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.     

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 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.     

Doppler spectrum of polarimetric scattering field from two-dimensional time-varying nonlinear sea surfaces

Polarimetric scattering model of second-order small-slope approxi-mation combined with “choppy wave" model (CWM) for describing nonlinear hydrodynamic interactions between ocean waves is utilized in this paper to investigate the influence of sea surface nonlinearities on backscattering coefficient as well as Doppler spectrum signatures including Doppler shift and spectral bandwidth. Simulation results show that at moderate to large incidence angles the Doppler shift and spectral bandwidth of the CWM nonlinear sea surfaces are significantly larger than those of linear sea surfaces, in particular at low grazing angles. In addition, Doppler signatures show distinct polarization dependence, and most importantly the cross-polarized Doppler signatures significantly differ from the co-polarized ones. It is also indicated that co-polarized Doppler shift increases obviously with wind speed increasing, whereas the cross-polarized Doppler shift looks less sensitive to wind speed variations. The difference of Doppler signatures between co- and cross-polarization is potentially valuable for ocean remote sensing applications, especially for observing very high winds.     

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.     

Investigation on the Doppler shifts induced by 1-D ocean surface wave displacements by the first order small slope approximation theory: comparison of hydrodynamic models

Based on the first order small slope approximation theory (SSA-I) for oceanic surface electromagnetic scattering,this paper predicts the Doppler shifts induced by wave displacements.Theoretical results from three distinct hydrodynamic models are compared:a linear model,the nonlinear Barrick model,and the nonlinear Creamer model.Meanwhile,the predicted Doppler shifts are also compared with the results associated to the resonant Bragg waves and the so-called long waves in the framework of the two-scale model.The dependences of the predicted Doppler shifts on the incident angle,the radar frequency,and the wind speed are discussed.At large incident angles,the predicted Doppler shifts for the linear and nonlinear Barrick models are found to be insensitive to the wind speed and this phenomenon is not coincident with the experimental data.The conclusions obtained in this paper are promising for better understanding the properties of time dependent radar echoes from oceanic surfaces.     

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.     

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.     

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.     

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.     

Convergence of solutions to the rough-surface scattering problem

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.     

Microwave Doppler spectra of sea return at small incidence angles: specular point scattering contribution

It is well known that the sea return echo contains contributions from at least two scattering mechanisms. In addition to the resonant Bragg scattering, the specular point scattering plays an important role as the incidence angle becomes smaller (≤20o). Here, in combination with the Kirchhoff integral equation of scattering field and the stationary phase approximation, analytical expressions for Doppler shift and spectral bandwidth of specular point scattering, which are insensitive to the polarization state, are derived theoretically. For comparison, the simulated results related to the two-scale method (TSM) and the method of moment (MOM) are also presented. It is found that the Doppler shift and the spectral bandwidth given by TSM are insufficient at small incidence angles. However, a comparison between the analytical results and the numerical simulations by MOM in the backscatter configuration shows that our proposed formulas are valid for the specular point scattering case. In this work, the dependences of the predicted results on incidence angle, radar frequency, and wind speed are also discussed. The obtained conclusions seem promising for a better understanding of the Doppler spectra of the specular point scattering fields from time-varying sea surfaces.     

基于十字变形结构的高效超宽带线性极化转换超表面

提出了一种基于十字变形结构超表面的极化转换器,在反射模式下获得了高效超宽带的交叉极化反射.在8.4到20.7GHz频段内交叉极化反射率大于-0.2dB,而共极化反射率小于-12dB,在谐振频率点处交叉极化反射率大于-0.03dB,而共极化反射率达到-60dB,即在谐振点处几乎可实现完全的交叉极化转换;相对带宽达84.5%,交叉极化的平均转换效率为96.7%;此外,利用电路板刻蚀制备了此极化转换器样品,实验测试其交叉极化反射率在工作频段内大于-1dB,而共极化反射率小于-10dB,实验结果与模拟结果吻合,验证了此超表面可以在超宽的频带内实现线极化电磁波的交叉极化转换.本文设计的超宽带极化转换超表面具有转换效率高和几何结构简单的优点,可被扩展到太赫兹甚至是可见光频段.     

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.     

Non-local small-slope approximation for wave scattering from rough surfaces

A new general analytical approach to solving the problems of wave scattering from rough surfaces, referred to as the non-local small-slope approximation (NLSSA), is suggested. It is formulated in the general form both for vector and scalar waves. This approach is valid for an arbitrary wavelength of radiation provided that the slopes of the undulations are small enough. The NLSSA represents a generalization of the small-slope approximation to situations where double scattering (in the optical sense) appears. It is demonstrated that with appropriate approximations the NLSSA of the lowest order reduces to the small-slope approximation of the second order.     

海面微波散射场多普勒谱特性研究

基于粗糙面电磁散射双尺度模型推导给出了海面微波散射场多普勒谱频移和谱宽的理论公式, 在该理论公式的推导过程中同时考虑了大尺度海浪的倾斜调制、遮蔽效应和曲率修正效应等因素的影响. 文中将理论公式计算结果与精确数值结果进行了比较, 并讨论了倾斜调制、遮蔽效应及曲率修正效应等因素对多普勒频移和谱宽的影响, 发现倾斜调制使水平极化散射回波多普勒频移显著增大, 从而导致水平极化回波多普勒频移比垂直极化回波多普勒频移大; 在中等入射角度区域, 遮蔽效应和曲率修正效应对多普勒谱并无显著影响, 而在掠射条件下, 遮蔽效应使得多普勒频移增大、谱宽变窄. 本研究对深入理解动态海面散射场频谱特性具有一定参考意义.     

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)     

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.     

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.     

沙丘粗糙面的二次极化电磁散射

研究了新月形沙丘粗糙面的二次极化电磁散射. 结合射线追踪理论, 由一次散射面元的反射场照射到二次散射面元, 采用基尔霍夫近似推导了二次散射面元的二次极化散射场. 计算结果表明二次极化散射结果在特定的角度和类型范围内有显著影响. 在电磁波射向背风坡时可以发现其同极化散射截面在入射角较大时大于其他入射方向的结果, 入射角在休止角附近时的交叉极化散射截面出现峰值, 以及前后狭长沙丘之间的二次极化散射特别突出. 本文结果可用于反演分析沙漠地区的风场信息. 关键词： 新月形沙丘 二次极化散射 射线追踪 休止角