# InSAR-CTPIM-Based 3D Deformation Prediction in Coal Mining Areas of the Baisha Reservoir, China
## 2 Methods
### Three-Dimensional CTPIM Modeling Based on Coordinate-Time Functions  
A three-dimensional deformation prediction model (3D-CTPIM) between dynamic surface subsidence and horizontal displacement with probabilistic integral parameters is constructed as follows [28]:  
\[W(x,y,ti)=\frac{W_{0}(x,ti)W_{0}(y,ti)}{w_{0}} \tag{4}\]  
\[U(x,y,\gamma,ti)=\frac{U_{0}(x,ti)W_{0}(y,ti)\cos\gamma}{w_{0}}+\frac{U_{0}(y, ti)W_{0}(x,ti)\sin\gamma}{w_{0}} \tag{5}\]  
where \(W(x,y,ti)\) and \(U(x,y,\gamma,ti)\) are the subsidence and horizontal displacement of any point on the surface of the mining area at the moment of workforce advancement \(t_{i}\), \(w_{0}\) is the maximum settlement under mining conditions, \(w_{0}=mq\cos\alpha\), where \(m\) indicates the mining thickness, \(q\) indicates the subsidence coefficient, and \(\alpha\) indicates the inclination angle of the mining area; \(\gamma\) indicates the angle between the advancing direction of the workface and a given direction, and the north-south displacement \(U_{N}\) can be obtained when the input value of the angle \(\gamma\) is \(\gamma_{N}\) in Equation (5), and the east-west displacement \(U_{E}\) (see Figure 2a) can be obtained when the input value of \(\gamma\) is \(\gamma_{E}\). \(W_{0}(x,t_{i})\), \(W_{0}(y,t_{i})\) and \(U_{0}(x,t_{i})\), \(U_{0}(y,t_{i})\) can be expressed as follows under different mining conditions:  
1. When the mining tendency of the mine is fully mined and the strike is not fully mined, the \(W_{0}(x,ti)\) in Equation (4) can be expressed as follows [16]: \[W_{0}(x,ti)= \begin{cases}\frac{w(x)}{2}\Big{[}erf(\sqrt{\pi})+erf\Big{(}\sqrt{ \pi}\frac{cmti-nH-r}{r}\Big{)}\Big{]},\ x\leq nH\\ \frac{w(x)}{2}\bigg{\{}erf\bigg{[}\sqrt{\pi}\Big{(}erf(\sqrt{\pi})+\frac{vmti -r-x+\frac{H}{2+bx}}{r}\frac{H(a+bx)}{3}\Big{)}\bigg{]}\bigg{\}},\ nH\leq x \leq 1.3r\\ =\frac{w_{0}}{2}\Big{[}erf\Big{(}\frac{\sqrt{\pi}}{r}\Big{)}+erf\Big{(}\sqrt{ \pi}\frac{vmti+H/\tan\omega-2x}{r}\Big{)}\Big{]},\ 1.3r\leq x\end{cases} \] (6) where \(x\) denotes the horizontal coordinate of any high-coherence point in the mining area, \(w(x)\) is the static subsidence, which can be expressed as \(w(x)=\frac{w_{0}}{2}\big{[}erf\big{(}\sqrt{\pi}\frac{x}{r}\big{)}+1\big{]}\); _eff_ is the probability integral function, which can be expressed as \(erf\bigg{(}\frac{\sqrt{\pi}}{r}x\bigg{)}=\frac{2}{\sqrt{\pi}}{\int_{0}^{\frac{ \sqrt{\pi}}{r}}}x^{-u^{2}}du\), where \(u\) is the integral parameter; \(r=\frac{H}{\tan\beta}\) is the main influence radius, where \(H\) is the mining depth, and \(\tan\beta\) is the tangent value of the main influence angle; \(nH\) is the start-up distance, \(n\) is the start-up coefficient, \(\omega\) is the overshooting static influence angle, \(v_{m}\) is the constant speed under uniform mining conditions, and the coefficients \(a\) and \(b\) can be expressed as \(a=\frac{117r-umH}{1.3r+nH}\), \(b=\frac{\omega-90}{1.3r+nH}\), respectively.\(U_{0}(x,t_{i})\) in Equation (5) can be expressed as follows: \[U_{0}(x,ti)=\sum_{0}^{i}\{\sigma(ti-tk)\left[U_{1}\bigg{(}x-\sum_{j=0}^{k}(vj \cdot tj)\bigg{)}\ -U_{1}\bigg{(}x-\sum_{j=1}^{k+1}(vj\cdot tj)\bigg{)}\right]\}\] (7) \[\sigma(tj)=1-\exp(-ctj)\] (8) where \(U_{1}(x)=d\ w_{0}\ \exp\left(-\pi\frac{x^{2}}{r^{2}}\right)\), \(d\) is the horizontal displacement constant, \(t_{j}\) is the mining time, and \(v_{j}\) is the average propulsion velocity. The \(\sigma(tj)\) in Equation (8) is the time function, where \(c\) is a time parameter that characterizes the change in horizontal movement with time. 2.