John A. DeSanto

I Analytical Techniques for Multiple Scattering from Rough Surfaces

Publisher Summary This chapter describes analytical techniques for multiple scattering from rough surfaces. All real surfaces are rough. The degree of roughness depends on both the geometry and the wavelength of the incident probe. The types of rough surfaces are generally classified into periodic rough surfaces such as diffraction gratings and nonperiodic surface variability, which includes random rough surfaces. The chapter discusses theoretical treatment of the latter class of surfaces, although, as can be seen, much of the development is in terms of a stochastic surface and stochastic equations, which in principle hold true for an arbitrary non-stochastic surface as well. In addition, it comprises of a mathematically oriented review of rough surface scattering theory. The basic concepts of vertical and horizontal scales of roughness are introduced and a very succinct review of the kinds of surface statistics required in subsequent analyses is provided. The chapter also presents a brief discussion of other recent multiple scattering approaches.

The reconstruction of shallow rough-surface profiles from scattered field data

A method for the reconstruction of a rough-surface profile s(x), from fixed frequency and fixed illumination angle data and assuming Dirichlet boundary conditions is described. The method is valid for surfaces of small roughness only. The resulting algorithm is FFT based and very simple. It is illustrated by numerical examples.

On the derivation of boundary integral equations for scattering by an infinite one-dimensional rough surface

A crucial ingredient in the formulation of boundary-value problems for acoustic scattering of time-harmonic waves is the radiation condition. This is well understood when the scatterer is a bounded obstacle. For plane-wave scattering by an infinite, rough, impenetrable surface S, the physics of the problem suggests that all scattered waves must travel away from (or along) the surface. This condition is used, together with Green’s theorem and the free-space Green’s function, to derive boundary integral equations over S. This requires careful consideration of certain integrals over a large semicircle of radius r; it is known that these integrals vanish as r→∞ if the scattered field satisfies the Sommerfeld radiation condition, but that is not the case here—reflected plane waves must be present. The integral equations obtained are Helmholtz integral equations; they must be modified for grazing incident waves. As such integral equations are often claimed to be exact, and are often used to generate benchmark n...

Exact spectral formalism for rough-surface scattering

The spectral amplitudes of the scattered and transmitted fields for plane-wave incidence on an arbitrary rough interface in one dimension are derived exactly and simply by using Green’s theorem. Results are stated in terms of integrals on values of the field and its normal derivative on the interface. The energy constraint is derived, and individual energy contributions in each region are also related to the surface-field values. The latter contributions can be calculated from coupled linear equations that are also derived using Green’s theorem. The interface separates media of different but constant densities and sound speeds (acoustics) or different dielectrics (electromagnetics).

Scattering from a sinusoid: derivation of linear equations for the field amplitudes

The problem of constructing the scattering field for plane‐wave incidence on a sinusoidal surface is considered. Above the highest excursion of the surface the scattered field is written in the usual way as a superposition of upgoing and decaying waves. Linear equations are derived for the coefficients of these waves. Both soft and hard boundaries are considered, as well as arbitrary incidence angle. The equations for the soft boundary condition are similar to those of Uretsky, but the derivation is simpler. The full results for the hard boundary condition are apparently new.

Rough surface scattering

Abstract A summary of the theoretical and computational approaches to rough surface scattering is presented. For the Dirichlet problem new computational results are presented for the behaviour of the normal derivative of the field on the surface as well as the behaviour of coherent and incoherent intensities as a function of angle and surface height. Examples are given for surface reconstruction using scattered data as a function of surface height and as the scattered data window is narrowed.

Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces

Abstract We discuss the scattering of acoustic or electromagnetic waves from one-dimensional rough surfaces. We restrict the discussion in this report to perfectly reflecting Dirichlet surfaces (TE polarization). The theoretical development is for both infinite and periodic surfaces, the latter equations being derived from the former. We include both derivations for completeness of notation. Several theoretical developments are presented. They are characterized by integral equation solutions for the surface current or normal derivative of the total field. All the equations are discretized to a matrix system and further characterized by the sampling of the rows and columns of the matrix which is accomplished in either coordinate space (C) or spectral space (S). The standard equations are referred to here as CC equations of either the first (CC1) or second kind (CC2). Mixed representation, or SC-type, equations are solved as well as SS equations fully in spectral space. Computational results are presented f...

On angular-spectrum representations for scattering by infinite rough surfaces

A plane acoustic wave insonifies an infinite rough surface. The reflected field is written as an angular-spectrum representation (plane-wave expansion), with an unknown amplitude function A. It is pointed out that A must be considered as a generalized function, and not as a continuous function. Various decompositions of A are suggested and analysed. Energy considerations lead to relations between the coefficients in these decompositions, generalizing some known results for scattering by periodic surfaces (gratings). It is shown that the reflected field must include at least one propagating plane wave.

Scattering by Rough Surfaces

Publisher Summary This chapter describes scattering by rough surfaces. All real surfaces are rough. Many length scales of the surface itself and the incident radiation characterize the roughness. General scaling parameters include the wavelength (λ) of the interrogating radiation, a vertical surface scale of height or rms height (h), one or two horizontal scales (L), surface slopes, curvatures, spectral parameters of the surface and a considerable range of parameters used by manufacturers to characterize specific surfaces and their finishes. These can include an average surface wavelength, various shape parameters such as ripples, average distances among zero crossings of the surface or surface extrema, polishing marks on surfaces, and so on. Rough surface scattering include acoustic and electromagnetic scattering from oceans, ice and terrain, atom surface scattering and scattering in computer chip micro topography. The scattering can be direct (given the topography, find the scattered field) or inverse (given some scattered field data reconstruct the surface or find some surface parameter of interest). A method of scattering based on a perturbation of the Greens function is used. The method reduces the propagator in the integral equation kernel in such a way that a convolution equation results. Finally, a recent application on scattering theories to oceanic scattering using modulated wave train surfaces has been used successfully to explain the phenomenon of sea spikes where the HH-to-VV polarization ratios exceed unity.

On the derivation of boundary integral equations for scattering by an infinite two-dimensional rough surface

A plane acoustic wave is incident upon an infinite, rough, impenetrable surface S. The aim is to find the scattered field by deriving a boundary integral equation over S, using Green’s theorem and the free-space Green’s function. This requires careful consideration of certain integrals over a large hemisphere of radius r; it is known that these integrals vanish as r→∞ if the scattered field satisfies the Sommerfeld radiation condition, but that is not the case here—reflected plane waves must be present. It is shown that the well-known Helmholtz integral equation is not valid in all circumstances. For example, it is not valid when the scattered field includes plane waves propagating away from S along the axis of the hemisphere. In particular, it is not valid for the simplest possible problem of a plane wave at normal incidence to an infinite flat plane. Some suggestions for modified integral equations are discussed.