## 2 Methods
### InSAR-DIB Dynamic Prediction Model and Parameter Solution  
Before using the DIB prediction model for surface prediction, it is necessary to determine its parameters. Generally, the surface deformation values obtained via traditional geodetic methods (leveling and GPS-RTK measurement) are used to invert the prediction parameters. This method not only requires significant effort and material resources but also has low efficiency overall. InSAR has attracted widespread attention from mining surveying scholars due to its high precision, all-day surface measurement, and low cost (especially with the launch of free data-source radar satellites). This paper combines DIB with InSAR technology to develop a surface prediction model suitable for Huainan's thick loose layer mining areas, referred to as \"InSAR-DIB.\" The idea of the model is described below. When the SAR satellite passes over a surface moving basin, it is assumed that the cumulative LOS value of any pixel \(i\) in the moving basin is \(r_{\text{ILOS}}\) (which can be obtained by accumulating the deformation in the LOS direction in each period). The subsidence value \(W_{\text{i}}\), the horizontal movement \(U_{\text{ISN}}\) in the north direction, and the horizontal movement \(U_{\text{iEW}}\) in the east direction of any pixel \(i\) are predicted based on the IB prediction model and parameter system. According to the SAR's imaging geometry principle, the predicted LOS value \(r_{\text{iLOS}}\) of any pixel \(i\) along the LOS direction can be expressed as  
\[r^{\prime}_{iLOS}=-U_{iSN}\sin\theta_{i}\cos(\alpha_{i}-\frac{3\pi}{2})-U_{ iEW}\sin\theta_{i}\sin(\alpha_{i}-\frac{3\pi}{2})+W_{i}\cos\theta_{i} \tag{16}\]  
where \(\theta_{\text{i}}\) is the incidence angle of the radar and \(\alpha_{\text{i}}\) is the heading angle of the satellites. Among these, \(W_{\text{i}}\), \(U_{\text{ISN}}\), and \(U_{\text{iEW}}\) can be calculated according to the mining subsidence prediction model (taking the rectangular shape of the working face as an example):  
\[\left\{ \begin{array}{l}W_{i}=W(x,y)=\frac{1}{W_{0}}W^{0}(x)W^{0}(y)\\ U_{iSN}=\frac{1}{W_{0}}\big{[}U^{0}(x)W^{0}(y)\cos\varphi_{SN}+W^{0}(x)U^{0}(y )\sin\varphi_{SN}\big{]}\\ U_{iEW}=\frac{1}{W_{0}}\big{[}U^{0}(x)W^{0}(y)\cos\varphi_{EW}+W^{0}(x)U^{0}(y )\sin\varphi_{EW}\big{]}\end{array} \right. \tag{17}\]where \(\varphi_{SN}\) is the angle between the working face direction and the north direction in a counterclockwise direction and \(\varphi_{EW}\) is the angle between the working face direction and the east-west direction in a counterclockwise direction. The estimated LOS moving residual \(v_{i}\) of any pixel in the SAR image is defined as  
\[v_{i}=r^{\prime}_{iLOS}-r_{iLOS} \tag{18}\]  
The parameters of the prediction model can be obtained through a multi-population genetic algorithm (MPGA) [36], and the corresponding parameter cost function is  
\[F(X)=\sum_{i=1}^{n}v_{i} \tag{19}\]  
The essence of an MPGA parameter search is to find a set of predicted parameters \(X\) so that the absolute sum of the differences between the predicted and measured values of the LOS movement of all of the pixels involved in the parameter search is minimized, that is, F(\(X\)) = min. However, due to the inherent defects of InSAR technology and the characteristics of mining subsidence movement and deformation, most of the data are incoherent. InSAR technology cannot monitor the cumulative surface deformation value, but instead monitors the deformation between two periods (the cumulative settlement from the initial sinking moment is unknown), and it can often only obtain the settlement value at the edges. From the description in the previous section, it can be seen that the DIB prediction model constructed in this paper can better fit the settlement information of the edges, so the use of edge deformation information can also fit well in the middle of the basin.