# Volume Loss Assessment with MT-InSAR during Tunnel Construction in the City of Naples (Italy)
## 3 Methodology
### Multi Temporal InSAR  
Interferometric synthetic aperture radar (InSAR) is a well-established technique suitable for detecting ground movements. It is based on the measurement of the phase shift \(\Delta\varphi\) between two radar signals, which is a fraction of the wavelength \(\lambda\). As InSAR satellites emit signals in the microwave region of the electromagnetic spectrum (\(\lambda\) = 3.1 cm for CSK satellites), it is possible to detect movements with the millimetric precision during the day and night and variable meteorological conditions. The limit of this application is due to spatial-temporal signal decorrelation, particularly in areas where there is a progressive change in the scattering properties of the targets, such as forests, harvesting fields, and so on. Indeed, interferometric coherence \(\gamma\) is a measure of the correlation between two radar signal and ranges from 0 to 1 (i.e., from total decorrelation to total correlation). Atmosphere artifacts are another limiting phenomena that delay radar signal, reducing the coherence of the pixels in the radar images. To overcome this limit, methods such as permanent scatter interferometry (PSI) [37] or small baseline subset (SBAS) [38] were developed in the early 2000s. For further technical insights, the reader is invited to refer to the following works: [21; 37; 39; 40]. This work uses the PSI method, consisting of detecting pixels that show high coherence to retrieve the displacement map (better known as velocity map, mm/year) of the area of interest, using the software SARPROZ [21]. A flowchart of the data processing is presented in Figure 6.  
A single Master PS linear analysis was adopted in this framework. PSs were selected using a coherence threshold of 0.5, which led to roughly 20,000 scatterers in the AoI, with a point density of 30,000 scatterers/km\({}^{2}\). For each scatterer, temporal series were interpolated using four linear regressions (see Figure 7), one per each following period. * 7 March 2018 to 2 July 2020 (i.e., from beginning of the analysis to prior to the first excavation);
* 2 July 2020 to 23 March 2021 (i.e., from the beginning to the end of the first excavation);
* 23 March 2021 to 7 September 2021 (i.e., from the end of the first excavation to the beginning of the second excavation);
* 7 September 2021 to 29 January 2022 (i.e., from the beginning of the second excavation to the end of the analysis). Figure 6: Flowchart of the data processing used in SARPROZ. Figure 7: Example of time series multi-linear interpolation. Each colored line represents a linear interpolation. Notice that this scatter lies on the longitudinal axis of the first excavated tunnel (coherence \(\gamma\) = 0.85). From those interpolations, line of sight (LOS) cumulative displacements were obtained on 2 July 2020, 23 March 2021, and 29 January 2022. By considering a reference point very close to the study area, the local topography, and the expected deformational processes, it is common to assume that the movements that occurred in these periods, owing to tunneling, are predominantly vertical [15; 18]. For this reason, LOS displacements can be converted into vertical displacements (settlements) according to Equation (8):  
\[\mathrm{d}_{\mathrm{v}}=\mathrm{d}_{\mathrm{LOS}}/\mathrm{cos}(\vartheta), \tag{8}\]  
where \(\mathrm{d}_{\mathrm{LOS}}\) is the displacement in the direction of the satellite line of sight (LOS) and \(\vartheta\) is the local incidence angle of the satellite LOS direction to the vertical. Nevertheless, it is crucial to assess the implications of this assumption.