## 2 The Knothe Time Function Model  
In 1952, [PERSON] used mathematical methods to describe the complex time-space process of surface subsidence caused by underground mining and made the following assumptions: The surface subsidence rate \(\frac{dW(t)}{dt}\) is proportional to the difference between the final surface subsidence \(W_{0}\) and the dynamic subsidence \(W(t)\) at time \(t\), which may be expressed as [17; 18; 19]:  
\[\frac{dW(t)}{dt}=c[W_{0}-W(t)] \tag{1}\]  
where \(c\) is a time influence parameter related to the overburdened rock mass physical and mechanical properties. This parameter is expressed in the unit 1/year.  
By separating variables and combining with boundary conditions \(t=0,\ W(t)=0\), the relationship between subsidence and time is obtained as follows:  
\[W(t)=W_{0}[1-\exp(-ct)] \tag{2}\]  
The Knothe time function model is widely used to predict the actual surface subsidence process due to its simple form and few undetermined parameters. However, with expanding research and practice, this model does not accurately describe the dynamic processes of the subsidence [19; 20].  
The first and second derivatives of Equation (2) are used to obtain the surface subsidence velocity and acceleration as follows:  
\[\left\{\begin{array}{l}V(t)=cW_{0}\exp(-ct)\\ a(t)=-c^{2}W_{0}\exp(-ct)\end{array}\right. \tag{3}\]  
Using Equations (2) and (3), the surface subsidence rule described by the Knothe time function model is shown in Figure 1. The figure shows that when \(t=0\), the surface subsidence velocity is at the maximum value \(cW_{0}\). With the passage of time, the velocity gradually decreases and finally approaches 0. That is to say, the surface subsidence gradually increases from 0, but the increasing range is shortened. The surface subsidence acceleration curve shows that from the beginning of surface subsidence to final stability, the whole subsidence process acceleration is always less than 0. In other words, surface subsidence velocity is always decreasing.  
Numerous research and field monitoring data show that the surface subsidence curve caused by coal mining is approximately an \"S\"-type curve [21; 22]. When the distance between set-up entry and the working face reaches the surface starting distance, the surface starts sinking. The subsidence velocity is 0 at that point. Then, the subsidence slowly increases, but subsidence velocity rapidly increases. As the working face continuously advances, the surface subsidence increases significantly, and the subsidence velocity gradually increases to the maximum value. When the working face has advanced a sufficient distance away from the entry set-up, the surface subsidence shape curve remains basically unchanged, and the subsidence continues to increase, but the velocity increase is very slow and eventually tends toward 0. Figure 1 shows the actual measured subsidence, subsidence velocity, and subsidence acceleration curve at the \(zZ\) monitoring point on the 35,101 working face in  
Figure 1: Surface point curve of subsidence, subsidence velocity, and subsidence acceleration: (**a**) based on Knothe time function model; (**b**) based on the field monitoring value.  
the Sandaogou mine [23] in Shanxi Province, China. Comparing Figure 1a,b, the Knothe time function model is clearly quite different from the actual situation in describing surface subsidence, subsidence velocity, and subsidence acceleration. The Knothe time function model is suitable only for describing the basic stable stage and cannot accurately predict the whole dynamic surface subsidence process [24].