Directional spreading function of the sea wave spectrum at short scale, inferred from multifrequency radar observations

Several models of the directional spreading function of the sea wave spectrum have been proposed in the literature for low wavenumbers. In this paper we propose a high-wavenumber extrapolation of those models, obtained by fitting a correcting term which vanishes at large scales (low wavenumbers). At short scales (high wavenumbers) the correcting term is constrained by multifrequency microwave observations of the normalized radar cross section sigma degrees from P band (frequency = 0.43 GHz) up to Ka band (frequency = 34.43 GHz), together with optical observations of the sea surface slope variance. Two formulations are given, one providing a high-wavenumber extrapolation to Apel's [1994] formulation and the other providing a high-wavenumber extrapolation to Donelan et nl.'s [1985] and Banner's [1990] formulation. The correcting term delta, expressed as a function of wavenumber k and wind speed U by means of six least squares fitted parameters, is found to vary strongly with k and slightly with U. Another simpler expression for delta, involving only the dependence with k by means of three fitted parameters, is therefore also proposed. According to our fitted model of the spreading function, there is a spectral region in the short gravity range where the sea spectrum shows only a weak dependence on the direction, in accordance with the previous models. However, unlike them, our model gives an increase of the anisotropy of the spreading function at higher wavenumbers, in such a way that the ratio between the cross-wind and along-wind spectral densities of the folded spectrum is reduced to no more than 35% at high wavenumbers (k approximate to 1000 rad/m). This increase of anisotropy at high frequency is in accordance with conclusions drawn by previous authors from radar backscatter data, which were, however, limited to narrow spectral bands since their analyses involved only single-frequency radar data.