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.     

Two families of non-local scattering models and the weighted curvature approximation

Abstract

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.     

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.     

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

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.     

A simplified asymptotic theory for ocean surface electromagnetic wave scattering

The normalized radar cross-section (NRCS) expression of the Local Curvature Approximation (LCA-1) is derived to first order. The polarization sensitivity of this model is compared to the Kirchhoff Approximation (KA), Two-Scale Model (TSM), Small Slope Approximation (SSA-1) and Small Perturbation Method (SPM-1) to first order in the backscattering configuration. Analytical comparisons and numerical simulations show that LCA-1 and TSM could be rewritten with the same formulation and that their polarization sensitivities are comparable. Comparisons with experimental data acquired in C- and Ku-band reveal that the polarization sensitivities of these models are not adequate. However, the NRCS azimuth modulation predicted by LCA-1 is found to be dependent on polarization and sea surface roughness. This property of the LCA-1 model yields to an azimuth modulation for the polarization ratio. Based on the surface curvature correction concept, a simplified electromagnetic model is proposed. The curvature correction is restricted to the resonant wave-number of the sea roughness spectrum. This is found to reproduce the polarization ratio given by experimental data versus incidence angle and wind speed.     

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.     

Scattering by two-dimensional rough surfaces: comparison between the method of moments,Kirchhoff and small-slope approximations

Abstract

We use a rigorous numerical code based on the method of moments to test the accuracy and validity domains of two popular first-order approximations, namely the Kirchhoff and the small-slope approximation(SSA), in the case of two-dimensional rough surfaces. The experiment is performed on two representative types of surfaces: surfaces with Gaussian spectrum, which are the paradigm of single-scale surfaces, and ocean-like surfaces, which belong to the family of multi-scale surfaces. The main outcome of these computations in the former case is that the SSA is outperformed by the Kirchhoff approximation(KA) outside the near-perturbative domain and in fact is quite unpredictable in that its accuracy does not depend only on the slope. For ocean-like surfaces, however, SSA behaves surprisingly well and is more accurate than the KA.     

A simplified asymptotic theory for ocean surface electromagnetic wave scattering

The normalized radar cross-section (NRCS) expression of the Local Curvature Approximation (LCA-1) is derived to first order. The polarization sensitivity of this model is compared to the Kirchhoff Approximation (KA), Two-Scale Model (TSM), Small Slope Approximation (SSA-1) and Small Perturbation Method (SPM-1) to first order in the backscattering configuration. Analytical comparisons and numerical simulations show that LCA-1 and TSM could be rewritten with the same formulation and that their polarization sensitivities are comparable. Comparisons with experimental data acquired in C- and Ku-band reveal that the polarization sensitivities of these models are not adequate. However, the NRCS azimuth modulation predicted by LCA-1 is found to be dependent on polarization and sea surface roughness. This property of the LCA-1 model yields to an azimuth modulation for the polarization ratio. Based on the surface curvature correction concept, a simplified electromagnetic model is proposed. The curvature correction is restricted to the resonant wave-number of the sea roughness spectrum. This is found to reproduce the polarization ratio given by experimental data versus incidence angle and wind speed.     

Scattering by two-dimensional rough surfaces: comparison between the method of moments, Kirchhoff and small-slope approximations

We use a rigorous numerical code based on the method of moments to test the accuracy and validity domains of two popular first-order approximations, namely the Kirchhoff and the small-slope approximation(SSA), in the case of two-dimensional rough surfaces. The experiment is performed on two representative types of surfaces: surfaces with Gaussian spectrum, which are the paradigm of single-scale surfaces, and ocean-like surfaces, which belong to the family of multi-scale surfaces. The main outcome of these computations in the former case is that the SSA is outperformed by the Kirchhoff approximation(KA) outside the near-perturbative domain and in fact is quite unpredictable in that its accuracy does not depend only on the slope. For ocean-like surfaces, however, SSA behaves surprisingly well and is more accurate than the KA.     

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.     

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.     

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.     

Predicted Doppler shifts induced by ocean surface wave displacements using asymptotic electromagnetic wave scattering theories

Sea surface motions can produce different measured Doppler shifts with respect to instrumental configurations (incidence angle, electromagnetic wavelength, polarization). Under Gaussian statistics for the sea surface elevation and in the general framework of asymptotic theories for ocean surface electromagnetic wave scattering, Doppler shifts can be predicted. The small-slope, Kirchhoff, local curvature and resonant curvature approximations are compared in the backscatter configuration. Predicted Doppler shifts for Kirchhoff and small-slope approximations in co-polarized configuration are insensitive to the polarization state. On the other hand, the local and resonant curvature solutions, through a phase perturbation formalism, yield to significant differences between co-polarized predicted Doppler shifts. Comparisons with data are shown to confirm the polarization and wind speed sensitivities.     

Predicted Doppler shifts induced by ocean surface wave displacements using asymptotic electromagnetic wave scattering theories

Sea surface motions can produce different measured Doppler shifts with respect to instrumental configurations (incidence angle, electromagnetic wavelength, polarization). Under Gaussian statistics for the sea surface elevation and in the general framework of asymptotic theories for ocean surface electromagnetic wave scattering, Doppler shifts can be predicted. The small-slope, Kirchhoff, local curvature and resonant curvature approximations are compared in the backscatter configuration. Predicted Doppler shifts for Kirchhoff and small-slope approximations in co-polarized configuration are insensitive to the polarization state. On the other hand, the local and resonant curvature solutions, through a phase perturbation formalism, yield to significant differences between co-polarized predicted Doppler shifts. Comparisons with data are shown to confirm the polarization and wind speed sensitivities.     

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.     

Tilt-invariant theory of rough-surface scattering: I

Abstract

The small-slope approximation (SSA) in rough-surface scattering theory uses the surface slope as a small parameter of expansion. But, from the physical point of view, the slope may not be a restrictive parameter because we can change the slope of a surface simply by tilting the coordinate system. We present the theory of rough-surface scattering in a coordinate-invariant form. The new method, tilt-invariant approximation (TIA), leads to a different expansion that does not require that the slope of a surface be small. For a small Rayleigh parameter this approximate solution provides the correct perturbation theory, for a large Rayleigh parameter it provides the Kirchhoff approximation with several correcting terms.     

Non-Perturbative Heat Kernel Asymptotics on Homogeneous Abelian Bundles

We study the heat kernel for a Laplace type partial differential operator acting on smooth sections of a complex vector bundle with the structure group G × U(1) over a Riemannian manifold M without boundary. The total connection on the vector bundle naturally splits into a G-connection and a U(1)-connection, which is assumed to have a parallel curvature F. We find a new local short time asymptotic expansion of the off-diagonal heat kernel U(t|x, x′) close to the diagonal of M × M assuming the curvature F to be of order t ⁻¹. The coefficients of this expansion are polynomial functions in the Riemann curvature tensor (and the curvature of the G-connection) and its derivatives with universal coefficients depending in a non-polynomial but analytic way on the curvature F, more precisely, on tF. These functions generate all terms quadratic and linear in the Riemann curvature and of arbitrary order in F in the usual heat kernel coefficients. In that sense, we effectively sum up the usual short time heat kernel asymptotic expansion to all orders of the curvature F. We compute the first three coefficients (both diagonal and off-diagonal) of this new asymptotic expansion.     

Convergence of solutions to the rough-surface scattering problem

Abstract

A new formulation of the rough-surface scattering problem is obtained that is closely linked to the Kirchhoff approximation. The governing equation is cast into a form amenable to solution by the method of successive approximations. The domain of convergence of this solution is established and is shown to apply also to the odd-ordered operator expansion, small-slope approximation and perturbation theory provided that the slope of the scattering surface is everywhere less than unity. The analysis is performed for scattering from one-dimensional pressure-release surfaces. Numerical examples are presented for sinusoidal and echelette gratings.