## 3 Rationale of the Doppler Centroid Anomaly (DCA) Method
### DCA Technique  
DCA is performed by subtracting from the estimated DC the term corresponding to a \"_stationary_\" scene [40, 56]. In a stationary scenario or when all targets in the scene are moving at the same speed as Earth's rotation, the SAR sensor uses the phase shifts measured in the sequence of transmitted pulses while the target is within the real beam of the antenna [15, 17]. The compensated DC of the stationary contribution due to Earth rotation is related to the radial component of the sea surface velocity. More specifically, given a target moving with a radial velocity \(v_{r}\), the Doppler Centroid Anomaly \(f_{DCA}\) is related to \(v_{r}\) as follows:  
\[f_{DCA}(x,r)=f_{DC}(x,r)-f_{DC0}(x,r)=\frac{2}{\lambda}v_{r}(x,r) \tag{2}\]  
where \(f_{DC}\) is the DC measured from the data, \(f_{DC0}\) is the DC corresponding to the stationary scene, i.e., the predicted Doppler shift arising from the relative velocity of the satellite and rotating Earth, \(\lambda\) is the sensor wavelength, and \(x\) and \(r\) are the azimuth and range coordinates, respectively.  
The radial velocity is the component of the (relative) velocity vector (achieved by subtracting the scene and sensor velocity vectors) along the radar-to-target line of sight (LOS) direction defined by the versor \(\hat{\mathbf{r}}\), i.e., \(v_{r}(x,r)=v(x,r)\cdot\hat{\mathbf{r}}\). Finally, for each pixel of the SAR image, \(v_{r}\) can be estimated as follows:  
\[v_{r}(x,r)=\frac{\lambda}{2}f_{DCA}(x,r) \tag{3}\]  
Figure 4 shows the block diagram of the DCA methodology used to estimate the \(v_{r}\) map [40, 56]. It is worth noting that the use of azimuth spectrum reduction techniques (e.g., Hamming filters [57]) can have a negative impact on DCA estimation since Doppler bandwidth filtering can potentially lead to the removal of a useful part of the signal.  
Figure 4: Block diagram describing the procedure for generating \(v_{r}\) maps from SAR data. In the yellow block on the left, an optional step is shown, which is used when raw data are available (depending on the SAR mission).  
Considering Equation (3), it is straightforward to derive the precision of the radial sea surface velocity estimate as follows:  
\[\sigma_{\Delta\Omega_{r}}=\frac{\lambda}{2}\sigma_{\Delta\Omega_{f_{D}}}=\frac{ \lambda}{2}a\frac{PRF}{\sqrt{M}} \tag{4}\]  
Of course, from Equation (4), it is evident that at the X-band, the standard deviation of the radial velocity is better than at the C-band.  
However, as earlier outlined the DC estimate is biased [55], i.e.,  
\[E[f_{\text{DC}A}]=f_{\text{DC}A,true}+f_{\text{DC}A,bias} \tag{5}\]  
where \(E[f_{\text{DC}A}]\) is the measured DCA value, \(f_{\text{DC}A,true}\) represents the true (unknown) DCA value, and \(f_{\text{DC}A,bias}\) is the bias term. Indeed, DC is estimated by searching for the maximum of the azimuth SAR power spectrum, which exhibits a pattern similar to the antenna power one. This circumstance leads to the presence of the biased term, diminishing the precision of the aforementioned methodology [55]. Accordingly, any region with relatively strong or weak intensity away from the center of the radar beam (e.g., at land-water boundaries) can bias the DC estimate. A novel technique based on the ML method was recently introduced in [33] to correct the DC biased estimates using simulated and Gaofen-3 SAR data.