## 2 Materials and Methods
### Subsidence Prediction and Parameter Inversion Methods  
The research on mining surface subsidence prediction parameter inversion comprises two main parts: the subsidence prediction model and the parameter inversion method [35]. The probability integration method stands as the most extensively employed approach for subsidence prediction in China. As its theory is well established, its mathematical model will not be reiterated in this paper. The probability integration method stands out for its simplicity in mathematical modeling and its requirement of fewer parameters [36; 37]. These parameters include deviation of inflection point (s\({}_{1}\), s\({}_{2}\), s\({}_{3}\), s\({}_{4}\)), mining influence propagation angle (\(\theta_{0}\)), horizontal movement coefficient (b), subsidence coefficient (q), and tangent of major influence angle (tan\(\beta\)). The best parameters for the probability integration model in the study area were obtained through a parameter inversion algorithm using the measured monitoring point movement data. The accuracy and reliability of surface movement deformation prediction are directly affected by the appropriateness of parameter selection. The BFGS algorithm is an iterative algorithm designed to tackle unconstrained optimization problems, serving as an analogue to [PERSON]'s method. It progressively seeks to determine the minima of the objective function by incrementally constructing an estimate of the inverse Hessian matrix of the objective function. This paper utilizes the BFGS algorithm for parameter inversion of the probability integration method, resulting in highly stable inversion outcomes. The accuracy of the BFGS algorithm's parameter inversion remains consistent even when faced with varying levels of random errors, demonstrating robust parameter estimation performance and strong anti-interference capabilities. Firstly, an objective function model, as depicted in Equation (1), incorporates the probability integration formulation and transforms it into an unconstrained optimization problem. \[f=\sum_{i=1}^{N}\left(W(x,y)-W_{0}\right)^{2}+\sum_{i=1}^{N}\left(U(x,y,\varphi )-U_{0}\right)^{2} \tag{1}\]  
In Equation (1), \(N\) is the number of ground observation points; \(W_{0}\) is the measured subsidence value; \(U_{0}\) is the measured horizontal movement value. The function \(W(x,y)\) represents the subsidence value at the position \((x,y)\), where \(x\) and \(y\) denote the horizontal and vertical coordinates of any point on the surface. Similarly, the function \(U(x,y,\varphi)\) denotes the horizontal shift value at the position \((x,y,\varphi)\), where \(x\) and \(y\) refer to the horizontal and vertical coordinates of any point on the surface, and \(\varphi\) represents the counterclockwise angle from the positive \(x\)-axis to the specified direction. The functions \(W(x,y)\) and \(U(x,y,\varphi)\) encompass the parameters necessary for the probability integration method. The iterative process of the BFGS algorithm commences by computing a new iterative solution as shown in Equation (2). \[X_{k+1}=X_{k}+\alpha_{k}d_{k} \tag{2}\]  
In Equation (2), \(X_{k}\) is the \(k\)th iterative solution of the unknown parameter; \(\alpha_{k}\) is the search direction for the \(k\)th iteration, which obeys [PERSON]'s step rule; \(d_{k}\) is the search direction for the \(k\)th iteration. Moreover, \(d_{k}\) can be expressed as in Equation (3),  
\[d_{k}=-A_{k}^{-1}j_{k}^{i} \tag{3}\]  
In Equation (3), \(j_{k}^{i}\) represents taking partial derivatives of Equation (1) with respect to the eight unknown parameters: s\({}_{1}\), s\({}_{2}\), s\({}_{3}\), s\({}_{4}\), \(\theta_{0}\), b, q, and tan\(\beta\); \(A_{k}\) is the Hessian approximation matrix of the objective function for the \(k\)th iteration, as shown in Equation (4),  
\[A_{k}=A_{k-1}+\frac{y_{k-1}y_{k-1}^{T}}{y_{k-1}^{T}s_{k-1}}-\frac{A_{k-1}s_{k-1}s _{k-1}^{T}A_{k-1}}{s_{k-1}^{T}A_{k-1}s_{k-1}} \tag{4}\]  
In Equation (4), \(y_{k-1}=\ddot{f}_{k}^{i}-\ddot{f}_{k-1}^{i}\).