## 2 Overview of InSAR Methodology
### Multi-Pass InSAR Techniques for the Retrieval of Surface Displacement Time-Series
The system of Equation (38) can be solved in the Least Squares (LS) sense:  
\[\mathbf{\Phi}=\mathbf{A}^{+}\cdot\Delta\mathbf{\Phi}=\left(\left(\mathbf{A}^ {T}\cdot\mathbf{A}\right)^{-1}\cdot\mathbf{A}^{T}\right)\cdot\Delta\mathbf{\Phi} \tag{40}\]  
where \(\mathbf{A}^{+}\) is the left inverse matrix. However, generally speaking, the matrix \(\mathbf{A}\) can also be rank deficient: This happens when SB InSAR data pairs lead to the presence of several independent subsets of SB interferograms separated by large baselines [71]. In this case, the matrix \(\mathbf{A}^{T}\cdot\mathbf{A}\) is rank-deficient and does not admit an inverse matrix, that is to say the number of LS solutions of the problem (38) is infinite. Among the infinite LS solutions, one solution is selected by applying the SVD method to matrix \(\mathbf{A}\). More specifically, the solution of the problem is the one that minimizes the norm of the residuals of the system of Equation (38) and, at the same time, the norm of the phase vector \(\mathbf{\Phi}\) (i.e., \(\left\|\mathbf{\Phi}\right\|_{2}\)). However, such a solution may introduce large discontinuities in the obtained results, thus leading to a non-physical sound solution (see Figure 12). A strategy to overcome this problem is at the base of the SBAS algorithm [71]: it consists in manipulating the system of Equations (38) in such a way to replace the phase unknowns \(\mathbf{\Phi}_{i}\), \(i=0,\ldots,N\) with the mean phase velocity between time-adjacent acquisitions \(V_{i}\), \(i=0,\ldots,N-1\), which are defined as follows:  
\[\mathbf{V}=\left[V_{0}=\frac{\Phi_{1}-\Phi_{0}}{t_{1}-t_{0}},V_{1}=\frac{\Phi_{2 }-\Phi_{1}}{t_{2}-t_{1}},\ldots,V_{N-1}=\frac{\Phi_{N}-\Phi_{N-1}}{t_{N}-t_{N-1 }}\right]^{T} \tag{41}\]  
Accordingly, the system of Equation (38) becomes:  
\[\mathbf{B}\cdot\mathbf{V}=\Delta\mathbf{\Phi} \tag{42}\]  
which is solved in the LS sense by applying the SVD method to the matrix \(\mathbf{B}\). Once the phase associated with each SAR acquisition, as well as the residual topography, are estimated, the phases are converted to deformation and the APS is computed and filtered out from the obtained deformation time-series. APS removal is achieved by exploiting the assumption that APS is correlated in space and uncorrelated in time. Thus, the processing of the atmospheric corrupted time-series is performed with a spatial low-pass (LP) filter and a time high-pass (HP) filter [71, 109]. The quality of retrieved LOS time-series is finally evaluated pixel-by-pixel by calculating the values of the temporal coherence factor, originally proposed in [110]. Residual orbital fringes are also estimated and filtered out in the SBAS processing chain by searching for (in each interferogram) any possible phase ramp, which can be directly related to errors in the knowledge of sensor position along its orbit. Such residual phase ramps are jointly analysed to correct the orbits state vectors. Finally, for pixels with high temporal coherence, the map of LOS mean deformation rate over the analysed period is computed. The accuracy of the displacement time-series retrieval by the SBAS algorithm in different geophysical contexts has been assessed and the results have been the subject of several investigations (see, for instance, [107, 116, 117, 118]). Experiments, based on the comparison between DInSAR deformation time-series and independent GPS/levelling measurements, show that the SBAS technique provides an estimate of the mean deformation velocity with a standard deviation of about 1 mm/year for a typical data set composed by 40 to 60 SAR images [117]. Moreover, the single displacement measurements, computed with respect to a reference point of known motion, show a sub-centimetric accuracy, with a standard deviation of about 5 mm, consistently in both the SAR/levelling and SAR/GPS comparisons. Different SB multi-pass DInSAR tools have been developed in the recent years for the reconstruction of ground deformation [119, 120, 121].