## 3 Sublook Generation and Temporal Sublook Coherence Evaluation
Since a pixel-wise product in the space-domain translates into a spectrum correlation in the frequency-domain, the TSC is estimated employing the coherence estimator through the temporal axis among the sublook collections available [25]:  
\[T\hat{S}C(i,j)=\frac{\left|\sum\limits_{n=1}^{N_{\text{sim}}}SL_{1}(i,j,n) \cdot SL_{2}^{*}(i,j,n)\right|}{\sqrt{\sum\limits_{n=1}^{N_{\text{sim}}}|SL_{1 }(i,j,n)|^{2}\cdot\sum\limits_{n=1}^{N_{\text{sim}}}|SL_{2}(i,j,n)|^{2}}} \tag{1}\]where \(SL_{1}\) and \(SL_{2}\) are the complex values from two pixels \((i,j)\) of the first and the second sublook for each acquisition image \(n\), and \(N_{im}\) refers to the total number of images. Notice how the spatial averaging used in the traditional approach [26] is now replaced by a temporal averaging [25]. Figure 5 represents the sketch of the TSC flow chart for a generic pixel \((i,j)\). Those targets exhibiting high values of TSC within the area of interest, will be directly referred to as SCSs and, consequently, will belong to the final selection of high-quality pixel candidates, or persistent scatterers, for the PSI processing [25]. At this stage, the accuracy of the TSC estimation needs to be commented on in further detail. The reliability of the TSC estimation depends on the number of independent samples involved in the estimation, _i.e._, the number of images [25]. In fact, the mathematical formulation to characterize the TSC estimation accuracy is totally equivalent to the one developed by [PERSON] and [PERSON] in [32] for the classical coherence estimator. The only difference is that now, the spatial averaging is replaced by a temporal one, as indicated above. The probability density function of the TSC magnitude estimator may hence be expressed as a function of the 'true' TSC absolute value and the number of temporal samples involved in the estimation [32]; see Figure 6. As happens with the coherence estimator, the TSC is a biased estimator. Its value is overestimated for low magnitudes of the 'true' TSC values and/or when the number of temporal samples, _i.e._, the number of SAR acquisitions involved in the estimation, is low. Since the estimation bias is smaller for high values of TSC, where the thresholds are typically set to detect the SCSs, a reliable estimation may be obtained even when a relatively reduced number of SAR images, around \(10\), is available. This fact provides the TSC estimator a competitive advantage with respect to the PS approach, which requires a larger number of images, typically more than \(20\), to reach reliable estimations of the \(D_{A}\) and, therefore, of the phase statistics. Figure 5: Sketch of the TSC flow chart for a generic pixel \((i,j)\). Figures 7 and 8 illustrate the performance of the TSC approach compared with the traditional phase quality estimators for both the ground-based and the orbital datasets described in Section 2, respectively. Figure 7a shows the TSC estimation, while Figure 7a,b shows the temporal mean coherence and the \(D_{A}\) maps, respectively, for the GB-SAR dataset. In parallel, Figure 8a,b,c show the same maps for the TerraSAR-X dataset. Since low \(D_{A}\) values correspond to the highest quality pixels, the color bar has been inverted for the latter. In addition, it is worth mentioning that when using large ML windows, especially with the new family of high-resolution X-band SAR sensors, many stable point-like targets surrounded by non-coherent clutter are unfortunately lost. In order to use the coherence estimator to make a cross-check with the other techniques, a \(5\times 5\) ML window has been used to preserve as much as possible the original resolution of images. This ML factor shows a good trade-off between the reliability of the coherence estimator and the preservation of the spatial resolution. Figure 6: (**a**) Expectation value and (**b**) standard deviation of the TSC estimation as a function of its “true” value, for different numbers of images.