A note on the theory of scattering from an irregular surface

Integral formulas are developed directly from vector field theory for scattering by a perfectly conducting irregular surface at very short wavelengths. It is shown that, in the optical limit, the back-scattered field has no cross polarized component. When the integrals are evaluated asymptotically by the method of stationary phase, it turns out that to a first approximation the back scattering cross section is proportional to the average number of specular points which are illuminated at a given angle of incidence and to the geometric mean of the principal radii of curvature at those points. The scattering problem is, thus, transformed to a problem in the statistical geometry of irregular surfaces.     

Rough Surface Scattering Based on the Specular Point Theory

The average number of specular points per unit area for a two-dimensionally rough surface is derived in terms of the surface statistics. Expressions for the average of the product of the principal radii of curvature of the specular point are also obtained. These quantities are then substituted into the average scattering cross section per unit area derived from the stationary-phase approach, and two probability models are considered. The results are general and apply to bistatic as well as backscatter, and are valid for finitely as well as perfectly conducting surface materials.     

Geometrical optics prediction of surface scattering statistics

A composite-roughness formulation of the geometrical optics approximation is applied to study the statistics of near-nadir electromagnetic scattering from the sea surface. For scattering from Gaussian random surfaces, the scattering cross section is dependent only on the probability density of surface slopes. The statistical distribution of the scattered intensity depends on both the slope probability density function and <|/spl Omega/|> $the mean absolute value of the surface curvature. The curvature is of interest because it provides a measure of capillary wave spectra. Numerical results are obtained for scattering from isotropic surfaces for a fixed number N of specular scatterers and for N Poisson distributed. Obtaining viable estimates of <|/spl Omega/|>, and hence of capillary wave spectra, from backscatter data at microwave frequencies may not be practical. Optical measurements for which individual point scatterers can be identified may, however, yield estimates of the surface curvature.     

Comparison of ? Obtained from the Conventional Definition with ? Appearing in the Radar Equation for Randomly Rough Surfaces

A comparison is made of the radar cross section of rough surfaces calculated in one case from the conventional definition and obtained in the second case directly from the radar equation. The objective of the analysis is to determine how well the conventional definition represents the cross section appearing in the radar equation. The analysis is executed in the special case of perfectly conducting, randomly corrugated surfaces in the physical optics limit. The radar equation is obtained by solving for the radiation scattered from an arbitrary source back to a colocated antenna. The signal out of the receiving antenna is computed from this solution and the result put into a form recognizeable as the radar equation. The conventional definition is obtained by solving a similar problem but for backscatter from an incident plane wave. It is shown that these two forms for ?' are the same if the observer is far enough from the surface; However, the usual far-field criteria are not sufficient. For the two cross sections to be the same, the observer must be far from the surface compared to the radii of curvature of the surface at the reflection (specular) points. Numerical comparison of the two cross sections has been made for normally distributed surfaces and the difference can be significant.     

Calculation of far-scattered fields by the method of stationary phase

The scattering problems for a nonconvex body can be analyzed by the physical-optics approximation. The method of stationary phase is a technique for evaluating the diffraction integral in the short-wavelength region. In case of the nonconvex body there exist several stationary points including the complex ones in the reduced phase integral. The radar cross section of the nonconvex body varies with frequency as a result of the interference of the scattered waves from each stationary point. For a moderately high frequency the solutions by the method of stationary phase agree with those obtained using the mode-matching method.     

On the use of finite surfaces in the numerical prediction of roughsurface scattering

Method of moments (MOM)-based Monte Carlo calculations are widely used in determining the average radar cross section of randomly rough surfaces. It is desirable in these numerical calculations to truncate the scattering surface into as short a length as possible to minimize the solution time. However, truncating the surface tends to change the solution for the surface fields near the truncation points and may alter the scattered far fields. In this paper, these end effect errors are examined for one-dimensional (i.e., grooved or corduroy) surfaces which are Gaussian distributed in height and have either a Gaussian or a Pierson-Moskowitz spectra. In the case of the Pierson-Moskowitz type surface, it is shown that a relatively short surface of 80-120 wavelengths can be used to obtain the average backscattered radar cross section for backscattering angles as large as 60° from normal. For a comparatively smooth Gaussian surface, on the other hand, its is shown that the truncation effects can be very significant at moderate backscattering angles. Also, great care should be taken when examining the scattering from Gaussian surfaces which are dominated by specular scattering. It is shown that in this situation, a very large number of calculations may be needed to obtain a good numerical average     

The Contribution of Wedge Scattering to the Radar Cross Section of the Ocean Surface

This letter considers the contribution to the radar cross section of the ocean surface due to scattering from edges for which the local radius of curvature is small compared with the radar wavelength. An analytic expression based on the method of equivalent currents is given for such scattering and is evaluated for several assumed sets of parameters. This contribution is shown to augment the Bragg scattering cross section in regions where the latter underestimates the measured radar cross section, while remaining smaller than the Bragg component elsewhere.     

A note on the scattering from a slightly rough surface

A Fourier transform approach is used to derive the bistatic radar scattering cross section of a slightly rough perfectly conducting infinite surface. A perturbation expansion is used to apply the boundary conditions, and the scattered fields are asympotically evaluated by means of the method of stationary phase. The resultant expression for the radar cross sectionsigma^{0}is shown to agree with that obtained using the method as outlined by Rice.     

Représentation ponctuelle de la contribution de la frontière d’ombre d’un objectif radar et son introduction dans le modèle des points brillants

In order to study the scattering of a monochromatic electromagnetic wave by a complex target (generally a plane), radar specialists usually replace the target by a collection of independent point contributors. The model obtained is called a discrete scatterers model because the majority of the scatterers are represented by specular points of reflection on the surface of the target. The possibility of optical perception of specular points and the apparent simplicity of the method make it very attractive, and this explains its success. The litterature about this technique tends to assume that it is known and understood, and therefore rarely explains the basics. The result is sometimes an impression of empiricism and a tendancy to mistake the optical problem for the easily ignored electromagnetic one which often leads to results so bad that they cannot be used even for the simplest of targets. In this article, the author explains the concept of point contributor in detail and show that the shadow boundaries on the surface of a target can also be represented by imaginary specular points and thus can be easily included in the model. The author studies the effect of imaginary points on the radar cross section of the targets, and illustrates the results obtained by applying the method to a few simple objects.     

Evaluation of reflected fields at caustic regions using a set of GO equivalent line currents

A set of equivalent electric and magnetic line currents is derived which supplements the geometrical optics (GO) solution in the far zone whenever one of the surface principal radii becomes very large. These hypothetical currents lie along the specular line of the surface and are shown to produce the same result as the stationary phase contribution of the physical optics integral. An example of a systematic application of such equivalent currents for the computation of the scattered field from a complex structure is also demonstrated.     

A reflection ansatz for surfaces with electrically small radii of curvature

Uniform reflection coefficients are developed for two- and three-dimensional, edge-like, perfectly conducting surfaces in the deep lit region. The uniformity is with respect to the electrical size of the radii of curvature at the surface's specular point. This uniformity allows one to physically interpret the reflected field from a smooth surface as one of the radii of curvature approaches zero as a diffracted field. The coefficients are heuristically generated from the exact scattered field for a two dimensional parabolic cylinder with plane wave illumination. The significant variables in this solution are the radii of curvature at the specular point and the distance between the specular point and the incident shadow boundaries in the principal planes. The field prediction accuracy of these reflection coefficients are critically examined through comparisons with reflected fields extracted from scattered fields of canonical surfaces.     

Scattering by S-shaped surfaces

When an S-shaped surface possesses no derivative discontinuities, techniques such as the geometrical theory of diffraction are not applicable. However, if the radius of curvature is relatively large at every point on the surface, the physical optics approximation may be employed. The authors present a uniform physical optics (UPO) solution which remains valid at caustics occurring when two or more specular points coalesce at the inflection point of the S-shaped surface. The solution is developed by approximating the surface with a localized cubic expansion, leading to exact expressions in terms of Airy integrals. In contrast to other solutions, the one given here requires only a knowledge of the stationary phase points and the first three derivatives of the surface-generating function at those points. A major effort is devoted to the validation of the UPO solution, and this is accomplished with numerical models of the S-shaped surface. It is found that the given UPO solution is quite accurate in the specular and nonspecular regions     

脉冲波入射时分形粗糙海面的双频散射截面和脉冲展宽

根据粗糙面基尔霍夫小斜率近似研究了脉冲波入射时实际海谱分布的一维分形海面的电磁散射。分析了毫米波入射时不同分维、入射角和入射中心频率下双频散射截面的散射角分布。结果表明分形海面的双频散射截面在镜反射方向有最大的相关带宽，随着海面分维的减小、入射中心频率和入射角的增加，该相关带宽是增大的。对于入射功率为δ函数时的散射波功率是一个具有一定脉冲展宽的散射脉冲，且脉冲展宽与相关带宽成反比关系。     

Curvature effects in ocean surface scattering

Curvature effects in EM scattering from ocean surface are described using a generalized curvature expansion of the fields at an elevated nonperfect conducting surface. The new expansion formalism allows us to describe analytically and in general, without separating into different scales, the scattering of EM waves from an undulated ocean surface. The model is exact to first order in curvature for nonshadowing imaging geometry, and obeys the law of reciprocity and tilt invariance. Explicit expressions for EM fields at the surface, including both the projection and the self induced fields, are derived up to first order in surface curvature. Analytic closed form expressions for the scattered fields are derived from the surface field solutions, and applied to the case of backscattering, providing a general expression for the normalized radar cross section. The analytic expression for the normalized radar cross section is implemented for a linear surface model using both the Eulerian and the Lagrangian frame of reference. The results show that the model is capable of describing the expected dependency on polarization, incidence angle, and wind field with minimal restrictions in terms of range of validity. Comparison of polarization ratio shows good agreement between the model and measurements from the Envisat ASAR instrument.     

High frequency approximations to the physical optics scatteringintegral

Discusses two high frequency (HF) approximations to the physical optics (PO) scattering integral for the far field radar backscatter from a general curved edged reflecting surface viewed at arbitary aspect. The PO scattering integral is first approximated as the sum of a specular effect and an edge effect, where the latter is represented explicitly as a certain line integral evaluated over the boundary edge of the reflector. A closed form result is then obtained by applying the method of stationary phase to the line integral. With the exception of singularities that can occur at caustics, or when the specular point falls on the boundary edge, these HF approximations are found to work reasonably well for smooth surfaces whose Gaussian curvatures have constant sign (positive or negative, but never zero)     

Computations of scattering cross sections for composite surfaces and the specification of the wavenumber where spectral splitting occurs

The scattering cross sections for composite random rough surfaces are evaluated using the full wave approach. They are compared with earlier solutions based on a combination of perturbation theory which accounts for Bragg scattering, and physical optics which accounts for specular point theory. The full wave solutions which account for both Bragg scattering and specular point scattering in a self-consistent manner are expressed as a weighted sum of two cross sections. The first is associated with a filtered surface, consisting of the larger scale spectral components, and the second is associated with the surface consisting of the smaller scale spectral components. The specification of the surface wavenumber that separates the surface with the larger spectral components from the surface with the smaller spectral components is dealt with in detail. Since the full wave approach is not restricted by the limitations of perturbation theory, it is possible to examine the sensitivity of the computed values for the backscatter cross sections to large variations in the value of the wavenumber where spectral splitting is assumed to occur.     

Scattering cross sections for composite rough surfaces using the unified full wave approach

The full wave approach is used to derive a unified formulation for the like and cross polarized scattering cross sections of composite rough surfaces for all angles of incidence. Earlier solutions for electromagnetic scattering by composite random rough surfaces are based on two-scale models of the rough surface. Thus, on applying a hybrid approach physical optics theory is used to account for specular scattering associated with a filtered surface (consisting of the large sonic spectral components of the surface) while perturbation theory is used to account for Bragg scattering associated with the surface consisting of the small scale spectral components. Since the full wave approach accounts for both specular point scattering and Bragg scattering in a self-consistent manner, the two-scale model of the rough surface is not adopted in this work. These unified full wave solutions are compared with the earlier solutions and the simplifying assumptions that are common to all the earlier solutions are examined. It is shown that while the full wave solutions for the like polarized scattering cross sections based on the two-scale model are in reasonably good agreement with the unified full wave solutions, the two solutions for the cross polarized cross sections differ very significantly.     

Scattering of Solar and Atmospheric Background Radiation from a Target

Light scattering property of environment is very important in theoretical study and application of the remote sensing. What's more, it is valuable for infrared radiation, imaging, and the detection of target and tracking. In this paper, solar and atmospheric background radiation, and their scattering property from target are discussed. BRDF (Biodirectional Reflectance Distribution Function) is a very important quantity that shows the radiation and reflection feature of target. According to electromagnetic radiant and scattering theories, the relationship between laser radar scattering cross section (LRCS) and BRDF is introduced. LOWTRAN model is an effective method of calculating the spectral distribution of solar and atmospheric radiation. Here it is applied to compute solar and atmospheric background radiation scattered from a target. The relative equations are deduced. Thus, the spatial and spectral distribution of scattering light is given. As a special example, for the Lambert's surface, the equations are simplified. As a result, the spatial and spectral distributions scattering radiation of solar and atmospheric background from a rough painted surface are present. The scattering of solar radiation plays a primary role in MIR region, but scattering of atmospheric background radiation is higher in LIR region. At the same time, there is obviously specular reflectance for solar radiation due to coherent scattering from rough surface.     

On the universal behavior of scattering from a rough surface forsmall grazing angles

It is shown that for scattering from a plane of an average rough surface, the scattering cross section of the range of small grazing angles of the scattered wave demonstrates a universal behavior. If the angle of incidence is fixed (in general, it should not be small), the diffuse component of the scattering cross section for the Dirichlet problem is proportional to &thetas;² where &thetas; is the (small) angle of elevation and for the Neumann problem it does not depend on &thetas;. For the backscattering case, these dependencies correspondingly become &thetas;⁴ and &thetas;⁰. The result is obtained from the structure of the equations that determine the scattering problem rather than the use of an approximation     

High-frequency asymptotic solutions for backscattering by cylinders with conic section directrixes

High-frequency asymptotic expansions are derived for electric and magnetic fields backscattered from a perfectly conducting smooth two-dimensional surface illuminated by a plane incident wave in two cases of TE and TM linear polarizations. Diffraction corrections up to the second order of the inverse large parameter p=ak (where a is a curvature radius at the specularly reflecting point, and k is a field wave number) to the geometrical optics fields, and specifically to their phases, backscattering cross sections (HH and VV for TE and TM polarizations, correspondingly), as well as the polarization ratio HH/VV, are derived for the specular points of a general form. These general results are applied to backscattering from cylinders with conical section directrixes (circle, parabola, ellipse, and hyperbola), and a number of new compact explicit equations are derived, especially for elliptic and hyperbolic cylinders illuminated at an arbitrary incidence angle relative to their axes of symmetry.