deformation of the area by solving the deformation of a large number of dense, high-coherence points in the area. Assuming that \(N+1\) SAR images of the study area are acquired and \(M\) differential interferograms are generated [20; 21], the phase of the \(i\)-th pixel on the \(m\)-th (\(1\leq m<M\)) differential interferogram can be expressed as follows [22]:  
\[\begin{array}{l}\delta\varphi_{m}^{i}=\delta\varphi_{def}^{i}+\delta\varphi_{ topo}^{i}+\delta\varphi_{orbit}^{i}+\delta\varphi_{atm}^{i}+\delta\varphi_{noise}^{i}+ \delta\varphi_{Non}^{i}\\ \approx\frac{4\pi}{\lambda}\Delta d^{i}+\frac{4\pi B_{i}}{\lambda R\sin\theta} \Delta H^{i}+\delta\varphi_{res}^{i}\end{array} \tag{1}\]  
where \(\delta\varphi_{def}^{i}\) denotes the deformation phase, \(\lambda\) denotes the radar wavelength, \(\Delta d(i)\) is the deformation component of the \(m\)-th differential interferogram, which is called the Low Pass (LP) deformation component, \(\delta\varphi_{topo}^{i}\) denotes the terrain phase \(\delta\varphi_{topo}^{i}=(4\pi B_{i}/\lambda R\sin\theta)\Delta H^{i}\), where \(B_{i}\) denotes the length of the vertical baseline, \(R\) is the distance between the target and the radar satellite location, \(\theta\) denotes the radar incidence angle, and \(\Delta H^{i}\) denotes the elevation correction value, which is regarded as an unknown parameter, \(\delta\varphi_{orbit}^{i}\) denotes the orbital error phase, and \(\delta\varphi_{res}^{i}\) denotes the residual phase, including atmospheric delay phase \(\delta\varphi_{atm}^{i}\), noise phase \(\delta\varphi_{noise}^{i}\), and High Pass (HP) deformation component \(\delta\varphi_{Non}^{i}\), where \(\Delta d^{i}=d(t_{\rm B}^{i})-d(t_{\rm A}^{i})\), \(d(t_{\rm B}^{i})\) and \(d(t_{\rm A}^{i})\) denote the deformation (\(t_{B}>t_{A}\)) components of the point relative to the start time \(t_{0}\) (i.e., \(d\) (\(t_{0}\)) = 0) at the image acquisition moments \(t_{B}\) and \(t_{A}\), respectively, which can be expressed as below [23; 24; 25]:  
\[\Delta d^{i}=\sum_{\begin{subarray}{c}t_{B}^{i}\\ t_{A}^{i}+1\end{subarray}}^{t_{B}^{i}}v_{k}(t_{k}-t_{k-1}) \tag{2}\]  
where \(v_{k}\) is the linear deformation rate at each interference epoch, \(k\) is the number of interference pairs, and Equation (1) can be written as follows:  
\[\delta\varphi_{m}^{i}=Bv_{k} \tag{3}\]  
The process of solving (3) is a least squares problem, where \(N\) linear velocities \(v\) and elevation corrections \(\Delta H^{i}\) need to be estimated according to Equations (1) and (2). Since the matrix \(B\) is a rank-loss matrix, the singular value decomposition algorithm [26] can solve the minimum van solution for the unknown parameters \(v_{k}\) and \(\Delta H^{i}\) in Equation (1), and then the deformation rate is integrated over each period, and the low-frequency deformation sequence can be calculated by substituting into Equation (3) [27]. The atmospheric delay phase component is a high-frequency signal in the time dimension and a low-frequency signal in the spatial dimension, and the nonlinear deformation is a low-frequency signal in the time dimension. Therefore, the time-dimensional high-pass filtering and spatial-dimensional low-pass filtering are used to obtain the high-frequency nonlinear deformation (HP) component, which is then accumulated with the low-frequency linear deformation (LP) component, and the total deformation volume of the point can be estimated. The principle of the SBAS-InSAR technique is shown in Figure 1.