## 3 Results and Discussions
### Wind Retrieval and Validation  
Bayesian methods were initially introduced and applied for scatterometer measurements [25]. The utilization of a variational approach for SAR wind field inversion was first suggested by [26], wherein the NRCS is employed alongside external wind speed and direction, typically acquired from numerical weather prediction (NWP) or reanalysis, as prior information. The reference [12] emphasized that incorporating Doppler shifts, in addition to backscatter and background winds, enhances inversion outcomes compared to Bayesian methods that solely consider NRCS. This improvement is particularly notable in cases where the background wind vector deviates from the actual wind vector, leading to a significant enhancement in wind direction accuracy. The inversion scheme used in this study is based on the premise that these three variables are weakly correlated [27], and can be applied independently to each wind cell. It is assumed that the background wind vector and the true wind vector errors follow a Gaussian normal distribution and are uncorrelated. Here, we combine NRCS, the background wind field, and the Doppler information to construct the cost function of SAR sea surface wind retrieval algorithm, as in Equation (5), and the sea surface wind vector solution is calculated by searching for the minimum value of the cost function. \[J=\left(\frac{\sigma(u,v)-\sigma_{obs}}{\Delta\sigma}\right)^{2}+\left(\frac{u -u_{m}}{\Delta u}\right)^{2}+\left(\frac{v-v_{m}}{\Delta v}\right)^{2}+\left( \frac{f_{D}-f_{Dm}}{\Delta f_{D}}\right)^{2} \tag{5}\]  
where \((u,v)\) is the wind vector space, \(\sigma(u,v)\) is the predicted NRCS using CMOD7 for a given \((u,v)\). \(\sigma_{obs}\) is SAR-observed NRCS. \((u_{m},v_{m})\) is the a priori wind vector. \(f_{D}\) is the SAR-observed DCA, and \(f_{Dm}\) is the prediction of the CDOP for a given \((u,v)\). \(\Delta u\), \(\Delta v\) are the Gaussian standard deviation errors for model wind vector, and \(\Delta\sigma\), \(\Delta f_{D}\) are the Gaussian standard deviation errors for the NRCS and the Doppler shift. To demonstrate the contribution of each term in the cost function, we assume a true wind speed of 7 m/s at a wind direction of 60\({}^{\circ}\), an incidence angle of 30\({}^{\circ}\), and an a priori wind speed of 12 m/s with a wind direction of 30\({}^{\circ}\). The Gaussian standard deviation errors for each term are considered as constants based on the radiometric accuracy of the S-1 SAR, with values of \(\Delta\sigma=0.1\sigma_{obs}\) and \(\Delta f_{D}=10\) Hz and \(\Delta u=\Delta v=\sqrt{3}\)m/s, respectively. Substituting these terms into the cost function, a space of wind speed (0-40 m/s) and wind direction (0-360\({}^{\circ}\)) are calculated. The cost function results are influenced by the NRCS term, Doppler term, and a priori model term as shown in Figure 7. The dark color indicates a lower cost function value, suggesting a more likely wind vector solution. When considering only NRCS or DCA for the cost function, multiple minima (black curves) indicate an under-constrained problem. The red \(\square^{\prime}\) represents the true wind vector, while '\(\times\)' marks the local minima of the cost function. When accounting for the NRCS term and the background wind vector term as in Figure 7d, the inversion results for wind speed and wind direction are 6.8 m/s and 41\({}^{\circ}\), respectively. With the addition of Doppler information,  
Figure 6: (**a**) Land coverage in the Gulf of Maine (red mask for land covered area); (**b**) radial variation of Doppler centroid anomaly on land; (**c**) land-corrected Doppler anomaly. the inversion yields a wind speed of 7.2 m/s and a wind direction of 54\({}^{\circ}\), signifying a significant improvement in direction accuracy, as shown in Figure 7f. According to Equation (5), we combine radar NRCS, background wind field, and Doppler information into the cost function. Figure 8 shows the comparison between the retrieved wind vector with DCA and the wind vector measured by buoys.