## 1 Introduction  
Coal, serving as a critical foundational energy source and industrial resource in China, plays a significant role in driving societal progress [1]. However, during the coal mining process, the initial stress equilibrium in the overlying rock layers above the coal seam gets disrupted [2]. As the mining face steadily advances, this deformation gradually propagates to the surface, resulting in the formation of geological features like cracks, terraces, and subsidence depressions [3]. Due to the influence of geological factors and other variables, surface movements and deformations persist for a certain duration after coal seam extraction has ceased. During this period, the impacts on groundwater continue to evolve over time [4, 5]. Additionally, structures on the surface, as well as the surrounding ecological environment within the mining area, are subjected to varying degrees of influence and damage during different postextraction phases [6, 7]. To ensure safe mining operations, promoting environmental preservation, and fostering sustainable development, forecasting dynamic surface subsidence and the implementation of essential protective measures tailored to distinct phases of surface deformations are of paramount importance [8, 9]. In order to gain a better understanding of the patterns of surface movement and deformation, many scholars both domestically and internationally have conducted extensive research on the time functions for dynamic prediction of surface movement and deformation over the past few decades. Polish scholar [PERSON] [10], in 1952, based on conclusions from soil compaction experiments, suggested a proportional relationship between the sinking speed of the surface at a particular moment, the final subsidence value of the surface, and the difference between the dynamic subsidence value at that moment. On this basis, the Knothe time function was derived. However, this time function falls short in effectively describing the sinking speed and acceleration of surface points, leading to disparities with actual surface subsidence processes. Consequently, many scholars have continuously refined the Knothe time function model and proposed new time function models, including [PERSON] [11, 12], [PERSON] [13, 14, 15, 16, 17], [PERSON] [18, 19, 20, 21], and MMF [22, 23, 24], among others. While these models partially reflect the dynamic process of surface movement and deformation to some extent, they also have their own limitations. Research findings indicate that the Usher model makes demanding geological conditions and is suitable for fully exploited areas characterized by substantial mining depth, significant thickness, and well-developed three zones of subsidence. However, it is not applicable in cases of insufficient exploitation, noncontinuous deformations, or complex conditions in coal seam mining areas. The Weibull model, on the other hand, exhibits a rapid increase in predicted subsidence during the initial phase of surface subsidence. When the subsidence rate reaches its maximum, the surface subsidence remains below half of the maximum surface subsidence value [25], which impacts the accuracy of predictions. In the case of the Richards model, the calculation of its parameters is challenging [26]. As for the MMF model, its predictions tend to be on the higher side. To achieve better predictive accuracy, data after the inflection point of the subsidence curve is required. This model exhibits a significant reliance on measured data, with increased observed data leading to enhanced predictive precision [27]. Due to the complexity of geological conditions in coal mining, multiple factors collectively influence the mining process. The individual models cannot fully account for the time-varying and contingency of overlying strata subsidence. Each individual model has its limitations; therefore, the combination of various individual models becomes necessary.