## 2 Methods  
### IB Prediction Model  
The rock movement and surface deformation of the thick loose layer in a mining area after coal seam mining differ from those observed in general geological mining conditions. This is mainly reflected in the following aspects: First, the maximum subsidence value of the working face is larger under the conditions of full or over-full mining in strike and dip. Second, the surface movement changes dramatically, the recession period is longer, the boundary converges slowly, and the range of surface movement is larger [33; 34]. The accuracy of the currently commonly used PIM is generally poor at the edges. Based on this, scholars in the field have proposed introducing the Boltzmann function and the improved Boltzmann function prediction model into the field of mining subsidence, and these functions have been applied effectively in the Huaian mining area [35; 36]. The main principles of the improved Boltzmann function prediction model are described in this manuscript. Consider the three-dimensional situation shown in Figure 1. The horizontal projections of the coal seam coordinate systems tO1s and xOy coincide. If we consider a unit \(B\)(s,t) with a width of \(ds\) and a length of \(dt\), a subsidence prediction formula for the main section of the surface movement basin can be determined by combining the Boltzmann prediction models of two different important influence radii in a certain proportion:  
\[W(x)=W_{0}\bigg{(}\frac{1-P}{\exp(-x/R_{1})+1}+\frac{P}{\exp(-x/R_{2})+1}\bigg{)} \tag{1}\]  
where the maximum surface subsidence value \(W_{0}\) is  
\[W_{0}=mq\cos\alpha \tag{2}\]  
\(R_{1}\) and \(R_{2}\) are two important influencing radii, \(P\) is the proportional coefficient, \(q\) is the subsidence factor, \(m\) is the thickness of the coal seam, and \(\alpha\) is the dip angle of the coal seam. According to the derivation principle of the PIM, it can be deduced that the subsidence value dW(x,y) caused by any point \(A\)(x,y) on the surface is  
\[dW(x,y)=W_{e}^{1}(x,y)\cdot(1-P)+W_{e}^{2}(x,y)\cdot P \tag{3}\]  
where  
\[W_{e}^{1}(x,y)=\frac{1}{R_{1}^{2}}\frac{\exp(-(x-s)/R_{1}-(y-t)/R_{1})}{(1+ \exp(-(x-s)/R_{1}))^{2}(1+\exp(-(y-t)/R_{1}))^{2}} \tag{4}\]  
\[W_{e}^{2}(x,y)=\frac{1}{R_{2}^{2}}\frac{\exp(-(x-s)/R_{2}-(y-t)/R_{2})}{(1+ \exp(-(x-s)/R_{2}))^{2}(1+\exp(-(y-t)/R_{2}))^{2}} \tag{5}\]  
\(W_{e}\)1 and \(W_{e}\)2 are the ground subsidence values caused by two different important influencing radii. As shown in Figure 1, if the mining range is O1 CDE, the length of O1C is \(D_{3}\), and the length of CD is \(D_{1}\), then the subsidence value of any point \(A\)(x,y) on the surface caused by the entire mining effort is  
\[\begin{split}& W(x,y)=W_{0}\int_{0}^{D_{3}}\int_{0}^{D_{1}}dW(x,y) dtds=W_{0}\int_{0}^{D_{3}}\int_{0}^{D_{1}}\frac{1}{R_{1}^{2}}\frac{\exp(-(x-s)/R_{1} -(y-t)/R_{1})(1-P)}{\left(1+\exp(-(x-s)/R_{1})\right)^{2}(1+\exp(-(y-t)/R_{1}) )^{2}}+\\ &\frac{1}{R_{2}^{2}}\frac{\exp(-(x-s)/R_{2}-(y-t)/R_{2})P}{\left( 1+\exp(-(x-s)/R_{2})\right)^{2}(1+\exp(-(y-t)/R_{2}))^{2}}dtds=\frac{1}{W_{0}} W^{0}(x)W^{0}(y)\\ &\text{where}\\ & W^{0}(x)=W_{0}\Bigg{(} \begin{array}{c}(1-P)\Big{(}\frac{1}{ \exp(-x/R_{1})+1}-\frac{1}{\exp(-(x-k_{3})/R_{1})+1}\Big{)}+\\ P\Big{(}\frac{1}{\exp(-x/R_{2})+1}-\frac{1}{\exp(-(x-k_{3})/R_{2})+1}\Big{)} \end{array} \Bigg{)} \tag{7}\]  
\[W^{0}(y)=W_{0}\Bigg{(} \begin{array}{c}(1-P)\Big{(}\frac{1}{\exp(-y/R_{1})+1} -\frac{1}{\exp(-(y-k_{1})/R_{1})+1}\Big{)}+\\ P\Big{(}\frac{1}{\exp(-y/R_{2})+1}-\frac{1}{\exp(-(y-k_{1})/R_{2})+1}\Big{)} \end{array}