## 4 Determination of Model Parameters
### Determination of Time Influence Parameter \(C\)  
According to Equation (7), the surface subsidence curve with different time influence parameter \(C\) is shown in Figure 5.  
Figure 4: The influence of model order \(n\) on surface subsidence acceleration. Figure 3: The influence of model order \(n\) on surface subsidence velocity. Comparing Figures 2 and 5, it can be seen that the impact of time influence parameter \(C\) on the surface subsidence curve is similar to the model order \(n\), both of which have no effect on the initial and final surface subsidence values, but only affect the time to reach the final subsidence value. The larger \(C\) is, the less time it takes for the surface subsidence to reach a stable state. When the model \(n\) is determined, the key to the improved time function model accuracy is the determination of \(C\). According to the probability integral method, when the mining range reaches the critical value of full mining, the maximum subsidence is approximately equal to \(0.98W_{0}\)[29; 30]. If we assume that the mining velocity is \(v\) and the critical gob dimension is \(L_{f}\), the critical time to achieve full mining can be expressed as \(\frac{L_{f}}{v}\). According to Equation (7), the time influence parameter \(C\) calculation formula is as follows:  
\[0.98W_{0}=W_{0}\bigg{[}1-\exp\bigg{(}-C\frac{1}{n}\bigg{(}\frac{L_{f}}{v} \bigg{)}^{n}\bigg{)}\bigg{]} \tag{14}\]  
Because the mining velocity is easily determined, the key to finding time influence parameter \(C\) is to determine the critical gob dimension when full mining is realized. Presently, three methods are used to calculate the critical gob dimension [31]:  
1. [label=(0)]
2. Bedrock decision method: Determines final surface subsidence based only on the bedrock with no loose layer consideration. Obviously, the \(L_{f}\) determined by this method is only suitable for a shallow loose layer, not applicable for mining areas with thick loose layers, such as those found in central and eastern China. This method's calculation model is shown in Figure 6a and \(L_{f}=\frac{2H_{j}}{tan\varphi_{j}}\) relating to the geometric relationship. 3. Comprehensive impact parameter method: Regards the loose layer and bedrock as a composite medium, so the bedrock and loose layers mining influence angle are equal. This method's calculation model is seen in Figure 6b, with \(L_{f}=\frac{2H_{0}}{tan\varphi_{j}}\) relating to the geometric relationship. This method ignores the mechanical properties difference between the loose layer and the bedrock, so errors exist. Generally speaking, the loose layer mining influence angle is larger than that of the bedrock. 4. Dual-medium method: The coal mining influence on the bedrock and loose layer deformation is considered comprehensively. The critical gob dimension obtained is close to the actual value. The calculation model is shown in Figure 6c. According to the geometric relationship, \(L_{f}\) is determined as follows: \[L_{f}=\frac{2H_{s}}{\tan\varphi_{s}}+\frac{2H_{j}}{\tan\varphi_{j}}\] (15)  
Figure 5: The surface subsidence curve with different time influence parameter \(C\). where \(H_{0}\) is the average seam depth, \(\varphi\) is the full subsidence angle, \(H_{j}\) is the bedrock thickness, \(\varphi_{j}\) is the bedrock full subsidence angle, \(H_{\rm{s}}\) is the loose layer thickness, and \(\varphi_{s}\) is the loose layer full subsidence angle. Substituting Equation (15) into Equation (14), the calculation formula of \(C\) is as follows:  
\[C=\frac{-n\upsilon^{n}\ln 0.02}{\left(\frac{2H_{\rm{s}}}{\tan\varphi_{s}}+\frac{ 2H_{j}}{\tan\varphi_{j}}\right)^{n}} \tag{16}\]  
A significant amount of monitoring data shows that it is not precise and that the corresponding surface subsidence is just half of \(W_{0}\) when the subsidence velocity reaches the maximum value. Therefore, there is an error in the value of model order \(n\). The time influence parameter \(C\) obtained according to Equation (16) will also produce a certain error.