Nonlocal Small Slope Approximation Research Articles

The SSA+ for scattering from dielectric surfaces at very low grazing angles

Previously we developed a practical model for scattering from randomly-rough surfaces at very low grazing angles for the Dirichlet problem which was found to give good numerical results. In this paper, we derive the expression for the bistatic scattering cross-section for the non-local small slope approximation for dielectric interfaces. We then extend our practical model to dielectric surfaces based on this result. We discuss numerical results for scattering at low forward grazing angles for a Gaussian roughness spectrum with an angle of incidence of 80○.

The small slope approximation<th>+<th>(S S A+) for rough surface scattering

To be practical, a rough surface scattering model must be easy to implement and easily incorporated into existing models. At the same time, it must be accurate enough to give useful results. Here ‘‘practical’’ is defined to mean (1) no more than N-D integration away from low grazing angles, where N represents the dimension of the surface, (2) no more than 2N-D integration at low grazing angles, and (3) accuracy to within a few dB. In previous work it was shown that at low forward grazing angles an approximation to the nonlocal small slope approximation (NLSSA) satisfies the first and third criteria but not the second. It was also shown that the NLSSA cross section can be written as the sum of the lowest-order SSA cross section and an additional term, the latter of which can be considered a correction term for the lowest-order SSA. In this paper the SSA+, which is based on the NLSSA but includes three approximations to the correction term, is presented. The SSA+ satisfies all three criteria for a ‘‘practical’’ scattering model at low forward grazing angles. Numerical results are presented. [Work supported by ONR.]

Two families of non-local scattering models and the weighted curvature approximation

There are several nonlocal scattering models available in the literature. Most of them are given with little or no mention of their expected accuracy. Moreover, high- and low-frequency limits are rarely tested. The most important limits are the low-frequency or the small perturbation method (SPM) and the high-frequency Kirchhoff approximation (KA) or the geometric optics (GO). We are interested in providing some insight into two families of non-local scattering models. The first family of models is based on the Meecham–Lysanov ansatz (MLA). This ansatz includes the non-local small slope approximation (NLSSA) by Voronovich and the operator expansion method by Milder (OEM). A quick review of this first family of models is given along with a novel derivation of a series of kernels which extend the existing models to include some more fundamental properties and limits. The second family is derived from formal iterations of geometric optics which we call the ray tracing ansatz (RTA). For this family we consider two possible kernels. The first is obtained from iteration of the high-frequency Kirchhoff approximation, while the second is an iteration of the weighted curvature approximation (WCA). In the latter case we find that most of the required limits and fundamental conditions are fulfilled, including tilt invariance and reciprocity. A study of scattering from Dirichlet sinusoidal gratings is then provided to further illustrate the performance of the models considered.

Toward a practical cross section for rough surface scattering

The lowest-order small slope approximation (SSA) provides a practical means of obtaining the scattering strength for rough surfaces. While results are accurate over a wide range of scattering angles, as surface slopes increase and for forward scattering beyond the specular angle, even higher-order SSA results become inaccurate. This is due, in part, to the lack of nonlocal interactions in the SSA. Voronovich introduced the nonlocal small slope approximation (NLSSA) as a generalization of the SSA to explicitly include nonlocal interactions. It has been shown that the NLSSA is more accurate than the SSA both for larger surface slopes and at low forward grazing angles. However, the computational cost precludes use of the NLSSA in practice. The NLSSA expression for the scattering strength can be separated into two terms. The first term is identical to the lowest-order SSA and, hence, the second term of the NLSSA can be considered a correction to the lowest-order SSA. In this paper, we make an approximation to the second term at forward scattering angles that reduces it to a simple algebraic expression. We present numerical results and discuss the region of validity for the reduced form of the equation. [Work supported by ONR.]

Radar images of rough surface scattering: comparison of numerical and analytical models

Rough surface scattering theories are investigated through analysis of radar images. Backscatter results from 10 GHz to 14 GHz under tapered wave illumination are considered for one-dimension (1D) random rough surface realizations which satisfy an impedance boundary condition. Back-projection tomography is applied to form two-dimensional (2D) synthetic aperture radar images from deterministic surface scattered field data at multiple incidence angles and frequencies. Numerical predictions of surface backscattered fields are obtained from an accelerated forward-backward (FB) method and the resulting images are compared with those obtained from approximate scattering theories such as the physical optics (PO) approximation, the small slope approximation (SSA), and the nonlocal SSA (NLSSA). The resulting radar images illustrate scattering sources associated with single and multiple scattering on the boundary, and a ray tracing analysis confirms the locations of time-delayed image points due to double reflections. For single scattering effects, the images demonstrate excellent agreement between analytical and numerical methods in both horizontal and vertical polarizations. For surfaces with RMS height 2.0 cm and correlation length 7.5 cm at normal incidence, multiple-scattering effects are observed and successfully captured when the lowest-order NLSSA is employed.

Numerical results for an approximate form of the nonlocal small slope approximation scattering strength

Voronovich introduced the nonlocal small slope approximation (NLSSA) as a generalization of the small slope approximation to explicitly include nonlocal interactions. He showed that the NLSSA generally accounts for double scattering in the high-frequency limit. Broschat and Thorsos presented numerical results for the lowest-order NLSSA scattering strength for two-dimensional pressure-release surfaces. Their results agreed well with Monte Carlo integral equation results and, in particular, were better than the higher-order SSA results at low forward grazing angles. However, the computational cost was extremely high, and results were unobtainable at low grazing angles. In this paper, we discuss the results obtained by making an ad hoc approximation to the lowest-order NLSSA scattering cross section that reduces the computational complexity of the integration substantially. In addition, we discuss the difficulties of the numerics and how they are handled. Numerical results for the scattering strength are presented for 2-D pressure-release surfaces. For the cases presented, the NLSSA and its approximate form give virtually the same results. Also, results at low grazing angles are obtainable and accurate with the approximate form. [Work supported by ONR.]

Reflection loss from a "Pierson-Moskowitz" sea surface using the nonlocal small slope approximation

The lowest-order nonlocal small slope approximation (NLSSA) reflection coefficient is derived, and numerical results are presented for one-dimensional (1D) surfaces, satisfying the Dirichlet boundary condition. Reduction to the perturbation expression occurs in the small height limit, and the first term of the reflection coefficient gives the Kirchhoff approximation (KA) result. Numerical results for the reflection loss at low grazing angles using a Pierson-Moskowitz spectrum are compared with those of other approximate methods as well as with exact integral equation (IE) results. For those cases when exact results are available, the NLSSA is found to give accurate results, comparable to those of the small slope approximation (SSA) and superior to those of classical perturbation theory (PT) and the KA.

A review of the SSA for rough surface scattering

The small slope approximation (SSA) for scattering from rough surfaces was introduced by Voronovich in the mid-1980s [A. G. Voronovich, Sov. Phys. JETP 62, 65–70 (1985)]. Numerical studies have shown that the SSA gives excellent results over a broad range of scattering angles. However, in the high-frequency limit the second-order (in the field) SSA scattering amplitude reduces to the geometrical optics result and, thus, to a single scattering model limited to local interactions with the surface. In this paper, small slope theory will be reviewed briefly, expressions for the bistatic scattering cross section will be presented, and numerical results will be shown for a number of different problems and several different roughness spectra. In addition, in an effort to extend the nonlocal scattering effects, Voronovich modified the SSA. He applied the same approach underlying the SSA, but he began by iterating the integral equation of the second kind [A. G. Voronovich, Waves Random Media 6, 151–167 (1996)] which led to the nonlocal small slope approximation (NLSSA). While the main focus of this talk will be the SSA, the NLSSA will be compared with the SSA for selected results. [Work supported by ONR.]

Coherent reflection loss from a Pierson–Moskowitz sea surface using the NLSSA

Numerical results for the incoherent bistatic scattering strength for the nonlocal small-slope approximation (NLSSA) for scattering from rough surfaces [A. G. Voronovich, Waves Random Media 6, 151–167 (1996)], using a Gaussian spectrum, were recently reported [S. L. Broschat and E. I. Thorsos, J. Acoust. Soc. Am. 100, 2702 (1996)]. The cases considered agreed well with exact integral equation results and demonstrated the potential of the NLSSA for accurate modeling of rough surface scattering. In this paper, results for the coherent reflection loss for one-dimensional surfaces are presented. Coherent loss is of particular importance in long-range acoustic propagation when the cumulative effect of multiple interactions with the surface can be significant. The lowest-order NLSSA reflection coefficient is derived, and numerical results are shown for the reflection loss at low grazing angles, using a Pierson–Moskowitz sea surface spectrum. The numerical results are compared with those of other approximate methods and agree well with exact integral equation results. [Work supported by ONR.]

A preliminary numerical study of the nonlocal small slope approximation

Modeling the effect of a rough air–sea interface on an acoustic signal is of considerable practical importance; for example, it is relevant in the detection of underwater mines, torpedoes, and submarines. Several promising theories have been developed for rough surface scattering. Among the most successful is the small slope approximation (SSA) [A. G. Voronovich, Sov. Phys. JETP 62, 65–70 (1985)]. Recent numerical studies [see S. L. Broschat and E. I. Thorsos, submitted to J. Acoust. Soc. Am. (1996), and references therein] have shown that the higher-order SSA gives excellent results over a broad range of scattering angles for rms surface slopes up to 1. However, at the lowest forward grazing angles the results become somewhat high. At these angles multiple interactions with the surface at distant points have an appreciable effect on scattering levels. Voronovich recently modified the SSA to extend the nonlocal scattering effects [A. G. Voronovich, Waves in Random Media 6, 151 (1996)]. In this paper, a preliminary numerical study of the nonlocal small slope approximation (NLSSA) is presented. The equations for the lowest-order bistatic scattering strength are derived, limiting cases are considered, and numerical results for a Gaussian spectrum are presented and compared with both original SSA and integral equation results. [Work supported by ONR.]

Non-local small-slope approximation for wave scattering from rough surfaces

Abstract A new general analytical approach to solving the problems of wave scattering from rough surfaces, referred to as the non-local small-slope approximation (NLSSA), is suggested. It is formulated in the general form both for vector and scalar waves. This approach is valid for an arbitrary wavelength of radiation provided that the slopes of the undulations are small enough. The NLSSA represents a generalization of the small-slope approximation to situations where double scattering (in the optical sense) appears. It is demonstrated that with appropriate approximations the NLSSA of the lowest order reduces to the small-slope approximation of the second order.