## 2 Fundamental Parameters of Probability Integral Method  
The theoretical basis of the probability integral method is the random medium theory. The fundamental parameters of the probability integral method include subsidence coefficient, horizontal movement coefficient, tangent of major influence angle, mining influence propagation angle and inflection point offset. The accuracy of these selected parameters is directly related to the prediction precision of surface subsidence in the mining areas.  
The subsidence coefficient \(q\) represents the ratio of maximum surface subsidence value and the projected length of coal seam mining thick in the vertical direction in full mining, as shown in Figure 1. The subsidence coefficient is a key parameter for surface movement and deformation prediction, and its calculation formula is as follows:  
\[q=\frac{W_{0}}{Mcos\alpha} \tag{1}\]  
where: \(W_{0}\) is the measured maximum surface subsidence value when the strike and tendency reach the full-mining;  
\(q\) is the surface subsidence coefficient;  
\(M\) is the coal seam mining thickness, m;  
\(\alpha\) is the coal seam dip angle, \({}^{\circ}\).  
Where: \(W_{0}\) is the measured maximum surface subsidence when the strike and tendency have reached full-mining, mm; \(H\) is the mining depth of coal seam, m; \(M\) is the mining thickness, m; \(W(x)\) is the surface subsidence function; \(s\) is the inflection point offset, m.  
The inflection point offset \(s\) refers to the offset distance of the inflection point of the subsidence curve from the mining boundary to the goal. This is due to the cantilever effect of the roof at the mining boundary. The value of the inflection point offset directly affects the shape and area of the surface subsidence basin, as shown in Figure 1.  
The horizontal movement coefficient b gives the ratio of maximum surface horizontal movement value and the maximum surface subsidence value. The calculation formula can be expressed as:  
\[\text{b}=\frac{U_{0}}{W_{0}} \tag{2}\]  
where: \(W_{0}\) is the measured maximum surface subsidence when the strike and tendency arrive the full-mining, mm; \(U_{0}\) is the measured maximum surface horizontal movement when the strike and tendency reach the full-mining, mm; \(b\) is the horizontal movement coefficient.  
The tangent of major influence angle \(tan\)\(\beta\) is the tangent of angle \(\beta\) between the line of main influence area boundary point and the mining boundary point and the horizontal line, as shown in Figure 2. The tangent of major influence angle is the major parameter to determine the area of surface movement and deformation, and the calculation formula is:  
\[tan\beta=\frac{H}{r} \tag{3}\]  
where:  
\(H\) is the mining depth of the coal seam, m;  
\(r\) is the main influence radius, m;  
\(tan\)\(\beta\) is the tangent of the major influence angle.  
Figure 1: Schematic diagram of the subsidence coefficient.  
Where: \(W_{0}\) is the measured maximum surface subsidence when the strike and tendency the full-mining, mm; \(r\) is the main influence radius, m; \(\beta\) is the main influence angle, \({}^{\circ}\); \(H\) is the mining depth of the coal seam, m; \(W_{(x)}\) is the surface subsidence function.  
The mining influence propagation angle is angle \(\theta\) that is formed by the line of the inflection point of the subsidence curve and the mining boundary point and the horizontal line under the condition of the full mining, which is the index of deviation degree of surface subsidence basin.