## 2 Materials and Methods
### Speckle Whitening of Complex SAR Data
#### 2.1.2 Problem Formulation  
In the following equation, we utilize the properties of SLC SAR images [33]. The observed scene is composed of a set of point scatterers. Let \(S^{w}(\mathbf{r})\) denote the discrete complex scattering coefficient and let \(\mathbf{r}\triangleq(r_{x},r_{y})\) denote the spatial 2D coordinate. Under the hypothesis of fully developed speckle, the term \(S^{w}(\mathbf{r})\) is a zero-mean, white symmetric circular Gaussian complex process, with variance \(\sigma(\mathbf{r})\), representing the radar reflectivity measured by the SAR system [29]. The complex-valued image found at the output of the SAR processor, equipped with a 2D separable tapering window, \(H(\mathbf{f})\), with \(\mathbf{f}\triangleq(f_{x},f_{y})\), can be regarded as the convolution of the discrete scattering coefficient with the PSF of the SAR system, \(h(\mathbf{r})\), which is the inverse Fourier transform of the tapering window, \(H(\mathbf{f})\):  
\[S(\mathbf{r})=S^{w}(\mathbf{r})*h(\mathbf{r}) \tag{1}\]  
Figure 1: Flowchart of SAR system (onboard sensor and on-ground processor) followed by optional whitening stage. which can be expressed in the Fourier domain as  
\[S(\mathbf{r})=\mathfrak{F}^{-1}\{\mathfrak{F}\{S^{w}(\mathbf{r})\}\cdot H(\mathbf{ f})\}. \tag{2}\]  
In Equations (1) and (2), we assume that the complete SAR system, including onboard acquisition and on-ground focusing, is modeled as a cascade of linear shift-invariant filters, defined by their PSFs. The SAR image with fully developed speckle is given by:  
\[|S(\mathbf{r})|^{2}=\sigma(\mathbf{r})\cdot u_{u}(\mathbf{r}) \tag{3}\]  
where \(u_{u}(\mathbf{r})\) is the fading term, usually modeled as an uncorrelated stochastic process with both mean and variance equal to one and a negative exponential PDF. Equation (3) describes the intensity format of a single-look image, i.e., the power of the backscattered signal. By definition [34], the signal-to-noise ratio (SNR) of a single-look SAR image in intensity format is zero dB. The model in Equation (3), however, is valid if the overall SAR system in Figure 1 features a PSF that is a discrete \(\delta(\mathbf{r})\) function. The case of polarimetric SAR (poISAR) is much more complex because speckles are not independent among the polarimetric channels [35], even though each channel is processed independently of the others. A more general formulation of the problem, including an autocorrelation function of the fading term, can be stated as  
\[|S(\mathbf{r})|^{2}\approx\sigma(\mathbf{r})\cdot u_{s}(\mathbf{r}) \tag{4}\]  
where \(u_{s}(\mathbf{r})\) denotes the fading term, which is spatially correlated due to the nonideal frequency response of the SAR system and is independent of the reflectivity term \(\sigma(\mathbf{r})\). According to [33], this approximation improves as the Fourier transform of the PSF of the SAR system extends further than the power spectrum of the reflectivity. Specifically, the correlation length of speckle should not be greater than that of the reflectivity. Equation (4) accounts for the spatial correlation of speckle, provided that the bandwidth of the system encompasses the power spectrum of the reflectivity. The goal of the spatial decorrelation of speckle is the removal of the effects of the frequency response of the SAR system to recover \(S^{w}(\mathbf{r})\) from \(S(\mathbf{r})\). Such a blind deconvolution [36] may be simplified if we assume that the frequency response of the SAR system is a band-limited lowpass filter with a cutoff frequency \(\mathbf{f}_{c}\):  
\[H(\mathbf{f})\approx 0\qquad\forall|\mathbf{f}|>|\mathbf{f}_{c}|. \tag{5}\]  
No assumptions on \(\sigma(\mathbf{r})\) are necessary. In [6], it was proven that an estimate of \(\hat{S}^{w}(\mathbf{r})\) is given by:  
\[\hat{S}^{w}(\mathbf{r})=\left\{ \begin{array}{cc}\mathfrak{F}^{-1}\{\mathfrak{F }\{S(\mathbf{r})\}\cdot[\hat{H}(\mathbf{f})]^{-1}\}&\quad\forall|\mathbf{f}| \leq|\mathbf{f}_{c}|\\ \\ 0&\quad\text{otherwise}\end{array} \right.