## 3 Results and Discussions
### Current Retrieval and Validation  
As introduced above, Doppler shift can be viewed as the radar's movement relative to the sea surface, composed of the platform motion, the sea surface wind field, and the motion induced by waves, as well as the influence of currents. The total contribution can be formulated as  
\[f_{cur}=f_{D}-f_{w} \tag{6}\]  
where \(f_{cur}\) denotes the residual Doppler shift associated with the surface radial current velocity. \(f_{D}\) is the geophysical Doppler shift after correction, which is used for wind retrieval in Equation (5). \(f_{w}\) is the sea-state contribution that needs to be removed from the corrected geophysical Doppler to obtain the surface current component. \(f_{w}\) is estimated using the retrieved wind speed and direction with the CDOP model. The surface current velocity is related to the residual Doppler shift by  
\[U_{D}=-\frac{\pi f_{cur}}{k_{e}\text{sin}\,\theta} \tag{7}\]  
where \(k_{e}\) is the incident radar wavenumber. \(U_{D}\) is assumed to be the line-of-sight velocity of the ocean surface current. Figure 9a depicts the radial current velocity of the SAR scene acquired on 26 September 2019. The red region in the figure represents surface velocity directed towards the satellite range line, indicating the current flowing northeastward. Beyond the Gulf Stream, the current velocity appears relatively uniform, ranging from 0.1 to 0.5 m/s, while within the Gulf Stream, velocities generally exceed 1 m/s. To validate the currents obtained through the above method, the spatial map of surface currents observed by HFR is plotted for comparison. In Figure 9b, the currents observed by HFR at 23:00 on the same day are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Metrics** & **NRCS + Model** & **NRCS + Model + Doppler** \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} Wind \\ speed \\ \end{tabular} } & Bias [m/s] & 0.34 & 0.33 \\ \cline{2-4}  & RMSE [m/s] & 1.45 & 1.45 \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Wind \\ direction \\ \end{tabular} } & Bias [\({}^{\circ}\)] & 7.25 & 6.94 \\ \cline{2-4}  & RMSE [\({}^{\circ}\)] & 35.84 & 35.74 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparison of different cost functions. Figure 8: Comparison of retrieved winds and buoy-measured winds for (**a**) wind speed and (**b**) wind direction. The performance metrics are annotated in each plot. The metrics in red in subplot (**b**) correspond to the results for wind speed higher than 4 m/s. The empty dots correspond to the points of wind speed lower than 4m/s. presented, revealing a distribution similar to the SAR-retrieved radial current velocity. Both images capture the Gulf Stream, where current velocities can reach 1.5 m/s. Figure 9c illustrates the comparison between the observed currents from the drift buoy projecting onto the radial direction of SAR observations and the inverted currents. In most cases, there is a strong consistency between the two datasets. However, due to potential significant variations in currents throughout the day, some outliers may be present. The overall correlation coefficient is 0.87, and after excluding individual outliers, the correlation coefficients and the metrics improve greatly. For example, the bias between the two sets of data is approximately 0.02 m/s. Another point worth noting is that the abnormally larger current velocity reaching up to 2 m/s observed by the drifters might be due to errors in the products. The overall RMSE of 0.32 m/s cannot be ignored; this might be due to the relatively long collocation time window of a duration within 3 h. Assuming a current velocity of 0.5 m/s, it will propagate out a distance of 5.4 km, which is larger than the spatial collocation criteria.