Gary S. Brown

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.

Issues related to the use of a Gaussian-like incident field for low-grazing-angle scattering

We study the suitability of a tapered plane-wave incident field, using both the Gaussian and the more advanced Thorsos tapers for low-grazing-angle rough-surface scattering problems as well as the problem of propagation in the presence of a rough surface. For surface scattering problems it is known that as the angle of incidence approaches grazing incidence the tapered beam waist should be made larger; several criteria relating these two parameters have been proposed for both the Gaussian and the Thorsos tapers. Our two-dimensional scattering simulations with the oceanlike Pierson–Moskowitz surfaces show that when the width of the Gaussian or the Thorsos taper is fixed, the backscatter cross section for TE polarization is characterized by a distinctive and consistent anomalous jump as grazing incidence is approached. This observation has led to a refined version of one of the above-mentioned beam waist–angle of incidence criteria and its robustness is demonstrated. The approximate (non-Maxwellian) nature of the Thorsos–Gaussian taper also becomes evident in over-surface-propagation simulations with use of the boundary integral equation method. A certain inconsistency was observed between the surface field that we obtained by first defining the Thorsos–Gaussian-tapered field on a vertical plane and then propagating it to the surface and that obtained by defining the same tapered field directly on the surface. This effect, not previously appreciated, may be of importance when the rough-surface effects are rigorously incorporated into the propagation problem. We conclude with a detailed derivation of the Thorsos taper that points out all the approximations involved in it and the resulting limitations.

Quasi-Specular Scattering from the Air-Sea Interface

Scattering of electromagnetic energy by an arbitrarily roughened surface comprises a very difficult boundary value problem (Beckmann and Spizzichino, 1963; Bass and Fuch, 1978; Desanto and Brown, 1986). When the surface shape changes with both time and space, this further complicates matters. Even describing the scattering as a random process does not simplify the analysis too much and the use of numerical methods has not proven to be very useful either. This state of affairs leads to a great deal of emphasis on approximate analysis methods to yield both quantitative predictions and physical insight (Bass, et al., 1968; Valenzuela, 1978; Brown, 1985). Chapters 11 and 12 both illustrate approximate scattering methodologies and this chapter comprises yet another, namely the quasi-specular scattering approximation.

A comparison of approximate theories for scattering from rough surfaces

Abstract Scattering from a perfectly conducting rough interface is developed in terms of a generalized angular spectrum of plane waves which includes both upward and downward traveling waves. This spectral formulation is used to examine the interrelationships between three classical approximate techniques in rough surface scattering, i.e., physical optics, boundary perturbation, and extinction theorem. These methods are also traced to conventional deterministic scattering analysis techniques. In particular, physical optics is related to the Magnetic Field Integral Equation, boundary perturbation is shown to result from the Electric Field Integral Equation subsequent to invoking the Rayleigh Hypothesis, and the extinction theorem is equivalent to the Extended Boundary Condition Method. The capabilities and limitations of these rough surface approximate techniques are developed using the angular spectrum approach, and they are discussed in detail.

Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces

Abstract This paper consists of two parts. In the first part, the solution of the Helmholtz equation under forward-scattering or propagation conditions is sought as a uniform asymptotic perturbation expansion using the method of multiple scales. It is then shown that the parabolic wave equation (PWE) solution is the zeroth-order term in this expansion. In the second part, the electric-field integral equation and the magnetic-field integral equation, derived under the PWE approximation, are solved for surface currents induced on a sinusoidal surface. The scattered fields produced by these currents are then calculated using the appropriate radiation integrals. Results are compared to those obtained using the method of ordered multiple interactions developed by Kapp and Brown.

A scattering result for rough surfaces having small height but arbitrary slope

Abstract An integral equation for the fluctuating part of the field scattered by a perfectly conducting randomly rough surface is derived. The Born term in this equation contains the Kirchhoff approximation and a new factor. This Born term is valid as long as the fluctuating field is small compared to the average scattered field. Translation of this field constraint to conditions on the rough surface shows that the new Born term will be accurate as long as the surface has a small surface height irrespective of the surface slopes, curvatures, etc. In the limit of small surface slopes, this result is shown to reduce to the classical Rice surface perturbation result. Applications of this new result are discussed.

A new approach to solving second kind integral equations

Many problems in applied electromagnetics can be formulated in terms of solving a boundary integral equation of the second kind. Much success has been achieved in solving these type equations through the Method of Moments (MOM) or related approaches by which the integral equation is reduced to solving a matrix equation. However, for problems involving long range interactions such as rough surface scattering, propagation over irregular terrain, or even large complicated structures, the MOM leads to intractable matrices. These problems have been overcome for two dimensional geometries through methods such as Forward-Backward or the Method of Ordered Multiple Interactions but the extension to three dimensions is still prohibitive. For this reason, there remains a need for a new approach to solving integral equations of the second kind.

Applying MOMI to a dielectric foliage layer above ground for propagation prediction

Past work has shown that solutions of the magnetic field integral equation (MFIE) or the combined field integral equations (CFIE) through the method of ordered multiple interactions (MOMI) can be used to provide predictions of irregular terrain loss for point-to-point links. This method uses the terrain profile along the point-to-point path to solve the two dimensional scattering problem. However, in comparison to measured data, paths with heavy foliage obstruction have not fit the predicted results very well. Usually, the predicted loss is less severe than that measured. However, diffraction and integral equation based models cannot easily include most of the statistically oriented volume scattering models for foliage. One way that foliage can be included in the model is to put a lossy layer over top of the ground at the foliage location. It is understood that such a model will not capture the time varying multiple scattering effects of foliage, but it can provide the basic mean or average attenuation and diffraction by the foliage.

UTD analysis of foliage penetration for use in propagation modeling

One of the key challenges in propagation modeling for point-to-point links is including the effects of foliage. Foliage is a volume scatterer and much of the work that has been done on modeling foliage is statistical in manner. However, such statistical models often do not fit easily into existing propagation models. Our past propagation modeling work has used MOMI to solve the scattering from a perfectly conducting or dielectric ground, and volume scattering media do not fit well into the two dimensional integral equation method. Similarly, such volumetric statistical models would not be a good fit for diffraction based propagation models. Thus, we sought a simple continuous medium which might adequately, though not perfectly, match the foliage layer attenuation and reflection.