## 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 \(\beta\) 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 s\({}_{1}\), s\({}_{2}\), s\({}_{3}\), and s\({}_{4}\), 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 \(\theta_{0}\). The results have a certain gap with the true value. The possible reasons for this are that \(\theta_{0}\) has a high collinearity with the inflection point offset (s\({}_{1}\), s\({}_{2}\), s\({}_{3}\), and s\({}_{4}\)) of the near-symmetric sedimentation basin, the IGGIII weight function mechanism is not applicable to \(\theta_{0}\), 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 \(\theta_{0}\) and can also achieve good results. **Author Contributions:** Conceptualization, methodology, validation, and writing, X.F. and R.Y. ; software, formal, investigation, resources, and data curation, J.D., M.Z. and S.C. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by Open Research Fund of Anhui Province Engineering Laboratory for Big Data Analysis and Early Warning Technology of Coal Mine Safety (NO.CSBD2022-ZD04); Anhui Provincial Key Laboratory Open Fund (NO.KLAHEI202307); Anhui University of Science and Technology Talent Introduction Research Startup Fund (NO.2022 yirc26); and Anhui Province Universities Natural Science Research Project (NO. 2023 AH051190). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The original contributions presented in the study are included in the article material. **Acknowledgments:** Thanks to X.F for providing the laboratory environment.