# A New Treatment of Rough Surface Scattering

## Author

**C. Macaskill** and **P. Cao**

## Status

Research Report 96-19

To appear in Proc. Roy. Soc. London, Ser. A

Date: 31 July 1995, revised 18 January 1996

## Abstract

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

## Content

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