## 3 Rationale of the Doppler Centroid Anomaly (DCA) Method
### Doppler Centroid (DC) Estimation  
The DC methodology is grounded in the principles of spectral estimation. Specifically, it hinges upon analyzing the variations in the image spectrum caused by the motion of the observed target, i.e., the sea surface. If the area under observation is stationary, the Doppler shift will be zero. For distributed targets, the azimuth signal is a superposition of many point-target responses. Therefore, one needs to find the central value of the spectrum, i.e., the Doppler Centroid.  
DC is estimated using a sliding window throughout the entire focused SAR image. The dimensions of the estimation window are typically set in order to ensure a good balance between spectral and spatial resolution [40].  
Many methods have been used for the DC estimation, such as the Energy Balancing (EB) method [55], the Matched-Correlation (MC) method [31] and the Correlation Doppler Centroid Estimator (CDCE) [30]. In particular, in this work, we exploit the CDCE method, which is based on a time-domain approach consisting on the calculation of the correlation function between two image patches separated by \(M_{az}\) samples along the azimuth direction. Interested readers may directly refer to [30] to deepen the topic. In particular, Equations (16) and (18) in [30] make explicit the mathematical expression of the correlation function and the estimated DC frequency, respectively.  
A Maximum-Likelihood (ML) estimator meeting the Cramer-Rao bound was also discussed in [31]. It is self-evident that an accurate DC estimation is required for more reliable DCA estimation.  
Under the white Gaussian assumption for the signal and noise, assuming an area with homogeneous scattering, it was shown in [31] that the standard deviation (STD) of the DC estimate is given by  
\[\sigma_{DC}=a\frac{PRF}{\sqrt{M}} \tag{1}\]  
where \(a\) is a constant that varies for each of the above-mentioned techniques, \(PRF\) is the sensor Pulse Repetition Frequency, and \(M=M_{az}\times M_{rg}\) is the number of independent samples in the box used for the estimation (az and rg stand for the azimuth and range coordinates).  
Table 3 shows the values of \(a\) in Equation (1) for the different approaches presented in [31].  
DC estimates may also be affected by inaccuracies caused by gradients in surface backscatter along the azimuth direction and antenna mis-pointing, which have only been briefly commented on in the literature [16; 38]. A more detailed analysis of such phenomena was carried out in [32] referring to an ASAR dataset.  
About the DC estimate perturbances due to NRCS azimuthal variations, it must be considered that overwater the local NRCS varies enormously in space and time due to effects such as the weighting of advancing and receding Bragg resonance waves, which depend strongly on the wind direction, specular reflection and wave breaking, as well as tilt and hydro- and aero-dynamic modulations of the scattering elements [38].  
In addition, as reported in [55], the backscattering gradient along the azimuth direction within the DC estimation area may introduce a bias into the DC estimation.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline EB & MC & CDCE & ML \\ \hline \(a\) & 0.3985 & 0.3407 & 0.3407 & 0.2516 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The \(a\) parameter for the different DC estimation approaches. [31]Finally, in [17, 40], an undesired DC azimuthal component for ENVISAT data was modeled as a sinusoidal contribution taking place at a constant frequency of 31.72 Hz and subsequently compensated for by using notch filtering.