A practical second-order electromagnetic model in the quasi-specular regime based on the curvature of a ‘good-conducting’ scattering surface

Abstract

This letter presents an approximate second-order electromagnetic model where polarization coefficients are surface dependent up to the curvature order in the quasi-specular regime. The scattering surface is considered ‘good-conducting’ as opposed to the case for our previous derivation where perfect conductivity was assumed. The model reproduces dynamically, depending on the properties of the scattering surface, the tangent-plane (Kirchhoff) or the first-order small-perturbation (Bragg) limits. The convergence is assumed to be ensured by the surface curvature alone. This second-order model is shown to be consistent with the small-slope approximation of Voronovich (SSA-1+SSA-2) for perfectly conducting surfaces. Our model differs from SSA-1 + SSA-2 in its dielectric expression, to correct for a full convergence toward the tangent-plane limit under the ‘good-conducting’ approximation. This new second-order formulation is simple because it involves a single integral over the scattering surface and therefore it is suitable for a vast array of analytical and numerical applications in quasi-specular applications.     

Formal tilt invariance of the local curvature approximation

Abstract

Tilt invariance is a stringent but necessary condition that a second-order wave scattering model must satisfy in order to qualify for a broad range of applications. This invariance expresses the fact that the scattering model is unchanged whether the tilting of the scattering surface is implemented before or after its reduction to the limit of the small-perturbation method (SPM). Our scattering model is based on a second-order kernel which is quadratic in its lowest order with respect to successive derivatives of the rough surface. Hence, it is termed the local curvature approximation (LCA). We have previously demonstrated that the LCA is approximately tilt invariant in the quasi-specular and quasi-backscattering geometries. In this contribution, LCA is made formally tilt invariant up to first order in the tilting vector. It will be shown that this formal tilt invariance is achieved mainly through inclusion of polarization mixing due to out-of-plane tilt. Even though the LCA formally reduces to the SPM and Kirchhoff limits in addition to tilt invariance, its curvature kernel stays reasonably concise and practical to implement in both analytical and numerical evaluations. This curvature kernel may also be used in the other two formulations of our model, namely the non-local curvature approximation and the weighted curvature approximation.     

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.     

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

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.     

Bistatic scattering from an anisotropic sea surface: Numerical comparison between the first-order SSA and the TSM models

The first-order small-slope approximation (SSA-1) model is used for numerical predictions of the normalized radar cross section (NRCS) of an anisotropic ocean surface in bistatic configurations for the K_u-band radar frequency. The calculations were made by assuming the Elfouhaily et al. surface-height spectrum for fully developed seas. In the forward-backward case, the SSA-1 presents an agreement with the geometric optics limit of the Kirchhoff approximation results in the near-specular directions where it is well known that the last model works well. In the fully bistatic case, SSA-1 numerical results are compared with those of the two-scale model in several configurations as a function of wind speed, wind direction, incident/scattering angles and for co-and cross-polarization states. Good agreement between the two models is noted in the co-and cross-polarization case with a small difference of about 1-2 dB. But in certain configurations, the SSA-1 model tends to overestimate the radar cross section peak behaviour. This irregularity is discussed and interpreted. The main purpose of this paper is to analyse NRCS predictions based on the SSA-1 model in a fully bistatic configuration.     

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

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.     

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.     

Multiple scattering in the high-frequency limit with second-order shadowing function from 2D anisotropic rough dielectric surfaces: II. comparison with numerical results

This second part presents illustrative examples of the model developed in the companion paper, which is based on the first- and second-order optics approximation. The surface is assumed to be Gaussian and the correlation height is chosen as anisotropic Gaussian. The incoherent scattering coefficient is computed for a height rms range from 0.5λ 1λwhere λ is the electromagnetic wavelength), for a slope rms range from 0.5 to 1 and for an incidence angle range from 0 to 70°. In addition, simulations are presented for an anisotropic Gaussian surface and when the receiver is not located in the plane of incidence. For a metallic and dielectric isotropic Gaussian surfaces, the cross- and co-polarizations are also compared with a numerical approach obtained from the forward.backward method with a novel spectral acceleration algorithm developed by Torrungrueng and Johnson (2001, JOSA A 18). (Some figures in this article are in colour only in the electronic version)     

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.     

Polarization dependency of enhanced multipath radar backscattering from an ocean-like surface

Previous analyses of electromagnetic scattering from a two-scale ocean surface assumed the tilts on a large-scale surface to be small. This means that multiple scattering between large-scale roughnesses is insignificant. If the tilts are not small, multipath-enhanced backscattering may occur due to quasi-specular reflections between the opposite slopes of a large-scale surface component. We have considered the simplest situation; this involves one reflection from the large-scale component and one single-scattering from the small-scale component. The coherent addition of this process to the reciprocally reverse one creates multipath-enhanced backscattering. The relative gain in the HH backscattering cross section up to the level of the VV signal was obtained for surface-wave slopes of about 30° and for large incidence angles. This gain occurs because the VV signal experiences an extinction during reflection at incident angles close to the pseudo-Brewster angle. The presented model provides insight into one scattering mechanism that is possibly responsible for the departure of radar sea experimental data from predictions by the conventional two-scale model.     

Multiple scattering in the high-frequency limit with second-order shadowing function from 2D anisotropic rough dielectric surfaces: II. comparison with numerical results

Abstract

This second part presents illustrative examples of the model developed in the companion paper, which is based on the first- and second-order optics approximation. The surface is assumed to be Gaussian and the correlation height is chosen as anisotropic Gaussian. The incoherent scattering coefficient is computed for a height rms range from 0.5λ 1λwhere λ is the electromagnetic wavelength), for a slope rms range from 0.5 to 1 and for an incidence angle range from 0 to 70°. In addition, simulations are presented for an anisotropic Gaussian surface and when the receiver is not located in the plane of incidence. For a metallic and dielectric isotropic Gaussian surfaces, the cross- and co-polarizations are also compared with a numerical approach obtained from the forward.backward method with a novel spectral acceleration algorithm developed by Torrungrueng and Johnson (2001, JOSA A 18). (Some figures in this article are in colour only in the electronic version)     

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.     

Weighted curvature approximation: numerical tests for 2D dielectric surfaces

Abstract

The weighted curvature approximation (WCA) was recently introduced by Elfouhaily et al [7] as a unifying scattering theory that reproduces formally both the tangent-plane and the small-perturbation model in the appropriate limits, and is structurally identical to the former approximation with some different slope-dependent kernel. Due to the simplicity of its formulation, the WCA is interesting from a numerical point of view and the aim of the present paper is to establish its accuracy on some representative test cases. We derive statistical formulae for the coherent field and the cross-section in the case of stationary Gaussian random surfaces. We then specialize to the case of isotropic Gaussian spectra and perform numerical comparisons against rigorous method of moments (MoM)-based results on 2D dielectric surfaces. We show that the WCA remains extremely accurate in a roughness range where other first-order classical approximations (small-slope and Kirchhoff) clearly fail, at the same computational cost.

(Some figures in this article are in colour only in the electronic version)     

Wave diffraction by a concave statistically rough surface

We consider a statistically rough impedance surface that is concave on average in contrast to a plane. Backscattering from such a surface is considered based on the small perturbation theory method. The diffraction problem is divided into two parts which are considered separately: the problem of scattering by small roughness (assumed to be local) and the propagation of incident and scattered fields over a smooth large-scale concave surface. In contrast to the 'two-scale' scattering model, the zero-order unperturbed wavefield is not assumed to be specularly reflected from the local tangent plane to the smooth surface, but it is a solution of a corresponding diffraction problem. Two particular cases of smooth surfaces are considered: first, the inner surface of a concave cylinder with a constant radius and finite angular pattern, and second, a compound surface that consists of a coupled half-plane and the cylindrical surface mentioned above. In a geometrical optics limit and with propagation at low grazing angles, the analytical results for a zero-order (unperturbed) field are obtained for these two cases in the form of a series over multiple specular reflected fields. It is shown that these non-local processes lead to the essential increase in the backscattering cross section in comparison with the two-scale model and tangent-plane approach.     

Wave diffraction by a concave statistically rough surface

Abstract

We consider a statistically rough impedance surface that is concave on average in contrast to a plane. Backscattering from such a surface is considered based on the small perturbation theory method. The diffraction problem is divided into two parts which are considered separately: the problem of scattering by small roughness (assumed to be local) and the propagation of incident and scattered fields over a smooth large-scale concave surface. In contrast to the ‘two-scale’ scattering model, the zero-order unperturbed wavefield is not assumed to be specularly reflected from the local tangent plane to the smooth surface, but it is a solution of a corresponding diffraction problem. Two particular cases of smooth surfaces are considered: first, the inner surface of a concave cylinder with a constant radius and finite angular pattern, and second, a compound surface that consists of a coupled half-plane and the cylindrical surface mentioned above. In a geometrical optics limit and with propagation at low grazing angles, the analytical results for a zero-order (unperturbed) field are obtained for these two cases in the form of a series over multiple specular reflected fields. It is shown that these non-local processes lead to the essential increase in the backscattering cross section in comparison with the two-scale model and tangent-plane approach.     

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.     

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.     

Three dimensional analyses of scattering by pressure-release sinusoidal surfaces

Scattering by pressure-release sinusoidal surfaces in three dimensions is analyzed using the Fresnel phase approximation and realistic source and receiver directivity approximations. Geometrical shadowing and second-order scattering are explicitly included to explore the validity of the Kirchhoff approximation. No restrictions on the surface heights and slopes are made. The "goodness" of the resulting expressions is verified by requiring the pressure scattered by a sinusoidal surface to reduce to the image solution as the surface amplitude goes to zero. The first-order scattered pressure achieves a very good approximation to the image solution, and the second-order scattered pressure goes to zero, as expected, under this requirement. The theory is compared with available experimental scattering measurements, and the agreement is good. Because the slopes on the experimental surface are very steep, shadowing corrections are indispensible to achieving accurate first and second order scattering results. Shadowing has a greater impact on the scattering prediction than the second-order scattering contribution. This suggests that the Kirchhoff approximation may be more robust when incorporated into a theory including a detailed shadowing treatment as well as the Fresnel phase approximation and a good directivity approximation.