## 2 Theory of Parameter Inversion
### Principles and Optimization Steps of the Improved Modular Vector Method  
The modular vector method, also known as the step acceleration method, is a technique utilized to expedite the optimization of the valley line (ridge line) direction. This method presents notable advantages, particularly in addressing unconstrained extreme value problems [34]. Figure 1: Schematic representation of spatial mining configuration. In the context of parameter inversion within the probability integral method using the modular vector method, precautions are taken to prevent the inversion results from converging toward local extrema due to inappropriate initial exploration value selection. To address this concern, a pattern recognition approach based on the fuzzy pattern recognition theory is employed. This approach utilizes mining subsidence pattern recognition techniques and adheres to a specified classification of similar phenomenon groups in mining subsidence and fundamental rock movement parameters. The preliminary determination of mining subsidence rock movement parameters for the mining area is based on the principle of selecting the closest matching parameters. The procedural steps for mining subsidence pattern recognition and parameter determination are as follows:  
1. Calculate the comprehensive deformation modulus of the rock mass using drilling data obtained from the specific mining area. 2. Compute the similarity criterion for the mining area in question. 3. Determine the average similarity criterion across all types of mining subsidence. 4. Calculate the proximity degree between the mining area in question and each type of mining subsidence. Following the principle of maximum proximity degree, identify the category to which the mining area belongs. Adopt the mining subsidence rock movement parameters corresponding to the determined mining subsidence category for the mining area under investigation. Following the prediction principle of the probability integral method, it is evident that the relationship between the subsidence value at any designated point \((x,y)\) on the surface and the predictive parameters of the probability integral method can be expressed as follows:  
\[f(x,y)=W(x,y;B) \tag{5}\]  
In the formula, \(B\) represents the predicted parameters of the probability integral method, which are the values subject to inversion. \(B=(q,\theta,tg\beta,S_{1},S_{2},S_{3},S_{4})^{T}\). Assume that \(F(x,y)\) represents the measured values at a specific point \((x,y)\) on the surface. Following the principle of least squares, the criterion for parameter inversion relies on minimizing the sum of squared errors between the predicted values and the measured values, which can be expressed as follows:  
\[\sum_{i=1}^{n}{v_{i}}^{2}=\min \tag{6}\]  
\[v_{i}=F(x,y)-f(x,y) \tag{7}\]  
In accordance with Equation (6), the general procedural steps for constructing the modular vector method for parameter inversion within the probability integral method are as follows:  
(1) Formulate the objective function as follows:  
\[\varepsilon(B)=\sum_{i=1}^{n}{v_{i}}^{2} \tag{8}\]  
In the equation, \(\varepsilon(B)\) represents the predicted error function. (2) Determine the initial parameter values: \(B_{1}=(q,\theta,tg\beta,S_{1},S_{2},S_{3},S_{4})^{T}\). (3) Specify the exploration step size and termination step size for each parameter. (4) Compute the error function value \(\varepsilon(B_{1})\) at the initial reference point \(B_{1}\), then adjust the modulus vector of the first parameter in \(B_{1}\) to obtain the new modulus vector point \(B_{11}\). The calculation of \(B_{11}\) is in accordance with Equation (9). \[B_{11}=\left\{ \begin{array}{c}B_{1}+\Delta_{11}\quad,\varepsilon(B_{1}+ \Delta_{11})<\varepsilon(B_{1})\\ B_{1}-\Delta_{11}\quad,\varepsilon(B_{1}-\Delta_{1i})<\varepsilon(B_{1})< \varepsilon(B_{1}+\Delta_{11})\\ B_{1}\quad,\varepsilon(B_{1})<\min\{\varepsilon(B_{1}-\Delta_{1i}), \varepsilon(B_{1}+\Delta_{11})\}\end{array} \right.