## 2 Methods
### IB Prediction Model
\Bigg{)} \tag{8}\]  
where \(l_{3}\) is the calculated length of the working face strike, \(l_{3}=D_{3}-S_{3}-S_{4}\); \(l_{1}\) is the calculated length of the working face inclination, \(l_{1}=(D_{1}-S_{1}-S_{2})\sin(\theta_{0}+\alpha)/\sin(\theta_{0})\); \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) are the offset distances of the downhill turning point, the uphill turning point, the open-mining turning point, and the stop-mining line turning point, respectively; and \(\theta_{0}\) is the mining impact propagation angle. According to the relationship between subsidence and horizontal movement, the horizontal movement value of the main section of the surface movement basin is:  
\[U(x)=bW_{0}\frac{\exp(-x/R_{1})}{\left(1+\exp(-x/R_{1})\right)^{2}}(1-P)+bW_{0} \frac{\exp(-x/R_{2})}{\left(1+\exp(-x/R_{2})\right)^{2}}P \tag{9}\]  
In the above formula, \(b\) is the horizontal movement coefficient. The horizontal movement expression of any point along direction \(\varphi\) can then be obtained as follows:  
\[U(x,y,\varphi)=\frac{1}{W_{0}}\left[U^{0}(x)W^{0}(y)\cos\varphi+W^{0}(x)U^{0} (y)\sin\varphi\right] \tag{10}\]  
where  
\[U^{0}(x)=U(x)-U(x-l_{3}) \tag{11}\]  
\[U^{0}(y)=\left[U(y)+W(y)\cot\theta_{0}\right]-\left[U(y-l_{1})+W(y-l_{1})\cot \theta_{0}\right] \tag{12}\]  
The inclination \(i\)(x,y,\(\varphi\)), curvature \(k\)(x,y,\(\varphi\)), and horizontal deformation \(\varepsilon\)(x,y,\(\varphi\)) of the surface point \(A\)(x,y) along the \(\varphi\) direction can be obtained using similar methods. No further derivation is shown here. In the model, \(q\), \(\theta_{0}\), \(S_{1}\), \(S_{2}\), \(S_{3}\), \(S_{4}\), and \(b\) have the same meanings as those defined via the PIM, and the parameters are similar. In the new model, \(R_{1}\) and \(R_{2}\) are important influencing radii, with the main influence angle tangents being \(\tan\beta_{1}=H_{0}/4.13R_{1}\) and \(\tan\beta_{2}=H_{0}/4.13R_{2}\). \(H_{0}\) is the coal seam mining depth. Therefore, there are ten parameters in the new model: \(q\), \(P\),  
Based on the actual situation, the following relationship is proposed to describe the relationship between the subsidence rate \(q\), and the subsidence factor \(q\) under insufficient mining conditions:  
\[q^{\prime}=f_{\text{c}}(D_{3}/H_{0})\cdot q \tag{13}\]  
\[f_{\text{c}}(D_{3}/H)=1-\exp[-C_{0}(D_{3}/H_{0})] \tag{14}\]  
where \(f_{\text{c}}\) (\(D_{3}/H_{0}\)) is the correction function of the subsidence factor and \(C_{0}\) is the model parameter related to the geological mining conditions. After constructing a dynamic function model of the subsidence factor in the IB prediction model that changes with the advancement of the underground working face, (that is, Formula (14)), it can be used to replace all \(q\) in the IB model. The subsidence factor \(q\) is only related to the maximum subsidence value under full mining conditions. Therefore, after replacing \(q\) in the IB prediction model during mining with \(q\) in Formula (2), the maximum subsidence value \({W_{0}}^{\prime}\) is  
\[{W^{\prime}}_{0}=m\cdot q^{\prime}\cos\alpha=m\cdot f_{\text{c}}(D_{3}/H_{0}) \cdot q\cos\alpha \tag{15}\]  
Combined with the prediction model constructed above, all \(W_{0}\) in the three-dimensional deformation expression in the prediction model are replaced by the above formulas so as to construct a dynamic prediction model based on the change in the subsidence rate. It is known from the above that compared to the IB prediction model, the DIB prediction model has one more parameter, \(C_{0}\), so the model has a total of 11 parameters: \(P_{\text{DIB}}=[q,\,P,\,b,\,\theta_{0},\,S_{1},\,S_{2},\,S_{3},\,S_{4},\,\tan \beta_{1},\,\tan\beta_{2},\,C_{0}]\).