The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum

Abstract

The validity of the Kirchhoff approximation for rough surface scattering is examined by comparison with exact results obtained by solving an integral equation. The pressure release boundary condition is assumed. The field quantity calculated is the bistatic scattering cross section, which is obtained with a Monte Carlo technique. The accuracy of correcting the Kirchhoff scattering cross section for shadowing is also addressed. The surface realizations used are randomly rough with a Gaussian roughness spectrum and have height variations in only one direction. The surface correlation length is found to be the most important parameter in defining the valid region of the Kirchhoff approximation away from the low grazing angle region. A procedure is given that provides a quantitative measure of the accuracy of the shadow-corrected approximation when the root-mean-square (rms) slope angle of the surface y is •< 20* and when the incident grazing angle 0 is >• 2y. Examples with 0 •< 2), are also discussed.PACS numbers: 43.30.Gv, 43.30.Hw INTRODUCTIONThere are two well-known theoretical methods for calculating acoustic scattering from rough surfaces. One is based on the small roughness perturbation approximation •'2 (also known as the Rayleigh-Rice approximation) and the other is based on the Kirchhoff (or physical optics) approximation. 3'4 The perturbation approach is valid when the rootmean-square (rms) surface height h is small compared with the acoustic wavelength A. It has been applied, for example, to low-frequency acoustic scattering from the sea surface where the validity condition can be satisfied. 2'• The valid region of the Kirchhoff approximation, on the other hand, is not well established. The Kirchhoff approximation is generally assumed to apply to "gently undulating" surfaces-more specifically, to surfaces with radius of curvature large compared with a wavelength. Since the Kirchhoff approximation is not restricted to small values of h/A, it has been developed for applications in the moderate-to-high frequency range. 6'7 Scattering at low grazing angles is of obvious practical importance in underwater acoustics, yet the Kirchhoff approximation is usually considered suspect at low grazing angles and for backscattering in general. No procedure has existed for estimating the conditions for which the error in the scattered intensity would exceed some prescribed value, such as I or 2 dB. Therefore, the applicable region of the Kirchhoff approximation has remained uncomfortably nebulous.In this paper, the accuracy of the Kirchhoff approximation for randomly rough pressure release surfaces is examined through comparison with "exact" numerical results.The exact solution for the bistatic g scattered field is found using an integral equation technique similar to that used by Lentz 9 and Axline and Fung.•ø The solutions are exact in the sense that no approximations are made in the scattering physics: Effects of multiple scattering and shadowing, for example, are automatically included. Two independent solution method...

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