On the theory of Δk radar observations of ocean surface waves

We consider the theory of waves scattered from a moving, rough, and dispersive surface in the small perturbation limit. The first-order scattered field for a time-dependent surface is obtained in the far zone of scattering in terms of the two-dimensional spectral amplitude of the surface and its dispersion relation. We develop a rigorous Δk radar theory and show that the nonzero output of a Δk radar occurs only when the Bragg condition for each signal component is satisfied separately. The frequency correlation function of the scattered field is then proportional to the mean value of the product of the spectral amplitudes of the surface at the corresponding Bragg wavenumbers. The mean value of this product is nonzero only for surfaces that have a locally varying spectrum and is proportional to the Fourier transform (with the argument Δk) of the variation of the local spectrum with respect to the pattern position. Such variations may be caused by either amplitude or phase modulation of the surface structure. In the former case, our results are similar to the results of existing theory. The latter case of phase modulation of the surface (for example, internal waves interacting with capillary waves) cannot be explained by previous theory