## 4 Experiments and Analyses
### Forward Verification
#### 4.2.2 Developing the Tunnel Model
The comparison results are shown in Table 5.  
The mechanical parameters of the surrounding rock were inverted using the displacement back-analysis method, and these parameters were subsequently applied in a forward model to calculate displacements for vault settlement and peripheral convergence. The forward model's calculations yielded displacements of 17.98 mm for vault settlement and 7.02 mm for peripheral convergence. When compared with actual site monitoring data--17.12 mm for vault settlement and 6.74 mm for peripheral convergence--the relative errors were found to be 5.02% and 4.15%, respectively. This comparison demonstrates a close match between the forward model's predictions and the real-world measurements, evidencing the model's conservative accuracy. Thus, the mechanical parameters derived from the improved IGWO-BP model inversion are deemed to be highly accurate. While it is hypothesized that the IGWO-BP method may offer potential advantages in handling more complex non-elastic responses, this assumption has yet to be validated through experimental data. Therefore, the conclusions of this study are based on the current elastic material testing scenario, and the applicability of the method to non-elastic responses will require further experimental investigation in future research. In this study, the IGWO-BP (Improved Grey Wolf Optimizer with Backpropagation) method was chosen to handle elastic responses due to its advantages over traditional approaches. Although the rock mass response in the Lotus Tunnel is predominantly elastic, the IGWO-BP offers improved accuracy by overcoming the limitations of traditional methods, which rely on linear assumptions and are sensitive to initial conditions. By combining the global search capabilities of the Grey Wolf Optimizer with the nonlinear fitting abilities of the Backpropagation neural network, the IGWO-BP enhances parameter inversion accuracy, even in complex geotechnical scenarios. This was evident when the parameters obtained from the IGWO-BP were applied in forward simulations to calculate roof settlement and surrounding convergence displacement. The simulation predicted a roof settlement of 17.98 mm, with a relative error of only 5.02% compared to the measured value of 17.12 mm. These results demonstrate the  
Figure 8: YK37 + 330 cross section forward inspection model displacement cloud map. (**a**) Crown settlement; (**b**) Peripheral convergence. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Pile Number** & **Monitoring Items** & **Measured** & **Calculated** & \multirow{2}{*}{**Error/\%**} & \multicolumn{4}{c}{**Inversion Parameters**} \\  & & **Values/mm** & & **Values/mm** & & **E/GPa** & \(\mu\) & **c/kPa** & \(\varphi\)/(\({}^{\circ}\)) \\ \hline \multirow{2}{*}{YK37 + 330} & Crown Settlement & 17.12 & 17.98 & 5.02 & \multirow{2}{*}{0.32} & \multirow{2}{*}{0.26} & \multirow{2}{*}{156.14} & \multirow{2}{*}{28.52} \\  & Peripheral Convergence & 6.74 & 7.02 & 4.15 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison results between forward model displacement and measured displacement. method's effectiveness in handling elastic responses and providing reliable predictions in tunnel excavation. Beyond elastic responses, the IGWO-BP also excels in modeling non-elastic behaviors, particularly in complex conditions. The Backpropagation neural network captures nonlinear material relationships, making the method well suited for handling non-elastic responses [41, 42]. Additionally, the Grey Wolf Optimizer improves the method's global search capability, ensuring more robust and accurate results, even when material responses deviate from linear behavior.