Two dimensional bispectral estimates from ocean SAR images

Synthetic Aperture Radar (SAR) images of the ocean yield a lot of information on the sea-state surface providing that the mapping process between the surface and the image is clearly defined. However it is well known that SAR images exhibit non-gaussian statistics and that the motion of the scatterers on the surface, while the image is being formed, may yield to nonlinearities. The detection and quantification of these nonlinearities are made possible by using Higher Order Spectra (HOS) methods and more specifically, bispectrum estimation, The development of the latter method allowed us to find phase relations between different parts of the image and to recognise their level of coupling, i.e. if and how waves of different wavelengths interacted nonlinearly. This information is quite important as the usual models assume strong nonlinearities when the waves are propagating in the azimuthal direction (i.e. along the satellite track) and almost no nonlinearities when propagating in the range direction. In this paper, the mapping of the ocean surface to the SAR image is reinterpreted and a specific model (i.e. a Second Order Volterra Model) is introduced. The nonlinearities are thus explained as either produced by a nonlinear system or due to waves propagating into selected directions (azimuth or range) and interacting during image formation. It is shown that quadratic nonlinearities occur for waves propagating near the range direction while for those travelling in the azimuthal direction the nonlinearities, when present, are mostly due to wave interactions but are almost completely removed by the filtering effect coming from the surface motion itself (azimuth cut-off). An inherent quadratic interaction filtering (azimuth high pass filter) is also present. But some other effects, apparently nonlinear, are not detected with the methods described here, meaning that either the usual relation developed for the Ocean-to-SAR, transform is somewhat incomplete, although the mechanisms leading to its formulation seem to be correct, or that these nonlinearities cannot be detected in the classical bispectrum theory.