## 2 Overview of InSAR Methodology
### Multi-Pass InSAR Techniques for the Retrieval of Surface Displacement Time-Series
Figure 10: Sentinel-1 differential (wrapped) interferograms relevant to the 16 September 2015, Mw 7.7 Chile earthquake, superimposed on a Google Earth image and computed by using the SAR pairs: (**left**) 26 August–19 September 2015 (ascending orbit, Track 18); (**right**) 31 July–17 September 2015 (descending orbit, Track 156). The red star is the main shock epicentre. (Adapted with permission from [97], MDPI, 2017). Hereinafter, a few additional details about the rationale of the Small BAseline Subset (SBAS) approach will be provided. Figure 11: Comparison between the unfiltered (left column) and noise-filtered (right column) multi-look ASAR/ENVISAT interferograms relevant to the area of the Abruzzi region, Italy. (**a–c**) interferograms relevant to 30 October 2005–8 November 2009, 21 August 2005–6 June 2010 and 1 August 2004–12 April 2009 SAR data-pairs, characterized by perpendicular baseline values of 192, 145 and 395 m, respectively. (**d–f**) Noise-filtered multi-look interferograms corresponding to the ones in (**a–c**), respectively. (Adapted with permission from [68], IEEE, 2015). The starting point is the availability of a sequence of, say M, differential SAR interferograms. Measured interferometric phases are restricted to the \([-\pi,\ \pi]\) interval. However, to produce deformation time-series, the information conceived in the phase must be unrestricted. Phase Unwrapping (PhU) operations, essentially consisting in recovering the absolute phase field from the observed wrapped measurements, are employed to accomplish this task. Over the years, much work has been done on two-dimensional (space) PhU, which has resulted in many algorithms. They may be arranged in three main categories: minimum-norm [111; 112], branch-cut [113] and minimum cost flow (MCF) network [69] methods. More recently, the need for analysing long sequences of multi-temporal interferograms for (advanced) DInSAR applications has also promoted the development of new PhU approaches, based on the joint exploitation of the spatial and temporal relationships among the produced DInSAR interferograms [110; 114; 115]. Among them, a technique that is very well integrated within the SBAS procedure is the one proposed in [110], which is referred to as extended MCF (EMCF) and represents the space-time extension of the conventional MCF approach [69]. The sequence of unwrapped interferograms can be expressed, pixel-by-pixel, as follows:  
\[\Delta\varphi_{k}=\frac{4\pi}{\lambda}d_{k}-\frac{4\pi}{\lambda}\frac{b_{ \perp k}}{r\sin\vartheta}\Delta z+\Delta\varphi_{orb,k}+\Delta\varphi_{atmo,k} +\Delta\varphi_{scatt,k}\ \ k=1,\ldots,M \tag{37}\]  
where \(d_{k}\) and \(b_{\perp k}\) are the LOS deformation and the perpendicular baseline values of the \(k\)-th interferogram. \(\Delta z\) is the error of the used DEM and \(\Delta\varphi_{orb,k}\), \(\Delta\varphi_{atmo,k}\) and \(\Delta\varphi_{scatt,k}\) are the orbital inaccuracies, the APS and the scattering phase contributions relevant to the k-th interferogram. For the sake of simplicity, we have included in \(\Delta\varphi_{scatt,k}\) also the phase noise signal that corrupt the measure of the phase measurement. The system of Equations (37) can be reorganized using the matrix formalism as follows [71]:  
\[\mathbf{A}\cdot\mathbf{\Phi}=\Delta\mathbf{\Phi} \tag{38}\]  
where \(\mathbf{A}\) is an incidence-like matrix, related to the identified set of SB InSAR data-pair and \(\Delta\mathbf{\Phi}\) is the vector of unwrapped phases. The elements of matrix \(\mathbf{A}\) are defined as follows:  
\[\left\{ \begin{array}{c}A\left(j,IM_{j}\right)=11M_{j}\
eq 0\\ A\left(j,IS_{j}\right)=-1 IS_{j}\
eq 0\\ Otherwise\end{array} \right. \tag{39}\]  
where \(\mathbf{I}\mathbf{M}\) and \(\mathbf{IS}\) are the indices vectors of the master and slave time acquisitions concerning the \(N+1\) ordered times \([t_{0},t_{1},t_{2},\ldots,t_{N}]^{T}\).