The region of validity of perturbation theory

Abstract

We present a study of the region of validity of perturbation theory applied to rough surface scattering. We solve numerically the case of a periodic surface or grating varying in one dimension. For a statistical ensemble of gratings with a sufficiently long period one may obtain a good approximation of rough surface scattering. We use this to test the validity of perturbation theory.

Only the perfect conductor case was considered. We find that as the grazing angle becomes small the perturbation result for the TE (E horizontal) polarization remains valid, while for the TM (E vertical) polarization it breaks down. The results show that the perturbation results should be used carefully when being compared with experimental data at grazing angles.     

An iterative method for scattering from rough dielectric or conducting interfaces

We introduces an iterative method for scattering a two-dimensional scalar wave from a rough interface between two media. The method is applicable to the case of electromagnetic scattering from a rough metallic or dielectric surface that varies only in one dimension. The first iteration is equivalent to the Kirchhoff approximation, and the series converges in one step for a flat surface. We discuss the conditions for convergence, and find that they are similar to those which Meecham showed to be necessary in the Dirichlet case.     

Bistatic scattering and depolarization by randomly rough surfaces: application to the natural rough surfaces in X-band

Abstract

The scattering of waves by random rough surfaces has important applications in the remote sensing of oceans and land. The problem of developing a model for rough surfaces is very difficult since, at best, the scattering coefficient σ⁰ is dependent upon (at least) the radar frequency, geometrical and physical parameters, incident and observation angles, and polarization. The problem of electromagnetic scattering from a randomly rough surface is analysed using the Kirchhoff approximation (stationary phase, scalar approximation), the small-perturbation model and the two-scale models. A first major new consideration in this paper is the polarimetric signature calculations as a function of the transmitter location and receiver location for a bistatic radio-link. We calculate the like- and cross-polarized received power directly using the scattering coefficients, without calculating the Mueller matrix. Next, a study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models (in the bistatic case) is presented. Comparisons between the numerical calculations and the models are made for various surface rms heights and correlation lengths both normalized to the incident wavenumber (denoted by σ and L, respectively). By using these two parameters to form a two-dimensional space, the approximate regions of validity are then established. The second major new consideration is the development of a theoretical two-scale model describing bistatic reflectivity as well as the numerical results computed for the bistatic radar cross section from rough surfaces especially from the sea and snow-covered surfaces. The results are used to show the Brewster angle effect on near-grazing angle scattering.     

A heuristic model approximation for scattering from perfectly conducting one-dimensional random rough surfaces

Abstract

We propose a model for scattering from one-dimensional, perfectly conducting, slightly rough surfaces. A possible method for solving the scattering equations is examined which, with some assumptions, suggests the final result. The approximation is relatively simple and is comparable in computational effort with most first-order theories. We compare the bistatic scattering cross section for TE waves predicted by the present model for Gaussian randomly rough surfaces with numerical simulations and with some first-order theories. The comparison shows that the model is remarkably accurate for slightly rough surfaces and TE polarization.     

Travel-time statistics for signals scattered at a rough surface

Abstract

The travel time of signals reflected or refracted by a rough surface is investigated in the geometrical optics approximation. It is shown that surface roughness typically decreases the mean travel time in the case of large-scale roughness, when only one specularly reflecting point moves randomly around its unperturbed position, resulting in a negative travel-time bias (toward early echoes). In the opposite limiting case of multipath propagation, when many specular points exist on a random surface, the travel-time bias is always positive. General results are illustrated by two examples related to ocean remote sensing which involve sound scattering from the ocean surface and bottom.     

Acoustic scattering from a rough sea surface: the mean field by the integral equation method

Abstract

The scattering of an acoustic signal incident from below at low angles on a rough sea surface is treated by the integral equation method in the parabolic approximation. Equations are obtained allowing the mean scattered field to be calculated even when the surface causes a large phase modulation in the incident wave. Solutions are found using the method of Laplace transforms and some results are presented for a specific type of rough surface.     

Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach

Abstract

It is known that the angular correlation function of scattered fields from a randomly rough surface has a conspicuous enhancement on the memory line, and becomes much smaller away from the line. The finite-element method is employed to solve the problem of scattering from a scatter target situated over a randomly rough surface. Numerical results show that the width of the angular correlation function becomes broadened owing to the presence of the target. It can be used to detect the presence of a target placed over the rough surface.     

Extension of the local curvature approximation to third order and improved tilt invariance

The second-order local curvature approximation (LCA2) is a theory of rough surface scattering that reproduces fundamental low and high frequency limits in a tilted frame of reference. Although the existing LCA2 model provides agreement with the first order small perturbation method up to the first order in surface tilt, results reported in this paper produce a new formulation of the model that achieves consistency with perturbation theory to first order in surface height and arbitrary order in surface tilt. In addition, extension of the modified LCA to third order is presented, and allows the theory to match the second-order small perturbation method to arbitrary order in surface tilt. Crucial to the development of the theory are a set of identities involving relationships among the small perturbation method (i.e. low frequency) and Kirchhoff approximation (i.e. high frequency) kernels; a set of new identities obtained in our derivations is also presented. Sample results involving 3D electromagnetic scattering from penetrable rough surfaces, as well as 2D scattering from Dirichlet sinusoidal gratings, are provided to compare the new results with the existing LCA2 model and with other rough surface scattering theories.     

Bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase and scalar approximation with shadowing effect: comparisons with experiments and application to the sea surface

Abstract

In this paper, the bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase method and scalar approximation with shadowing effect is investigated. Both of these approaches use the Kirchhoff integral. With the stationary phase, the bistatic cross section is formulated in terms of the surface height joint characteristic function where the shadowing effect is investigated. In the case of the scalar approximation, the scattering function is computed from the previous characteristic function and in terms of expected values for the integrations over the slopes, where the shadowing effect is analysed analytically. Both of these formulations are compared with experimental data obtained from a Gaussian one-dimensional randomly rough perfectly-conducting surface. With the stationary-phase method, the results are applied to a two-dimensional sea surface.     

A coordinate–spectral method for rough surface scattering

Abstract

In this paper, we deal with the time-harmonic scattering by one-dimensional rough surfaces separating two homogeneous and isotropic media. The method is based on a rigorous integral formalism. The unknown of the integral equation is projected onto a Fourier basis while the equation itself is sampled as in a classical method of moments. The accuracy is tested against both other methods and experimental results. One of the main interests in choosing a Fourier basis lies in the ability to solve rigorously the scattering of a p polarized incident beam by a shallow metallic rough surface. The role of the surface waves is accurately taken into account and phenomena such as enhanced backscattering are well described. With this method, one can consider that the gap between the domain of validity of perturbation theories and the domain of practical use of rigorous methods is filled.     

A numerical evaluation of Rayleigh's theory applied to scattering by randomly rough dielectric surfaces

Abstract

We present a numerical simulation of scattering by one-dimensional randomly rough surfaces. It is based on the use of plane-wave expansions to describe the Melds on the surface (i.e. Rayleigh hypothesis). Accuracy and convergence properties of two different numerical implementations are studied. Some examples of results for a dielectric and a metallic Gaussian rough surface are shown to be in good agreement with calculations by a rigorous numerical method. The Rayleigh method appears to be a fast computation tool for dielectric surfaces with slopes of less than 0.2.     

Dispersion and damping of a surface plasmon polariton on a one-dimensional randomly rough metal surface

Abstract

By the use of the reduced Rayleigh equation for the amplitude of a surface plasmon polariton on a one-dimensional randomly rough metal surface that is in contact with vacuum, we calculate the dispersion and damping of the surface electromagnetic wave to the lowest nonzero order in the rms height of the surface. It is found that the frequency of the surface plasmon polariton is depressed by the surface roughness. The attenuation of the surface plasmon polariton in the long wavelength limit is due primarily to its scattering into other surface plasmon polaritons, while in the short wavelength limit it is due primarily to its roughness-induced scattering into volume electromagnetic waves in the vacuum. The energy mean free path of the surface plasmon polariton is shorter on a randomly rough metal surface than it is on a lossy planar metal surface, and the surface plasmon polariton is more tightly bound to a rough surface than to a planar one.     

Enhanced backscattering and blazing effect from gratings,quasi-gratings and randomly rough surfaces

Abstract

The blazing effect is probably the most important property of diffraction gratings used for spectroscopic purposes. On the other hand, the enhanced backscattering phenomenon has been generally studied in the framework of scattering from randomly rough surfaces. Using numerical results from rigorous theories, it will be shown that these phenomena, which have very different origins, should have more precise definitions. In a special case of a randomly rough surface formed by random corners, it will be shown that the effects of these phenomena are sometimes very difficult to distinguish.     

Universal behaviour of scattering amplitudes for scattering from a plane in an average rough surface for small grazing angles

Abstract

It is shown that for scattering from a plane in an average rough surface, the scattering cross section of the range of small grazing angles of the scattered wave demonstrates a universal behaviour. If the angle of incidence is fixed (in general it should not be small), the diffusive component of the scattering cross section for the Dirichlet problem is proportional to θ² where θ is the (small) angle of elevation, and for the Neumann problem it does not depend on θ. For the backscattering case these dependences correspondingly become θ⁴ and θ°. The result is obtained from the structure of the equations that determine the scattering problem rather than by use of an approximation.     

Comment on ‘X-ray scattering from a randomly rough surface’

Abstract

Recently, in this journal, Leskova and Maradudin published a new method for calculating x-ray scattering from a rough surface. In this comment their results will be compared with those obtained by other methods, especially the distorted-wave Born approximation. It is concluded that their results are useful in the limit of small transverse correlation length of the surface roughness. For intermediate values of the correlation length the results of both approximations are equivalent, being valid for small root-mean-square roughness and/or large incident perpendicular wavevector. In the case of large correlation length, neither approximation is correct. It is noted that the new method includes cross-polarization effects.     

An iterative solution of the rough-surface scattering problem

Abstract

An iterative solution to the problem of scattering from a one-dimensional rough surface is obtained for the Dirichlet boundary condition. The advantages of this method are that bounds for convergence of the solution can be established and that the solution may readily be iterated to sufficiently high order in the interaction to examine the rate at which it converges. Absolute convergence of the iterative solution is also a sufficient condition for the convergence of the operator expansion method for surfaces on which the slope is everywhere less than unity. A numerical example of scattering from an echelette grating is considered, and bounds for convergence established. It is found that for scattering from such surfaces the rate at which the iterative solution converges decreases as the surface slope is increased. Corresponding results are found for the operator expansion method.     

Wave diffraction by rough interfaces in an arbitrary plane-layered medium

Abstract

The problem of electromagnetic wave scattering by a slightly rough interface in an arbitrarily layered medium is solved by a small-perturbation method. The bistatic amplitude of scattering as well as the scattering cross sections for statistically rough surfaces are calculated for linear polarized waves. Along with scattering into up-going waves in a homogeneous medium and scattering cross sections in down-going waves into a layered medium, scattering amplitudes from a rough interface in the arbitrarily layered medium are obtained.     

Numerical simulation methods for rough surface scattering

Abstract

Numerical methods are of great importance in the study of electromagnetic scattering from random rough surfaces. This review provides an overview of rough surface scattering and application areas of current interest, and surveys research in numerical simulation methods for both one- and two-dimensional surfaces. Approaches considered include numerical methods based on analytical scattering approximations, differential equation methods and surface integral equation methods. Emphasis is placed on recent advances such as rapidly converging iterative solvers for rough surface problems and fast methods for increasing the computational efficiency of integral equation solvers.     

Rigorous solutions for electromagnetic scattering from rough surfaces

Abstract

First, the rough surface scattering problem is formulated from a statistical point of view. Then, different numerical schemes that permit one to solve Maxwell equations without approximation are presented for the three-dimensional scattering problem. Particular attention is paid to boundary integral methods and to the numerical techniques developed to handle large linear systems when short-range interactions dominate. Lastly, several important connected issues that require further numerical and theoretical improvements are discussed.