# Multi-Temporal InSAR Structural Damage Assessment: The London Crossrail Case Study
## 4 Results and Discussion  
Approximately 228,000 persistent scatterers have been identified over a 25 km\({}^{2}\) area in London (Figure 1) for an overall density of 9000 PS/km\({}^{2}\). Figure 3 shows how the typical transverse settlement profile caused by the tunneling-induced loss of ground can be recognized from the cumulative displacement field. Time-series analysis of points located over different areas reveal subsidence varying from 2 cm to 3.5 cm, providing a precise assessment tool able to monitor the current progress \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Category of Damage** & **Damage Class** & **Approximate Crack Width** & **Limiting Tensile Strain Levels (\%)** \\ \hline \multirow{2}{*}{Aesthetic damage} & Negligible & Up to 0.1 mm & 0-0.05 \\  & Very slight & Up to 1 mm & 0.05-0.075 \\  & Slight & Up to 5 mm & 0.075–0.15 \\ \hline \multirow{2}{*}{Functional damage, affecting serviceability} & Moderate & 5 to 15 mm or a number of & \multirow{2}{*}{0.15–0.3} \\  & Severe & 15 to 25 mm, but also depends & \\  & & on number of cracks & \\ \hline Structural damage, affecting stability & Very severe & Usually \textgreater{} 25 mm, but depends & \\  & & on number of cracks & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Damage classification of building, based on limiting tensile strain levels and expected crack width [24; 44]. of excavation works (Figure 1a-d). The PS density enabled by the high-resolution X-band CSK constellation in urban areas allows to capture subsidence profiles of sections transverse to the tunnel axis (Figures 3 and 4). The comparison between InSAR PS and precise levelling points (PLP) on the ground (Figure 3) indicates a reasonable match between the two datasets (5.5 mm standard deviation for PS with temporal coherence higher than 0.8 and 2.5 mm for PS with temporal coherence higher than 0.9). The graph in Figures 3 and 4 includes the Gaussian curve typically used to model tunneling-induced greenfield displacements [40]. The soil-structure interaction effect is measured by the difference between the greenfield and building displacements (Figure 4), which are responsible for the actual damage. Figure 4 also shows the displacements of the PS located on top of the building. To assess the building damage within the framework of the relative stiffness method we implemented the following procedure: first we validated the greenfield model with the PS tunnel cross section shown in Figure 3. Because relatively small horizontal strains are transferred to buildings [24; 25; 28], for each building we selected PS cross sections localized on the roof and projected them onto the vertical. We interpolated the PS points using a modified Gaussian model [46; 47] (Figure 4) and calculate the \(\Delta/L\) values (see Figure 2). We derived the sagging and hogging modification factors by dividing these values by the corresponding (\(\Delta_{gr}/L_{gr}\)), obtained through fitting the PLP data with an appropriate Gaussian curve, as illustrated in Figure 3. Figure 5 shows the modification factors resulting from the analysis of 14 sample buildings. The modification factors were obtained for the sagging and hogging part of each structure settlement trough, according to Equation 1. Due to their relative high stiffness with respect to the soil stiffness, the buildings tend to reduce the curvature of the greenfield settlement profile, leading to modification factors smaller than 1 for the majority of the analyzed cases. The modification factors can then be used to directly approximate the building strain which is then used to classify the building damage. Figure 3: Example of ground- and MT-InSAR-based displacements at the ground level (adjacent to the building). The EB and WB tunnel markers indicate the centerline of the Eastbound and Westbound tunnels respectively. Figure 6 shows the building tensile strains based on the InSAR monitoring data in relation to their corresponding greenfield values.