## 2 Materials and Methods
### Trust-Region Reflective Algorithm (TRF) for Parameters  
For the parameter calculation of the time function model, this paper employed the nonlinear least squares fitting method to fit the model's predicted time function with the observed data. The objective of this fitting process was to minimize the sum of squared residuals between the observed data and the model's predicted values in order to seek the best-fitting parameters. There are various optimization methods for finding the optimal parameters in nonlinear least squares fitting, such as the Gauss-Newton method [54; 55], the TRF algorithm [40; 56; 57], and the Levenberg-Marquardt algorithm [58; 59]. The Gauss-Newton method transforms the nonlinear fitting problem into a linear least squares problem for solving. The TRF algorithm adopts a trust-region approach, converting the nonlinear least squares problem into a subproblem for iteratively solving the trust region. In each iteration, it establishes a trust region to control the update step size and direction of parameters and adjusts the step length through reflection and contraction. The Levenberg-Marquardt algorithm is commonly used to solve minimum unconstrained optimization problems, combining the advantages of gradient descent and [PERSON]'s methods. However, it is sensitive to the choice of initial values. This algorithm demonstrates high accuracy and fast convergence only when the initial values are in the vicinity of the global optimum. Poor initial value selection may lead to convergence issues or getting stuck in local optima. In contrast, the TRF algorithm offers good stability and convergence performance in nonlinear optimization problems. It ensures convergence to the optimal solution in a finite number of steps even if the initial parameter values are not chosen rationally. It can quickly approach the best solution while avoiding getting trapped in local optima. In this paper, the TRF algorithm was chosen for finding the optimal parameters. When utilizing the TRF algorithm to solve for the parameters of the time function model, the first step is to transform the optimization problem of the model parameters into a nonlinear least squares optimization problem. The objective function is defined as follows:  
\[\left\{ \begin{array}{l}\min\!f(U)=\sum\limits_{i=1}^{N}\left[W_{i}(U)-W_{i} \right]^{2}\\ \text{ }s.t.\ lb\leq U\leq ub\end{array} \right. \tag{5}\]  
In the formula, \(W_{i}(U)\) represents the computed \(i\)-th subsidence value, \(W_{i}\) is the \(i\)-th measured subsidence value, and \(i\) corresponds to the number of observations for each point, \(i=1,2,\ldots,N\). \(U\) is the parameter vector to be inverted, and \(ub\) and \(lb\) denote the upper and lower bounds of \(U\). When the time function model includes parameter \(c\), \(U=\left[W_{m}\quad a\quad b\quad c\right]^{\text{T}}\); when it does not include parameter \(c\), \(U=\left[W_{m}\quad a\quad b\right]^{\text{T}}\). In this paper, we assumed the function model included parameter \(c\). The objective function is transformed into a trust region model subproblem as follows:  
\[\left\{ \begin{array}{l}\min\!m_{k}(p)=f(U_{k})+g_{k}{}^{T}p+\frac{1}{2}p^{T} H_{k}p\\ s.t.\ \|p\|\leq\Delta_{k}\end{array} \right. \tag{6}\]  
In the formula, \(k\) represents the \(k\)-th iteration, \(m_{k}(p)\) is the quadratic model, \(f(U_{k})\) represents the function value of \(f(U)\) at the point \(U(k)\), \(g_{k}\) is the gradient of \(f(U)\) at the current point \(U(k)\), \(H_{k}\) represents the Hessian matrix, and \(\Delta_{k}\) is the trust region radius. \(p\) is the solution to the subproblem and the iteration step (i.e., the iteration update \(p_{k}\), including both step length and direction). The calculations for \(g_{k}\) and \(H_{k}\) are as follows:  
\[g_{k}=\left(\frac{\partial f}{\partial W_{m}}\quad\frac{\partial f}{\partial a }\quad\frac{\partial f}{\partial b}\quad\frac{\partial f}{\partial c}\right) \tag{7}\]\[H_{k}=\left(