## 3 Simulation Experiments and Statistical Analyses of Results
### 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:  
\[p_{i}=\left\{ \begin{array}{l}1,\;\left|\widetilde{v_{i}}\right|\leq\text{c} \\ \frac{\text{c}}{\left|\widetilde{v_{i}}\right|},\;\left|\widetilde{v_{i}} \right|>\text{c}\end{array} \right. \tag{21}\]
2. L1 method:  
\[p_{i}=\frac{1}{\left|\widetilde{v_{i}}\right|} \tag{22}\]
3. Geman-McClure method:  
\[p_{i}=\frac{1}{\left(1+\widetilde{v_{i}}^{2}\right)^{2}} \tag{23}\]  
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  
\[\widetilde{v_{i}}=\frac{v_{i}}{\sigma_{0}} \tag{24}\]  
\(v_{i}\) is the residual of the ith observation, and \(\sigma_{0}\) is the unit weight medium error, calculated as  
\[\sigma_{0}=\sqrt{\frac{V^{T}PV}{n-m}} \tag{25}\]  
Figure 2: Schematic diagram of the simulation working face. 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 Figures 3 and 4. From Figure 3, it can be seen that the results of the inversion of the three main parameters q, b, and tan \(\beta\) in the IGGIII method converge consistently across iterations, while the other three methods are caught in the local minimum problem, and the results of  
Figure 4: The inversion results of parameters s\({}_{1}\), s\({}_{2}\), s\({}_{3}\) and s\({}_{4}\) of the four methods. Figure 3: The inversion results of parameters q, b and tan\(\beta\) of the four methods. the inversion of the three main parameters fluctuate to different degrees. From Figure 4, it can be seen that the stability of inversion of s\({}_{1}\), s\({}_{2}\), s\({}_{3}\), and s\({}_{4}\) 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 \(\beta\)--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.