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

Abstract

The validity of the perturbation approximation for rough surface scattering is examined (1) by comparison with exact results obtained by solving an integral equation and (2) through comparison of low-order perturbation predictions with higher-order predictions. The pressure release boundary condition is assumed, and the field quantity calculated is the bistatic scattering cross section. A Gaussian roughness spectrum is used, and the surfaces have height variations in only one direction. It is found, in general, that the condition kh≪1 (k is the acoustic wavenumber, h is the rms surface height) is insufficient to guarantee the accuracy of first-order (or higher-order) perturbation theory. When the surface correlation length l becomes too large or too small with h held fixed, higher-order perturbation terms can make larger contributions to the scattering cross section than lower-order terms. An explanation for this result is given. The regions of validity for low-order perturbation theory are also given. These results resolve (for the Gaussian roughness spectrum) the long-standing discrepancy between the first-order perturbation cross section and the cross section obtained with the Kirchhoff approximation for conditions when both approximations have generally been considered accurate. It is shown explicitly that, for some cases, the Kirchhoff prediction is the correct one. Finally, it is found that the perturbation expansion for the scattering cross section reduces to a power series expansion in the rms slope s when kl≪1. This observation provides additional insight into the connection between the small slope assumption and the validity of low-order perturbation theory.

Models for Scattering from Rough Surfaces

“…The SPM has been compared to more accurate numerical simulations in (Thorsos & Jackson, 1989;1991) for one-dimensional rough surfaces with a Gaussian roughness spectrum. Under these conditions the authors show that the first-order SPM gives accurate results for kσ << 1 and 1 kl ≈ .…”

Time-domain sensing of targets buried under a Gaussian, exponential, or fractal rough interface

“…Scattering from a surface with an exponential density function (non-Gaussian) has been discussed in [13]. The most widely applied type of rough surface model is characterized by a Gaussian autocorrelation function [5]. For a surface with variance , the autocorrelation is expressed as (1) with the parameter termed the correlation length.…”

“…In practice, the random process (surface) is generated in the Fourier domain, by passing a Gaussian white-noise process through a filter with a spatial-frequency response corresponding to the desired rough-surface spectrum [5].…”

A comparison of perturbation theory and the small-slope approximation for acoustic scattering from a rough interface for a Biot medium

“…The scattering cross section quantifies the energy scattered from the rough interface between the upper fluid half space and the Biot medium and is therefore related to the scattered displacement field . It can be written in terms of the -matrix [17] (4) where , , , the subscripts and indicate the angles of incidence and scatter, respectively, the angle brackets indicate ensemble averages, an asterisk indicates the complex conjugate, and the right-hand side of (4) is proportional to . The -matrix can be expanded in a perturbation series in powers of where is the incident wavenumber.…”

“…Since the numerical results presented in the next section are for 1-D surfaces, the remaining equations are given for 1-D surfaces with and . Equation (13) becomes (14) The scattering cross section for 1-D surfaces is given by [17] (15)…”