## 4 Phase Statistics and Pixel Selection Candidates  
In order to to establish an adequate threshold to carry out a pixel section of persistent scatterers, the relationship between the TSC metrics and the pixels' phase standard deviation should be calculated. As indicated in [25], this relationship may be obtained through the following simulation. First, a deterministic point-like scatterer is simulated in the spectrum domain, generating an ideal flat-shape function of unitary amplitude. This spectrum is hence corrupted by adding uncorrelated complex circular Gaussian noise in order to degrade its quality. The process is repeated for each image of a simulated dataset. At this point, the standard deviation of the added noise is gradually incremented from \(0.05\) to \(0.95\), and for each case, \(5000\) realizations are performed. The TSC is then calculated, and finally, the spectrum is inverse transformed to the temporal domain in order to evaluate the mean and temporal standard deviation of the target's phase [25]. Figure 9a,b shows the relationship between the estimated TSC and the pixels' phase standard deviation considering \(10\) images and \(30\) images, respectively. As seen in the analysis carried out during the estimation accuracy (Figure 6), a larger number of samples involved in the estimation yields to smaller error bars in the phase statistics estimation. In order to establish equivalent thresholds during the pixel selection step and, thus, make a fair comparison among all methods, the same relationship is calculated for the coherence stability and the \(D_{A}\) approaches, as follows. Figure 9c illustrates the relationship between the estimated coherence and the phase standard deviation following the formulation presented by [PERSON] and [PERSON] in [32]. Likewise, Figure 9d,e, shows the relationship between the phase standard deviation and the amplitude dispersion index \(D_{A}\) derived by the numerical simulation stated in [6] considering \(10\) and \(30\) images being available, respectively. The values of dispersion in the estimation are represented by the vertical bars, while the solid line represents its mean value. Notice how, again, the quality of the \(D_{A}\) estimation depends on the number of images available, the error bars for the case of having \(10\)SAR images available being considerably larger. At this stage, all estimators have been related to the target's phase statistics. An adequate threshold may now be established in order to ensure a specific phase stability during the PSI processing. Figures 10 and 11 show a comparison in terms of pixels selected between the TSC approach and the classical techniques for the GB-SAR and the TerraSAR-X datasets, respectively. In order to ensure a fair comparison, equivalent thresholds over the phase quality estimators have been established in order to provide the same requirements in terms of phase dispersion according to Figure 9. A threshold of \(0.82\), which corresponds to a phase standard deviation of roughly \(15^{\circ}\) (see Figure 9a) has been fixed to the TSC approach. In parallel, a threshold of \(0.65\) considering a \(5\times 5\) ML [32] (see Figure 9c) has been fixed to the mean coherence, and finally, a threshold of \(0.25\) has been applied to the \(D_{A}\)[6] (see Figure 9d) to produce the same response in terms of phase standard deviation. Figure 12 gives the same information in terms of the number of pixels selected. Notice how the pixels selected with the coherence and the TSC approach correspond with the persistent scatterers present in the area of interest for both datasets, since they perfectly fit the road, the man-made structures and the few rocky and bare surfaces within the illuminated scenario. Notice the resolution loss inherent to the coherence approach, which leads to a reduced number of pixels compared with the TSC and \(D_{A}\) approaches (roughly four-times less).