# Dynamic Prediction Model for Progressive Surface Subsidence Based on MMF Time Function
###### Abstract  
It is imperative to timely and accurately predict the progressive surface subsidence caused by coal mining in the context of precision coal mining. However, the existing dynamic prediction methods that use time functions still have limitations, especially in the description of the moments of initiation and maximum subsidence velocity, which hinder their wide application. In this study, we proposed the MMF (Morgan-Mercer-Flodin) time function for predicting progressive surface subsidence based on the model assumptions and formula derivations. MMF time function can resolve the limitations in the description of the moments of initiation and maximum subsidence velocity perfectly. Afterward, we established the dynamic prediction model by combining the probability integral method with the MMF time function. Finally, using the measured subsidence data of working panel 22101 as an example, the accuracy and reliability of the dynamic prediction model was verified. The average RMSE and average relative RMSE (RRMSE) of prediction progressive subsidence using MMF time function are 46.65 mm and 4.63%, respectively. The accuracy is optimal compared with other time functions (for the average RMSE, Logistic time function is 80.57 mm, Gompertz time function is 79.77 mm, and Weibull time function is 90.61 mm; for the average RRMSE, Logistic time function is 7.66%, Gompertz time function is 7.73%, and Weibull time function is 8.62%). The results show that the method proposed in this paper can fully meet the requirements of practical engineering applications, achieve accurate dynamic prediction during the coal mining process, and provide good guidance for surface deformation and building protection. mining subsidence; dynamic prediction model; MMF time function; probability integral method; coal planning production +
Footnote †: journal: Applied  
002333  
of surface maximum subsidence points is more than one). The amount of subsidence in the flat bottom is essentially the same and the horizontal deformation in this region is minimal (the values of \(\epsilon_{\text{total}}\) are around the building in Figure 1b). In Figure 1, the building in this area has been affected or even damaged due to the dynamic deformation that the area underwent (\(\epsilon_{1}\rightarrow\epsilon_{1+2}\rightarrow\ldots\rightarrow\epsilon_{ \text{total}}\)). The horizontal deformation around the building has dynamic values which include 0, \(+\epsilon\) (tensile deformation), \(-\epsilon\) (compression deformation) and \(\epsilon_{\text{total}}\). Although this dynamic deformation does not act for a long time, there is a risk that the building may be affected by the deformation and rendered unusable. It is therefore important to determine the amount of ground movement and deformation over different time periods in order to take the necessary protective measures. From the analysis of the above example, it is clear that the actual problem cannot be adequately solved based only on the final state of the surface movement [4; 7]. Therefore, the dynamic prediction of mining subsidence has important practical engineering applications [8; 9; 10]. Moreover, in the context of precision coal mining [11], the prediction  
Figure 1: Dynamic changes about surface subsidence and horizontal deformation caused by coal mining (from position 1 to position 7): (**a**) Dynamic changes of surface subsidence, the curve of \(W_{1}\) is formed when the working panel advances from open-off cut to position 1, the curve of \(W_{1+2}\) is formed when advancing to position 2, the others are the same, the curve of \(W_{\text{total}}\) is the final subsidence when advancing to position 7; (**b**) Dynamic changes of horizontal deformation.