## 1 Introduction  
The surface subsidence and deformation caused by underground mining is a complex four-dimensional problem varying with a continuous time-space function [1, 2]. With progressive mining of the working face, the surfaces' subsidence undergoes three stages of subsidence, namely, slow deformation stage, rapid deformation stage, and basic stable stage. The slow deformation stage only accounts for 10%-15% of the final subsidence and causes minor ground deformation. The rapid deformation stage is the principal subsidence corresponding to about 75% of the final subsidence and causes major ground deformation. The basic stable stage may vary from 5% to 10% of the final subsidence [3]. The deformation that occurs in each of these stages leads to various forms of damage, such as that to buildings, water facilities, and roads. Geological disasters resulting from ground subsidence cause huge economic losses in China every year. Researching the ground subsidence characteristics caused by underground mining and accurately predicting the deformation caused by ground subsidence is one of the most important aspects of protecting buildings and structures above the gob [4, 5, 6, 7]. Surface subsidence is a widespread problem frequently caused by underground mining. In the past several decades, a number of studies have focused on using the geometric theory prediction method,profile function prediction method, mechanical model prediction method, interferometric synthetic aperture radar (InSAR) method, GPS (Global Positioning System) method, and numerical simulation prediction method to estimate the ground subsidence and deformation caused by underground coal mining. [PERSON] et al. [8] proposed a one-parameter Knothe's function model based on a growth law and a two-parameter Sroka-Schober's function based on double use of the growth law considering the convergence of extracted excavations and delaying activity of overlain rock. [PERSON] et al. [9] proposed a new functional influence method and additional modeling subsidence asymmetry functions for the extraction of edge effects, with the method's accuracy examined by four cases. [PERSON] et al. [10] extended an existing classical dynamic subsidence prediction model, proposing new potential research avenues offered by cellular automata (CA) models. The model verified the deformation measurements collected above a coalmine panel in the Huainan mining area in Anhui Province, China. [PERSON] et al. [11] proposed a new subsidence prediction approach using FEM (Finite Element Models) and satellite measurements and gave two case studies as examples for the application of the proposed approach. [PERSON] et al. [12] established a model for calculating the Knothe time function parameters based on a full subsidence angle, verifying the proposed model's reliability and applicability by comparing the predicted and measured values. [PERSON] et al. [13] proposed using intensity tracking technique and the small baseline subset (SBAS) interferometry technique to obtain complete monitoring results. The comparison between experiment monitoring results and GPS measurements showed that the proposed method has the capability to detect and measure the large gradient deformation. [PERSON] et al. [14] proposed a method for completely retrieving three-dimensional (3-D) mining-induced displacements with OT (Offset Tracking)-derived observations of LOS (Line-of-Sight) deformation from a single amplitude pair of SAR (Single Amplitude Pair), and this method had been successfully used in the Daliuta coal mining area. [PERSON] et al. [15] analyzed surface subsidence above the Velenje Coal Mine's underground mining sites using a modified sigmoid function (logistic function surrogate) and considered the time estimation of the next and final epoch measurement. [PERSON] et al.