## 5 PSI by Means of TSC: Results and Discussion
### Interferometric Processing Chain Considerations and CPT Overview
#### 5.1.3 The Coherent Pixels Technique  
Among the multiple advanced PSI techniques developed in the last decade by the SAR community [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], the CPT [18, 40] has been employed in this work to carry out the PSI processing. This technique has been widely exploited during the last decade for studying the temporal evolution of a large number of ground displacements caused by human activities and natural hazards, using SAR data collected by space-borne sensors. The CPT allows the estimation of the linear and non-linear components of displacement, the topographic error of the digital elevation model (DEM) used in the generation of the differential interferograms and the APS from a set of differential interferograms [18, 40]. In this framework, the CPT has been recently adapted to work with the zero-baseline fully-polarimetric GB-SAR data provided by the RiskSAR-X sensor. In addition, the TSC approach has been integrated in the pixel selection block, which worked before with the coherence stability and PS approaches. Once the pixel selection is carried out, the first step of the CPT algorithm consists of performing a Delaunay triangulation of the pixel candidates selected. This strategy allows one to work with phase increments between pixels instead of absolute phases, which present some advantages. On the one hand, the absolute phase of individual pixels is not of practical utility due to the presence of different phase offsets among the set of differential interferograms. On the other hand, it allows minimizing the APS present in interferograms. Finally, if a good distribution of pixel candidates is assumed, the absolute phase increments may be supposed in most of the interferograms to be lower than \(\pi\) radians. The phase unwrapping step may hence be skipped at this stage. Under the more general approach considering non-zero baseline interferograms, which applies to space-borne SAR data, the interferometric phase increment \(\Delta\varphi_{m,n}\) between two connected points \(P_{m}\) and \(P_{n}\) by the triangulation is expressed for the \(i-th\) interferogram as [18, 40]:  
\[\Delta\varphi_{m,n}\left(T_{i},B_{n,i}\right)=\frac{4\pi}{\lambda}\cdot T_{i} \cdot\left(v(x_{m},y_{m})-v(x_{n},y_{n})\right)+\frac{4\pi}{\lambda}\cdot \frac{B_{n,i}}{R_{i}\sin\theta_{i}}\cdot\left(\varepsilon(x_{m},y_{m})- \varepsilon(x_{n},y_{n})\right)+\Delta\varphi_{m,n}^{res} \tag{4}\]  
where \((x_{m},y_{m})\) and \((x_{n},y_{n})\) refer to the coordinates of the nodes forming the \(arc_{m,n}\), \(v(x_{m},y_{m})-v(x_{n},y_{n})\) and \(\varepsilon(x_{m},y_{m})-\varepsilon(x_{n},y_{n})\) account for the increment of linear displacement rate and topographic error, \(\lambda\) indicates the wavelength, \(T_{i}\) and \(B_{n,i}\) are the temporal and spatial baselines, respectively, \(R_{i}\) is the sensor to target distance, \(\theta_{i}\) the incidence angle and \(\Delta\varphi_{m,n}^{res}\), referred to as residual phase, refer to the atmospheric, non-linear and noise components of the phase. For GB-SAR sensors operating under a zero-baseline configuration, the topographic error phase term is not considered. Likewise, there is not any APS contribution in the residue, since it was compensated for previously. The interferometric phase increment expression seen in Equation (4) is hence simplified in the following way:  
\[\Delta\varphi_{m,n}\left(T_{i}\right)=\frac{4\pi}{\lambda}\cdot T_{i}\cdot\left( v(x_{m},y_{m})-v(x_{n},y_{n})\right)+\Delta\varphi_{m,n}^{res} \tag{5}\]  
where, now, \(\Delta\varphi_{m,n}^{res}\) accounts only for the non-linear and noise components of the phase. In order to retrieve the displacement rate, a linear model is defined for each arc of the triangulation. The model is then adjusted to data through the minimization of a cost function, referred to as model adjustment function (MAF). Once the linear displacement rate is obtained, the MAF is evaluated in order to measure the quality of each arc.