Bistatic Scattering Cross Section Research Articles

The small slope approximation for acoustic scattering from a rough interface between a fluid and a fluid-saturated porous solid

The small slope approximation, introduced by A. G. Voronovich in the 1980s, gives a systematic expansion in terms of a generalized surface slope. Previous numerical studies indicate that the lowest-order term provides a reasonably accurate approximation for a number of different problems of practical importance. In this work we consider the small slope approximation for acoustic scattering from a rough interface between a fluid and a fluid-saturated porous solid in the context of Biot theory. Biot theory is of continued interest in the study of sediment acoustics; it predicts a slow compressional wave in the fluid-filled porous medium in addition to the conventional compressional and shear waves found in an elastic solid. The lowest-order small slope approximation is derived for Biot theory, and an expression for the bistatic scattering cross section is presented for plane-wave scattering from a randomly rough interface between a fluid and a fluid-saturated porous solid. Numerical results for the scattering strength are discussed for both the small slope and perturbation methods. [Work supported by ONR.]

On bistatic sea surface scattering: Field measurements and modeling

The bistatic scattering cross section of the sea surface, σ, is studied, along with a model for σ and its comparison with field data. The data are horizontal spatial coherence and ensemble-averaged intensity, which represent integral measures of sea surface bistatic scattering, and the model for σ is used to generate these same properties for comparison with the field data. The data are from an experiment conducted in shallow waters off southern Florida, using a sound frequency of 30 kHz. Directional wave measurements were made with a wave buoy positioned within 100 m of the acoustic measurements, with the environment characterized by rms wave heights of O(10) cm and wind speeds of 1–4 m/s. In the model σ is divided into two components: σr associated with scattering from the rough, air/sea interface, and σb associated with scattering from near-surface bubbles. The second-order small slope approximation is used to compute σr, which is a much improved approach over the traditionally used composite roughness model. The primary advantage in the small slope approximation was the resulting smooth behavior in σr over a broad range of scattering angles. Directional wave data obtained by the wave buoy were converted to an estimate of the 2-D spatial correlation function of sea surface roughness, C(ξ,final_sigma), for use in the scattering calculations. An analysis of the effective correlation properties of C(ξ,final_sigma) suggested that an isotropic correlation function C(ρ), based on the directionally averaged wave-number spectrum, would be equally effective in the scattering calculations. Model-data agreement was quite satisfactory, regardless of whether C(ξ,final_sigma) or C(ρ) was used in the scattering calculations.

Scattering by an elastic sphere embedded in an elastic isotropic medium

The scattering of acoustic waves by an elastic sphere embedded in an elastic isotropic medium is investigated. Expressions for the scattered waves are given in terms of monostatic and bistatic scattering cross sections. The resonances of the solid sphere were determined numerically in the individual normal mode amplitudes; dispersion curves for the phase velocities of the circumferential waves were also obtained. Computations and experimental results for an aluminum sphere embedded in Plexiglas were in good agreement.

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.]

Near-field scattering through and from a two-dimensional fluid–fluid rough interface

A general analytical expression for the time-dependent mean-square incoherent field scattered from or through (penetrating) a 2-D fluid–fluid rough interface for a narrow-band incident plane-wave source is derived and expressed in terms of the second moment of the rough interface T-matrix. This analytical expression is independent of the scattering solution technique, and for distances greater than only a few wavelengths from the interface, is equivalently expressed in terms of the bistatic scattering cross section per unit area per unit solid angle (differential cross section) of the rough interface. Using this rigorously derived result, the scattered field for a narrow-band point source is heuristically derived. This derivation leads to the usual sonar equation in the limit as the narrow-band signal approaches the cw (continuous wave) case. First-order perturbation calculations for the case of a baseband Gaussian shaped source pulse illustrate narrow-band pulse dispersion effects of the incoherent field for forward scattering into a lossy sediment. For the case of incidence below the critical grazing angle, first-order perturbation computations also show that the incoherent field scattered through a rough interface can be much greater than the zeroth-order field (coherent) transmitted below the corresponding flat-surface depending on loss and receiver depth. These computations for the first-order mean square incoherent field penetrating the rough interface are compared to the results for the flat-surface case, for both plane-wave and point sources.

The Local Spectral Expansion Method for Scattering From Perfectly Conducting Surfaces Rough in One Dimension

We present a new analytical approach for scattering from perfectly conducting surfaces that are rough in one dimension. The method is based on expanding the Helmholtz equation using a set of local basis functions. The transformed equations contain the field and its normal derivatives at the surface as source functions. The first order approximate solution, sought for slightly rough surfaces, is obtained using the local tangent plane approximation at the surface. It is shown analytically that the new method reduces to the first order perturbation method in the proper limit. A systematic series solution is proposed using perturbation theory in conjunction with the present formulation. The second order result is given and analyzed. The results of the present theory are not reciprocal and a reciprocity rule is formulated to overcome this disadvantage. We compare numerically the bistatic scattering cross section for Gaussian-randomly rough surfaces predicted by the first order solution with numerical results and other first order theories. It is shown that the present formulation is overall more accurate than other existing theories.

Scattering of electromagnetic waves on a magnetodielectric with chiral properties

The numerical method that we proposed earlier for solving the problem of diffraction of electromagnetic waves on three-dimensional magnetodielectric bodies of any shape in the resonance frequency range is generalized to the case of objects having chiral properties. The results of numerical calculations of bistatic scattering cross sections of spherical and cylindrical chiral objects are reported. The influence of the values of the chirality parameters and of electrical and magnetic absorption on the scattering diagrams is examined.

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.

Sampling criteria for resonant antennas and scatterers

From a knowledge of the characteristic constant of an antenna or scatterer, simple expressions are found for the modal bandlimits and near-field sample spacing in planar, cylindrical and spherical coordinates required to reconstruct the fields everywhere outside the antenna or scatterer (except possibly within a small fraction of a wavelength from discontinuities in the surface of the antenna or scatterer). For resonant antennas and scatterers the modal bandwidths can be much larger and the required sample spacings much smaller than for nonresonant antennas and scatterers. The modal bandwidths also determine upper bounds on the gain and effective area of antennas, and on the total and bistatic scattering cross sections of scatterers. The maximum possible radius of the significant reactive fields of an antenna or scatterer is shown to equal the radius of the maximum possible effective area of an antenna, or the radius of the maximum possible total scattering cross section of a scatterer. For resonant antennas and scatterers, this radius can be much larger than the radius of the sphere circumscribing the physical antenna or scatterer.

Near‐field scattering through and from a 2‐D fluid‐fluid rough interface.

A general analytical expression for the time‐dependent field intensity scattered from or through (penetrating) a 2‐D fluid‐fluid rough surface due to a narrow‐band incident plane wave and a narrow‐band point source are derived and expressed in terms of the bistatic scattering cross section per unit area of the rough surface. Even though the scattering cross section is defined as a far‐field quantity, this near‐field result is general and exact for the special case of a continuous wave source and incident plane wave. Dispersion of a pulse is a function of medium parameters, the incident and scattered directions, as well the bandwidth and center frequency of the source signal. First‐order perturbation calculations for the case of a Gaussian pulse illustrate intensity pulse dispersion effects due to forward scattering into a lossy sediment. [Work supported by ONR.]

The effect of gradients on high-frequency bottom scattering

The first-order perturbation expression for the bistatic scattering cross section of a rough surface [Moe and Jackson, J. Acoust. Soc. Am. (to be published)] is used to study the effect of gradients on high-frequency bistatic bottom scattering. Strong gradients on scales of several centimeters are often observed in shallow-water environments. It is shown that upward refraction or reflection by strong gradients can lead to significant enhancement of high-frequency scattering. This enhancement occurs over a wide range of bistatic angles, but is strongest when the incident and scattered grazing angles are equal. These effects can be explained in terms of the reflection coefficient of the corresponding mean (flat) surface. [Work supported by ONR.]

Copolarized and cross‐polarized enhanced backscattering from two‐dimensional very rough surfaces at millimeter wave frequencies

Wideband millimeter wave experiments from 75–100 GHz on the scattering from two‐dimensional very rough conducting surfaces are presented. The two‐dimensional very rough surfaces are manufactured using a computer‐numerical‐controlled milling machine so that the surface statistics are precisely controlled. The surfaces have both Gaussian roughness statistics and Gaussian surface correlation functions. Bistatic scattering experiments on surfaces with either isotropic or anisotropic correlation functions are performed. Copolarized and cross‐polarized bistatic scattering cross sections are measured for both transverse electric and transverse magnetic incidence at 20°. For isotropic surfaces, backscattering enhancement exists for both copolarized and cross‐polarized returns and is found to be a function of the surface rms slope. In addition, a strong frequency dependence is observed across the 25‐GHz bandwidth in the cross‐polarized returns. To investigate the effect of surface correlation anisotropy, scattering experiments on anisotropic rough surfaces are also performed. It is found that the bistatic scattering cross section for an anisotropic surface is a function of the effective correlation length projected along the plane of scattering. Results on the bistatic scattering experiments presented here serve as a motivation to further pursue more elaborate and complete scattering experiments in order to advance research on scattering from very rough surfaces.

Acoustic scattering from a fluid–elastic-solid interface using the small slope approximation

In this paper the small slope approximation is applied to acoustic scattering from a randomly rough fluid–elastic-solid interface. Expressions for the zeroth-, first-, and second-order bistatic scattering cross sections are derived. Numerical results are obtained for the zeroth-order small slope approximation for Gaussian and modified power law surface roughness spectra and are compared with those of first-order perturbation theory and the Kirchhoff approximation. The environmental parameters used correspond to those of water–granite, water–basalt, or water–sediment interfaces for lossless media. The small slope results show the complex structure associated with elastic wave scattering, including critical angle and Rayleigh angle structure. For the modified power law, the small slope results agree with those of Monte Carlo simulations performed by Berman [J. Acoust. Soc. Am. 89, 623–636 (1991)]. The study includes a comparison of scattering strengths both with and without the shear wave component. The importance of the shear wave component for a sufficiently rigid solid is illustrated.

Combined random rough surface and volume scattering based on Monte Carlo simulations of solutions of Maxwell's equations

The scattering of electromagnetic plane waves by one‐dimensional lossy dielectric random rough surfaces containing densely distributed two‐dimensional dielectric scatterers is studied. The finite element formulation for random rough surface scattering is extended to include random volume scattering in the lower half‐space. An advantage of this method is that the scattering by the discrete scatterers below the dielectric rough surface can be taken into account with only a small increase in computational complexity. The Monte Carlo simulation results show that the presence of volume scattering can have significant effects on the backscattering intensity and bistatic scattering cross section depending on the statistics of the rough surface, the permittivity of the lower half‐space, and properties of the discrete scatterers such as sizes, permittivities, and concentrations.

The local parabolic approximation for rough surface scattering.

A method, based on a technique introduced by A. V. Belobrov and I. M. Fuks [Radiophys. Electron. (Izv. VUZ Radiofiz.) 29 (12), 1083–1089 (1986)], is studied for the case of scalar wave scattering from one-dimensional randomly rough surfaces with a Gaussian roughness spectrum. Using a local parabolic approximation of the rough surface, the surface source density is obtained. The small parameter in such an approximation is the inverse of the radius of the surface curvature multiplied by the wave number and the cube of the cosine of the local angle of incidence. The bistatic scattering cross sections per unit length are derived for the Dirichlet and Neumann boundary conditions. The first term for each is that of the Kirchhoff approximation; successive terms account for diffraction effects. Numerical results for both soft and hard surfaces are obtained and compared with other available numerical results. [Work supported by NSF and ONR.]

Far‐field considerations in surface and bottom scattering

It is sometimes argued that the concept of scattering cross section is inapplicable for most surface and bottom scattering measurements, because the acoustic receiver is invariably in the near‐field zone of the ensonified patch and scattering cross section is defined in the far‐field limit. Winebrenner and Ishimaru [IEEE Trans. Antennas Propag.AP34, 847–849 (1986)] have shown that the correlation length of the surface field defines the relevant far‐field region for stochastic scattering. This sets a less stringent constraint than the far‐field criterion based on the ensonified patch size and greatly extends the domain of applicability of the scattering cross‐section concept. Using a standard formalism that includes near‐field phase terms, it is shown how the scattering cross section can be used even in the near field of the ensonified patch. The result is a proof of the method commonly used in reverberation calculations: One integrates over the ensonified patch with the appropriate propagation loss factors and bistatic scattering cross section as integration variables. This assumes that the far‐field criterion based on the correlation length is satisfied. [Work supported by ONR.]

Full wave analysis for rough surface diffuse, incoherent radar cross sections with height-slope correlations included

The bistatic scattering cross sections are derived for rough one-dimensional perfectly conducting surfaces using the full wave approach. The surfaces are characterized by four-dimensional Gaussian joint probability density functions for heights and slopes. Thus, correlations between the rough surface heights and slopes are accounted for in the analysis. Convergence of the formal series solution is considered. Self-shadowing effects are included. The full-wave solutions are compared with the small perturbation solutions, which are polarization dependent, and the specular point (physical optics) solutions, which are independent of polarization. Both the physical optics and the small perturbation solutions can be obtained from the full-wave solution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Numerical studies of rough surface scattering models

In recent years a number of rough surface scattering models have been introduced that attempt to “bridge the gap” between the classical perturbation and Kirchhoff approximations. Among these are the phase perturbation technique and the small slope approximation. Investigations have shown that for pressure-release surfaces with a Gaussian roughness spectrum, the phase perturbation model is accurate in cases when the classical approximations fail. Monte Carlo calculations by Betman for the second-order small slope approximation show excellent agreement with results obtained using the Rayleigh-Fourier method both for wind-driven sea surfaces and for the fluid-solid boundary condition. In this work, the first- and second-order small slope approximation reflection coefficients and bistatic scattering cross sections are derived for the Dirichlet (pressure-release) problem. The first-order results are shown to reduce to those of the perturbation and Kirchhoff methods for the reflection coefficient and to that of the perturbation method for the bistatic scattering cross section. In addition, phase perturbation results are calculated for a multiscale (Pierson-Moskowitz) surface spectrum. For both models, numerical results for one-dimensional, pressure-release surfaces with a Pierson-Moskowitz spectrum are compared with exact numerical results obtained using a Monte Carlo technique.

Statistical parameters of interference noise above a randomly uneven interface

Methods of statistical diffraction theory have been used to investigate the parameters of interference noise which arises in a tropospheric channel due to interfering reflections from an underlying surface. It has been assumed that the explicit form of the bistatic scattering cross section of radio waves by a randomly rough surface is known. The systematic errors in measuring the signal parameters, the energy spectra, and the variances of the fluctuations in the values of the parameters relative to their average values have been determined. The spectrum of the fluctuations in the bearing of a point source moving horizontally above a rough surface having mildly sloping roughnesses has been calculated.

Scattering of acoustic and electromagnetic waves by an airfoil

The finite-difference approach based on the generalized scattering amplitude concept is applied to the scattering of acoustic and electromagnetic waves by a two-dimensional airfoil. The transformed Helmholtz equation and associated boundary conditions, in terms of the generalized scattering amplitude, are solved in a numerically generated, body-fitted, curvilinear coordinate system. These are obtained by grid generation procedures commonly used for airfoil aerodynamic computations. Numerical results are obtained for normal incidence of acoustic and electromagnetic waves on a modified NACA 4418 airfoil with a chord length approximately equal to the wavelength of incident waves. The directivity of the scattered sound pressure intensity in the far field is presented in the form of bistatic scattering cross sections. Induced surface current of electromagnetic scattering is also presented. The results of the finite-difference method are in good agreement with those obtained by the method of moments.