## 3 Case Study
### Results and Discussion  
#### 3.4.1 ABGPR-MCS Reliability Analysis Results  
Firstly, the geo-mechanical parameters in Table 2 were sampled 1000 times by the Latin hypercube method. Secondly, these 1000 sample sets are substituted into the FEM model, and simulation calculations are performed to obtain the corresponding displacement data of each monitoring point in Figure 3. Third, the proposed ABGPR algorithm is used to establish one model to surrogate the numerical calculation. Finally, we enter and update the optimized surrogate model with the displacement data from each measurement site in Table 1 to produce the appropriate mean value, variance, and confidence interval of geo-mechanical parameters, as shown in Table 3.  
Each FEM simulation calculation time using ABAQUS (three-step construction method, excavation length 12 m and each excavation length 1 m) is about 1 h, while the calculation time of the surrogate model is generally only 30-40 s.  
Figure 4: \(R2\) score values corresponding to different kernel functions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **E/GPa** & \(\mu\) & \(c\)/MPa & \(\varphi^{c}\) \\ \hline Mean & 0.3306 & 0.338 & 0.1896 & 32.795 \\ Standard deviation & 0.001875 & 0.001 & 0.001775 & 0.17575 \\ Confidence interval & [0.3269, 0.3344] & [0.336, 0.340] & [0.1860, 0.1931] & [32.444, 33.147] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculation results of surrounding-rock mechanical parameters. On using the ABGPR-MCS reliability analysis method, it is first necessary to set the maximum number of tests. The amount of random sampling of parameters should be determined in combination with the target failure probability of the tunnel engineering. Assuming that the failure probability \(P_{\text{f}}\) of the tunnel is 0.5% and the error level is 0.1, the total number of random samples \(N\) is 20,000, obtained using Formula (17) [32]. \[N>\left(\frac{1}{P_{\text{f}}}-1\right)\cdot 10^{2}\approx\frac{10^{2}}{P_{ \text{f}}} \tag{17}\]  
According to the statistical parameters of the surrounding rock in Table 3, 30,000 rounds of random sampling are carried out via the MCS method. A frequency distribution histogram is generated by the statistical analysis of sampling results, as shown in Figure 5. Comparing the randomly generated parameter sample values with the fitted normal curve, we note from Figure 5 that the statistical parameters conform to the normal distribution of the original inversion results. It can be concluded that the random sampling of each group meets the requirements of the ABGPR-MCS algorithm. The substitution of FEM simulation for tunnel excavation and support was completed on the above-mentioned 30,000 sets of geo-mechanical parameters using the optimized sur  
Figure 5: Frequency distribution histogram of surrounding-rock parameters. rogate model used by Python 3.7, respectively, to obtain the corresponding vault settlement and horizontal displacement data of the tunnel. For monitoring point A, based on the performance function of the initial support structure \(Z=0.1-u_{\text{v}}\) denoted by Formula (14), the number of \(Z<0\) is 55, and the number of \(Z>0\) is 29,945; using the approximate probability method and Formula (13), its failure probability is 0.1833%, and its reliability index is 2.906. In the same way, we can obtain the reliability indices of measuring lines BC, AB, AC, AD, and AE, respectively, as 3.004, 2.713, 2.713, 2.734, and 2.734. 4.2 Comparison of Numerical Calculation and Surrogate Model Calculation Based on ABGPR, Kriging, and XGBoost for Monitoring Point Displacement  
From Figure 6, we note that for monitoring points A, E, and H, the displacement calculation values of each monitoring point based on the ABGPR surrogate model are relatively close to their numerical calculation values, while the error based on the XGBoot surrogate model is the largest. The error based on the Kriging surrogate model is between the error based on the ABGPR surrogate model and the error based on the XGBoot surrogate model.