The failure of specular limit formulae for Kirchhoff integrals associated with Gaussian surfaces with ocean-like spectra

We analyse the use of asymptotic expansions for Kirchhoff and related integrals for scattering from Gaussian surfaces, for the limit of high Rayleigh parameter (high frequency, near-normal incidence). An extension of the traditional analysis of this type has been given very recently by Tatarskii and Tatarskii. We show that such expansions are invalid for typical frequencies of interest (L-, C-, X-band radar) when the correlation function arises from a class of spectra with ocean-like characteristics, and present alternative numerical results for such integrals. The near-normal-incidence behaviour of these integrals can in fact be fitted quite well by a parametrized Gaussian, but with parameters not directly related to the distribution of slopes. This suggests that measurements of the cross section near normal incidence may not be the reliable indicator of surface-slope parameters that they are thought to have been. These results also illustrate the dangers of using asymptotic expansions in cases where the relevant parameters are insufficiently large for the asymptotic method to be valid.