## 2 Overview of InSAR Methodology
### InSAR Technique for the Estimation of Surface Displacements
The APS varies gradually over space and is modelled as a long-wavelength component in the unwrapped phase [83]. Experiments also showed that changes in the tropospheric thickness are responsible for additional phase error contributions that are correlated with the terrain topography [85]. They are due to heterogeneities of the troposphere that appear on interferograms with large topographic variations. Accordingly, because these contributions to APS are highly correlated with the topography, they can be extracted and compensated in sequences of interferograms by searching for those residual phases that are highly correlated with the terrain height, as synthesized from a DEM of the area [83; 86]. Furthermore, an additional contribution to the path delay is related to ionospheric heterogeneities, which can significantly differ from the tropospheric artefacts [87; 88; 89; 90]. Simple models that describe the APS phase statistics in single interferograms and a noise co-variance model for InSAR time-series is provided in [91]. Figures 7-10 show a selection of DInSAR interferograms generated from SAR data acquired from RADAR instruments mounted on board the first- and second-generation of SAR missions. In particular, Figure 7,b shows two differential SAR interferograms relevant to the Big Island of the Hawaii, U.S., which is dominated by the deformation due to the Kilauea volcano eruptions. The first interferogram (left side) was produced by processing two SAR images collected by the ASAR instrument on board the ESA ENVISAT satellite [92] on 3 September 2007 and 8 October 2007, with a VV polarization and a perpendicular baseline of 240 m. The second interferogram was obtained starting from two RADARSAT-1 images collected on 28 July 2001 and 25 March 2002, respectively: RADARSAT-1 was an Earth Observation satellite working at C-band managed by the Canadian Space Agency, operational from November 1995 to March 2013 [93]. Figure 8a shows an Advanced Land Observation Satellite (ALOS)-1 differential interferogram relevant to Mt. Etna volcano, Italy, superimposed on amplitude SAR image of the area. Acquisition dates are 22 March and 7 May 2010. Figure 8b shows the map of ground deformation (in LOS) corresponding to the interferogram shown in Figure 8a. The maximum detectable deformation along the satellite LOS direction was about 23 cm, corresponding approximately to the two fringes that are apparently visible in the interferogram. Note that ALOS-1, managed by the Japanese Aerospace Exploration Agency (JAXA), was operational from January 2006 to May 2011 and worked at L-band  
Figure 8: Mt. Etna volcano, Italy InSAR experiments: (**a**) ALOS-1 differential (wrapped) interferogram computed from the SAR acquisitions collected on 22 March and 7 May 2010, i.e., one repeat cycle (46 days) covering the seismic event. (**b**) LOS ground deformation map corresponding to the interferogram in (**a**). (Adapted with permission from [94], Elsevier, 2012). Figure 7: Hawaii Big Island InSAR experiments: (**a**) ENVISAT/ASAR differential interferogram relevant to the SAR data-pair 3 September-8 October 2007. (**b**) RADARSAT-1 differential interferograms generated from the two SAR data acquired on 28 July 2001 and 25 March 2002. (23.6 cm-wavelength) [95]. As a representative of the use of COSMO-SkyMed SAR data for InSAR applications, Figure 9 shows one differential interferogram (in radar coordinates) relevant to the coastal area of Shanghai produced by SAR images collected on 8 January and 9 February 2014. COSMO-SkyMed is a constellation of four satellites for Earth Observation managed by the Italian Space Agency and operating at X-band (3-cm-wavelength) [96]. Finally, two differential interferograms showing the deformation induced by 16 September 2015, Mw 7.7 Chile earthquake [97] are reported in Figure 10. The interferograms are generated from two SAR data pairs acquired on 26 August-19 September 2015 (Figure 10a) and on 31 July-17 September 2015 (Figure 10b) by the Sentinel-1 mission [98].