Analytical expressions for correlation functions and Kirchhoff integrals for Gaussian surfaces with ocean-like spectra

We present closed-form expressions for the correlation functions associated with Gaussian surfaces with certain ocean-like spectra. These correlation functions may be used to derive asymptotic expansions for the Kirchhoff integral with a wide range of validity. Some comparisons with numerical simulations are also presented. Our analysis establishes that the Kirchhoff integral is well approximated by a Bragg scattering model at very high incidence angles (that is, well away from normal incidence) even when conditions for perturbation theory do not apply. We can compute the nonlinear corrections to Bragg explicitly-these corrections grow as one approaches normal incidence. A novel feature of our analysis is the use of the computer mathematics system Mathematica to construct the relevant asymptotic series. These results eliminate the need for extensive amounts of numerical fast Fourier transform (FFT) computation, and may also be used to simplify computations of scattering cross sections from more complex surfaces with spectra that are perturbations of those we have considered.