# Dynamic Prediction Model for Progressive Surface Subsidence Based on MMF Time Function
## 2 Models and Methods
### Establishment of MMF Time Function  
Based on the analysis of dynamic surface movement shown in Figure 2, we found that the subsidence velocity at time t is correlated with the average subsidence velocity and the potential subsidence ratio (\((W_{0}(x_{p})-W(t))/W_{0}(x_{p})\)), shown in Equation (13):  
\[\overset{\bullet}{W}(t)=\frac{dW(t)}{dt}=b\cdot\frac{W(t)-W(0)}{t-0}\cdot[ \frac{W_{0}(x_{p})-W(t)}{W_{0}(x_{p})}]=b\cdot\frac{W(t)}{t}\cdot[\frac{W_{0}( x_{p})-W(t)}{W_{0}(x_{p})}] \tag{13}\]  
where \(b\) is a parameter related to geological and mining technical parameters. Rearranging Equation (13),we have:  
\[\frac{W_{0}(x_{p})}{W(t)\cdot\big{[}W_{0}(x_{p})-W(t)\big{]}}dW(t)=\frac{b}{t}dt \tag{14}\]  
Integrating both sides of Equation (14):  
\[\int\frac{W_{0}(x_{p})}{W(t)\cdot\big{[}W_{0}(x_{p})-W(t)\big{]}}dW=\int\frac {b}{t}dt \tag{15}\]  
Decomposing the rational fraction of Equation (15):  
\[\int[\frac{1}{W(t)}+\frac{1}{W_{0}(x_{p})-W(t)}]dW=\int\frac{b}{t}dt \tag{16}\]  
Integrating both sides of Equation (16):  
\[\ln W(t)-\ln[W_{0}(x_{p})-W(t)]+C_{1}=b\ln t \tag{17}\]  
where \(C_{1}\) is a constant. Let \(C_{1}=\ln a\), we have:  
\[\ln W(t)-\ln[W_{0}(x_{p})-W(t)]+\ln a=\ln t^{b} \tag{18}\]  
Removing the logarithmic symbol of Equation (18):  
\[\frac{W(t)\cdot a}{W_{0}(x_{p})-W(t)}=t^{b} \tag{19}\]  
Adding 1 to both sides of Equation (19) and simplifying:  
\[W(t)=\frac{W_{0}(x_{p})\cdot t^{b}}{a+t^{b}}=W_{0}(x_{p})\cdot\varphi(t) \tag{20}\]  
The form of \(W(t)\) in Equation (20) is similar to the Morgan-Mercer-Flodin growth model [43]. \(\varphi(t)\) in Equation (20) is the MMF time function for the surface dynamic subsidence prediction proposed in this paper. The subsidence velocity and acceleration expressions of MMF time function are shown in Equations (21) and (22), respectively:  
\[v(t)=\frac{W_{0}(x_{p})\cdot a\cdot b\cdot t^{b-1}}{\left(a+t^{b}\right)^{2}} \tag{21}\]  
\[a(t)=\frac{W_{0}(x_{p})\cdot a\cdot b\cdot t^{b-2}}{\left(a+t^{b}\right)^{3} }[(b-1)(a+t^{b})-2 bt^{b}] \tag{22}\]  
The curves of the MMF time function are shown in \[\frac{W_{0}(x_{p})}{W(t)\cdot\big{[}W_{0}(x_{p})-W(t)\big{]}}dW(t)=\frac{b}{t}dt \tag{14}\]0  
Let \(a(t)=0\), and then \(t_{1}=0\), \(t_{2}=\sqrt[b]{a(b-1)/(b+1)}\) and \(t_{3}=\infty\). When \(0\leq t\leq\sqrt[b]{a(b-1)/(b+1)}\), \(a(t)>0\); when \(\sqrt[b]{a(b-1)/(b+1)}<t<\infty\), \(a(t)<0\). It means the subsidence velocity increases at the interval \([0,\sqrt[b]{a(b-1)/(b+1)}]\) and decreases at the interval \((\sqrt[b]{a(b-1)/(b+1)},\infty)\), where the subsidence velocity reaches the maximum. It is proven that there is an inflection point in the curve of the MMF time function. Moreover, the inflection point is not fixed and is correlated with the parameters \(a\) and \(b\), this characteristic is more accord with the practical situation of subsidence engineering. Let \(b=5.5\), \(a=1.0\times 10^{7}\), \(1.0\times 10^{8}\), \(1.0\times 10^{9}\), \(1.0\times 10^{10}\), respectively; the effect of the variation of parameter \(a\) is on the curves of subsidence and velocity is analyzed, as shown in Figure 6a,b. Let \(a=1.0\times 10^{8}\), \(b=4.5\), \(5.0\), \(5.5\), \(6.0\), respectively; the variation of parameter \(b\) is on the curves of subsidence and subsidence velocity is analyzed, as shown in Figure 6c,d. As shown in Figure 6, the variation of parameters \(a\) and \(b\) has the opposite effect on the curves. For a surface point, as the parameter \(a\) increases, the subsidence velocity decreases and the time required from the beginning of subsidence to stabilization increases significantly. As the parameter \(b\) increases, the subsidence velocity of increases significantly, and the time required from the beginning of subsidence to the stabilization is significantly shortened. Therefore, it can be considered that parameters \(a\) and \(b\) reflect the subsidence velocity together, and the values of these two parameters depend on the nature of the overlying rock layers in the mining area.