## 5 Model Validation  
To verify the improved Knothe time function model accuracy and reliability established in this paper, based on the surface movement observation station field monitoring data above the 1301 working face in Guotun coal mine from Heze city, Shandong Province, China, the monitoring data and the model predicted results were compared and analyzed. The surface movement observation stations' layout is shown in Figure 7, and the spacing between each observation station is 30 m. Due to a long observation time span and a large amount of monitoring data, clearer monitoring and prediction data comparison and analysis were achieved by randomly selecting only four stations, z38, z41, z44, and z47, as the research study objects for improved and traditional Knothe time function model accuracy analysis in surface subsidence prediction. The working face 1301 elevation is from \(-\)748.9 to \(-\)817 m; the coal seam thickness is 1 to 3.6 m, with the average being 2.9 m, and the coal seam dip angle is 12\({}^{\circ}\); the roof is managed by full caving methodology. Due to mining area limitations, the observation point along the strike direction sinks to a flat bottom, with no flat bottom phenomenon along the inclined surface, that is to say, the working face strike is full mining, and the working face incline is not. Table 1 shows the monitoring data of z38, z41, z44, and z47 for each period. The traditional and improved Knothe time function models were, respectively, used to predict the surface subsidence and subsidence velocity. The two models' prediction comparison results are shown in Figures 8 and 9. Figure 6: Calculation model of critical dimension for full mining: (**a**) bedrock decision method; (**b**) comprehensive impact parameter method; (**c**) dual-medium method. As can be seen from the monitoring data in Table 1 and Figure 8, the duration of the rapid deformation stage of z37, z41, z44, and z47 monitoring sites was 310 days, 303 days, 276 days, and 198 days, respectively. Moreover, the duration of the rapid deformation stage of z44 and z47 monitoring sites was shorter than that of the slow deformation stage and basic stable stage. However, the data  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Observation** & **Observation** & **Relative** & **z38** & **z41** & **z44** & **z47** \\
**Times** & **Date** & **Observation** & **Settlement** & **Settlement** & **Settlement** \\  & & **(Time/d)** & **(Value/mm)** & **(Value/mm)** & **(Value/mm)** & **(Value/mm)** \\ \hline
1 & 2010/10/18 & 0 & 0 & 0 & 0 & 0 \\
2 & 2010/12/27 & 69 & \(-\)5.4 &  
of the four monitoring sites indicated that the dynamic surface subsidence curve was approximately \"\(S\)\"-type. Therefore, the proposed model in the manuscript can be used to predict the vertical surface deformation. In addition, due to the lack of surface horizontal movement monitoring data and the focus of the manuscript is the vertical surface deformation. The horizontal deformation law of the surface will be discussed in the following research. Figures 8 and 9 demonstrate that the improved Knothe time function model prediction accuracy for surface subsidence is significantly better than the Knothe time function model, and can comprehensively reflect the whole dynamic surface subsidence velocity process from beginning acceleration to attenuation. The prediction results were consistent with the monitoring data. The prediction value of the Knothe time function model was quite different from the monitoring data. The improved Knothe time function model parameters obtained by the least square method are shown in Table 2.  
Standard deviation and relative standard deviation can measure the deviation between monitored and predicted values [20; 31]. To further verify the improved Knothe time function model accuracy, the standard deviation and relative standard deviation of the monitoring value and the predicted value of four monitoring points were analyzed without considering the monitoring error.