## 4 Numerical Analysis for Verification
### Sensor Layout  
In the process of field-monitoring tunnels, budgetary constraints and practical conditions often limit the deployment of sensors within the tunnel itself. As discussed in this section, we explored four distinct sensor layout schemes (denoted as Schemes 1-4) to verify the effectiveness and robustness of the proposed method for estimating the vertical displacement in a tunnel under realistic field monitoring conditions. Schemes 1 and 2 entail sensor arrangements involving five monitoring units, while Schemes 3 and 4 focus on sensor arrangements with seven monitoring units. For Schemes 1 and 3, three sensors were arranged on each monitoring unit and the local sensor arrangement is as shown in Section II of Figure 12. For Schemes 2 and 4, three sensors were arranged on part of the monitoring units and only the top sensor (Point 1) was reserved for the remaining monitoring units, as also illustrated in Figure 12. Other details regarding the four sensor layouts are provided in Table 5.  
The load from case C3 (Section 4.1) was applied in this analysis. Initially, the coupled strain on the corresponding monitoring unit was extracted according to different sensor layout schemes. For schemes 1 and 3, the responses of the unmonitored units were then calculated via curve interpolation using the monitoring unit, then strain separation and displacement estimation were carried out according to the proposed method. As there was only one monitoring point on certain units in schemes 2 and 4, the strain data for points 2 and 3 were firstly obtained on each monitoring unit with reference to the sensor layout scheme. Next, the strain responses of all units on the investigated tunnel were obtained via interpolation method using the measured strain. Finally, strain separation and displacement estimation were conducted using the proposed method. Figure 13 shows the REs between the separated vertical bending strain and the actual strain with different sensor layout schemes. Table 6 lists the MAVPE \begin{table}
\begin{tabular}{c c c c} \hline \hline Noise & 2\% & 4\% & 6\% \\ \hline MAVPE & 3.79 & 3.31 & 3.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: MAVPEs (%) of identified vertical displacement at different noise levels. Figure 12: Sensor layout diagram. for the different schemes. By comparing scheme 1 to scheme 3, the sensor arrangement containing seven monitoring sections appears to ensure that the RE of strain separation remains stable throughout the tunnel and does not exceed 1%; the sensor arrangement containing five monitoring sections causes the RE of strain separation to fluctuate slightly throughout the tunnel, though the MAVRE still does not exceed 3%. The MAVRE is slightly smaller in the case of scheme 4 compared to scheme 2, indicating that the separated strain of scheme 4 is closer to the true value than that of scheme 2. Although the absolute RE values of a few units in scheme 4 exceed 5%, these units are located near the support and thus have a small response, so they would not substantially impact the displacement identification in practice. Figure 14 shows the REs between the estimated displacement and actual displacement for the four sensor layout schemes. The MAVPEs for the identified vertical displacement under various schemes are listed in Table 7. The MAVPEs for schemes 1 and 3 are consistently maintained at approximately 5%, while the MAVPEs for schemes 2 and 4 fall below 8%. However, the units experiencing MAVPE are primarily situated near the supports, where the strain responses are relatively small. In contrast, the REs for units located in the mid-span region are generally contained within 3%; their strain responses exert a significant impact on the tunnel structure. The differences in the maximum and minimum REs between schemes 1 and 3 are all less than 1%, which indicates that the identified displacements under the two schemes are basically consistent.