# High Performance Computing in Satellite SAR Interferometry: A Critical Perspective
## 2 SAR Interferometric Processing
### SAR Interferometry Fundamentals  
InSAR today represents a mature and well-known technique for the detection and monitoring of Earth's surface deformations [14], which are connected with different natural phenomena (e.g., earthquakes, volcanic eruptions, flooding, landslides, etc.) or triggered by human activities (e.g., instabilities of public infrastructures in urban texture). InSAR techniques rely on the computation of the phase difference between pairs of SAR images collected at distinctive epochs, namely \(t_{1}\) and \(t_{2}\), and from slightly different orbital positions. A sketch of the SAR interferometric acquisition geometry is reported in Figure 1.  
For every SAR pixel of the scene, the complex-valued signal returns related to the two SAR images can be mathematically expressed as [15]:  
\[ \begin{split} i_{1}&=\gamma_{1}\exp\Bigl{[}-j\frac{ 4\pi}{\lambda}\tau_{1}\Bigr{]}\\ i_{2}&=\gamma_{2}\exp\Bigl{[}-j\frac{4\pi}{\lambda} \tau_{2}\Bigr{]}\end{split}  \tag{1}\]  
where \(\gamma_{1}\) and \(\gamma_{2}\) are the values of the reflectivity function at the imaged SAR pixel in the two interfering SAR images, \(r_{1}\) and \(r_{2}\) are the slant range distances between the sensor platforms and the imaged target on the ground at the two relevant acquisition times, and \(\lambda\)  
Figure 1: Sketch of the interferometric SAR acquisition geometry: \(S_{1}\) and \(S_{2}\) represent the two sensor satellites, \(b\) is the (spatial) baseline (i.e., the distance between the two satellites), \(b_{\perp}\) is the perpendicular baseline, and \(z\) is the topographic height. The dashed line represents the topographic profile after a ground displacement has happened. The segment highlighted in red represents the projection of the three-dimensional displacement vector along the line-of-sight (LOS). is the operational sensor wavelength. The phase difference (interferometric phase) between the two SAR images, namely \(\Delta\varphi\), can be decomposed in different terms [15]:  
\[\Delta\varphi=\frac{4\pi}{\lambda}\frac{b_{\perp}}{r_{1}\sin\theta}z+\frac{4\pi} {\lambda}d_{LOS}+\Delta\varphi_{atm}+n \tag{2}\]  
where \(z\) is the topographic height, and \(d_{LOS}\) is the radar line-of-sight projection of the 3D ground displacement occurred between the two acquisition times. Note that \(\theta\) is the incidence angle of the radar signal, \(r_{1}\) is the range to the primary (reference) antenna, \(b_{\perp}\) is the perpendicular baseline of the considered interferometric SAR data pair, the interferometric atmospheric phase \(\Delta\varphi_{atm}\) accounts for the difference between the atmospheric phase components in primary and secondary acquisitions over the same area, and \(n\) accounts for the decorrelation phase artefacts [16; 17]. From Equation (2), it is evident that InSAR can compute the topography of the observed area; however, the most attractive feature of this technology is its ability to estimate ground displacements (i.e., topography variations) with a precision in the order of the wavelength [15]. This is of paramount importance in many application fields and is the fundamental reason for the considerable popularity achieved by this technique. Although initially designed to investigate a single ground deformation episode [14], InSAR technology received a further boost over the last twenty years with the development of _multi-temporal interferometric SAR_ (MT-InSAR) approaches [18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. In the MT-InSAR case, a ground displacement time-series is obtained by processing a sequence of SAR data collected repeatedly over the investigated zone. Some processing steps must be carried out to compute the prescribed phase difference(s) and extract the corresponding topography and/or ground displacement information. A possible computational scheme for ground deformation measurements, including time series computation, is presented in Figure 2. In the following, we briefly describe each stage of the InSAR processing emphasizing the computational aspects.