# Prediction of Delayed Surface Subsidence Based on the Improved Knothe-n Model
## 1 Introduction  
Land subsidence is a process that accompanies all mineral and groundwater exploitation. The mining of underground coal seams leads to gravity-driven subsidence and damage to the overlying rock strata. The vertical movement deformation generated within the rock strata is also gradually transferred to the surface, causing surface subsidence. [PERSON] et al. proposed the Analogous Hyperbolic Settlement Model (AHSM) to describe the movement and damage of internal load while employing a unique finite element method to validate the strata movement induced by mining [1]. After the active phase of surface subsidence ends, residual and persistent surface subsidence persists due to the slow release of stress, posing a challenging problem in rock engineering [2], which may lead to the destruction of new buildings, railroads, highways, water conservancy facilities, etc. on the ground, and substantial economic losses [3, 4, 5]. Before land reuse in the coal mining subsidence area, it is necessary to predict residual subsidence in the area. A comprehensive understanding of the surface subsidence process over time is needed to accurately predict the residual subsidence [6, 7, 8]. The time function models determining the dynamic subsidence caused by coal mining mainly include the Knothe model, the Boltzmann model [9], the Weibull time series function [10], and the Logistic model [11]. Based on the time function proposed by Polish scholar [PERSON] in 1952 [12], [PERSON] et al. [13] proposed the segmented Knothe time function model for dealing with the deficiencies of the Knothe time function. [PERSON] al. [14] extended the application range of the segmented Knothe time function model and optimized its model parameterization method [15]. [PERSON] et al. [16] used the probability integration model-related theory to construct a model for solving the time function parameters \(c\) and \(\tau\), and further optimized the segmented Knothe time function model. [PERSON] and [PERSON] [17,18] proposed an improved power exponential Knothe time function model. [PERSON] et al. [19] applied the monitoring results of InSAR to fit the parameters of the power exponential Knothe time function model. [PERSON] et al. [20] further modified the traditional Knothe time function model by considering the non-linear mechanical properties of the overlying strata, as well as the non-linear change characteristics of rock strata in bending, breaking, collapse, and surface movement deformation with time after coal seam mining.  
In view of this, the prediction of multi-parameter models is more accurate, as they are based on a greater number of parameters. However, the difficulty associated with determining these parameters reduces the practicality of the model [21]. The Knothe-n time function model proposed by [PERSON] [22] is utilized for segmental correction to propose a simple and more accurate prediction model. This improved model often requires a large amount of observation data for validation. To overcome difficulties in human and material resources required for obtaining a large amount of subsidence data, FLAC3D-based numerical simulation [23] is performed to capture the surface residual deformation after underground mining of coal seams, and to further validate the accuracy of the improved time-functional model for subsidence prediction. Finally, a novel Knothe-n segmentation function is obtained.