## 2 Overview of InSAR Methodology
### InSAR Technique for the Estimation of Surface Displacements  
SAR interferometry nowadays is mostly used for the detection/monitoring of surface changes occurring between passages of the RADAR sensor over the same scene. In such a case, as a slight change across the two SAR acquisition times happens in the imaged scene, an additive term in the interferometric phase associated with the RADAR line of sight (LOS) component of the surface displacement arises, in addition to the phase dependency on the topography. By the inspection of the imaging geometry depicted in Figure 6, we get:  
\[\Delta\varphi=-\frac{4\pi}{\lambda}\cdot\frac{b_{\perp}}{r\sin\vartheta}z+ \frac{4\pi}{\lambda}d_{LOS} \tag{33}\]  
where \(d_{LOS}\) represents the projection of the occurred deformation with respect to sensor LOS direction. For the sake of convenience, we neglected the presence of the flat earth phase contribution. Figure 6: InSAR geometry for the estimation of the displacement of Earth’s surface. To measure the interferometric phase term associated with the displacement, the interferometric phase contribution pertinent to the underlying topography in Equation (33) has to be removed. Precisely, the differential SAR interferometry (DInSAR) mainly consists in the synthesis of a simulated topographic phase screen from an available Digital Elevation Model (DEM) of the area and to subtract, pixel-by-pixel, these synthetic fringes, leaving only the term associated with the displacement. Computed differential SAR interferograms, however, contain, in addition to the deformation component, some (unwanted) phase terms arising from certain inaccuracies on the knowledge of the actual topographic pattern and of the orbital parameters. In particular, the variation of the interferometric phase can be expressed, more in general, in the form:  
\[\Delta\varphi=\Delta\varphi_{displ}+\Delta\varphi_{topo}+\Delta\varphi_{orb}+ \Delta\varphi_{atmo}+\Delta\varphi_{scatt}+\Delta\varphi_{noise} \tag{34}\]  
where:  
\(\Delta\varphi_{displ}=\frac{4\pi}{\Lambda}d_{LOS}\) accounts for a possible displacement of the scatterer on the ground between observations, where \(d_{LOS}\) denotes the projection of the relevant displacement vector on the LOS;  
\(\Delta\varphi_{topo}=-\frac{4\pi}{\Lambda}\cdot\frac{b_{1}}{r\sin\vartheta}\Delta z\) represents the residual-topography-induced phase due to a non-perfect knowledge of the actual height profile (i.e., the DEM errors \(\Delta z\));  
\(\Delta\varphi_{orb}\) accounts for residual fringes due to the use of inaccurate orbital information in the synthesis of the topographic phase;  
\(\Delta\varphi_{atm}\) denotes the phase components due to the variation of propagation conditions (pertinent to the change in the atmospheric and ionospheric dielectric constant) between the two master/slave acquisitions;  
\(\Delta\varphi_{scatt}\) denotes the phase components due to changes in scattering behaviour;  
\(\Delta\varphi_{noise}\) includes all the phase noise contributions (see Section 2.2). Finally, another source of misinterpretation about the (wanted) deformation signal is the presence of phase unwrapping errors, due to an incorrect estimate of \(2\pi\)-multiple integers to be added to the (measured) InSAR phase to recover the full (unrestricted) phase [69]. Additional information on the phase unwrapping procedures is provided in Section 2.3. Overall, limits and potential of InSAR technique are addressed in [44]. Some additional remarks on the error budget of DInSAR measurements are now in order. First of all, we remark that for DInSAR applications, the (wanted) signal to be recovered is the terrain displacement \(d_{LOS}\) (see Equations (33) and (34)), whereas the additional phase terms that appear on the right-hand side of Equation (34) evidently represent error sources, namely \(\Delta\varphi_{artifacts}\).