## 2 Overview of InSAR Methodology
### InSAR Technique for the Estimation of Height Topography  
In this review, we concentrate on the repeat-pass across interferometry. As earlier anticipated, such technology allows the measurement of geomorphological characteristics of the ground, such as the topography height and the modifications of the Earth's surface, due to earthquakes, volcano eruptions or other geophysical phenomena. Historically, InSAR has been mostly used for the measurement of the ground topography [38; 39; 40]. To understand how InSAR works, let us refer to the imaging geometry depicted in Figure 1 and suppose the involved RADAR system acquires two SAR images over the same area. The first SAR image (i.e., commonly referred to as master image) is taken from the orbital position labelled to as \(M\). The second image (i.e., the so-called slave image) is captured from the orbital position marked as \(S\), located at a distance b (referred to as baseline) from \(M\). Notice that in the plane orthogonal to azimuth direction the two SAR antennas are separated by the baseline vector \(\mathbf{b}\), which can be decomposed into either its parallel/perpendicular \((b_{//},b_{\perp})\) or horizontal/vertical \((b_{h},b_{v})\) components, respectively. Taking into account geometric considerations and by direct inspection of Figures 1 and 2, it is easy to demonstrate the validity of the following mathematical relationships:  
\[\left\{ \begin{array}{c}b_{\perp}=b\cos(\vartheta-\alpha)=b_{h}\cos\vartheta+ b_{v}\sin\vartheta\\ b_{//}=-b\sin(\vartheta-\alpha)=b_{h}\sin\vartheta-b_{v}\cos\vartheta\\ b_{h}=b\cos\alpha=b_{\perp}\cos\vartheta+b_{//}\sin\vartheta\\ b_{v}=b\sin\alpha=b_{\perp}\sin\vartheta-b_{//}\cos\vartheta\end{array} \right. \tag{1}\]  
where \(\vartheta\) and \(\alpha\) are the satellite side-looking angle and the inclination angle of the baseline vector with respect to the horizontal plane, respectively. Equations (1) are fundamental to calculate the sensitivity of InSAR measurements to deformation measurements, as subsequently outlined in this Section. If we assume that the RADAR system has an infinite bandwidth, the master and slave complex-valued SAR images can be mathematically represented, pixel-by-pixel, as follows:  
\[\hat{\gamma}_{1}=\gamma_{1}\exp\left[-j\frac{4\pi}{\lambda}r\right] \tag{2}\]  
\[\hat{\gamma}_{2}=\gamma_{2}\exp\left[-j\frac{4\pi}{\lambda}(r+\delta r)\right] \tag{3}\]  
where \(\gamma_{1}\) and \(\gamma_{2}\) are the complex reflectivity functions of the master and slave data, respectively and \(\lambda\) denotes the effective RADAR wavelength. The interferometric phase map (called an interferogram) is formed, on a pixel-by-pixel basis, starting from the two co-registered (complex-valued) master and slave SAR images. Notice that the misalignment of the second image is precisely due to the path difference between the two RADAR signals. Figure 1: SAR interferometric configuration. Precisely for this reason, it is needed to properly co-register the images to represent the two SAR images in the same reference geometry [41; 42]. In Equations (2) and (3), for the sake of simplicity, we have not considered the presence of any possible noise signals that corrupt the SAR signals. A comprehensive overview of the sources of noise associated with SAR images falls beyond the scopes of this review paper. Nonetheless, the Section 2.2 addresses the problem of noise corrupting interferometric SAR signal. Let us first discuss the rationale of the interferometry SAR technique.