## 2 Methodology
### RSM Model
#### 2.3.2 Numerical Analysis of RSM Model  
The response values were analyzed by Design-Expert V12 software and the best fitting summary indicated that for the displacement (s) and acceleration (a) response of entrance point A of small-radius curved tunnel, the application of a quadratic model was recommended, and the adequacy of the developed model was tested using analysis of variance (ANOVA) methods. Tables 4 and 5 summarize the ANOVA of the responses and show the significance and adequacy measures of the models. The adequacy measures include the F-value and \(p\)-value of the model, the coefficient of determination R\({}^{2}\), the adjusted coefficient R\({}^{2}\), the prediction coefficient R\({}^{2}\) and the signal-to-noise ratio. Among them, F-value is used to evaluate the differences between groups and indicates the significance of the fitting equation of the RSM model. \(p\)-value is an indicator to measure the difference between the control group and the experimental group [21]. Figure 3: Layout of monitoring point of small-radius curve tunnel. \begin{table}
\begin{tabular}{c c c c c c} \hline
**No.** & **Depth (m)** & **Angle (deg)** & **R (m)** & **s (m)** & **_a (m/s\({}^{2}\))_** \\ \hline  
5 & 325 & 0.11911 & 7.442 \\ \hline \end{tabular}
\end{table} 
Table 3: The design matrix of Point B. Table 4 shows the ANOVA results arising from the simplified linear model of the displacement response at point A. The F-value of the model is 1.7995 \(\times\)\(10^{4}\), suggesting a significant model. The \(p\)-value of the model is less than 0.0001, indicating the extremely low likelihood of an F-value attributed to noise in this model. The prediction coefficient R\({}^{2}\) of 0.9993 corresponds well with the adjustment coefficient R\({}^{2}\) of 0.9999. The difference between the prediction coefficient R\({}^{2}\) and the adjustment coefficient R\({}^{2}\) is within 0.2, which is consistent with the results drawn from the extensive statistical documentation and fits with the corresponding literary results. Furthermore, the signal-to-noise ratio of 370.57 is greater than the 4 specified in the relevant statistical literature, indicating that there is sufficient signal in the model and that the model has sufficient accuracy, and that there is no need to remove the relatively insignificant term to simplify the solution of the equation. The F-values of the designed  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & \ \begin{tabular}{c} **Sum of** \\ **Squares** \\ \end{tabular}  & \begin{tabular}{c} **Mean** \\ **Square** \\ \end{tabular}  & **F-Value** & \(p\)**-Value** & **Significance 1** \\ \hline Model & 0.0002 & 0.0000 & 1.7995 \(\times\)\(10^{4}\) & \textless{}0.0001 & *** \\ U-Depth & 0.0002 & 0.0002 & 1.5870 \(\times\)\(10^{5}\) & \textless{}0.0001 & *** \\ V-  
parameter variables shown in the model are available to denote the sequence of factors influencing the displacement response, and an ANOVA on Table 4 shows the buried depth of small-radius curved tunnel to be the greatest factor affecting displacement, with an F-value of 1.5870 \(\times\) 10\({}^{5}\). The angle of the curved tunnel and the radius of the curve do not offer significant implications for the displacement response. Table 5 presents the ANOVA results derived from the quadratic model of the acceleration response at point A. The model exhibits an F-value of 12.36 and a \(p\)-value of 0.0016, indicating the F-value of the model has a 0.16% probability of being caused by noise, so the model is significant. The prediction coefficient R\({}^{2}\) of 0.7908 is well-aligned with the adjustment coefficient R\({}^{2}\) of 0.8657, with a difference between the two coefficients within 0.2, in agreement with the relevant statistical literature. The signal-to-noise ratio of 10.9503 exceeds the value of 4 specified in the literature, confirming that the model features low noise and is suitable for parametric studies.