# Dynamic Prediction Model for Progressive Surface Subsidence Based on MMF Time Function
## 2 Models and Methods
### Determination of parameters for MMF Time Function  
There is a close relationship between the parameters in dynamic prediction models and the geological and mining technical conditions, and the dynamic subsidence is different under different geological and mining technical conditions [8; 12; 14; 28]. Therefore, it is necessary to analyze the statistical relationship between model parameters and geological and mining technical parameters in order to guide the actual modeling work. The geological and mining parameters include mining thickness \(M\), average mining depth \(H_{0}\), loose layer thickness \(h\), advance velocity of a working panel \(v\) and subsidence coefficient \(q\). The parameters of MMF time function are \(a\) and \(b\). In order to obtain the statistical relationship between the two parameters (\(a\), \(b\)) and geological and mining technical parameters (\(M\), \(H_{0}\), \(h\), \(v\)), the geological and mining technical parameters of sixteen working panels were selected for research. MMF time function was also applied to measured data  
Figure 5: Dynamic subsidence, velocity and acceleration of MMF time function. of the maximum subsidence point (MSP) on the sixteen strike main sections of the working panels [8,14]. The method for determining parameters \(a\) and \(b\) is nonlinear least squares algorithm based on MATLAB curve fitting toolbox. The relevant MMF time function parameters of the maximum subsidence points and geological and mining technical parameters of each working panel are shown in Table 1. Considering the magnitude variance between parameter \(a\) and parameter \(b\), the relationship between the logarithm value of parameter \(a\) (In \(a\)) and geological mining conditions is statistically analyzed. For the exploitation cases in Table 1, where \(q\) is greater than 1, which can be caused by reactivation of goats. The reactivation of goats is related to the ratio of the thickness of the loose layer to the thickness of the bedrock; the larger the ratio, the more likely the goats are activated. Furthermore, when the thickness of the loose layer exceeds 100 m, the soil has additional settlement (settlement caused by consolidation and compression of the soil and water table drop), which also results in q greater than 1. According to Table 1, we found that the MMF model parameters (\(a\), \(b\)) are mainly related to the ratio of bedrock thickness to mining thickness \((H_{0}-h)/M\), advance velocity \(v\) and overburden properties. The smaller the ratio of bedrock thickness to mining thickness (i.e., the larger \(M/(H_{0}-h)\)), the larger the advance velocity; the smaller the average solidity coefficient of the overlying rock layers, the more intense the surface movement and  
Figure 6: Variation of surface subsidence and subsidence velocity curves with parameters \(a\) and \(b\) based on MMF time function; (**a**) Variation of parameter \(a\) on MMF subsidence curve; (**b**) Variation of parameter \(a\) on MMF subsidence velocity curve; (**c**) Variation of parameter \(b\) on MMF subsidence curve; (**d**) Variation of parameter \(b\) on MMF subsidence velocity curve. deformation, and accordingly the larger the values of \(a\) and \(b\) of MMF time function. The relationship between parameters \(a\) and \(b\) and \(M/(H_{0}-h)\), \(v\) and \(q\) is shown in Figure 7.  
According to Figure 7, the statistical relationship of parameters \(a\), \(b\) and \(M/(H_{0}-h)\), \(v\) and \(q\) is:  
\[\left\{ \begin{array}{l}\ln a=157.7452\frac{M}{H_{0}-h}vq+17.6518\\ b=48.6275\frac{M}{H_{0}-h}vq+4.3393\end{array} \right. \tag{23}\]  
In Equation (23), the subsidence coefficient \(q\) reflects the nature of the overburden. If the overburden is softer, \(q\) is larger, while the overburden is harder, \(q\) is smaller; other parameters are the same above.