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

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 for scattering from various periodic surfaces. The results include examples with grazing incidence, a very rough surface and a highly oscillatory surface. The examples vary over a parameter set which includes the geometrical optics regime, physical optics or resonance regime, and a renormalization regime.

The objective of this study was to determine the best computational method for these problems. Briefly, the SC method was the fastest, but it did not converge for large slopes or very rough surfaces for reasons we explain. The SS method was slower and had the same convergence difficulties as SC. The CC methods were extremely slow but always converged. The simplest approach is to try the SC method first. Convergence, when the method works, is very fast. If convergence does not occur with SC, then SS should be used, and failing that CC.     

Theoretical and computational aspects of scattering from periodic surfaces: one-dimensional transmission interface

Abstract

We consider the scattering from and transmission through a one-dimensional periodic surface. For this problem, the electromagnetic cases of TE and TM polarization reduce to the scalar acoustic examples. Three different theoretical and computational methods are described, all involving the solution of integral equations and their resulting discrete matrix system of equations for the boundary unknowns. They are characterized by two sample spaces for their discrete solution, coordinate space and spectral space, and labelled by the sampling of the rows and columns of the discretized matrices. They are coordinate-coordinate (CC), the usual coordinate-space method, spectral-coordinate (SC) where the matrix rows are discretized or sampled in spectral space and spectral-spectral (SS) where both rows and columns are sampled in spectral space. The SS method uses a new topological basis expansion for the boundary unknowns.

Equations are derived for infinite surfaces, then specialized and solved for periodic surfaces. Computational results are presented for the transmission problem as a function of roughness, near-grazing incidence as well as many other angles, density and wavenumber ratios. Matrix condition numbers and different sampling methods are considered. An error criterion is used to gauge the validity of the results.

The computational results indicated that the SC method was by far the fastest (by several orders of magnitude), but that it became ill-conditioned for very rough surfaces. The CC method was most reliable, but often required very large matrices and was consequently extremely slow. It is shown that the SS method is computationally efficient and accurate at near-grazing incidence and can be used to fill a gap in the literature. Extensive computational results indicate that both SC and SS are highly robust computational methods. Spectral-based methods thus provide viable computational schemes to study periodic surface scattering.     

Theoretical and computational aspects of scattering from periodic surfaces: one-dimensional transmission interface

We consider the scattering from and transmission through a one-dimensional periodic surface. For this problem, the electromagnetic cases of TE and TM polarization reduce to the scalar acoustic examples. Three different theoretical and computational methods are described, all involving the solution of integral equations and their resulting discrete matrix system of equations for the boundary unknowns. They are characterized by two sample spaces for their discrete solution, coordinate space and spectral space, and labelled by the sampling of the rows and columns of the discretized matrices. They are coordinate-coordinate (CC), the usual coordinate-space method, spectral-coordinate (SC) where the matrix rows are discretized or sampled in spectral space and spectral-spectral (SS) where both rows and columns are sampled in spectral space. The SS method uses a new topological basis expansion for the boundary unknowns.

Equations are derived for infinite surfaces, then specialized and solved for periodic surfaces. Computational results are presented for the transmission problem as a function of roughness, near-grazing incidence as well as many other angles, density and wavenumber ratios. Matrix condition numbers and different sampling methods are considered. An error criterion is used to gauge the validity of the results.

The computational results indicated that the SC method was by far the fastest (by several orders of magnitude), but that it became ill-conditioned for very rough surfaces. The CC method was most reliable, but often required very large matrices and was consequently extremely slow. It is shown that the SS method is computationally efficient and accurate at near-grazing incidence and can be used to fill a gap in the literature. Extensive computational results indicate that both SC and SS are highly robust computational methods. Spectral-based methods thus provide viable computational schemes to study periodic surface scattering.     

Theoretical and computational aspects of scattering from periodic surfaces: two-dimensional perfectly reflecting surfaces using the spectral-coordinate method

We consider the scattering from a two-dimensional periodic surface. From our previous work on scattering from one-dimensional surfaces (1998 Waves Random Media 8 385) we have learned that the spectral-coordinate (SC) method was the fastest method we have available. Most computational studies of scattering from two-dimensional surfaces require a large memory and a long calculation time unless some approximations are used in the theoretical development. By using the SC method here we are able to solve exact theoretical equations with a minimum of calculation time.

We first derive in detail (part I) the SC equations for scattering from two-dimensional infinite surfaces. Equations for the boundary unknowns (surface field and/or its normal derivative) result as well as an equation to evaluate the scattered field once we have solved for the boundary unknowns. Special cases for the perfectly reflecting Dirichlet and Neumann boundary value problems are presented as is the flux-conservation relation.

The equations are reduced to those for a two-dimensional periodic surface in part II and we discuss the numerical methods for their solution. The two-dimensional coordinate and spectral samples are arranged in one-dimensional strings in order to define the matrix system to be solved.

The SC equations for the two-dimensional periodic surfaces are solved in part III. Computations are performed for both Dirichlet and Neumann problems for various periodic sinusoidal surface examples. The surfaces vary in roughness as well as period and are investigated when the incident field is far from grazing incidence ('no grazing') and when it is near-grazing. Extensive computations are included in terms of the maximum roughness slope which can be computed using the method with a fixed maximum error as a function of the azimuthal angle of incidence, the polar angle of incidence and the wavelength-to-period ratio.

The results show that the SC method is highly robust. This is demonstrated with extensive computations. Furthermore, the SC method is found to be computationally efficient and accurate for near-grazing incidence. Computations are presented for grazing angles as low as 0.01°. In general, we conclude that the SC method is a very fast, reliable and robust computational method to describe scattering from two-dimensional periodic surfaces. Its major limiting factor is high slopes and we quantify this limitation.     

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.     

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.     

Directional hemispherical radiative properties of random dielectric rough surfaces

In this paper the directional hemispherical reflectivity and transmissivity of one-dimensional, randomly rough, dielectric surfaces are determined by the use of the integral method. This method is derived from electromagnetic theory without any restrictive hypotheses. Since this exact approach is computationally very intensive, a geometric optics approximation method is also developed. Curves displaying radiative properties versus the correlation length for a constant mean square deviation of the surface from flatness are presented. In this respect, the influence on the validity of the approximate method of multiple scattering, the shadowing effect and the real index of refraction of the dielectric have been investigated. Transverse electric and transverse magnetic polarized incident plane waves are considered. For the latter, our interest is focused on the influence of roughness on the reflected and transmitted intensities for an angle of incidence close to the Brewster angle.     

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.     

Gaussian beam scattering from arbitrarily shaped objects with rough surfaces

Abstract

The scattered field of Gaussian beam scattering from arbitrarily shaped dielectric objects with rough surfaces is investigated for optical and infrared frequencies by using the plane wave spectrum method and the Kirchhoff approximation, and the formulae for the coherent and incoherent scattering cross sections are obtained theoretically based on geometrical optics and tangent plane approximations. The infrared laser scattering cross sections of a rough sphere are calculated at 1.06 μm, and the influence of the beam size is analysed numerically. It is shown that when the beam size is much larger than the size of the object, the results in this paper will be close to those of an incident plane wave.     

Radar scatter cross sections for two-dimensional random rough surfaces – full wave solutions and comparisons with experiments

Abstract

In this paper, the full wave expressions for the radar scattering cross sections for two-dimensional random rough surfaces are obtained. The rough-surface height/slope correlations are accounted for in this analysis. Analytical and numerical comparisons of the full wave solution with the small perturbation and physical optics solutions are made for isotropic, homogeneous random rough surfaces with Gaussian probability density function. The full wave results are also compared with experimental results.     

Modeling the nadiral low-frequency pulse return of one-dimensionally rough surfaces

Measurements of radar pulse return waveforms are known to provide information on properties of the observed surface, and are commonly used in oceanographic altimetry. In such applications, the convolution model (also called the “Brown model”) is widely applied for waveform analysis. This model describes pulse return waveforms as a multiple convolution of the “flat surface impulse response”, the surface's height probability density function, and the radar's point target response. The flat surface impulse response is typically determined by an integration of scattering contributions from incremental surface elements weighted by a geometrical optics (GO) prediction of the normalized radar cross-section of the elemental surfaces.

Recent interest in the analysis of pulse return waveforms at VHF and lower frequencies for ice sheet sensing applications motivate reconsideration of the convolution model. While the ultimate goal of this effort is the development of a model to be utilized for interpreting VHF radar measurements over ice sheets, it is important first to establish the validity of the convolution model for these applications. Such an investigation, which involves the comparison of convolution model predictions with those of a method that does not require a separation of surface length-scales into “elemental” and large-scale regions, is most easily performed for one-dimensional surfaces.

This paper describes a derivation of the convolution model for one-dimensionally rough surfaces that is applicable at low frequencies, primarily through the replacement of GO surface scattering coefficients with those of a physical optics theory. The method is validated by comparing its predictions with a Monte Carlo physical optics approach. Results show the convolution model to provide reasonable estimates of the pulse return waveform, so that a similar method can be utilized to develop a convolution model for two-dimensional surface pulse return waveforms in ice-sheet sensing applications. The results also suggest the possibility of retrieving surface profile statistical information from waveform measurements.     

New regime map of the geometric optics approximation for scattering from random rough surfaces

The electromagnetic wave scattering from random roughness surfaces is a technologically important but challenging problem. There is a significant amount of interest in understanding the radiative properties of the rough surfaces for diverse applications. On one end, the rigorous models require the solutions of complex formulations of the Maxwell's equations or rely on various numerical schemes, which typically are computationally intensive. On the other hand, it has been found that the geometric optics (GO) ray tracing approximation method produces reasonably accurate radiative property predictions in some cases and with little computational effort. However, the latter ignores the wave interference and polarization effects, which are important when the wavelength is on the same order or larger than the geometrical length scale. It is therefore important to quantify the accuracy of the GO approximation. This study reports a new regime map based on the comparisons of the GO and finite-difference time-domain solutions.     

Pulse broadening of enhanced backscattering from rough surfaces

Abstract

Recently, we presented a study of pulse scattering by rough surfaces based on the first-order Kirchhoff approximation which is applicable to rough surfaces with RMS slope less than 0.5 and correlation distance l?λ. However, there has been an increased interest in enhanced backscattering from rough surfaces, study of which requires inclusion of the second-order Kirchhoff approximation with shadowing corrections. This paper presents a theory for the two-frequency mutual coherence function in this region and shows that the multiple scattering on the surface gives rise to an additional pulse tail in the direction of enhanced backscattering. The theory predicts pulse broadening approximately 20% greater than that caused by single scattering alone for a delta-function incident pulse and typical surface parameters. Analytical results are compared with Monte Carlo simulations and millimetre-wave experiments for the one-dimensional rough surface with RMS height 1λ and correlation distance 1λ, showing good agreement.     

Numerical,analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement

Abstract

This paper Presents numerical simulations, theoretical analysis, and millimeter wave experiments for scattering from one-dimensional very rough surfaces. First, numerical simulations are used to investigate the effects of roughness spectrum, height variation, interface medium, polarization, and incident angle on the backscattering enhancement. The enhanced backscattering due to rough surface scattering is divided into two cases; the RMS height close to a wavelength and RMS slope close to unity, and RMS height much smaller than a wavelength with surface wave contributions. Results also show that the enhancement is sensitive to the roughness spectrum. Next, a theory based on the first- and second-order Kirchhoff approximation modified with angular and propagation shadowing is developed. The theoretical solutions provide a physical explanation of backscattering enhancement and agree well with the numerical results. In addition to the scattering of a monochromatic wave, the analytical results of the broadening and lateral spreading of a pulsed beam wave scattering from rough surfaces are also discussed. Finally, the existence of backscattering enhancement from one-dimensional very rough conducting surfaces with exact Gaussian statistics and Gaussian roughness spectrum is verified by a millimeter-wave experiment. Experimental results which show enhanced backscattering for both TE and TM polarizations for different angles of incidence are presented.     

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

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.     

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.     

Transmission scatter cross sections across two-dimensional random rough surfaces – full wave solutions and comparison with numerical results

Abstract

In this paper, the full wave expressions for the bistatic transmission scattering cross sections across two-dimensional random rough surfaces are obtained. The full wave analysis accounts for the surface height/slope correlations. Analytical and numerical comparisons of the full wave solutions with the small perturbation and physical optics solutions are made for isotropic random rough surfaces. The full wave results are also compared with the numerical results based on Monte Carlo simulations of one-dimensional random rough surfaces. Detailed consideration is given to illustrating the relationship between these full wave solutions and the original full wave solutions including the impact of accounting for the height/slope correlations in this analysis.     

Emissvity of metallic microcontoured surfaces

In this paper, we study the diffraction of electromagnetic radiation by a periodic micro-rough surface separating vacuum from a metal with a finite conductivity. We submit the integral method to the surface impedance boundary condition. Thus the numerical implementation is greatly reduced. We compare the numerical emissivities obtained by this approach to those we have calculated through the rigorous multilayer modal method. This enables us to show that the mentioned approximate method has two regions of validity: one corresponding to fairly flat surfaces and the other to very deep surfaces. It is well known that both the Kirchhoff approximation and the constant flat boundary impedance approximation are also valid for fairly flat surfaces. Our investigation aims also to establish whether these two approximate methods lead to the same results, and whether the integral method submitted to the surface impedance condition has a larger domain of validity. Concerning deep surfaces with a period smaller than the wavelength, we introduce the homogenization process in order to study its accuracy.Finally, this work permitted to identify three different regimes depending on the surface slopes: the simple scattering regime, the homogenization regime and the intermediate regime. For the latter, if the period is in the order of the wavelength, then we will show that the emissivity can be exalted.