Parameter Inversion of Probability Integral Model Based on GA–BFGS Hybrid Algorithm

Abstract

The probability integral method is the primary technique for predicting mining-induced subsidence in China, and its predictive accuracy strongly depends on the precision of the model parameters. To improve the accuracy and stability of parameter inversion and to overcome the convergence randomness of the Genetic Algorithm (GA) in global search, as well as the tendency of the BFGS quasi-Newton method (BFGS) to converge to local optima in non-convex optimization problems, a hybrid GA–BFGS optimization algorithm is proposed for inverting the parameters of the probability integral model. This hybrid approach combines the global exploration capability of GA with the fast local refinement of BFGS, resulting in a more efficient and robust parameter optimization process. Simulation results under ideal conditions without model error demonstrate that the proposed GA–BFGS algorithm outperforms pattern search (PS), GA, and BFGS in terms of inversion accuracy, convergence stability, and robustness to noise and outliers. In engineering applications, the inversion accuracy is reduced compared with simulation experiments, which can be attributed to complex geological conditions and inherent model uncertainties. Therefore, further improvements in subsidence prediction accuracy require not only refined inversion algorithms but also the development of more accurate prediction models that explicitly account for site-specific geological and mining conditions.

1. Introduction

Coal remains a vital component of the global energy structure, particularly in China, where it continues to dominate primary energy consumption, even as the world transitions toward cleaner energy sources. In 2023, global coal consumption reached 8.7 billion tons, a 2.6% increase over the previous year, while coal accounted for 55.3% of China’s primary energy consumption, growing by 5.6% [1,2,3]. Large-scale underground mining has caused severe environmental impacts, including ground subsidence, surface cracking, water accumulation, and structural damage, posing significant threats to ecological security and sustainable development [4,5]. Accurate prediction of mining-induced subsidence is, therefore, crucial for both scientific understanding and engineering practice.

The probability integral method (PIM), derived from stochastic medium theory, is widely used for subsidence prediction [6,7]. Its predictive accuracy relies on precise parameter inversion, motivating the development of swarm-based, multi-population genetic, improved snake, and hybrid heuristic algorithms [8,9,10,11,12,13]. While these approaches enhance global search, they often exhibit slow convergence, instability, or susceptibility to local optima. Local optimization methods, such as the BFGS quasi-Newton algorithm, converge rapidly but may fail on non-convex objectives [14,15,16,17,18]. Similar geoscience challenges have been addressed using Bayesian back analysis for unsaturated hydraulic parameters [19] and probabilistic consolidation analysis via subset simulation [20], highlighting the need for robust global–local optimization strategies.

To overcome these limitations, this study proposes a hybrid GA–BFGS framework for inverting the eight PIM parameters. By combining the Genetic Algorithm’s global exploration with BFGS local refinement, the method thereby improves convergence stability, inversion accuracy, and robustness to noise and model uncertainty. This framework extends prior BFGS and evolutionary optimization strategies, providing a theoretically grounded and practically implementable solution for complex subsidence prediction problems in both idealized simulations and real engineering applications [21,22].

2. Theoretical Foundation and Algorithm Framework

2.1. Probability Integral Model

The Probability Integral Model (PIM) is widely used for predicting ground subsidence induced by underground coal mining. Its fundamental equations describe the vertical subsidence and horizontal displacement caused by coal seam extraction.

The vertical subsidence W(x, y) at any point (x, y) over the working face is expressed as:

$$\mathrm{W}(\mathrm{x},\mathrm{y})={\displaystyle \frac{1}{\mathrm{m}\mathrm{q}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\alpha }}}{\mathrm{C}}_{\mathrm{x}}\cdot {\mathrm{C}}_{\mathrm{y}}$$

(1)

where m is the coal seam thickness, q is the subsidence factor, and cosα denotes the dip angle of the coal seam. C_x and C_y represent the subsidence components in the strike and dip directions at coordinates x and y, respectively. The corresponding expressions are given by:

$$\left\{\begin{array}{c}{\mathrm{C}}_{\mathrm{x}}={\displaystyle \frac{1}{2}}\mathrm{m}\mathrm{q}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\alpha }\left(\mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\mathrm{x}}{\mathrm{r}}}\right]-\mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\left(-\mathrm{l}+\mathrm{x}\right)}{\mathrm{r}}}\right]\right)\\ {\mathrm{C}}_{\mathrm{y}}={\displaystyle \frac{1}{2}}\mathrm{m}\mathrm{q}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\alpha }\left(\mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\mathrm{y}}{{\mathrm{r}}_1}}\right]-\mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\left(-\mathrm{L}+\mathrm{y}\right)}{{\mathrm{r}}_2}}\right]\right)\end{array}\right.$$

(2)

In Equation (2), r denotes the main influence radius, while r₁ and r₂ represent the main influence radii in the dip and rise directions, respectively. L and l are the calculated lengths of the working face in the dip and strike directions, respectively, which are determined as:

$$\mathrm{r}={\displaystyle \raisebox{1ex}{$\mathrm{H}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{T}\mathrm{a}\mathrm{n}\mathrm{\beta }$}\right.},$$

(3)

$$\mathrm{L}=\left({\mathrm{D}}_3-{\mathrm{s}}_1-{\mathrm{s}}_2\right){\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\left[\mathrm{\theta }+\mathrm{\alpha }\right]}{\mathrm{s}\mathrm{i}\mathrm{n}\left[\mathrm{\theta }\right]}},$$

(4)

$$\mathrm{l}={\mathrm{D}}_1-{\mathrm{s}}_3-{\mathrm{s}}_4,$$

(5)

where H is the mining depth, tanβ is the tangent of the main influence angle, D₁ and D₃ are the lengths of the working face in the dip and strike directions, respectively. s₁, s₂, s₃, and s₄ denote the offset distances of the inflection points at the lower dip boundary, upper dip boundary, left boundary, and right boundary, respectively, and θ represents the mining influence propagation angle.

Similarly, the horizontal displacement at any point (x, y) based on the probability integral method can be expressed as:

$$\mathrm{U}(\mathrm{x},\mathrm{y},\mathrm{\phi })={\displaystyle \frac{1}{\mathrm{m}\mathrm{q}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\alpha }}}({\mathrm{D}}_{\mathrm{x}}\cdot {\mathrm{C}}_{\mathrm{y}} \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\phi }+{\mathrm{D}}_{\mathrm{y}}\cdot {\mathrm{C}}_{\mathrm{x}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\phi }),$$

(6)

where φ denotes the angle measured counterclockwise from the positive x-axis. D_x and D_y represent the horizontal displacements in the principal strike and principal dip sections, respectively, and are given by:

$$\left\{\begin{array}{c}{\mathrm{D}}_{\mathrm{x}}=\mathrm{b} \mathrm{m}\mathrm{q}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\alpha }\left(-{\mathrm{e}}^{-{\displaystyle \frac{\mathrm{\pi }{\left(\mathrm{l}-\mathrm{x}\right)}^2}{{\mathrm{r}}^2}}}+{\mathrm{e}}^{-{\displaystyle \frac{\mathrm{\pi }{\mathrm{x}}^2}{{\mathrm{r}}^2}}}\right)\\ {\mathrm{D}}_{\mathrm{y}}=\mathrm{b} \mathrm{m}\mathrm{q}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\alpha }\left({\mathrm{e}}^{-{\displaystyle \frac{\mathrm{\pi }{\mathrm{y}}^2}{{{\mathrm{b}}_1}^2}}}-\mathrm{b}{\mathrm{e}}^{-{\displaystyle \frac{\mathrm{\pi }{\left(-\mathrm{L}+\mathrm{y}\right)}^2}{{{\mathrm{b}}_2}^2}}}\right)\end{array}\right.,$$

(7)

where b is the horizontal movement coefficient, and b₁ and b₂ are the horizontal movement coefficients in the lower dip and upper dip directions, respectively.

2.2. Principle of the BFGS Algorithm

The BFGS algorithm is a widely used quasi-Newton optimization method. It iteratively constructs an approximation of the Hessian matrix using successive changes in the parameter vector and the corresponding gradients, and this approximation is subsequently employed to compute a Newton-type search direction that guides the solution toward a local optimum.

By combining Equations (1) and (6), the objective function f(x) is defined as a least-squares function that minimizes the discrepancy between predicted and observed surface displacements:

$$\begin{array}{c}\mathrm{f}({\mathrm{x}}_{\mathrm{i}})={\displaystyle \frac{1}{N}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\{{[\mathrm{W}({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}})-{\mathrm{W}}_{\mathrm{i}}]}^2+{[\mathrm{U}({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}},{\mathrm{\phi }}_{\mathrm{i}})-{\mathrm{U}}_{\mathrm{i}}]}^2\}}={\left(-{\mathrm{W}}_{\mathrm{i}}+{\displaystyle \frac{1}{4}}\mathrm{m}\mathrm{q}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\alpha }\left(\begin{array}{l}\mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\mathrm{x}}{\mathrm{r}}}\right]-\\ \mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\left(-\mathrm{l}+\mathrm{x}\right)}{\mathrm{r}}}\right]\end{array}\right)\left(\begin{array}{l}\mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\mathrm{y}}{\mathrm{r}1}}\right]-\\ \mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\left(-\mathrm{L}+\mathrm{y}\right)}{\mathrm{r}2}}\right]\end{array}\right)\right)}^2+\hfill \\ {\left(-{\mathrm{U}}_{\mathrm{i}}+{\displaystyle \frac{1}{2}}\mathrm{b}\mathrm{w}0\left(\left(-{\mathrm{e}}^{-{\displaystyle \frac{\mathrm{\pi }{\left(\mathrm{l}-\mathrm{x}\right)}^2}{{\mathrm{r}}^2}}}+{\mathrm{e}}^{-{\displaystyle \frac{\mathrm{\pi }{\mathrm{x}}^2}{{\mathrm{r}}^2}}}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left[\mathrm{\phi }\right]\left(\begin{array}{l}\mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\mathrm{y}}{\mathrm{r}1}}\right]-\\ \mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\left(-\mathrm{L}+\mathrm{y}\right)}{\mathrm{r}2}}\right]\end{array}\right)+\left(-{\mathrm{e}}^{-{\displaystyle \frac{\mathrm{\pi }{\left(\mathrm{L}-\mathrm{y}\right)}^2}{\mathrm{b}{2}^2}}}+{\mathrm{e}}^{-{\displaystyle \frac{\mathrm{\pi }{\mathrm{y}}^2}{\mathrm{b}{1}^2}}}\right)\left(\begin{array}{l}\mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\mathrm{x}}{\mathrm{r}}}\right]-\\ \mathrm{E}\mathrm{r}\mathrm{f}\left[{\displaystyle \frac{\sqrt{\mathrm{\pi }}\left(-\mathrm{l}+\mathrm{x}\right)}{\mathrm{r}}}\right]\end{array}\right)\mathrm{s}\mathrm{i}\mathrm{n}\left[\mathrm{\phi }\right]\right)\right)}^2,\hfill \end{array}$$

(8)

where N is the total number of surface monitoring points; W(x_i, y_i) and W_i denote the predicted and observed subsidence at point i, respectively; and U(x_i, y_i, φ_i) and U_i represent the predicted and observed horizontal displacements.

Starting from an initial estimate x_k, the BFGS algorithm updates the parameter vector at each iteration according to:

$${\mathrm{X}}_{\mathrm{k}+1}={\mathrm{X}}_{\mathrm{k}}+{\mathrm{\alpha }}_{\mathrm{k}}{\mathrm{d}}_{\mathrm{k}},$$

(9)

where d_k is the search direction and α_k is the step size determined by the Wolfe conditions. The search direction is obtained by:

$${\mathrm{d}}_{\mathrm{k}}=-{\mathrm{B}}_{\mathrm{k}}^{-1}{\mathrm{g}}_{\mathrm{k}}^{\mathrm{i}},$$

(10)

$$\left\{\begin{array}{c}\mathrm{f}({\mathrm{x}}_{\mathrm{k}} +{\mathrm{\alpha }}_{\mathrm{k}}{\mathrm{d}}_{\mathrm{k}})\le \mathrm{f}({\mathrm{x}}_{\mathrm{k}})+{\mathrm{\sigma }}_1\nabla \mathrm{f}{({\mathrm{x}}_{\mathrm{k}})}^{\mathrm{T}}{\mathrm{d}}_{\mathrm{k}}\\ \nabla \mathrm{f}{({\mathrm{x}}_{\mathrm{k} }+{\mathrm{\alpha }}_{\mathrm{k}}{\mathrm{d}}_{\mathrm{k}})}^{\mathrm{T}}{\mathrm{d}}_{\mathrm{k}}\ge {\mathrm{\sigma }}_2\nabla \mathrm{f}{({\mathrm{x}}_{\mathrm{k}})}^{\mathrm{T}}{\mathrm{d}}_{\mathrm{k}}\end{array}\right.,$$

(11)

where 0 < σ₁ < σ₂ < 1. The approximate Hessian matrix B_k is updated using the standard BFGS update formula:

$${\mathrm{B}}_{\mathrm{k}+1}={\mathrm{B}}_{\mathrm{k}}-{\displaystyle \frac{{\mathrm{B}}_{\mathrm{k}}{\mathrm{s}}_{\mathrm{k}}{\mathrm{s}}_{\mathrm{k}}^{\mathrm{T}}{\mathrm{B}}_{\mathrm{k}}}{{\mathrm{s}}_{\mathrm{k}}^{\mathrm{T}}{\mathrm{B}}_{\mathrm{k}}{\mathrm{s}}_{\mathrm{k}}}}+{\displaystyle \frac{{\mathrm{y}}_{\mathrm{k}}{\mathrm{y}}_{\mathrm{y}}^{\mathrm{T}}}{{\mathrm{y}}_{\mathrm{k}}^{\mathrm{T}}{\mathrm{s}}_{\mathrm{k}}}},$$

(12)

in Equation (10), ${\mathrm{g}}_{\mathrm{k}}^{\mathrm{i}}(\mathrm{i}=1~8)$ denotes the partial derivative of the objective function with respect to the eight estimated parameters q, b, tanβ, θ, s₁, s₂, s₃, and s₄. The specific expressions are provided in Appendix A.

2.3. Basic Steps of GA–BFGS Hybrid Algorithm for Parameter Inversion

The basic steps of the GA–BFGS hybrid algorithm for parameter inversion are illustrated in Figure 1.

Step 1: Data input and objective function construction. Read the geological parameters of the mining area and the surface deformation monitoring data and construct the objective function involving only the parameters to be estimated.

Step 2: Initialization of GA parameters. Set the population size N, maximum number of iterations K_max1, crossover probability P_c, mutation probability P_m, and termination tolerance ε₁. Initialize the population within the predefined parameter bounds.

Step 3: GA-based global optimization. Perform genetic operations, including selection, crossover, and mutation, to conduct a global search in the parameter space. Evaluate the fitness value f(x_i) of each individual and iterate until the termination criterion |f(x_i)| ≤ ε₁ is satisfied or the maximum number of iterations is reached. The resulting solution x_i is taken as a global approximate optimum.

Step 4: Initialization of BFGS using GA results. Use the GA-derived solution x_i as the initial estimate for the BFGS algorithm. Set the termination tolerance ε₂, maximum number of iterations K_max2, and initialize the approximate Hessian matrix as H₀ = I.

Step 5: Convergence check. Compute the gradient Gi = ▽f(x_i). If |G_i| ≤ ε₂, terminate the iteration; otherwise, proceed to the next step.

Step 6: BFGS-based local optimization. Compute the search direction d_i, determine the step size α_i, and update the parameter vector x_i+1 and the approximate Hessian matrix H_i+1. Return to Step 5 and repeat until convergence is achieved, yielding the final locally optimal solution.

3. Simulation Experiment

Parameter inversion in the probability integral model is influenced not only by the performance of the optimization algorithm but also by inherent structural discrepancies between the actual subsidence basin and its idealized theoretical representation. These discrepancies make it difficult to distinguish whether inversion errors arise from model limitations or algorithmic performance. To eliminate the influence of model errors and enable a fair comparison among different algorithms, a series of simulation experiments was conducted. Under predefined geological and mining conditions, theoretical subsidence and horizontal displacement values were calculated using known model parameters and treated as reference (ground-truth) data. Parameter inversion was then performed using the PS, GA, classical BFGS, and GA–BFGS algorithms to evaluate and compare their inversion performance.

The simulation settings were as follows: strike length D₃ = 600 m, dip length D₁ = 410 m, mining thickness m = 6.0 m, coal seam dip angle α = 3°, and mining depth H = 300 m. Forty-four monitoring points were arranged along the strike direction and twenty-five along the dip direction, as shown in Figure 2. The probability integral model parameters were set as follows: subsidence factor q₀ = 0.75, horizontal movement coefficient b₀ = 0.3, tangent of the main influence angle tanβ₀ = 2.0, mining influence propagation angle θ₀ = 75°, and inflection point offsets s₁₀,₂₀,₃₀,₄₀ = 60 m.

For algorithm comparison, the termination step size of the PS algorithm was set to 10⁻⁶. The GA employed a population size of 100 with a maximum of 200 iterations, a crossover probability of 0.9, and a mutation probability of 0.001. The BFGS algorithm adopted a quasi-Newton scheme with a maximum of 1000 iterations and an optimality tolerance of 10⁻⁷. In the hybrid GA–BFGS approach, GA was first applied for global exploration, followed by BFGS for local refinement. For each inversion, the initial values of the parameters were randomly generated within the following ranges: q ∈ [0.1; 0.9], b ∈ [0.2; 0.4], tan β ∈ [1.0; 3.0], θ ∈ [60; 90], s₁ ∈ [40; 160]; s₂ ∈ [40; 160]; s₃ ∈ [40; 160]; s₄ ∈ [40; 160].

3.1. Impact of Random Errors on Parameter Inversion

In practical mining applications, observational data inevitably contain random noise due to limitations in instrument precision, human operation, and other external factors. To evaluate the robustness of different algorithms against random errors, Gaussian noise with an amplitude of 20 mm was added to the reference deformation data, generating a corresponding noisy dataset. Parameter inversion was then performed using the PS, GA, BFGS, and GA–BFGS algorithms. To ensure statistical reliability and reduce the influence of random variability, 200 independent samples were generated for this noise level. In each trial, all algorithms used randomly initialized parameters within identical value ranges to eliminate the influence of initial conditions.

Table 1. Statistical results of parameter inversion for four algorithms (add random errors).

Parameter Inversion	0 mm				20 mm
	PS	GA	BFGS	GA–BFGS	PS	GA	BFGS	GA–BFGS
MAD(|q − q0|)	0.000	0.005	0.000	0.000	0.000	0.006	0.000	0.000
RE(q)/%	0.00	0.68	0.00	0.00	0.02	0.82	0.02	0.01
MAD(|b − b0|)	0.000	0.004	0.000	0.000	0.000	0.004	0.000	0.000
RE(b)/%	0.00	1.29	0.00	0.00	0.06	1.19	0.06	0.06
MAD(|tanβ − tanβ0|)	0.000	0.031	0.000	0.000	0.001	0.025	0.001	0.001
ER(Tanβ)/%	0.00	1.57	0.00	0.00	0.05	1.24	0.05	0.05
MAD(|θ − θ0|)/°	1.802	0.294	0.000	0.000	0.022	0.324	0.023	0.025
RE(θ)/%	2.40	0.39	0.00	0.00	0.03	0.43	0.03	0.03
MAD(|s1 − s10|)/m	17.132	6.434	6.515	0.028	16.562	6.314	6.378	0.069
RE(s1)/%	28.55	10.72	10.86	0.05	27.60	10.52	10.63	0.12
MAD(|s2 − s20|)/m	17.145	6.362	6.510	0.028	16.584	6.095	6.360	0.070
RE(s2)/%	28.58	10.60	10.85	0.05	27.64	10.16	10.60	0.12
MAD(|s3 − s30|)/m	8.547	5.928	6.652	0.009	7.320	5.857	6.752	0.026
RE(s3)/%	14.25	9.88	11.09	0.02	12.20	9.76	11.25	0.04
MAD(|s4 − s40|)/m	8.540	5.747	6.651	0.009	7.320	5.967	6.743	0.025
RE(s4)/%	14.23	9.58	11.09	0.02	12.20	9.95	11.24	0.04
UWSD/m	13.54	6.12	6.58	0.02	11.94	5.98	6.56	0.05

Note: The unit weight standard deviation is calculated by s₁, s₂, s₃, and s₄.

summarizes the inversion results under both noise-free and noisy conditions, including statistical indicators such as the mean absolute deviation (MAD), relative error (RE), and unit weight standard deviation (UWSD).

As shown in Table 1, the following conclusions can be drawn. (1) The GA–BFGS algorithm achieved the highest inversion accuracy. Under the applied noise conditions, the eight parameters estimated by GA–BFGS closely matched the reference values, with relative errors for all parameters remaining within 0.2%. In contrast, the PS, GA, and BFGS algorithms exhibited larger errors when estimating secondary parameters (i.e., s₁, s₂, s₃, s₄). (2) For the 20 mm noise level, the relative errors of the primary parameters (q, b, tanβ, θ) estimated by all four algorithms remained below 3%. The increase in noise did not significantly degrade inversion accuracy, indicating that the probability integral model itself exhibits strong inherent resistance to random noise. (3) Overall, the GA–BFGS algorithm provided the best fitting performance, with a significantly lower unit weight standard deviation than the other algorithms. The PS algorithm showed the largest errors, while the GA and BFGS algorithms exhibited intermediate performance.

In summary, the hybrid GA–BFGS algorithm demonstrates superior inversion accuracy and robustness compared with individual optimization algorithms, making it more suitable for parameter inversion in mining subsidence prediction.

3.2. Impact of Gross Errors on Parameter Inversion

To further evaluate the robustness of the GA–BFGS algorithm against gross errors, an additional simulation dataset was generated by introducing gross errors into the noisy data described in Section 3.1. Specifically, 5–7 monitoring points (approximately 7–10% of all points) were randomly selected in each dataset and contaminated with gross errors ranging from 50 to 200 mm, corresponding to 5–20 times the unit weight standard deviation. Parameter inversion was then performed using the PS, GA, BFGS, and GA–BFGS algorithms. The inversion results of the eight parameters over 200 samples were visualized (Figure 3) and statistically summarized (

Table 2. Statistical results of parameter inversion for four algorithms (including gross errors).

Parameter Inversion	PS	GA	BFGS	GA–BFGS
MAD(|q − q0|)	0.002	0.008	0.002	0.002
RE(q)/%	0.28	1.03	0.28	0.28
MAD(|b − b0|)	0.002	0.007	0.002	0.002
RE(b)/%	0.68	2.29	0.73	0.61
MAD(|tanβ − tanβ0|)	0.012	0.041	0.014	0.013
ER(Tanβ)/%	0.60	0.60	0.70	0.63
MAD(|θ − θ0|)/°	0.254	0.677	0.261	0.267
RE(θ)/%	0.34	0.90	0.35	0.36
MAD(|s1 − s10|)/m	19.291	7.158	6.760	0.705
RE(s1)/%	32.15	11.93	11.27	1.17
MAD(|s2 − s20|)/m	19.186	6.731	6.918	0.716
RE(s2)/%	31.98	11.22	11.53	1.19
MAD(|s3 − s30|)/m	7.761	6.045	6.470	0.250
RE(s3)/%	12.93	10.08	10.78	0.42
MAD(|s4 − s40|)/m	7.758	6.123	6.487	0.264
RE(s4)/%	12.93	10.21	10.81	0.44
UWSD/m	14.43	6.56	6.66	0.52

).

Based on Table 2 and Figure 3, several observations can be made. (1) The GA–BFGS algorithm exhibited the strongest resistance to gross errors. Although the introduction of gross errors degraded the inversion accuracy of all algorithms, the GA–BFGS algorithm was least affected. The relative errors of all eight parameters ($\mathrm{q},\mathrm{b},\mathrm{t}\mathrm{a}\mathrm{n}\mathsf{\beta },\mathsf{\theta },{\mathrm{s}}_1,{\mathrm{s}}_2,{\mathrm{s}}_3,{\mathrm{s}}_4$) were controlled within 1.2%, significantly outperforming the other three algorithms. (2) The GA–BFGS algorithm also demonstrated superior stability. As shown in Figure 3, the parameter estimates obtained by GA–BFGS exhibited clear clustering across the 200 samples, indicating strong convergence consistency. In contrast, the GA showed more dispersed distributions for the primary parameters, while the PS and BFGS algorithms exhibited large fluctuations in the secondary parameters, with the PS algorithm being the most unstable.

Overall, by effectively integrating global exploration and local optimization, the proposed GA–BFGS algorithm demonstrates significant advantages in inversion accuracy, result stability, and robustness against gross errors.

4. Engineering Case Study and Result Analysis

4.1. Overview of the Study Area

To verify the applicability of the GA–BFGS algorithm in practical engineering, a case study was conducted on the 1613(1) working face of Guqiao Mine in Huainan City. The working face employed the full caving method for roof management and was mined from March to December 2017. Geological and mining parameters were as follows: dip length of the coal seam measures 251 m, strike length is 1528 m, average seam thickness is 2.8 m, mining height is 2.9 m, dip angle varies between 0° and 6°, and the average mining depth is 668 m. The monitoring system comprised one half-strike line and one dip line, including 6 control points and 115 monitoring points. Due to external disturbances, 87 observation points were retained for analysis, corresponding to a data loss rate of 24.3% (Figure 4).

4.2. Stability Analysis of the GA–BFGS Algorithm

Using the observed surface deformation data, parameter inversion of the probability integral model was performed using the PS, GA, BFGS, and GA–BFGS algorithms. To reduce sensitivity to initial values and mitigate the randomness inherent in intelligent optimization algorithms, each algorithm was independently executed 50 times, with all eight parameters reinitialized in each run. The initial parameter values were randomly generated within the following ranges: q ∈ [0.5; 1.5], b ∈ [0.2; 0.8], tan β ∈ [1.0; 3.0], θ ∈ [70; 90], s₁ ∈ [−100; 100]; s₂ ∈ [−100; 100]; s₃ ∈ [−100; 100]; s₄ ∈ [−100; 100].

Figure 5 shows the distribution of parameters obtained from the 50 independent trials, and

Table 3. Statistical results of parameter inversion for four algorithms.

Method	q	b	tanβ	θ/°	s1/m	s2/m	s3/m	s4/m
PS	1.37	0.40	1.38	81.98	42.84	−29.32	41.39	9.40
GA	1.24	0.42	1.46	83.50	−8.05	1.61	25.33	24.22
BFGS	1.34	0.41	1.44	82.27	5.71	2.94	22.43	28.19
GA–BFGS	1.34	0.41	1.44	82.27	6.58	2.06	24.78	25.849

presents the corresponding average values. Based on Figure 5 and Table 3, the following conclusions can be drawn: (1) The GA–BFGS algorithm exhibits the highest stability in parameter inversion. Although stability for secondary parameters (s₁, s₂, s₃, and s₄) is slightly lower than in simulation experiments, its overall inversion performance remains superior to the PS method, standalone GA, and standalone BFGS algorithm. (2) For primary parameters (q, b, tanβ, and θ), both BFGS and GA–BFGS demonstrate strong consistency and stability. The GA exhibits larger fluctuations and more scattered results, reflecting lower stability, consistent with findings from simulation experiments.

4.3. Accuracy Analysis of the GA–BFGS Algorithm

Using the inversion results in Table 3, each algorithm’s parameters were incorporated into the probability integral model to construct corresponding subsidence prediction models. Predicted subsidence at observation points were then compared against actual measurements (Figure 6). Observations from Figure 6 are as follows:

(1)

Both GA–BFGS and BFGS algorithms achieved the best fitting performance. Their predicted subsidence curves closely match the measured data, with accuracy surpassing that of PS and standalone GA. Errors are minimal near the edges of the subsidence basin but increase near the basin center.

(2)

Central-region errors are primarily attributed to inherent limitations of the probability integral model. Despite high overall accuracy, the model systematically underestimates subsidence in the central area, highlighting a limitation in fully capturing the basin’s deformation characteristics.

(3)

Although the inversion algorithms perform effectively, improving prediction accuracy ultimately depends on enhancing the model structure. The GA–BFGS algorithm demonstrates high stability and reliability; however, introducing more accurate surface deformation models is necessary to reduce systematic errors and improve both parameter inversion and predictive performance.

Figure 6. Fitting performance of parameters inverted by different algorithms.

Figure 6. Fitting performance of parameters inverted by different algorithms.

5. Conclusions

(1)

The hybrid GA–BFGS algorithm effectively inverts the parameters of the probability integral model and mitigates large errors and instability observed in the inversion of inflection point offsets with other algorithms (PS, GA, BFGS). Under ideal conditions without model error, inverted parameters closely match actual values.

(2)

Simulation results indicate that GA–BFGS achieves higher accuracy in parameter inversion than PS, GA, or BFGS, maintaining robust estimation even in the presence of noise. In engineering applications, predicted subsidence curves from GA–BFGS align more closely with measured data, confirming its practical effectiveness.

(3)

Compared with simulation experiments, the advantages of GA–BFGS in engineering applications are somewhat reduced, particularly in areas with significant surface deformation. This reduction indicates that the probability integral model itself has inherent limitations. Therefore, improving parameter inversion methods alone is insufficient; refining the model structure is essential to enhance deformation pattern prediction and overall accuracy.