# Dynamic Prediction Model for Progressive Surface Subsidence Based on MMF Time Function
###### Abstract
the curve of \(\epsilon_{1}\) is formed when the working panel advances from open-off cut to position 1, the curve of \(\epsilon_{1+2}\) is formed when advancing to position 2, the others are the same, the curve of \(\epsilon_{\text{total}}\) is the final horizontal deformation when advancing to position 7.  
subsidence amount at a certain time under water, buildings, railways or roads has higher accuracy requirements during the process of coal mining. Accurate dynamic prediction is crucially important. Most existing dynamic prediction models are modelled using static prediction model with different time functions [12; 13; 14]. In China, the static prediction model for mining subsidence is the probability integral method (PIM) [4; 15; 16]. The key to improving the accuracy of dynamic prediction is to determine the optimal time function [12; 17]. Many scholars have conducted relevant studies. In 1952, [PERSON] proposed the Knothe time function, which describes the dynamic subsidence in the form of a differential equation [18], but it has drawbacks in describing surface subsidence velocity and acceleration [10; 14; 19]. Considering the flaws of the Knothe time function, [PERSON] proposed a Sroka-Schboer's function considering the delay of the overburden movement and surface subsidence [20; 21; 22], but it is difficult to obtain the parameters. [PERSON] and [PERSON] proposed the segmented Knothe time function [23], which does not match the actual values at the maximum and segmented points. [PERSON] presented the power exponent of Knothe time function [24]. [PERSON] also proposed an improved Knothe time function [19] where the prediction accuracy was also improved. However, the physical meanings of the parameters at the exponential position of two-time functions are not clear. [PERSON] and [PERSON] developed an optimized segmented Knothe time function [25], but the moment at which the maximum subsidence velocity occurred, it needed to be determined empirically. In addition to Knothe time function and its improved forms, there are other time functions such as Logistic time function [9; 13; 26], Gompertz time function [27; 28], Weibull time function [10; 29] and so on. When synthesizing the above-mentioned studies related to the mining subsidence dynamic prediction based on time functions, it can be found that they almost focus only on a few single points locally, rather than the progressive surface subsidence. As a result, the reliability and practicality of these models are somewhat lacking and do not serve the fundamental purpose of accurately predicting mining subsidence. In this study, the features of the ideal time function are described, the characteristics and shortcomings of the existing time functions are summarized, and through model analysis and formula derivation it is concluded that the MMF function can accurately describe the dynamic process of surface subsidence. Then, the MMF time function is combined with the PIM to construct a dynamic prediction model for progressive surface subsidence. Finally, an engineering example is applied to verify the accuracy and validity of the proposed model, and the validation is good. This work can be used for the dynamic prediction to provide scientific reference for practical engineering, ensure the safe and effective mining of coal resources and build protection.