## 3 Methods
### Estimation Methodology for SAR SSW Speed and Its Uncertainty
It means that the examined configurations in \(\Sigma^{\prime}\times\Theta^{\prime}\times\Phi^{\prime}\) give the same maximum absolutevalue of \(\Delta\)W(\(\varepsilon(\Delta\sigma_{0})\),\(\varepsilon(\Delta\theta)\),\(\varepsilon(\Delta\varphi)\)) as the one obtained from the all the possible estimates in \(\Sigma\times\Theta\times\Phi\). Hence, the maximum absolute value of the estimates \(\Delta\)W derived from (4) for all the selected configurations is assumed as the SAR SSW speed uncertainty:  
\[\Delta\text{W}_{\text{max}}=\text{max}\{\big{|}\Delta\text{W}(\varepsilon( \Delta\sigma_{0}),\varepsilon(\Delta\theta),\varepsilon(\Delta\varphi))\big{|}, \text{ with : }(\varepsilon(\Delta\sigma_{0}),\varepsilon(\Delta\theta), \varepsilon(\Delta\varphi))\in\ \Sigma^{\prime}\ \times\ \Theta^{\prime}\ \times\ \Phi^{\prime}\} \tag{5}\]  
It is worth noting that the maximum uncertainty in SAR SSW speed, i.e., \(\Delta\text{W}_{\text{max}}=\text{f}(\sigma_{0},\Delta\sigma_{0},\Delta\varphi,\varphi,\Delta\varphi,\Delta\varphi,``\text{GMF}'',``\text{J}'')\), is affected by several parameters, which are:  
1. The uncertainties \(\Delta\sigma_{0}\), \(\Delta\theta\) and \(\Delta\varphi\). The two former are evaluated within each ROI as the standard deviation of the calibrated SAR backscatter (i.e., \(\Delta\sigma_{0}=\text{STD}(\sigma_{0}^{\text{ROI}})\)) and the SAR incidence angle (i.e., \(\Delta\theta=\text{STD}(\theta^{\text{ROI}})\)), whereas the corresponding mean values are given by \(\sigma_{0}\) and \(\theta\), respectively. The latter uncertainty is instead represented by the error associated with the SSW direction \(\varphi\), both estimated for the same ROI. The directional estimate \(\beta^{\text{ROI}}\) (i.e., \(\varphi=\beta^{\text{ROI}}\)) and the marginal error \(\text{ME}_{\alpha}^{\text{ROI}}\) (i.e., \(\Delta\varphi=\text{ME}_{\alpha}^{\text{ROI}}\)) are provided by the MS LG-Mod method, as described in [40]. 2. The average values \(\sigma_{0}\), \(\theta\) and the mean directional estimate \(\varphi\). They are assumed as the true values of the corresponding parameters within the ROI, with \(\sigma_{0}=\text{MEAN}(\sigma_{0}^{\text{ROI}})\), \(\theta=\text{MEAN}(\theta^{\text{ROI}})\) and \(\varphi=\text{DIRMEAN}(\varphi^{\text{ROI}})\), respectively. The operator _DIRMEAN_ computes the directional mean, thus assuming that directions \(\varphi^{\text{ROI}}\) must be considered as axial data in the ROI [40]. The triplet (\(\sigma_{0}\),\(\theta\),\(\varphi\)) represents the 'true state' for which the corresponding SAR SSW speed uncertainty is considered to be null. 3. The specific GMF used to model the SAR sigma sought as a function of the SSW speed and direction, the SAR geometry, frequency and transmitting-receiving polarizations. 4. The cost function J adopted for the minimization procedures. Without limiting the generality of the foregoing, a square function was chosen for this study as expressed by (2) and (3). 5. The approximation error for SAR SSW speed retrievals. In this work, the inversion procedure provides wind speed values in the range [0, 35] m/s with an approximation error of 0.1 m/s. The latter represents the step used in the iterative procedure to minimise the cost function. Figure 2 illustrates the scheme of the methodology for the estimation of SAR SSW speed and its uncertainty, and the MS LG-Mod processing flowchart. The latter may provide not only the SSW direction and its uncertainty, but also SAR NRCS and the incidence angle, i.e., all SAR-derived input parameters for the proposed estimation methodology.