## 3 Simulation Experiments and Statistical Analyses of Results
### Comparison of IGGIII-BFGS Algorithm with BFGS Algorithm
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Gross Error** & **Position of** & **Parameter Name** & **q** & **b** & **tan \(\beta\)** & **s\({}_{1}\)** & **s\({}_{2}\)** & **s\({}_{3}\)** & **s\({}_{4}\)** \\ \cline{3-10}
**Number** & **Gross Error** & **Truth Value** & **0.85** & **0.25** & **2.15** & **50** & **50** & **50** & **50** \\ \hline \multirow{8}{*}{1} & \multirow{4}{*}{Near the maximum} & 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\% \\  & & \multirow{4}{*}{subsidence value} & 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\% \\  & & BFGS & 0.848 & 0.251 & 2.179 & 51.506 & 48.738 & 46.252 & 53.405 \\  & & \multirow{4}{*}{Near the reflection points} & \multirow{4}{*}{Near the 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\% \\  & & \multirow{4}{*}{Near the edges} & \multirow{4}{*}{Near the edges} & \multirow{4}{*}{Near the relative error} & 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\% \\ \hline \multirow{8}{*}{2} & \multirow{4}{*}{Near the maximum} & BFGS & 0.815 & 0.259 & 2.181 & 48.649 & 46.716 & 53.040 & 46.067 \\  & & \multirow{4}{*}{subsidence value} & 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\% & 1.18\% & 1.18\% \\  & & & 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\% \\  & & & 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\% \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of inversion parameters and relative errors for different roughness cases. 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.