## 2 Overview of InSAR Methodology
### InSAR Technique for the Estimation of Height Topography
Taking into account Equations (2) and (3), we obtain, for each RADAR pixel, the phase difference between the two SAR images by merely multiplying the first image (master) by the complex conjugate of the second image (slave) and, then, by extracting its phase term:  
\[C=\hat{\gamma}_{1}\hat{\gamma}_{2}^{*}=|\hat{\gamma}_{1}\hat{\gamma}_{2}|\exp \left[j\left(\frac{4\pi}{\lambda}\delta r+\measuredangle\gamma_{1}-\measuredangle \gamma_{2}\right)\right] \tag{4}\]  
where the symbol (*) denotes the complex conjugate operation and the symbol \(\measuredangle\) represents the phase extraction operation. From Equation (4), assuming that the electromagnetic-wave scattering mechanism on the ground has not changed between the two passages of the sensor over the illuminated area (this requirement could be directly accomplished if a single-pass configuration was considered), we get:  
\[C=\hat{\gamma}_{1}\hat{\gamma}_{2}^{*}=|\hat{\gamma}|^{2}\exp\left[j\frac{4 \pi}{\lambda}\delta r\right] \tag{5}\]  
and its full phase term (i.e., not restricted to the [\(-\pi\), \(\pi\)] interval) represents the so-called interferometric phase:  
\[\varphi=\frac{4\pi}{\lambda}\delta r \tag{6}\]  
Taking into account the acquisition geometry illustrated in Figures 1 and 2 and in the application of the cosine's rule to the MST triangle depicted in Figure 1[41], it is straightforward to demonstrate that:  
\[r+\delta r=\sqrt{b^{2}+r^{2}-2 br\sin(\vartheta-\alpha)}\simeq r^{\prime}-b\sin (\vartheta-\alpha) \tag{7}\]  
Figure 2: Interferometric baseline geometry. The baseline vector can efficiently be decomposed in the perpendicular/parallel components (**a**) and the horizontal/vertical components (**b**), respectively. Accordingly, the interferometric phase, see (6), can be written as:  
\[\varphi\simeq-\frac{4\pi}{\lambda}b\sin(\vartheta-\alpha)=\frac{4\pi}{\lambda}b_{//} \tag{8}\]  
where \(b_{//}\) is the parallel baseline component, see Equation (1). At this stage, the problem to be addressed is the recognition of the interferometric phase dependency upon the surface topography and the subsequent estimation of the error budget of the InSAR-based topographic measurements. To this aim, let us refer again to the geometry of the problem sketched in Figure 1 and we expand Equation (8) around the angular position \(\vartheta=\vartheta_{0}\), which represents the side-looking angle that corresponds to the case when Earth's surface is flat (i.e., \(z=0\)). Therefore:  
\[\varphi\simeq-\frac{4\pi}{\lambda}b[\sin(\vartheta_{0}-\alpha)+\cos(\vartheta _{0}-\alpha)(\vartheta-\vartheta_{0})] \tag{9}\]  
The target height, namely \(z\), is related to the side-looking angle \(\vartheta\) and to the satellite height \(H\) as \(z=H-r\cos\vartheta\). Hence:  
\[\hat{\gamma}_{2}=\gamma_{2}\exp\left[-j\frac{4\pi}{\lambda}(r+\delta r)\right] \tag{3}\]0  
Finally, the interferometric phase can be seen as composed of two distinctive terms, as follows:  
\[\hat{\gamma}_{2}=\gamma_{2}\exp\left[-j\frac{4\pi}{\lambda}(r+\delta r)\right] \tag{3}\]1  
where the mathematical relation \(b_{\perp}=b\cos(\vartheta-\alpha)\) (see (1)) has been used. The first term on the right-end side of Equation (11) represents the so-called flat-Earth phase contribution, i.e., the phase that is obtained when the topographic profile is entirely flat (\(z=0\)). The second term relates the interferometric phase to the surface height \(z\). By Equation (11), it is simple to understand that the interferometric phase is proportional to the height through the value of the perpendicular baseline of the interferometric data pair.