The Kirchhoff approximation: A new iterative solution

This paper presents radar-altimeter scattering models for each of the diagenetic zones of the Greenland ice sheet. AAFE radar-altimeter waveforms obtained during the 1991 and 1993 NASA multi-sensor airborne altimetry experiments over Greenland reveal that the Ku-band return pulse changes significantly with the different diagenetic zones. These changes are due to varying amounts of surface and volume scattering in the return waveform.In the ablation and soaked zones, where surface scattering dominates the AAFE return, geophysical parameters such as rms surface height and rms surface slope are obtained by fitting the waveforms to a surface-scattering model. Waveforms from the percolation zone show that sub-sruface ice features have a much more significant effect on the return pulse than the surrounding snowpack. Model percolation waveforms, created using a combined surface- and volume-scattering model and an ice-feature distribution obtained during the 1993 field season, agree well with actual AAFE waveforms taken in the same time period. Using a combined surface- and volume-scattering model for the dry-snow-zone return waveforms, the rms surface height and slope and the attenuation coefficient of the snowpack are obtained. These scattering models not only allow geophysical parameters of the ice sheet to he measured but also help in the understanding of satellite radar-altimeter data.

A theory of scattering from very rough surfaces is presented. Both the surface rms height and the correlation distance are close to a wavelength and the rms slope is close to unity. The theory is based on the first- and modified second-order Kirchhoff approximations. The second-order scattering includes the incident and scattering shadowing and the angular and propagation tapering shadowing. The angular shadowing limits the angular spectrum of the second-order scattering within those intercepted by the surface, while the propagation shadowing limits the propagation distance within those intercepted by the surface. The calculations are made for the rms height equal to 1.5, 1.0, and 0.5 wavelengths and the correlation distance equal to 1.8 wavelengths and show close agreement with the Monte Carlo simulation of the exact integral equation. The results show that the first-order Kirchhoff scattering is dominant for rms height equal to 0.5 wavelengths, but for rms heights equal to 1.0 and 1.5 wavelengths, the second-order scattering becomes comparable to the first-order and produces the enhanced backscattering. Therefore, this theory provides a possible physical explanation for the enhancement. The range of validity of the theory is also discussed. [Work supported by ARO and NSF.]

The problem of electromagnetic scattering from a randomly rough surface is analyzed using the Kirchhoff approximation (stationary phase, scalar approximation), the small perturbation model and the two-scale model. The authors show that physical and geometrical optics approximations can be used for a rough sea surface. A small perturbation model gives a satisfactory solution. Unfortunately, it only applies to a smooth sea surface. Therefore, these methods have been shown to be limited in their regions of validity. Comparisons between the numerical calculations and the models are made for various surface rms heights and correlation lengths both normalized to the incident wave number (denoted by k/spl sigma/ and kL, respectively). The two-scale model proposed in this paper is the method which tries to conciliate the two previous aspects. One application is made in the ocean surface case. A set of experimental data, describing simultaneous bistatic radar cross section and forward radar cross section measurements of sea clutter taken with a moderately high resolution X-band scatterometer system operating near grazing incidence for both horizontal and vertical polarizations over a range of low grazing bistatic angles, is used but not presented in this paper to examine the applicability of bistatic scattering models.

Wave scattering from rough surface is important in many diverse areas, including radio communications, microwave remote sensing, underwater acoustics. The recent interest in bistatic radar, or the study of radio links above the sea surface, has led to the need for information concerning bistatic reflectivity of clutter viewed by such systems, particularly for near grazing incidence. The problem of developing a model for a rough surface is very difficult since at best /spl sigma/ is dependent upon (at least) radar frequency, geometrical parameters, physical parameters, incident and observation angles, and polarization. Therefore, it is necessary to restrict the areas of applicability of such models. A first major new consideration in this paper is to study the domain of validity of the Kirchhoff and small-perturbation rough surface scattering models (in the bistatic case). Comparisons between the numerical calculations and the models are presented for various surface rms height and correlation length both normalized to the incident wave number (denoted by k/spl sigma/ and kL, respectively). These two parameters form a two-dimensional space, into which the approximate regions of validity are established. The second new major consideration is the development of the theoretical composite model describing bistatic reflectivity. One application is implemented for the ocean surface case. Numerical results for the cross sections are compared to those found using Kirchhoff approximation (Physical Optics), small-perturbation model, and the phase perturbation technique.

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.

High spectral components contained in surfaces with non-Gaussian correlation functions are believed to increase the magnitude of backscattering enhancement. In this paper, we present experimental and numerical studies of bistatic scattering from one-dimensional surfaces with Gaussian and power-law roughness spectral densities. Rough surfaces with an rms height of σ = 3 mm and correlation lengths of l = 3, 6 and 9 mm were fabricated and measured with a millimetre-wave scatterometer. The corresponding rms slopes were m = 1.41, 0.707 and 0.47, respectively. Both experimental and numerical simulation data show that surfaces obeying the power-law roughness spectral density have stronger backscattering enhancement than those with a Gaussian roughness spectral density for the same surface rms slope. Good agreement between experiments and numerical simulations was obtained. In addition, the numerical studies were conducted to obtain the scattering characteristics of surfaces with Gaussian and power-law roughness ...

This paper discusses the measurement of the finish of diamond-turned surfaces by differential light scattering. Experimental scattering data are analyzed by electromagnetic theory to give the two-dimensional power spectral density of the surface roughness. These spectral densities are direct functional measures of the surface quality, and may be characterized in terms of topographic finish parameters. These parameters can then be used to specify surface finish, to predict scattering under a variety of conditions, and to aid in studies of other functional properties of these surfaces. Scattering spectra are separated in-to three groups corresponding to three classes of surface roughness: periodic tool marks and one- and two-dimensional random roughness. Periodic tool marks give rise to discrete diffraction lines in the scattering spectrum and are characterized by their surface periods and their Fourier amplitudes. Random one- and two-dimensional roughness give rise to one- and two-dimensional continua underlying the diffraction lines and are characterized by band-limited values of the rms surface heights and slopes, and transverse length parameters. Using HeNe light, vertical roughnesses are measured from a fraction of an Angstrom to several hundred Angstroms, for transverse spatial wavelengths from a fraction of a micron to several hundred microns. We re-view experimental techniques for making these measurements with emphasis on the scatterometer developed in our laboratory, which uses a fixed source-detector geometry and a rotating sam-ple. Results are illustrated by a number of scattering spectra taken with this instrument.

This paper presents a theory of scattering from very rough metallic and dielectric surfaces using the first- and second-order Kirchhoff approximations (KA) modified with the angular and propagation shadowing. The shadowing functions limit the single and double scattered waves which are illuminated and not shadowed by the surface. The theoretical results are compared with the Monte Carlo simulations showing the range of validity of the theory. The theory is applicable to the range where the RMS height is close to a wavelength and the RMS slope is close to unity, and the second medium is lossy. The second-order scattering includes two waves travelling in opposite directions on the surface, giving a physical explanation of the enhanced backscattering.

In bistatic radar observations, reflected echoes from the surface of a target planet can be analyzed to infer its surface statistics and near-surface constituents. In this work, a preliminary inspection of two X-band bistatic radar observations gathered by the Cassini spacecraft about Titan’s polar regions is presented. Profiles of relative dielectric constant and root-mean-square (rms) surface slope are provided as outputs of the analysis, discussed, and compared with the present knowledge of Titan geomorphology. For the assessment of the rms slope, proportional to the spectral broadening of reflected echoes, a basic fitting procedure was applied to the received spectra using a Gaussian template, to later evaluate the full-width half-maximum of the fitting curve. The dielectric constant was computed from the power ratio between orthogonally circularly polarized components of signal reflections from Titan. Dielectric constant estimates are, on average, consistent with the expected materials covering the dry surfaces of the planet, while slightly low values were found over the seas. The rms slopes are generally low compared to past bistatic observations of other targets. Titan’s north polar seas are revealed to feature an unprecedented smoothness, with 0.01^circ of slope as an upper bound. Similar values were inferred for isolated spots in the southern pole, hinting at the possible presence of basins filled with liquid hydrocarbons. The main issues with the analysis are emphasized throughout the document, and some ideas for future work are presented in the conclusions.

The actual rough sea surface is described by Pierson-Moscowitz (PM) spectrum and Monte Carlo method; the composite electromagnetic scattering from the double rectangular crosssection conductor columns above the rough sea surface is investigated using the hybrid methodthe Method of Moment with the Kirchhoff approximation. Angular distribution curves of composite scattering coefficient are obtained; and this coefficient that varies with the parameter of sea surface, rectangular cross-section conductor columns and incident waves is calculated in detail. The characteristics of the composite scattering coefficient from the double rectangular cross-section conductor columns above the rough sea surface are also obtained. Results show that the hybrid methodthe Method of Moment with the Kirchhoff approximation, compared to the traditional Method of Moment, can obtain higher accuracy. Moreover, the hybrid method can effectively reduce the computation time and memory footprint, and the performance becomes more favorable with the increase of the sizes of rough surface.

Based on the physical acoustic method (the Kirchhoff's approximation) and planar element method, the problem of the echo intensity of the target in a shallow water waveguide is studied. The physical acoustic method is a kind of high frequency approximate calculation method, which ignores the contribution to the scattering field of the shadow of the target. The planar elements method is a method of surface integral numerical discretization based on the physical acoustic method, which is to approximate the target surface with a large number of small planar polygons (or plates and surfaces). In this paper, the target echo is decomposed into the direct wave, sea surface reflection wave and sea bottom reflection wave. The direct wave is obtained by using the planar elements method directly, which is mono-static scattering. The sea surface and sea bottom reflection wave is analyzed by the mirror image method firstly, and the reflection waves are obtained by using the planar elements method, which include mono-static and bi-static scattering. The total scattering field of the target in an homogeneous wave guide is the summation of all the waves mentioned above. The target strength of the BeTSSi-Sub (Benchmark Target Strength Simulation Submarine) simple submarine in free field, and in a shallow water homogeneous wave guide, are calculated at different directions. The BeTSSi-Sub model is a benchmark model and is created and meshed using Ansys soft. In the computational examples, we suppose that the water depth is 100m, the sea surface is the absolute soft interface, and the bottom is the absolute hard interface. The target strength of a submarine under water 20m to 80m was calculated at 1 kHz. The results show that the echo intensity of the target in a shallow water waveguide is different from that of the free field due to the influence of upper and lower interface, and the echo intensity of the target is continuously changing with the depth of target.

One of the most popular methods for calculating the scattering of light from rough surfaces is the Kirchhoff approximation. A recent modification of the Kirchhoff approximation allows calculation of the scattering of electromagnetic radiation from rough surfaces with very high, or even infinite, slopes, which occurs frequently in, for example, imaging or testing of printed circuits or remote sensing applications. The modification involves a simple change in the way the surface normal is described which removes the explicit dependence of the resulting integral equations on the surface slope, and changing the sampling of the surface, removing the requirement of small slopes for application of the method. In this paper we present the application of the method to vector electromagnetic scattering, using the Stratton-Chu equation, from 2D rough metallic surfaces with infinite slope. The single and double scatter contributions to the total scattered intensity are calculated numerically with this new formulation. Geometrical shadowing is included by using ray tracing directly on the surface. The theoretical basis and preliminary numerical results of the application of this method to vector electromagnetic scattering from simple rough metallic surfaces (plane surfaces with single grooves or with a small number of identical grooves) with infinitely sloped structure are presented.

The coherent and incoherent scattering are composed in the backscattering characteristics of arbitrarily shaped dielectric object with rough surface both in laser and THz bands. If the radius of curvature at any point of the surface is much greater than the incident wavelength which is also longer than the surface height fluctuation and RMS of surface slope, the Kirchhoff approximation and Physical optics method, as well as the stationary phase evaluation are invited here to deduce the analytical expression of coherent backscattering cross section of rough dielectric object. Basically, the coherent cross section can be viewed as the combination of the RCS of corresponding smooth and perfectly conducting object, the Fresnel reflection coefficient of dielectric surface and the characteristic function of rough surface. Thus, the scattering expression of rough conducting object, smooth dielectric object and the rough dielectric object can be logically obtained. Using the tangent plane approximation, the surface of the object is divided into a series of patches, and then the incoherent component is achieved by integrating over the illuminated area combined with the covering function. Based on the Physical optics approximation and GPU parallel computing, the coherent scattering component of smooth conducting object, the incoherent component of rough object and its corresponding backscattering cross section can be easily computed. In this paper, we numerically simulate the backscattering characteristics in laser and THz bands of rough dielectric sphere and other complex rough dielectric targets respectively, meanwhile, we also analysis the influence of dielectric coefficient and roughness concentration on the results of the backscattering cross section.

Convergence rates of iterative solutions of algebraic matrix Riccati equations

Compared with scattering from a rough surface only, composite scattering from a target above a rough surface has become so practical that it is a subject of great interest. At present, this problem has been solved by some numerical methods which will produce an enormous calculation amount. In order to overcome this shortcoming, the reciprocity theorem (RT) and the method of equivalent edge currents (MEC) are used in this paper. Due to the advantage of RT, the difficulty in computing the secondary scattered fields is reduced. Simultaneously, MEC, a high-frequency method with edge diffraction considered, is used to calculate the scattered field from the cone-cylinder target with a high accuracy and efficiency. The backscattered field and the polarization currents of the rough sea surface are evaluated by the Kirchhoff approximation (KA) method and physical optics (PO) method, respectively. The effects of the backscattering radar cross section (RCS) and the Doppler spectrum on the size of the target and the windspeed of the sea surface for different incident angles are analysed in detail.