# Prediction of Delayed Surface Subsidence Based on the Improved Knothe-n Model
## 3 Knothe-n Time Function Model Improvement
### Knothe-n Segmented Time Function  
The analysis of the structural features of the Knothe time function model confirms the different significance of the parameters \(\eta\) and \(n\). The parameter \(n\) does not affect the beginning and termination moments of the surface subsidence but only affects the process of its medium-term changes. As \(n\) decreases, the time at which surface subsidence reaches its maximum rate is delayed. As \(\eta\) decreases, the development rate of the surface subsidence becomes faster, and the time required for subsidence stabilization becomes shorter when the other conditions are constant. To enhance the general applicability of the time function, the fitting of parameters \(n\) and \(\eta\) needs to be conducted based on the monitoring data from subsidence points in different subsidence areas. The function should be segmented and adjusted using the maximum subsidence values. Concerning the Yutianbao coal mining subsidence area, the monitored data in terms of subsidence and time at monitoring point WC1 are taken for fitting the parameters \(n\) and \(\eta\) by using the least-squares method. The fitting results are \(\eta=1\times 10^{-7}\) and \(n=3\). The simulated maximum surface subsidence at WC1 is \(1\times 10^{3}\) mm. The parameters are substituted into Equation (2) for calculation and plotting of surface subsidence curves, as presented in Figure 6.  
Figure 6 demonstrates that the surface subsidence after 612 days reaches a maximum value of 1000 mm. The monitoring data from the existing monitoring points in the Yutianbao mining area reveal that the duration of the recession stage does not coincide with the measured data. To be closer to the actual situation of the project, combining the segmented  
Figure 6: Surface subsidence curve for \(\eta=1\times 10^{-7}\) and \(n=3\).  
Knothe time function model proposed by [PERSON] [13] with the variation patterns observed in the measured data, the subsidence process is divided into three stages: the initiation stage, active stage, and decay stage, corresponding to the time when the maximum subsidence rate occurs. Finally, by amplifying the time variation through the exponent \(n\), the function better conforms to the actual deformation pattern, resulting in the improved segmented Knothe-n function, as shown in Equation (9).  
\[w(t)=\left\{\begin{array}{cc}0\;0<t\leq t_{0}\\ 0.5w_{m}(e^{-\eta(\tau-t)^{n}}-e^{-\eta\tau^{n}})&t_{0}<t\leq\tau\\ 0.5w_{m}(2-e^{-\eta(t-\tau)^{n}}-e^{-\eta\tau^{n}})&\tau<\;t\leq T\end{array}\right. \tag{9}\]  
where \(t_{0}\) is the initiation time of surface subsidence, \(d\); \(\tau\) is the time corresponding to the maximum subsidence rate at ground points, \(d\); and \(T\) is the duration from coal seam extraction to the stabilization of surface subsidence, \(d\).  
The corrected time function curve at \(\eta\) = \(1\times 10^{-7}\) and \(n\) = 3 is calculated by Equation (9), as shown in Figure 7.  
To verify the effectiveness of the function when it is used for the recession stage, the continuous 37-month monitoring data at monitoring point WC1 are compared with the predicted data obtained from the time function curve, as indicated in Figure 8.  
Figure 8 shows that the overall change rule of the curve is in good agreement with the measured data. The root-mean-square error between the predicted and measured values is 0.91, which indicates that the improved Knothe-n segmented time function model can more accurately characterize the relationship between surface subsidence and time than the single Knothe-n function, and also confirms the correctness and applicability of the present model.  
Figure 8: Comparison of predicted and measured subsidence data at WC1 from July 2015 to July 2018.  
Figure 7: Corrected time function curve at \(\eta\) = \(1\times 10^{-7}\) and \(n\) = 3.