###### Abstract
The spatial propagation of error is of great importance for modelling environmental processes, and ideally a map of an essential variable should be accompanied by a map of the spatial distribution of the related error. In other words, such a map would provide users a local estimation of accuracy. As is well-known, a methodology for SSW speed estimation at high spatial resolution is represented by the so-called scatterometry-based approach, which relies on the exploitation of spaceborne SAR data through the inversion of a geophysical model function (GMF). The latter model stands as a semi-empirical relationship among the calibrated SAR amplitude, i.e., normalised radar cross section (NRCS) or alternatively named sigma nought (\(\sigma_{0}\)), the mean wind vector (speed and direction), and the mean radar incidence angle, all evaluated in a wind vector cell (WVC). A number of GMFs, some of them initially developed only for the C-band scatterometers [19, 20, 21], have been presented in the literature and also exploited by means of SAR images acquired at different frequency bands [22, 23, 24, 25, 26, 27, 28, 29, 30]. A particularly critical point is the requirement of SSW direction as a priori information in the GMF-based inversion scheme. A general way of obtaining such input direction is by the spatial resampling of the SSW direction provided from NWP model data [31], although they can show unpredictable errors in coastal areas [10]. SSW direction may also be inferred directly from wind-induced linear patterns (i.e., wind streaks, WSs [32], and boundary layer rolls, BLRs [33]) onto SAR amplitudes, and consequently SAR SSW speed may be derived at the same high spatial resolution, as well as at the same acquisition time of the SSW direction extracted from those SAR signatures. Among the several methods for SSW direction extraction from SAR data [34, 35, 36, 37, 38, 39], the local gradient-modified (LG-Mod) method [11] and its multi-scale (MS) version [40], developed by the same authors of this work, are suitable to locally provide not only the mean wind direction but also its related uncertainty within each examined WVC. The capability of the LG-Mod algorithm to furnish a spatial distribution of both SSW direction and the related uncertainty can be fruitfully exploited, whereas NWP models give all the wind direction estimates with a constant maximum value of the uncertainty (e.g., 20\({}^{\circ}\) for ECMWF [12]). As shown in [11; 40], each wind direction map can be accompanied by an associated uncertainty map. As a consequence, following the above-cited approach, the spatial distribution of both the SSW speed, along with the corresponding uncertainty, may be generated, in turn, in order to fulfil the further lack of wind accuracy information, especially near the coast. The main aim of the study is to address an effective and simple operational methodology to fully evaluate the uncertainty in SAR SSW speed estimation when a GMF-based inversion is applied. The coupling between the unique ability of the LG-Mod algorithm to produce a spatial distribution of the SSW direction uncertainty and the methodology presented here can yield a spatial distribution of the SSW speed uncertainty as well. Moreover, focusing on the more recent C-band VV-polarised C-band model 5.N (CMOD5.N) [23] and C-band model 7 (CMOD7) [25] GMFs, a sensitivity analysis of the SSW speed uncertainty depending on both the uncertainties and the mean values of NRCS, incidence angle and SSW direction is carried out at different WVC dimensions (i.e., 5 km, 10 km, and 15 km). A few authors have approached the problem concerning the uncertainty of SAR SSW speed inferred through a GMF inversion. In [12], SAR backscattering information is combined with some a priori information coming from high-resolution limited area models to retrieve the most probable wind vector, assuming that the NRCS, as well as the external wind (i.e., zonal U and meridional V wind components), contain errors and that these are well characterised in terms of Gaussian error standard deviations.