## 3 Wind Retrieval from SAR  
The dominant contribution to the radar backscatter comes from the wind generated surface gravity-capillary waves through the resonant Bragg wave scattering mechanism [31; 32]. Since the caps of these waves tend to align perpendicular to the local wind, the radar backscatter is also wind direction-dependent. Therefore, SAR backscatter can be used in principle to derive sea-surface wind information. Several empirically derived backscatter models, known as Wind Geophysical Model Functions (W-GMF), have been proposed in the literature [33; 34; 35] to relate the 10 m wind vector to SAR observations. They are generally defined as  
\[\sigma^{0}=\mathrm{B0}\left[1+\mathrm{B1}\cos\phi+\mathrm{B2}\cos(2\phi) \right]^{n} \tag{15}\]  
where \(\sigma^{0}\) is the backscattering coefficient measured on the SAR image, \(\phi\) is the wind direction relative to the antenna pointing (\(\phi=0\) indicates an upwind, i.e., a wind aligned to the range and blowing towards the radar), and the coefficients B0, B1 and B2 depend on the wind speed \(w\), the local incidence angle, and the polarization and frequency of the radar beam. The Bi parameters were determined statistically after drawing comparisons with wind data from ECMWF atmospheric modeling outputs, or from operational buoy measurements. The exponent \(n\) could be 1.6 or 1, depending on the model considered. In this work, the GMF used was developed at IFREMER and is referred to as CMOD-IFREMER [36].  
Unfortunately, the inverse relationship of Equation (15) is an inherently underdetermined problem for single-view measurement instruments, as happens for SAR imaging. Indeed, Equation (15) establishes that the backscatter is a monotonically increasing function of the wind speed and that is sinusoidal with \(\phi\). In particular, fixing the wind speed, \(\sigma^{0}\) is maximum for the wind blowing towards (or away from) the radar (\(\phi=0^{\circ},180^{\circ}\)) and minimum for the wind blowing perpendicular to the radar, see Figure 2.  
As a consequence, both wind speed and direction cannot be estimated from a single \(\sigma^{0}\) measurement so that some a priori wind information is needed. For this research, a downscaling of the 10 m wind fields obtained from the non-hydrostatic mesoscale model Weather Research and Forecast-Advanced Research (WRF-ARW) version 3.3.1 [37] were considered as a priori wind vector information. The choice fell on this wind database since it has been already assessed for the Mediterranean basin. Indeed, [33] studied the performance of the third-generation wave model WAVEWATCH III (WWIII) in the Mediterranean by using as a wind forcing term the downscaled WRF data and validating their simulation results through buoys data provided by the Rete Ondametrica Nazionale (RON) [38], as better specified in Section 6.  
Following [39], the SAR retrieved wind vector components are computed by minimizing the cost function:  
\[J=\left(\frac{\sigma^{0}(u,v)-\sigma^{0}_{m}}{\Delta\sigma}\right)^{2}+\left( \frac{u_{ww}-u}{\Delta u}\right)^{2}+\left(\frac{v_{ww}-v}{\Delta v}\right)^{2} \tag{16}\]  
where \(\sigma^{0}_{m}\) is the SAR measured backscatter, \(\sigma^{0}\) is the backscatter value computed by solving Equation (15) with wind vector of components \(u\) and \(v\), \(\Delta\sigma=0.078~{}\sigma^{0}_{m}\), and \(\Delta u=\Delta v=\sqrt{3}~{}~{}m/s\). Finally, \(u_{ww}\) and \(v_{ww}\) are the reference wind components provided by the WWIII hindcast. Cost function minimization is performed by looking at the global minimum of Equation (16) computed for velocities from 0 m/s to 25 m/s with a 0.05 m/s spacing and wind direction from \(0^{\circ}\) to \(360^{\circ}\) and spacing of \(1^{\circ}\).