## 2 Materials and Methods
### Tunnel Point Cloud Displacement Detection
#### 2.3.3 Displacement Key Metrics  
To validate the precision and accuracy of the displacement analysis method, key metrics were executed by comparing data collected through the SILSS at different times in the same position. The main metrics include the average shield ring point cloud displacement, centerline displacement, and shield ring ovality change for deformation detection, each providing critical insights into the tunnel's structural integrity.  
The average displacement, \(\overline{d}\), the mean of all the displacement values across the matched shield ring points, was calculated [42]. If there are \(n\) matched points, the average displacement is given by Equation (7).  
\[\overline{d}=\frac{1}{n}\sum\limits_{i=1}^{n}d_{i} \tag{7}\]  
where \(n\) is the total number of point displacement values in the dataset and \(\sum\limits_{i=1}^{n}d_{i}\) is the summation of all \(d_{i}\) values from \(i=1\) to \(n\).  
The centerline in tunnel deformation analysis represents a calculated central axis along the tunnel shield ring's length, serving as a reference for assessing displacements and detecting structural shifts. This line traces the tunnel's designed path, allowing any deviations from the intended alignment to be identified. In practice, the centerline is derived from point cloud data by calculating the centroid of each shield ring and connecting these centroids along the shield ring's length. Centerline displacement refers to the shift in the tunnel's center of mass between scans taken at different times. If \(C_{0}\) and \(C_{i}\) represent the centroids of corresponding shield rings in the point clouds from the initial scan and the subsequent scans, respectively, the centerline displacement is determined using the Euclidean distance, as shown in Equation (6).  
Then, ovality assesses the tunnel's shape by comparing the major (\(a\)) and minor (\(b\)) axes of an ellipse fitted to the tunnel shield ring. The fitting is performed using Principal Component Analysis (PCA) on the 2D projections of the points onto the \(xz\) plane. If \(a\) and \(b\) represent the lengths of the major and minor axes, respectively, the ovality, \(O\), is calculated [43; 44] according to Equation (8).  
\[O=\mid a-b\mid \tag{8}\]  
Ovality change quantifies how the tunnel's shield ring shape has changed over time. If \(O_{0}\) is the ovality at \(P_{0}\) and \(O_{i}\) is the ovality at \(P_{i}\), the change in ovality, \(\Delta O\), is computed using Equation (9).  
\[\Delta O=O_{i}-O_{0} \tag{9}\]  
The key metrics provide valuable information about the tunnel's structural integrity, helping engineers and decision-makers identify potential issues and take appropriate corrective actions. By applying this methodology, the tunnel's structural behavior can be comprehensively assessed, allowing for early detection of potential risks and facilitating timely maintenance or reinforcement actions to ensure the safety and longevity of the tunnel infrastructure.  
The accuracy of the proposed method was evaluated by comparing the data from SILSS, particularly the \(P_{i}\) dataset, with data from the Leica Pegasus Two Ultimate scanner using metrics such as average point cloud displacement and ovality change. The identical workflows were applied to ensure consistent and fair validation, providing insights into the precision and applicability of SILSS for tunnel deformation monitoring.