## 3 Methods  
### Estimation Methodology for SAR SSW Speed and Its Uncertainty  
A C-band VV-polarised GMF may be modelled as follows:  
\[\sigma_{0}{}^{\rm GMF}(\theta,\rm W,\varphi)=B_{0}(\rm W,\theta)\left[1+B_{1}( \rm W,\theta)\rm cos(\varphi)+B_{2}(\rm W,\theta)\rm cos(2\varphi)\right]^{n} \tag{1}\]  
where \(\theta\), \(\rm W\) and \(\varphi\) are the local magnitudes that represent respectively the SAR incidence angle, the SSW speed at the reference height of 10 m, and the azimuth angle (or SSW relative direction) between the SAR look direction and the SSW direction within a WVC. The coefficients \(\rm B_{i}\) (i = 0,..., 2) are experimentally computed and tuned using a mixed dataset, typically comprising SAR measurements co-located with wind data from in situ stations, scatterometers and NWP models. The exponent n is 1.6 for the CMOD5.N [23]. Although the CMOD7 [25] was developed as the successor of the CMOD5.N, it does not have a closed-form expression as (1). The mismatch between the CMOD5.N and the backscatter measurements at low wind speeds was fixed in the CMOD7 by exploiting an independently developed ASCAT C-band GMF, namely C2013, which performs particularly well for low winds [46]. This yields that the CMOD7 is provided by tabled values of NRCS for a number of specific triplets (\(\rm\Theta\),\(\rm W\),\(\varphi\)). Whatever the exploited GMF, SAR SSW speed \(\rm W\) may be estimated from the minimization of a cost function J, defined as follows [42]:  
\[\rm J(W)=\left[\sigma_{0}-\sigma_{0}{}^{\rm GMF}(\theta,\rm W,\varphi)\right]^ {2} \tag{2}\]  
where \(\sigma_{0}\), \(\theta\) and \(\varphi\) are the NRCS, the incidence angle and the SSW relative direction respectively, all measured as average values within a ROI. \(\sigma_{0}{}^{\rm GMF}\) represents the NRCS modelled by means of the GMF adopted for the SSW speed retrieval. From (2), the dependence of the function J on all the parameters may be explicitly expressed as \(\rm J(W)=\rm J(\sigma_{0},\rm\theta\),\(\rm W\),\(\varphi\),\"GMF\"). Consequently, if the uncertainties in NRCS, incidence angle and SSW relative direction within the ROI, i.e., \(\Delta\sigma_{0}\), \(\Delta\theta\) and \(\Delta\varphi\), respectively, are taken into account, the uncertainty in SAR SSW speed within the ROI itself may be experimentally estimated as \(\Delta\rm W=W^{\prime}-W\), whereas \(\rm W\) and \(\rm W^{\prime}\) are inferred from the inversion (minimization) problem in (2), and in the following expression (3), respectively:  
\[\rm J(W^{\prime})=\left[(\sigma_{0}+\varepsilon(\Delta\sigma_{0}))-\sigma_{0}{ }^{\rm GMF}(\theta+\varepsilon(\Delta\theta),\rm W^{\prime},\varphi+ \varepsilon(\Delta\varphi))\right]^{2} \tag{3}\]  
where \(\varepsilon(\Delta\sigma_{0})\), \(\varepsilon(\Delta\theta)\) and \(\varepsilon(\Delta\varphi)\) represent a generic error that affects the NRCS, the incidence angle and the SSW relative direction, and that occurs in the range \([-\Delta\sigma_{0},\Delta\sigma_{0}]\), \([-\Delta\theta,\Delta\theta]\) and \([-\Delta\varphi,\Delta\varphi]\), respectively. The SAR SSW speed uncertainty that corresponds to the generic error triplet (\(\varepsilon(\Delta\sigma_{0})\), \(\varepsilon(\Delta\theta)\),\(\varepsilon(\Delta\varphi)\)) may be expressed by the notation:  
\[\Delta\rm W=argmin_{W^{\prime}}\{J(\sigma_{0}+\varepsilon(\Delta\sigma_{0}), \theta+\varepsilon(\Delta\theta),\rm W^{\prime},\varphi+\varepsilon(\Delta \varphi),\rm``GMF\")\}-argmin_{W}\{J(\sigma_{0},\rm\theta,\rm W,\varphi,``GMF\")\} \tag{4}\]  
To obtain an estimation of the SAR SSW speed uncertainty, a number of configurations of the error triplet (\(\varepsilon(\Delta\sigma_{0})\),\(\varepsilon(\Delta\theta)\),\(\varepsilon(\Delta\varphi)\)) (\(\in\Sigma^{\prime}\times\Theta^{\prime}\times\Phi^{\prime}\)) need to be considered, provided that they are fully representative of the space \(\Sigma\times\Theta\times\Phi\) of all possible error triples.