## 4 Doppler Modeling of the Wind Component  
The topic of evaluating the influence of the wind influence on the Doppler centroid has been already addressed in the literature, see f.i. [4; 40]. We focus here on the investigations carried out in [28], which deal with the problem of using the SAR DCA as an additional source of information for wind retrieval. In particular, the goal in [28] was to integrate the DCA measurement in classical approaches, such as the one described in the previous section, for wind inversion of models for the Normalized Radar Cross Section (NRCS) [36]. In doing such an integration, the authors proposed and empirically derived Doppler Geophysical Model Function (D-GMF) relating the wind speed and direction to the Doppler anomaly trough parameters associated with the SAR observation geometry and characteristics. This empirical function, which is summarized in the following, was derived by analyzing a match-up database between the wind speed and direction provided by ECMWF and Doppler measurements derived by ASAR acquisitions for both VV and HH polarizations. The analysis emphasizes that the Doppler shift due to sea surface wind can be described by the wind speed (\(w\)), the wind direction (\(\phi\)  
Figure 2: \(\sigma^{0}\) computed with CMOD-IFREMER, see Equation (15), as a function of wind speed and direction, for 23\({}^{\circ}\) of incidence angle. Results for three different wind speed values are shown: 7 m/s (blue line), 5 m/s (green line), 2 m/s (red line). \(\phi=0^{\circ}\) indicates a wind blowing towards the antenna. with respect to the antenna look angle in degrees, the incidence angle of the impinging radiation in degree (\(\vartheta\)) and the radar polarization (\(pol\)), i.e.,  
\[f_{W\_DCA}=CDOP(\phi,w,\vartheta,pol) \tag{17}\]  
Expression (17) is developed in [28] as  
\[f_{W\_DCA}=\alpha_{pol}F[X(\phi,w,\vartheta,pol)]+\beta_{pol} \tag{18}\]  
where \(\alpha_{pol}\) and \(\beta_{pol}\) are two coefficients depending on polarization, and  
\[F(x)=(1+e^{-x})^{-1} \tag{19}\]  
the argument of the function \(F\) is in Equation (18) is defined as  
\[X(\phi,w,\vartheta,pol)=\gamma_{0}^{pol}+\sum_{i=1}^{11}\gamma_{i}^{pol}F\Big{[} \Gamma_{i}^{pol}\Big{(}\overline{\phi},w,\vartheta\Big{)}\Big{]} \tag{20}\]  
where  
\[\Gamma_{i}^{pol}\Big{(}\overline{\phi},w,\vartheta\Big{)}=\omega_{i,0}^{pol} +\omega_{i,1}^{pol}V_{1}^{pol}\Big{(}\overline{\phi}\Big{)}+\omega_{i,2}^{pol} V_{2}^{pol}(w)+\omega_{i,3}^{pol}V_{3}^{pol}(\vartheta) \tag{21}\]  
\[V^{pol}=\left( \begin{array}{c}V_{1}^{pol}\\ V_{2}^{pol}\\ V_{3}^{pol}\end{array} \right)=\left( \begin{array}{c}\overline{\phi}\lambda_ {20}^{pol}+\lambda_{21}^{pol}\\ \vartheta\lambda_{00}^{pol}+\lambda_{01}^{pol}\\ w\lambda_{10}^{pol}+\lambda_{11}^{pol}\end{array} \right) \tag{22}\]  
with \(\overline{\phi}=360^{\circ}-\phi\) if \(\phi<180^{\circ}\) and \(\overline{\phi}=\phi\) elsewhere. Notice that \(\omega^{pol},\quad\lambda^{pol}\) and \(\gamma^{pol}\) are polarization-dependent coefficients whose values can be found in [28]. The model described by [28] and here summarized was validated on two cases of complex meteorological situation and it is found that the high sensitivity of the Doppler to the wind direction is useful to retrieve more realistic wind patterns in the case of complex and rapidly changing meteorological situations. More detail about the results and the areas of study can be found in [28]. Figure 3a shows the absolute value of the Dopper Anomaly (\(f_{W\_DCA}\)), evaluated according to Equations (18)-(22), as a function of wind speed, the latter ranges from 1 m/s to 18 m/s. For the ENVISAT case, the incidence angle is equal to \(23^{\circ}\) and the polarization is VV. In particular, in this figure are shown the upwind case (green continuous line), which is the case of wind blowing toward the antenna, and the downwind (red dashed line) case, which is the case of wind blowing away from the antenna.