## 2 Materials and Methods
### InSAR Processing  
Let \(g_{1}(k,l)\) and \(g_{2}(k,l)\) denote the SLC SAR data at pixel position \((k,l)\) in two co-registered observations that constitute an interferometric pair, \(g=\Re[g]+j\cdot\Im[g]\), where \(\Re[g]\) and \(\Im[g]\) are the zero-mean in-phase and quadrature components of the complex envelope. Let us define the interferogram as the complex cross-correlation function, namely the normalized complex covariance, between \(g_{1}\) and \(g_{2}\)[42]  
\[C(k,l)=\frac{E[g_{1}(k,l)\cdot g_{2}^{*}(k,l)]}{\sqrt{E[|g_{1}(k,l)|^{2}]\cdot E [|g_{2}(k,l)|^{2}]}} \tag{7}\]  
where \(g^{*}\) denotes the complex conjugate of \(g\) and \(E[\cdot]\) is the expectation operator. The complex cross-correlation (7) can be written as  
\[C(k,l)=\gamma(k,l)\cdot e^{-j\phi(k,l)}. \tag{8}\]  
The modulus \(0\leq\gamma(k,l)\leq 1\) is the interferometric coherence between the two images at pixel \((k,l)\). The phase \(\phi(k,l)\) is the interferometric phase, which is the relative phase between the two images. Coherence provides information about the confidence of the estimated interferometric phase and is useful for assessing the quality of interferograms. The higher the coherence, the lower the phase noise, that is, the scattering phase centers in the two observations are more similar. Thus, coherence is typically high in steady targets, such as man-made structures (buildings and infrastructure), whereas it is lower in areas characterized by rapid changes in scattering, such as vegetated areas. Equation (7) can be estimated by replacing the expectation with the average over a local window of \(L\) neighboring pixels (\(L\)-look coherent multilooking). In this case, the phase \(\phi(k,l)\) is the maximum likelihood (ML) estimator of the interferometric phase. The extremely high noise level in SLC data makes the size of the local window, estimated using Equation (7), crucial. If the window is too large, it will degrade the spatial resolution of the estimated coherence and phase maps. If the window is too small, it will be ineffective at reducing the noise in the estimated interferometric phase [43]. In addition, small windows would overestimate coherence. To avoid this drawback, unbiased estimators, e.g., [44], can be adopted. Another source of error in the estimated coherence map depends on the spatial autocorrelation function of the speckle. Equation (7) represents the complex covariance between the two images, normalized by the product of the standard deviations (square roots of the variances) of the moduli of the individual images. While the former is insensitive to the spatial correlation because the measurements are taken across two different observations, the calculation of both variances is biased because of the spatial autocorrelation function. The underestimation of variance occurs because the local estimation domain is not ergodic in variance because the two processes are autocorrelated. A balance between spatial resolution and estimation accuracy can be achieved by using non-local (NL) estimators, which overcome the limitations of a local processing window. Non-Local Filtering of Interferogram  
The NL mean (NLM) filter relies on the concept of estimating the noise-free image as a weighted combination of surrounding noisy pixels, where the weights account for the \"similarity\" between the processed pixel \(n\) and its surrounding pixel \(m\). The weights are assumed to depend on the Euclidean distance between local pixel patches centered at \(n\) and \(m\). Analogously to Section 2.1.2, \(m\) and \(n\) denote spatial 2D coordinates. For SAR images, weights have been tailored to the speckle statistics. The probabilistic patch-based (PPB) filter [45] extends the NLM filter to deal with SAR images as a weighted maximum likelihood estimator (WMLE).