Robust Parameter Inversion and Subsidence Prediction for Probabilistic Integral Methods in Mining Areas

Abstract

Surface subsidence induced by coal mining poses severe threats to global ecosystems and infrastructure. A critical challenge in subsidence prediction lies in the sensitivity of existing probabilistic integral parameter inversion methods to gross errors, leading to unstable predictions and compromised reliability. To address this limitation, we propose the IGGIII-BFGS algorithm that integrates robust estimation with unconstrained optimization method, enhancing resistance to gross errors during parameter inversion. Through systematic comparison of four robust estimation methods (Huber, L1, Geman–McClure, IGGIII) fused with BFGS, the IGGIII-BFGS method demonstrated superior stability and accuracy, reducing relative errors in key parameters (subsidence factor q, horizontal displacement coefficient b, and tangent of major influence angle tan β) to near-zero levels. Validation on the Huainan mining case study showed that the IGGIII-BFGS method achieved a 25.8% reduction in subsidence RMSE compared to standard BFGS, with predicted curves exhibiting strong agreement with field measurements. This advancement enables precise forecasting of subsidence and horizontal displacement, which hold significant value for the sustainable development of the surface ecological environment and social stability.

1. Introduction

China is a country rich in coal and less in oil and gas. Coal plays a central role in China’s energy structure [1,2]. However, the disasters associated with coal mining have brought serious challenges to the ecology and social stability of mining areas [3]. Therefore, it is particularly important to study the patterns and characteristics of surface changes caused by coal mining [4].

The most commonly used of the existing coal mining prediction methods is the probability integral method, which establishes a probabilistic integral nonlinear model, derives the formula, and then calculates and analyzes the surface deformation law [5,6]. The formula contains parameters that need to be inverted beforehand, and the specific formula can be obtained only after it is derived. Therefore, probability integral parameter prediction is a key link for this method to achieve the prediction of mining subsidence, and the inversion accuracy directly affects the validity of surface deformation prediction.

Probability integral parameter prediction mainly includes the linearity approximative method, modular vector method, intelligent optimization algorithm, etc. [7,8]. Guo [9] inverted the parameters by introducing robust estimation theory, which reduces the interference of heteroscedasticity and coarseness and overcomes the problem of parameter divergence in fitting the linearity approximative method. Ge [10] applied the modular vector method for solving the unconstrained extreme value problem to obtain the predicted mining subsidence parameters and solved the problem of using the measured mining data of arbitrary shape working face and using the dynamic measured data to obtain parameters. However, these two methods have significant limitations: they have high requirements for the initial values of parameters and are prone to fall into local optimal solutions during the iterative process. In recent years, intelligent optimization algorithms have attracted extensive attention from scholars and become a research hotspot. Experts and scholars are committed to the innovation and improvement of intelligent optimization algorithms, such as particle swarm optimization algorithm [11], genetic algorithm [12], and whale optimization algorithm [13], taking them as important tools to solve the problem of parameter inversion, which have improved the prediction accuracy and computational efficiency [14,15]. However, it still has inherent flaws, such as the attenuation of convergence speed in the later stage of iteration and the poor stability of parameter inversion. In recent years, optimization methods have been gradually applied to the field of parameter prediction due to their superlinear convergence and strong stability [16]. Yang [17] introduced the optimization algorithm into the probability integral parameter estimation and transformed the parameter inversion under the probability integral model into an unconstrained optimization problem. The parameter inversion results are significantly better than those of traditional optimization algorithms (linearity approximative method and modular vector method) and intelligent optimization algorithms, which show better parameter estimation performance and stronger stability and improve the accuracy of parameter estimation. However, in the case of gross error, the BFGS algorithm will find it difficult to jump out of the local minimum point and cannot find the global optimal solution, and the estimated parameter value deviates from the true value, which affects the accuracy of parameter prediction. In response to the problem that the existing parameter estimation methods have poor robustness, Wang [18] developed a robust CA-rPSO algorithm with adaptive weighting, achieving highly robust parameter optimization. Lou [19] proposed a robust ridge estimation model, combining ridge estimation and robust theory to address ill-conditioned matrices and gross errors, improving parameter stability and accuracy. The two aforementioned improvement methods are based on the combination of intelligent optimization algorithm and linearity approximative method with robust method. Moreover, the BFGS algorithm outperforms the intelligent optimization algorithm and linearity approximative method in terms of convergence speed and stability. Therefore, embedding the robust estimation theory into the BFGS framework and constructing a robust BFGS algorithm have significant theoretical significance and engineering value for achieving high-precision and high-reliability parameter inversion of probability integrals.

To address the limitations of existing studies, this paper proposes a robust estimation parameter inversion algorithm based on the BFGS method, the IGGIII-BFGS method, which improves the BFGS parameter inversion method by substituting the least-squares correction weight with the IGGIII weight function to improve the resistance to gross errors, further improving the accuracy of the parameter results.

2. Research Methodology

2.1. Principle of the Probability Integral Method

The probability integral method is a mining subsidence mobile deformation prediction method based on stochastic medium theory, which considers that the surface subsidence belongs to random events [20]. It proposes to divide the whole mining area into many tiny units, and the surface subsidence is the sum of the effects of each unit on the surface superimposed on each other and can be solved directly by the integration method [21,22,23].

Take the origin o of the unit coordinate system o-xyz as the center of the mining unit, and the unit subsidence basin-wide expression on the neighboring domain dGA (area microproduct) of point A(x,y,z) is

$$W_e=f\left(x^2\right)f\left(y^2\right)={\displaystyle \frac{1}{r^2}}{e}^{-\pi {\displaystyle \frac{x^2+y^2}{r^2}}}$$

(1)

In the formula presented, $f\left(x^2\right)$ and $f\left(x^2\right)$ denote the probability density function that follows a normal distribution; $r$ denotes main influence radius; and the calculation formula is

$$r={\displaystyle \frac{H}{\tan \beta }}$$

(2)

In the formula presented, H denotes the mining depth of the working face along the strike; tan β denotes the tangent of major influence angle.

W_e divide the entire mining area to obtain the expression for the subsidence full-basin integral:

$$W\left(x,y\right)=W_0\int {\int }_s{W}_{e}dS$$

(3)

In the formula presented, dS is the area of coal seam micro-original mining unit, and W₀ is the maximum subsidence value.

Therefore, the formula model for calculating subsidence at any point on the surface is

$$W\left(x,y\right)=W_0\times C_{xm}\times C_{ym}$$

(4)

In the formula presented, $C_{xm}$ and $C_{ym}$ denote $x$ (strike) and $y$ (inclination) directions, respectively; $W_0$ denotes the maximum subsidence value, which is calculated by the following formula:

$$W_0=m\times q\times \cos \alpha$$

(5)

In the formula presented, $m$ denotes mining thickness in the vertical tangential direction; $q$ denotes subsidence coefficient; and $\alpha$ is the dip angle of coal seam.

$$C_{xm}={\displaystyle \frac{1}{\sqrt{\pi }}}\times \left({\int }_{-\sqrt{\pi }\times {\displaystyle \frac{x}{r}}}^{\mathrm{\infty }}{e}^{-{\lambda }^2}d\lambda -{\int }_{-\sqrt{\pi }\times {\displaystyle \frac{x-l}{r}}}^{\mathrm{\infty }}{e}^{-{\lambda }^2}d\lambda \right)$$

(6)

$$C_{ym}={\displaystyle \frac{1}{\sqrt{\pi }}}\times \left({\int }_{-\sqrt{\pi }\times {\displaystyle \frac{y}{r_1}}}^{\mathrm{\infty }}{e}^{-{\lambda }^2}d\lambda -{\int }_{-\sqrt{\pi }\times {\displaystyle \frac{y-L}{r_2}}}^{\mathrm{\infty }}{e}^{-{\lambda }^2}d\lambda \right)$$

(7)

In the formula presented, $d\lambda$ denotes the unit thickness of the coal seam; $l$ is the calculated length of the working face along the strike; $L$ is the calculated length of the tendency working face; and their calculation formulas are

$$l=D_3-s_3-s_4$$

(8)

$$L=\left(D_1-s_1-s_2\right)\times {\displaystyle \frac{\sin \left({\theta }_0+\alpha \right)}{\sin {\theta }_0}}$$

(9)

In the formula presented, $D_1$ is the working face tendency length; $D_3$ is the working face strike length; and ${\theta }_0$ is the propagation angle of extraction; and $s_1$, $s_2$, $s_3$, and $s_4$ are deviation of the inflection points of the downhill boundary, uphill boundary, left boundary, and right boundary, respectively.

Similarly, the horizontal movement of any point on the surface is given by

$${U\left(x,y\right)}_{\phi }=U_{ax}\times C_{ym}\times \cos \phi +U_{ay}\times C_{xm}\times \sin \phi$$

(10)

In the formula presented, $\phi$ is the angle value from the positive counterclockwise of the x-axis to the specified direction; and $U_{ax}$ and $U_{ay}$ are the horizontal movement values of the projective point positions on the principal section of the strike and tendency, respectively.

2.2. Robust Estimation Method

When gross errors cannot be avoided, robust estimation methods can ensure more reliable results by fully utilizing valid information, excluding potentially harmful information and minimizing the impact of gross errors [24,25].

The IGG (Institute of Geodesy and Geophysics) III method is one of the more commonly used robust estimation methods in recent years [26]. In the traditional least-squares method, all the residuals (the difference between the observed value and the estimated value) are treated equally. The IGGIII method introduces a specific weight function, which appropriately adjusts the size of the residuals through a weight adjustment mechanism [27]; the specific weight function helps to weaken the adverse effect of gross errors on the estimation of unknowns to a certain extent and thus improves the stability of the estimation [28,29]. The specific-weight IGGIII method is

$$\begin{array}{c}{p}_i=\left\{\begin{array}{cc}{p}_i,& \left|\widetilde{v_i}\right|\le k_0\hfill \\ p_i*{\displaystyle \frac{k_0}{\left|\widetilde{v_i}\right|}}{\left({\displaystyle \frac{k_1-\left|\widetilde{v_i}\right|}{k_1-k_0}}\right)}^2,& k_0<\left|\widetilde{v_i}\right|\le k_1\hfill \\ 0,& k_1\le \left|\widetilde{v_i}\right|\hfill \end{array}\right.\end{array}$$

(11)

In the formula presented, $k_0$ and $k_1$ denote conciliation coefficients: $k_0$ usually takes 1.0~1.5, and $k_1$ usually takes 2.5~3.0 [30,31]; $\widetilde{v_i}$ denotes standardized residual; and the specific formula is shown below:

$$\widetilde{v_i}={\displaystyle \frac{v_i}{{\sigma }_0}}$$

(12)

$v_i$ denotes the residual of the ith observation, ${\sigma }_0$ is the unit weight medium error, and the specific formula is shown below:

$${\sigma }_0=\sqrt{\frac{V^{T}V}{n-m}}$$

(13)

In the formula presented, $V$ denotes the residual vector, $n$ denotes the number of observations, and $m$ denotes the number of unknowns.

2.3. BFGS Parameter Inversion Algorithm

The BFGS (Broyden–Fletcher–Goldfarb–Shanno) method transforms the objective function into an unconstrained optimization problem by constructing the objective function, determining the Hessian matrix approximation array through the first derivatives, then iterating afterward, continuously updating the search direction and the step size to determine the variables, and then gradually approximating the minimum value of the objective function, thereby achieving robust and stable parameter inversion [32,33]. The specific steps are as follows:

(1) To approximate the minimum value, the objective function f is first constructed according to the probability integral formula. The objective function formula is as follows:

$$f=\sum _{i=1}^N{\left[W\left(x,y\right)-W_i\right]}^2+\sum _{i=1}^N{\left[{U\left(x,y\right)}_{\phi }-U_i\right]}^2$$

(14)

In the formula presented, N is the number of ground observation points; $W\left(x,y\right)$ and ${U\left(x,y\right)}_{\phi }$ are the predicted values of subsidence and horizontal movement of the ith point; $W_i$ is the subsidence measurement value; and $U_i$ is the horizontal movement measurement value.

(2) At which time, the gradient of each probability integral parameter, the partial derivative $G_k$, is calculated. Using the BFGS update rule, the search direction d is calculated in each iteration step. Where the initial Hessian matrix $H_0$ is the identity matrix,

$$d=-H_k\times G_k$$

(15)

(3) Armjio line search is utilized to determine the step size $\alpha$, to obtain feasible and significantly improved points in the search direction. The Armjio line search rule is as follows:

$$f\left(X_k+\alpha d_k\right)\leqq f\left(X_k\right)+c\alpha {G_k}^T{d}_k$$

(16)

In the formula presented, $c$ ∈ (0,1), and $X_k$ is the parameter vector after the kth iteration.

(4) The current iteration point $X_{k+1}$ is updated:

$$X_{k+1}=X_k+\alpha \times d$$

(17)

(5) The gradient $G_{k+1}$ is updated, the BFGS is set to find the convergence parameter ε, and the BFGS computation iteration is stopped when the norm of the gradient is less than ε stops from the BFGS computation iteration.

(6) The Hessian matrix approximation array is updated by BFGS update rule.

$$H_{k+1}=H_k+\left(1+{\displaystyle \frac{{Y_k}^{\mathrm{T}}\times H_k\times Y_k}{v}}\right)\times {\displaystyle \frac{S_k\times {S_k}^{\mathrm{T}}}{v}}-{\displaystyle \frac{S_k\times {Y_k}^{\mathrm{T}}\times H_k+H_k\times Y_k\times {S_k}^{\mathrm{T}}}{v}}$$

(18)

In the formula presented, $Y_k=G_{k+1}-G_k{,S}_k=X_{k+1}-X_k$, if ${S_k}^{\mathrm{T}}\times Y_k>0$, then $v={Y_k}^{\mathrm{T}}\times S_k$, otherwise $H_{k+1}=H_k$.

(7) Let k = k + 1, and go back to step (2).

2.4. Robust BFGS Parameter Inversion Algorithm

Probabilistic integral parameter inversion for parameterization with the IGGIII-BFGS algorithm uses a stepwise BFGS weight selection iteration method:

(1) Set the initial value of the parameters to be obtained according to the experience value of the probability integral parameter differentiated by stratum properties: q₀, tan β₀, ${\theta }_{0_0}$, $s_{1_0}$, $s_{2_0}$, $s_{3_0}$, $s_{4_0}$, b₀, and the initial Hessian approximation array H₀, the iterative convergence accuracy ε.

(2) Calculate the predicted subsidence value based on parameter vector X, compute the residual difference between the predicted value and the observed value, update the weight matrix using the IGGIII weight function, and then build the objective function f₁:

$${\mathrm{f}}_1=\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left[W_i\left(\mathrm{X}\right)-{\mathrm{W}}_i\right]}^T\times {\mathrm{P}}_{\mathrm{W}}\times \left[W_i\left(\mathrm{X}\right)-{\mathrm{W}}_i\right]$$

(19)

In the formula presented, N is the number of ground observation points; P_W is the weight matrix corresponding to the residuals; X is a vector of parameters to be solved consisting of the parameters q₀, tan β₀, ${\theta }_{0_0}$, $s_{1_0}$, $s_{2_0}$, $s_{3_0}$, and $s_{4_0}$; and the function $W_i\left(\mathrm{X}\right)$ with independent variable X can be obtained by substituting the working face location data and geological mining parameters into the subsidence prediction formula.

The BFGS algorithm is utilized to calculate the partial derivatives cyclically and further calculate the Hessian approximation array, search direction d, and step size α to update the parameters until the partial derivatives satisfy the convergence conditions to stop the loop. Then, determine whether the parameters satisfy the convergence condition; if not, re-update the weight array and the objective function to find the parameters until they satisfy the convergence conditions and output parameter vector X.

(3) Substitute parameter X, the working face position data, and the geological mining parameters into the horizontal movement formula to obtain function U (b) with the independent variable b. Establish objective function f₂, and calculate parameter b using the same method.

$${\mathrm{f}}_2=\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left[U_i\left(\mathrm{b}\right)-{\mathrm{U}}_i\right]}^T\times {\mathrm{P}}_{\mathrm{U}}\times \left[U_i\left(\mathrm{b}\right)-{\mathrm{U}}_i\right]$$

(20)

In the formula presented, ${\mathrm{U}}_i$ is the measured value of horizontal movement, and P_U is the weight matrix corresponding to the residual of horizontal movement. Function $U_i\left(\mathrm{b}\right)$ with independent variable b can be obtained by substituting the working face location data and geological mining parameters into the subsidence prediction formula.

The specific steps are shown in Figure 1:

3. Simulation Experiments and Statistical Analyses of Results

3.1. Design of Simulation Experimental Area and Experimental Data

To verify the accuracy and stability of the inversion parameters of the IGGIII-BFGS algorithm, a simulation experiment area is established to discuss the performance of the inversion of the parameters of each method after adding gross errors to the simulated observations.

Firstly, the geological mining conditions and probability integral parameters of the mining face are designed, with mining thickness in the vertical tangential direction m = 4 m, the dip angle of coal seam α = 6°, the working face tendency length D₁ = 300 m, strike length D₃ = 500 m, and mining depth H = 300 m. The values of the probability integral parameters are as follows: subsidence coefficient q = 0.85, horizontal displacement factor b = 0.25, tangent of major influence angle tan β = 2.15, propagation angle of extraction θ₀ = 86°, and offset distance of inflection points s₁ = s₂ = s₃ = s₄ = 50 m. There are 13 monitoring points on the strike line of the designed working face (E1 to E13) and 13 monitoring points on the tendency line (N1 to N13), totaling 26 monitoring points. The locations of the simulated working face and monitoring points are shown in Figure 2.

According to the designed parameter values in the probabilistic integral model, the subsidence and horizontal movement values of the above 26 monitoring points were calculated as the true values. Afterward, some gross errors are artificially added to them to compare the anti-interference ability of parameter inversion of each method.

3.2. Comparison of Commonly Used Robust Estimation Methods

In order to explore the accuracy and stability of different robust estimation methods combined with the BFGS algorithm, three commonly used robust estimation methods are selected to compare with the IGGIII method, and the weight functions of the three commonly used robust estimation methods are shown below:

1. Huber method:

$$p_i=\left\{\begin{array}{c}1,\left|\widetilde{v_i}\right|\le \mathrm{c}\\ {\displaystyle \frac{c}{\left|\widetilde{v_i}\right|}},\left|\widetilde{v_i}\right|>\mathrm{c}\end{array}\right.$$

(21)

2. L1 method:

$$p_i={\displaystyle \frac{1}{\left|\widetilde{v_i}\right|}}$$

(22)

3. Geman–McClure method:

$$p_i={\displaystyle \frac{1}{{\left(1+{\widetilde{v_i}}^2\right)}^2}}$$

(23)

Of the three formulas given above, c is the conciliation coefficient that takes into account the balance of robustness and efficiency, and $\widetilde{v_i}$ is the standardized residual, calculated as

$$\widetilde{v_i}={\displaystyle \frac{v_i}{{\sigma }_0}}$$

(24)

$v_i$ is the residual of the ith observation, and ${\sigma }_0$ is the unit weight medium error, calculated as

$${\sigma }_0=\sqrt{{\displaystyle \frac{V^{T}PV}{n-m}}}$$

(25)

In the formula presented, $n$ is the number of observations, and $m$ is the number of unknowns.

Combining the above IGGIII method and three robust estimation methods with the BFGS algorithm, respectively, under the above simulation experimental conditions, the true value of the above simulation experiment area design is added to the random error with a median error of 10 mm, 50 parameter inversion experiments are carried out, respectively, a point near the inflection point and a point near the maximum subsidence value are randomly selected for each experiment, and 200 mm gross error is added to the parameter inversion, resulting in the parameter inversion results of the four robust estimation methods applied in the BFGS method, as shown in Figure 3 and Figure 4.

From Figure 3, it can be seen that the results of the inversion of the three main parameters q, b, and tan β in the IGGIII method converge consistently across iterations, while the other three methods are caught in the local minimum problem, and the results of the inversion of the three main parameters fluctuate to different degrees. From Figure 4, it can be seen that the stability of inversion of s₁, s₂, s₃, and s₄ is worse than that of the first three main parameters, but all four parameters fluctuate within a certain range.

To further validate and compare the accuracy of the inversion parameters obtained by combining four robust estimation methods with the BFGS algorithm, we re-conducted 50 experiments by adding random errors with a median error of 10 mm to the true value of the above simulated experimental area design. In each experiment, we randomly selected one point near the inflection point and one point near the maximum subsidence value, adding 300 mm of gross error for the parameter inversion. This approach allowed us to derive the parameter inversion results for the four robust estimation methods applied within the BFGS algorithm. The results from the 50 inversions, both with 200 mm and 300 mm gross error added, were averaged, and the relative errors of these averages were calculated. The findings are presented in Figure 5.

From Figure 5, it is evident that the IGGIII method achieved the highest accuracy in inverting the three primary parameters—q, b, and tan β—across both experimental groups, with relative errors reaching as low as 0.0%. The inversion accuracy of the remaining methods—Huber, L1, and Geman–McClure—decreases in that order. Additionally, Figure 5 indicates that the inversion accuracies for parameters s₁, s₂, s₃, and s₄ across the four methods show only slight margin in general.

In summary, the IGGIII method, when combined with the BFGS algorithm, demonstrates the highest accuracy and stability among the inversion parameters.

3.3. Comparison of IGGIII-BFGS Algorithm with BFGS Algorithm

The gross error rate in the observed data is generally not more than 10%. Therefore, based on the true values, random errors with a median deviation of 10 mm were added, and the gross error of 200 mm is added to one to two maximum subsidence values, edges, and near the inflection points in each experiment as the new test data. To reduce randomness in the experimental results, 20 inversions of the BFGS algorithm and IGGIII-BFGS algorithm were performed for each experiment, and the mean values of the parameters and their relative errors of the inversions of the two algorithms are given in

Table 1. Comparison of inversion parameters and relative errors for different roughness cases.

Gross Error Number	Position of Gross Error	Parameter Name	q	b	tan β	s1	s2	s3	s4
		Truth Value	0.85	0.25	2.15	50	50	50	50
1	Near the maximum subsidence value	BFGS	0.832	0.255	2.168	50.654	47.160	51.804	47.709
		relative error	2.13%	1.90%	0.84%	1.31%	5.68%	3.61%	4.58%
		IGGIII-BFGS	0.850	0.250	2.150	51.614	48.386	50.881	49.119
		relative error	0.00%	0.00%	0.00%	3.23%	3.23%	1.76%	1.76%
	Near the inflection points	BFGS	0.848	0.251	2.179	51.506	48.738	46.252	53.405
		relative error	0.28%	0.35%	1.35%	3.01%	2.52%	7.50%	6.81%
		IGGIII-BFGS	0.850	0.250	2.150	52.928	47.092	52.415	47.557
		relative error	0.02%	0.03%	0.01%	5.86%	5.82%	4.83%	4.89%
	Near the edges	BFGS	0.849	0.250	2.155	49.722	50.237	50.537	49.577
		relative error	0.10%	0.15%	0.25%	0.56%	0.47%	1.07%	0.85%
		IGGIII-BFGS	0.850	0.250	2.150	49.209	50.789	48.670	51.330
		relative error	0.00%	0.00%	0.00%	1.58%	1.58%	2.66%	2.66%
2	Near the maximum subsidence value	BFGS	0.815	0.259	2.181	48.649	46.716	53.040	46.067
		relative error	4.11%	3.60%	1.45%	2.70%	6.57%	6.08%	7.87%
		IGGIII-BFGS	0.850	0.250	2.150	48.966	51.033	49.412	50.589
		relative error	0.00%	0.00%	0.00%	2.07%	2.07%	1.18%	1.18%
	Near the inflection points	BFGS	0.839	0.255	2.200	50.739	49.287	46.984	52.516
		relative error	1.29%	1.87%	2.30%	1.48%	1.43%	6.03%	5.03%
		IGGIII-BFGS	0.849	0.250	2.154	48.528	51.474	47.781	48.284
		relative error	0.00%	0.00%	0.00%	2.94%	2.95%	4.44%	3.43%
	Near the edges	BFGS	0.845	0.252	2.181	47.605	52.162	49.261	51.671
		relative error	0.55%	0.86%	1.44%	4.79%	4.32%	1.48%	3.34%
		IGGIII-BFGS	0.850	0.250	2.150	50.397	49.602	53.034	46.967
		relative error	0.00%	0.00%	0.00%	1.58%	1.58%	2.66%	2.66%

.

Table 1 shows that when the number of roughness is one, the maximum relative errors of the three main parameters q, b, and tan β using BFGS inversion are 2.13%, 1.90%, and 1.35%, respectively; when the number of roughness is two, the maximum relative errors of the three main parameters q, b, and tan β using BFGS inversion are 4.11%, 3.60%, and 2.30%, respectively; and the maximum relative error of the three main parameters q, b, and tan β using IGGIII-BFGS inversion for the three parameters is 0.00% in all cases. For the four parameters s₁, s₂, s₃, and s₄, the results of the inversion by the two methods are not much different in general.

It can also be seen from Table 1 that the effect on the inversion accuracy of parameters q and b is greatest when the roughness appears near the maximum subsidence value, and the effect on the inversion accuracy of parameters q and b is smallest when the roughness appears at the edge. The effect on the inversion accuracy of tan β is greatest when the roughness appears near the inflection point, and the effect on tan β is smallest when the roughness appears at the edge.

Figure 6 shows the comparison of subsidence value and horizontal movement value. Two points are selected near the maximum subsidence value to join the 200 mm roughness, respectively, using two methods of parameter inversion experiments to obtain the parameter mean value, using the parameter mean value of the expected subsidence value of the plotted curve and then compared with the measured value. It can be seen that the curves predicted by the IGGIII-BFGS method nearly coincide with the observed values, which is better relative to the BFGS method. The absolute value of the error between the subsidence value and the horizontal movement value predicted by the IGGIII-BFGS algorithm and the true value are all within 10 mm, the absolute value of the error of BFGS algorithm exceeds 10 mm in many points, and some points even exceed 90 mm. The root mean square error (RMSE) is the square root of the ratio of the sum of the squares of the deviations of the predicted values from the observed values to the ratio of the number of observed points, in which the root mean square error of the subsidence values and the root mean square error of the horizontal shifts for the BFGS method are 39.1 mm and 14 mm, respectively, while the root mean square error of the subsidence and the root mean square error of the horizontal movement for the IGGIII-BFGS method are 1.34 mm and 5.19 mm, respectively. The absolute value errors and root mean square errors of the IGGIII-BFGS algorithm are both smaller than those of the BFGS algorithm.

In summary, it can be proved that the IGGIII-BFGS algorithm can effectively resist the interference of roughness compared with the BFGS algorithm, and the inversion obtains more accurate parameters and improves the accuracy of subsidence prediction.

3.4. Comparison of IGGIII-BFGS Algorithm with PSO Algorithm

We added a comparative experiment between the PSO algorithm and the IGGIII-BFGS algorithm, randomly selecting one observation point, respectively, at the maximum settlement point and the inflection point. We added a gross error of 200 mm to the settlement value of each point to generate the perturbation dataset and then selected the more classic and widely applied intelligent optimization algorithm—the PSO algorithm—comparing it with the IGGIII-BFGS algorithm. The two algorithms, respectively, carried out 20 inversions of probability integral parameters. The mean value and relative error of the inversion parameters were calculated. The comparison of the mean values and errors of the inversion parameters of the two algorithms is shown in

Table 2. Comparison of IGGIII-BFGS algorithm with PSO algorithm.

Parameter	PSO Algorithm			IGGIII-BFGS Algorithm
	Inversion Mean	Absolute Value of Error	Relative Error	Inversion Mean	Absolute Value of Error	Relative Error
q	0.812	0.038	4.53%	0.850	0.000	0.00%
b	0.259	0.009	3.54%	0.250	0.000	0.02%
tan β	2.224	0.074	3.42%	2.150	0.000	0.00%
θ0	86.413	0.413	0.48%	85.953	0.047	0.05%
s1	42.599	7.401	14.80%	50.045	0.045	0.09%
s2	53.110	3.110	6.22%	49.000	1.000	2.00%
s3	33.857	16.143	32.29%	49.138	0.862	1.72%
s4	67.206	17.206	34.41%	50.408	0.408	0.82%

.

Table 2 shows that for most parameters, the absolute error and relative error of the IGGIII-BFGS algorithm are significantly lower than those of the PSO algorithm. The IGGIII-BFGS algorithm is less affected by gross errors and can predict parameters more stably and accurately. In contrast, the PSO algorithm has large error fluctuations in some parameters and is less stable when dealing with gross error interference. Overall, the IGGIII-BFGS algorithm performs better in parameter prediction.

4. Engineering Examples and Results of Statistical Analyses

4.1. Overview of the Mining Area

To further validate the performance of the IGGIII-BFGS algorithm, the engineering application example of the Banji coal mine is selected to compare the accuracy of parameter inversion between the BFGS algorithm and the IGGIII-BFGS algorithm. Banji coal mine is located in Huainan City, Anhui Province, and 110801 working face is selected for the experimental analysis.

The 110801 first mining face adopts the integrated mechanized coal mining method of strike-longwall-based backward one-time full-height mining, and the roof is managed by the all-caving method. This working face has the characteristics of thick loose layer, large mining depth, and thin bedrock. Trial mining began on 1 April 2021, and mining ended on 15 October 2022, with 562 days of mining time, an average strike length of 1300 m and a tendency length of 279 m, with the top plate of the coal seam mostly sandy mudstone and the bottom plate mostly mudstone and sandy mudstone, with an average inclination angle of 8°, an average coal thickness of 2.2 m, and an average mining thickness of 2.5 m, an average thickness of the loose layer in the monitoring area of 600 m, and an average mining depth of 767 m. The schematic diagram of the 110801 working face is shown in Figure 7.

4.2. Results and Analyses

There are 177 observation points available in working face 110801, including 96 observation points on the strike observation line and 81 observation points on the inclination observation line. The subsidence values of the 177 observation points are obtained as the initial data, 5% of the observation points are randomly selected to join the 100 mm gross error, and the probability integral parameter inversion is performed by using the BFGS algorithm and the IGGIII-BFGS algorithm, respectively. The two algorithms were inverted 15 times, respectively, and the mean value was taken as the final inversion parameter. The subsidence value was predicted according to the obtained parameter, and the predicted curve was plotted. The results are shown in Figure 8, and the results of the parameter inversion and the root mean square error are shown in

Table 3. Comparison of the predicted results of the parameters of the two methods.

Method	q	tan β	θ0	s1	s2	s3	s4	RMSE/mm
BFGS	1.530	1.665	93.825	5.023	0.235	62.479	59.481	109.482
IGGIII-BFGS	1.513	1.808	93.825	15.809	4.605	71.095	72.479	81.278

.

According to Figure 8, it can be seen that the trends of the subsidence curves obtained by the two methods are basically the same as the measured values, but there is an obvious difference in the predicted curves in the area of maximum subsidence of the subsiding basin. The expected curves of the IGGIII-BFGS method in the area of maximum subsidence of the subsiding basin are more closely matched with the measured values, while the expected curves of the BFGS method are shifted upward significantly, and the expected curves of both methods at the inflection point are shifted to a smaller extent from the measured values. The predicted curves of the two methods at the inflection point also have a small degree of deviation from the measured values. Comparing the absolute value of the errors between the predicted values and observed values of the two methods, it can be seen that the absolute value of the errors of the BFGS method are between 0 and 300 mm, and the absolute value of the errors of the IGGIII-BFGS method are between 0 and 200 mm, the errors of the BFGS method are larger than that of the IGGIII-BFGS method in most cases, and the errors of both methods are larger near the maximum subsidence value and the inflection point. According to Table 3, it can be seen that the parameter inversion results of the two methods are still different, and the difference is bigger compared with the simulation results, in which the difference between the inversion results of q, tan β, and θ₀ is relatively small; the difference between the inversion results of the four parameters of s₁, s₂, s₃, and s₄ is bigger; and the root mean square error of the BFGS method is 109.482 mm and that of the IGGIII-BFGS method is 81.278 mm. However, the root mean square error of the IGGIII-BFGS method is still smaller than that of the BFGS method, and it can be seen that the predicted results of the IGGIII-BFGS are more accurate.

Overall, due to the complexity of the measured data in the actual engineering, which leads to its incomplete conformity to the probability integral prediction curve, compared with the simulation experiments, the accuracy of the prediction of the inversion parameters by both methods is reduced. However, the IGGIII-BFGS algorithm demonstrates superior practical performance. The RMSE decreases from 109.482 mm (BFGS) to 81.278 mm (IGGIII-BFGS), representing a 25.8% improvement. It is proved that the prediction results obtained by using the IGGIII-BFGS method have good validity and reliability. Despite the complex geology, IGGIII-BFGS still maintains a closer consistency with on-site measurements, meets the accuracy requirements of general engineering, ensures reliable settlement management in high-risk areas, and reduces the maintenance costs of mining-related infrastructure.

5. Conclusions

In this paper, the robust estimation theory is applied to the process of probabilistic integral parameterization of the BFGS algorithm, and the IGGIII-BFGS parameter inversion algorithm is proposed to reduce the influence of gross error on the parameter inversion and to improve the accuracy of the parameter inversion. The precision and stability of the method are examined through simulation experiments and engineering examples, and the following conclusions are drawn:

(1) Comparing the resistance to a gross error of different robust estimation methods combined with the BFGS algorithm, the accuracy of the IGGIII weight function combined with the BFGS algorithm for the inversion of the main parameters q, b, and tan β still reaches 0.0%, and the overall inversion accuracy and stability are better than that of the other three methods.

(2) It is proved by simulation experiments that the IGGIII-BFGS algorithm has a stronger resistance compared to the BFGS algorithm for parameterization, for the surface movement observations containing gross errors, parameter prediction, and subsidence prediction can be more accurately carried out by using the IGGIII-BFGS algorithm.

(3) Although the performance advantage of the IGGIII-BFGS method diminishes when applied to real inversion data compared to simulation experiments, it still demonstrates superior effectiveness over the standard BFGS method. Specifically, it more accurately processes meaningful information, mitigates the impact of noise or outliers, and yields more optimal parameter estimates. The IGGIII-BFGS method demonstrates improved performance over the standard BFGS method when applied to real inversion data. As shown in Table 3, the IGGIII-BFGS approach yields a lower root mean square error (RMSE) of 81.278 mm, compared to 109.482 mm for the BFGS method, indicating higher overall prediction accuracy. Additionally, the IGGIII-BFGS method provides notably different and potentially more robust estimates for parameters such as s₁, s₂, s₃, and s₄, suggesting an enhanced ability to manage data variability and outlier influence.

(4) This paper also has some limitations in the study. These are seen in the simulation experiment using the IGGIII-BFGS method to find the parameters of the propagation angle of extraction θ₀. The results have a certain gap with the true value. The possible reasons for this are that θ₀ has a high collinearity with the inflection point offset (s₁, s₂, s₃, and s₄) of the near-symmetric sedimentation basin, the IGGIII weight function mechanism is not applicable to θ₀, etc. To solve the above limitations, subsequent studies can focus on using Bayesian regularization or constrained optimization frameworks, optimizing the fusion of equivalence weights and BFGS Hessian update rules to suppress the collinearity effect. In this paper, the IGGIII-BFGS method directly uses the BFGS method to solve θ₀ and can also achieve good results.