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.     

Energy conservation of the scattering from one-dimensional random rough surfaces in the high-frequency limit

Energy conservation of the scattering from one-dimensional strongly rough dielectric surfaces is investigated using the Kirchhoff approximation with single reflection and by taking the shadowing phenomenon into account, both in reflection and transmission. In addition, because no shadowing function in transmission exists in the literature, this function is presented here in detail. The model is reduced to the high-frequency limit (or geometric optics). The energy conservation criterion is investigated versus the incidence angle, the permittivity of the lower medium, and the surface rms slope.     

Extension of the local curvature approximation to third order and improved tilt invariance

The second-order local curvature approximation (LCA2) is a theory of rough surface scattering that reproduces fundamental low and high frequency limits in a tilted frame of reference. Although the existing LCA2 model provides agreement with the first order small perturbation method up to the first order in surface tilt, results reported in this paper produce a new formulation of the model that achieves consistency with perturbation theory to first order in surface height and arbitrary order in surface tilt. In addition, extension of the modified LCA to third order is presented, and allows the theory to match the second-order small perturbation method to arbitrary order in surface tilt. Crucial to the development of the theory are a set of identities involving relationships among the small perturbation method (i.e. low frequency) and Kirchhoff approximation (i.e. high frequency) kernels; a set of new identities obtained in our derivations is also presented. Sample results involving 3D electromagnetic scattering from penetrable rough surfaces, as well as 2D scattering from Dirichlet sinusoidal gratings, are provided to compare the new results with the existing LCA2 model and with other rough surface scattering theories.     

Extension of the local curvature approximation to third order and improved tilt invariance

The second-order local curvature approximation (LCA2) is a theory of rough surface scattering that reproduces fundamental low and high frequency limits in a tilted frame of reference. Although the existing LCA2 model provides agreement with the first order small perturbation method up to the first order in surface tilt, results reported in this paper produce a new formulation of the model that achieves consistency with perturbation theory to first order in surface height and arbitrary order in surface tilt. In addition, extension of the modified LCA to third order is presented, and allows the theory to match the second-order small perturbation method to arbitrary order in surface tilt. Crucial to the development of the theory are a set of identities involving relationships among the small perturbation method (i.e. low frequency) and Kirchhoff approximation (i.e. high frequency) kernels; a set of new identities obtained in our derivations is also presented. Sample results involving 3D electromagnetic scattering from penetrable rough surfaces, as well as 2D scattering from Dirichlet sinusoidal gratings, are provided to compare the new results with the existing LCA2 model and with other rough surface scattering theories.     

Scattering on rough surfaces with alpha-stable non-Gaussian height distributions

We study the electromagnetic scattering problem on a random rough surface when the height distribution of the profile belongs to the family of alpha-stable laws. This allows us to model peaks of very large amplitude that are not accounted for by the classical Gaussian scheme. For such probability distributions with infinite variance the usual roughness parameters such as the RMS height, the correlation length or the correlation function are irrelevant. We show, however, that these notions can be extended to the alpha-stable case and introduce a set of adapted roughness parameters that coincide with the classical quantities in the Gaussian case. Then we study the scattering problem on a stationary alpha-stable surface and compute the scattering coefficient under the first-order Kirchhoff and small-slope approximations. An analytical formula is given in the high-frequency limit, which generalizes the well known geometrical optics approximation. Some numerical results are given and discussed.     

二维粗糙海面的光散射及其红外成像

首先根据JONSWAP海面功率谱模型数值模拟出二维粗糙海面，采用几何光学近拟与基尔霍夫（Kirchhoff）标量近似计算了二维海面的光散射，计算中将每一面元看成一具有微粗糙度的粗糙面而不是近似地当作平面，并利用投影法与射线追踪法数值计算了一定入射角和散射角下的遮挡函数，有效地提高了海面光散射计算的精确性。最后利用太阳光的光谱辐照度数值模拟了海面的3μm-5μm红外散射图像，对于红外探测器抑制海面反射太阳光造成的亮带干扰具有一定的参考价值。     

Multiple scattering in the high-frequency limit with second-order shadowing function from 2D anisotropic rough dielectric surfaces: I. Theoretical study

Abstract

In this paper the first- and second-order Kirchhoff approximation is applied to study the backscattering enhancement phenomenon, which appears when the surface rms slope is greater than 0.5. The formulation is reduced to the geometric optics approximation in which the second-order illumination function is taken into account. This study is developed for a two-dimensional (2D) anisotropic stationary rough dielectric surface and for any surface slope and height distributions assumed to be statistically even. Using the Weyl representation of the Green function (which introduces an absolute value over the surface elevation in the phase term), the incoherent scattering coefficient under the stationary phase assumption is expressed as the sum of three terms. The incoherent scattering coefficient then requires the numerical computation of a ten- dimensional integral. To reduce the number of numerical integrations, the geometric optics approximation is applied, which assumes that the correlation between two adjacent points is very strong. The model is then proportional to two surface slope probabilities, for which the slopes would specularly reflect the beams in the double scattering process. In addition, the slope distributions are related with each other by a propagating function, which accounts for the second-order illumination function. The companion paper is devoted to the simulation of this model and comparisons with an ‘exact’ numerical method.     

Local and non-local curvature approximation: a new asymptotic theory for wave scattering

Abstract

We present a new asymptotic theory for scalar and vector wave scattering from rough surfaces which federates an extended Kirchhoff approximation (EKA), such as the integral equation method (IEM), with the first and second order small slope approximations (SSA). The new development stems from the fact that any improvement of the ‘high frequency’ Kirchhoff or tangent plane approximation (KA) must come through surface curvature and higher order derivatives. Hence, this condition requires that the second order kernel be quadratic in its lowest order with respect to its Fourier variable or formally the gradient operator. A second important constraint which must be met is that both the Kirchhoff approximation (KA) and the first order small perturbation method (SPM-1 or Bragg) be dynamically reached, depending on the surface conditions. We derive herein this new kernel from a formal inclusion of the derivative operator in the difference between the polarization coefficients of KA and SPM-1. This new kernel is as simple as the expressions for both Kirchhoff and SPM-1 coefficients. This formal difference has the same curvature order as SSA-1 + SSA-2. It is acknowledged that even though the second order small perturbation method (SPM-2) is not enforced, as opposed to the SSA, our model should reproduce a reasonable approximation of the SPM-2 function at least up to the curvature or quadratic order. We provide three different versions of this new asymptotic theory under the local, non-local, and weighted curvature approximations. Each of these three models is demonstrated to be tilt invariant through first order in the tilting vector.     

Multiple scattering in the high-frequency limit with second-order shadowing function from 2D anisotropic rough dielectric surfaces: I. Theoretical study

In this paper the first- and second-order Kirchhoff approximation is applied to study the backscattering enhancement phenomenon, which appears when the surface rms slope is greater than 0.5. The formulation is reduced to the geometric optics approximation in which the second-order illumination function is taken into account. This study is developed for a two-dimensional (2D) anisotropic stationary rough dielectric surface and for any surface slope and height distributions assumed to be statistically even. Using the Weyl representation of the Green function (which introduces an absolute value over the surface elevation in the phase term), the incoherent scattering coefficient under the stationary phase assumption is expressed as the sum of three terms. The incoherent scattering coefficient then requires the numerical computation of a ten- dimensional integral. To reduce the number of numerical integrations, the geometric optics approximation is applied, which assumes that the correlation between two adjacent points is very strong. The model is then proportional to two surface slope probabilities, for which the slopes would specularly reflect the beams in the double scattering process. In addition, the slope distributions are related with each other by a propagating function, which accounts for the second-order illumination function. The companion paper is devoted to the simulation of this model and comparisons with an 'exact' numerical method.     

The weighted curvature approximation in scattering from sea surfaces

A family of unified models in scattering from rough surfaces is based on local corrections of the tangent plane approximation through higher-order derivatives of the surface. We revisit these methods in a common framework when the correction is limited to the curvature, that is essentially the second-order derivative. The resulting expression is formally identical to the weighted curvature approximation, with several admissible kernels, however. For sea surfaces under the Gaussian assumption, we show that the weighted curvature approximation reduces to a universal and simple expression for the off-specular normalized radar cross-section (NRCS), regardless of the chosen kernel. The formula involves merely the sum of the NRCS in the classical Kirchhoff approximation and the NRCS in the small perturbation method, except that the Bragg kernel in the latter has to be replaced by the difference of a Bragg and a Kirchhoff kernel. This result is consistently compared with the resonant curvature approximation. Some numerical comparisons with the method of moments and other classical approximate methods are performed at various bands and sea states. For the copolarized components, the weighted curvature approximation is found numerically very close to the cut-off invariant two-scale model, while bringing substantial improvement to both the Kirchhoff and small-slope approximation. However, the model is unable to predict cross-polarization in the plane of incidence. The simplicity of the formulation opens new perspectives in sea state inversion from remote sensing data.     

Topical Issue on Wave Scattering from Rough Surfaces and Related Phenomena (Part 1)

Many fast asymptotic models of electromagnetic scattering from a single rough interface have been developed over the last few years, but only a few have been developed on stacks of rough interfaces. The specific case of very rough surfaces, compared to the incident wavelength, has not been treated before, which is the context of this paper. The model starts from the iteration of the Kirchhoff approximation to calculate the fields scattered by a rough layer, and is reduced to the high-frequency limit in order to rapidly obtain numerical results. The shadowing effect, important under grazing angles, is taken into account. The model can be applied to any given slope statistics. Then, the model is compared with a reference numerical method based on the method of moments, which validates the model in the high-frequency limit for lossless and lossy inner media.     

Bistatic scattering from one-dimensional random rough homogeneous layers in the high-frequency limit with shadowing effect

Many fast asymptotic models of electromagnetic scattering from a single rough interface have been developed over the last few years, but only a few have been developed on stacks of rough interfaces. The specific case of very rough surfaces, compared to the incident wavelength, has not been treated before, which is the context of this paper. The model starts from the iteration of the Kirchhoff approximation to calculate the fields scattered by a rough layer, and is reduced to the high-frequency limit in order to rapidly obtain numerical results. The shadowing effect, important under grazing angles, is taken into account. The model can be applied to any given slope statistics. Then, the model is compared with a reference numerical method based on the method of moments, which validates the model in the high-frequency limit for lossless and lossy inner media.     

High-frequency solitons in media with induced scattering from damped low-frequency waves with nonuniform dispersion and nonlinearity

The dynamics of high-frequency field solitons is considered using the extended nonhomogeneous nonlinear Schrödinger equation with induced scattering from damped low-frequency waves (pseudoinduced scattering). This scattering is a 3D analog of the stimulated Raman scattering from temporal spatially homogeneous damped low-frequency modes, which is well known in optics. Spatial inhomogeneities of secondorder linear dispersion and cubic nonlinearity are also taken into account. It is shown that the shift in the 3D spectrum of soliton wavenumbers toward the short-wavelength region is due to nonlinearity increasing in coordinate and to decreasing dispersion. Analytic results are confirmed by numerical calculations.     

On the high-frequency limit of the second-order small-slope approximation

Small-slope approximation (SSA) is a scattering theory that is supposed to unify both the small-perturbation model and the Kirchhoff approximation (KA). We study and compute the second-order small-slope approximation (SSA2) in a high-frequency approximation (SSA2-hf) that makes it proportional to the first-order term, with a roughness-independent factor. For the 3D electromagnetic problem we show analytically that SSA2-hf actually meets KA in the case of perfectly conducting surfaces. This no longer holds in the dielectric case but we give numerical evidence that the two methods remain extremely close to each other for moderate scattering angles. We discuss the potential applications of SSA2-hf and give some 2D numerical comparison with rigorous computations.     

Regions of validity of geometric optics and Kirchhoff approximations for reflection from Gaussian random rough dielectric surfaces

In this paper, the effects of characteristics of incident light and the geometrical parameters to the reflectivity of dielectric Gaussian random rough surfaces are presented. The behaviors of the reflectivity vs. several parameters are quantified using approximate methods: the geometric optics approximation (GOA) and the Kirchhoff approximation (KA) and an exact method called integral method (IM). Finally, we determine the limits of validity of approximate methods by comparisons with IM results. The regions of validity of approximate methods depending of many geometrical and physical parameters: roughness, Brewster and shadowing effects, multiple reflections, surface materials, and nature of polarization. The broader domain of validity (DV) is for KA, at normal TE-polarized incident light, for the higher dielectric permittivity. However, the narrowed DV is for GOA, at normal TM-polarized incident light for lower dielectric permittivity.     

Free-carrier absorption and dielectric losses in crystals with dipole-impurities

In the one-band isotropic effective-mass approximation and the first Born approximation of the scattering the high-frequency electrical conductivity, dielectric losses and free-carrier absorption in crystals are calculated when the scattering mechanism is connected with the screened electrical dipoles. As in the case of low-frequency conductivity [1] it is shown that in some cases the scattering from dipole impurity centres may be essential and even more effective for the free-carrier absorption and dielectric losses than the other scattering mechanisms.     

Analytical comparison between the surface current integral equation and the second-order small-slope approximation

Abstract

This paper is the third in a series discussing a new approximate bistatic model for electromagnetic scattering from perfectly conducting rough surfaces. Our previous approach supplemented the Kirchhoff model through the addition of new terms involving linear orders in slope and surface elevation differences that arise naturally from a second iteration of the surface current integral equation. This completion of the Kirchhoff was shown to provide the correct first-order small perturbation method (SPM-1) in the general bistatic context. The agreement with SPM-1 was achieved because differences of surface heights are no longer expanded in powers of surface slope. While consistent with SPM, our previous formulation fails to reconverge toward the Kirchhoff model, at some incidence and scattered angles, when the illuminated surface satisfies the high frequency roughness condition. This weakness is also shared with the first-order small slope approximation (SSA-1) which is structurally equivalent to our previous formulation where the polarization is independent of surface roughness. The second-order small slope approximation (SSA-2), which satisfies the SPM-1 and second-order small perturbation method (SPM-2) limits by construction, was shown by Voronovich to converge toward the tangent plane approximation of the Kirchhoff model under high frequency conditions. In the present paper, we show that, in addition to the linear orders in our previous model, one must now include cross-terms between slope and surface elevation to ensure convergence toward both high frequency and small perturbation limits. With the inclusion of these terms, our new formulation becomes comparable to the SSA-2 (second-order kernel) without the need to evaluate all the quadratic order slope and elevations terms. SSA-2 is more complete, however, in the sense that it guarantees convergence toward the second-order Bragg limit (SPM-2) in the fully dielectric case in addition to both SPM-1 and Kirchhoff. Our new generalization is shown to explain correctly extra depolarization in specular conditions to be caused by surface curvature and surface autocorrelation for incoherent and coherent scattering, respectively. This result will have large repercussions on the interpretation of bistatically reflected signals such as those from GPS.     

On the high-frequency limit of the second-order small-slope approximation

Abstract

Small-slope approximation (SSA) is a scattering theory that is supposed to unify both the small-perturbation model and the Kirchhoff approximation (KA). We study and compute the second-order small-slope approximation (SSA2) in a high-frequency approximation (SSA2-hf) that makes it proportional to the first-order term, with a roughness-independent factor. For the 3D electromagnetic problem we show analytically that SSA2-hf actually meets KA in the case of perfectly conducting surfaces. This no longer holds in the dielectric case but we give numerical evidence that the two methods remain extremely close to each other for moderate scattering angles. We discuss the potential applications of SSA2-hf and give some 2D numerical comparison with rigorous computations.     

An iterative method for scattering from rough dielectric or conducting interfaces

Abstract

We introduces an iterative method for scattering a two-dimensional scalar wave from a rough interface between two media. The method is applicable to the case of electromagnetic scattering from a rough metallic or dielectric surface that varies only in one dimension. The first iteration is equivalent to the Kirchhoff approximation, and the series converges in one step for a flat surface. We discuss the conditions for convergence, and find that they are similar to those which Meecham showed to be necessary in the Dirichlet case.