# Dynamic Prediction Model for Progressive Surface Subsidence Based on MMF Time Function
## 2 Models and Methods
### Dynamic Prediction Model for Progressive Surface Subspace  
The dynamic prediction model based on MMF function can be determined by multiplying the surface subsidence of each point by the MMF time function. During the process of calculating, the mining area is divided into \(n\) independent mining units according to the time series. First, the surface subsidence generated by each unit is calculated. Then, the surface subsidence generated by all units is superimposed to obtain the total surface subsidence.  
The start of mining is defined as the start time, and the advance velocity of each unit is \(v_{1},v_{2},\ldots,v_{n}\), and the advance times of each unit are \(t_{1},t_{2},\ldots,t_{n}\), respectively, then the size of each unit is \(v_{1}t_{1}\), \(v_{2}t_{2}\),..., \(v_{n}t_{n}\); after the working panel is fully extracted, the sum of surface subsidence generated by all mining units is shown in Figure 9. \(W_{1}\), \(W_{2}\), \(W_{3}\),..., \(W_{n}\) is the subsidence contributed by each mining unit, and \(W_{t}\) is the total subsidence affected by mining. Considering the amount of surface subsidence \(W_{t}\) at time \(t\), when the first mining unit is mined, it experiences the longest time from 0 to \(t\), and has the maximum contribution to the surface subsidence, as \(W_{1}\) in Figure 9. Accordingly, after the second mining unit is mined, it experiences the time from \(t_{1}\) to \(t\), it has the contribution to the surface subsidence, as \(W_{2}\) in Figure 9. After the kth unit is extracted (\(t-t_{1}-t_{2}-\ldots-t_{k-1}\)), it has the minimal contribution to surface subsidence, like \(W_{n}\) in Figure 9. After all \(n\) units are extracted, the surface subsidence is \(W_{t}\) (Figure 9).  
When the coal seam is horizontal (\(\alpha=0^{\circ}\)), the dynamic prediction model along the strike main section is  
\[\begin{array}{ll}W(x,t)&=\varphi(t)W_{1}^{0}(x)+\varphi(t-t_{1})W_{2}^{0}( x)+\cdots+\varphi(t-t_{1}-t_{2}-\cdots t_{n})W_{1}^{0}(x)\\ &=\varphi(t)[W(x-s_{3})-W(x-v_{1}t_{1})]+\varphi(t-t_{1})[W(x-v_{1}t_{1})-W( x-v_{1}t_{1}-v_{2}t_{2})]\\ &+\varphi(t-t_{1}-t_{2})[W(x-v_{1}t_{1}-v_{2}t_{2})-W(x-v_{1}t_{1}-v_{2}t_{2}- v_{3}t_{3})]\\ &+\cdots+\varphi(t-t_{1}-t_{2}-\cdots t_{n-1})\{W(x-v_{1}t_{1}-\cdots-v_{n-1}t _{n-1})-W[(x-(v_{1}t_{1}+v_{2}t_{2}+\cdots+v_{n}t_{n}-s_{4})]\}\end{array} \tag{25}\]  
Figure 8: (**a**) Coordinate system along strike main section; (  
where \(W(x-\Sigma v_{t}i_{t})\) can be calculated according to the first formula in Equation (24), and \(s_{3}\), \(s_{4}\) are the inflection point offsets of strike left and right directions. \(\varphi(t)\) is MMF time function in Equation (20).  
When the inclined coal seam is mined, the dynamic prediction model along the inclined main section is:  
\[W(y,t)=\varphi(t)W_{1}^{0}(y)+\varphi(t-t_{1})W_{2}^{0}(y)+\cdots+\varphi(t-t_ {1}-t_{2}-\cdots t_{n})W_{n}^{0}(y) \tag{26}\]  
where \(W_{i}^{0}(y)\) is the second formula in Equation (24) and \(\varphi(t)\) is MMF time function in Equation (20).  
Equation (25) and (26) are the progressive dynamic prediction model along strike and inclined main sections, respectively.