A New Treatment of Rough Surface Scattering


C. Macaskill and P. Cao


Research Report 96-19
To appear in Proc. Roy. Soc. London, Ser. A
Date: 31 July 1995, revised 18 January 1996


This paper treats the problem of the scattering of a narrow beam at a randomly rough surface where the height distribution is Gaussian and the correlation of heights is Gaussian. The field at the surface is assumed to satisfy a Dirichlet boundary condition.

The problem is formulated using the standard integral equation approach. In the case when the rms height is very large, numerical solution of this equation for a realization of the surface involves the solution of a very large matrix problem, particularly if the beam is scattered to other parts of the surface that are at a considerable distance from the initial illumination.

Here we provide a straightforward technique, motivated by ray theory, that allows only those parts of the surface that are actually illuminated at any stage to be treated. Thus for the first reflection of the beam one solves an integral equation over the initially illuminated area. This solution is used to determine the parts of the surface which give rise to doubly scattered waves, and one then solves the integral equation over the newly illuminated area, and so on, until all reflections have been accounted for. This procedure can be carried out with no explicit reference to any ray model, working simply with the discrete matrix form of the integral equation. In particular, for 1-D surface variation, a strategy of successive sweeps from left to right and back again (for a beam incident from the left) is shown to be successful. Any standard iterative or direct solution technique can be used for the sequence of sub-problems so introduced. The general method is also applicable to scattering from surfaces varying in two dimensions.

Key phrases

rough surface scattering. random surfaces. integral equations. iterative techniques.

AMS Subject Classification (1991)

Primary: 78A45
Secondary: 45A05, 78-08


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