Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface: II. Multiple scattering

The integral equation model (IEM) has been developed over the last decade and it has become one of the most widely used theoretical models for rough-surface scattering in microwave remote sensing. In the IEM model the shadowing function is typically either omitted or a form based on geometric optics with single reflection is used. In this paper, a shadowing function for one-dimensional rough surfaces which incorporates multiple scattering, finite surface length and both monostatic and bistatic configurations is developed. For any uncorrelated process, the resulting equation can be expressed in terms of the monostatic statistical shadowing function with single reflection, derived in the preceding companion paper. The effect of correlation between the surface slopes and heights for a Gaussian surface is studied to illuminate the range over which such correlations can be ignored. It is found that while the correlation between surface slopes and heights in the monostatic statistical shadowing function with single reflection can be ignored, when calculating the average shadowing function with double reflection the correlation between slopes and heights between points must be incorporated.     

Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface: II. Multiple scattering

Abstract

The integral equation model (IEM) has been developed over the last decade and it has become one of the most widely used theoretical models for rough-surface scattering in microwave remote sensing. In the IEM model the shadowing function is typically either omitted or a form based on geometric optics with single reflection is used. In this paper, a shadowing function for one-dimensional rough surfaces which incorporates multiple scattering, finite surface length and both monostatic and bistatic configurations is developed. For any uncorrelated process, the resulting equation can be expressed in terms of the monostatic statistical shadowing function with single reflection, derived in the preceding companion paper. The effect of correlation between the surface slopes and heights for a Gaussian surface is studied to illuminate the range over which such correlations can be ignored. It is found that while the correlation between surface slopes and heights in the monostatic statistical shadowing function with single reflection can be ignored, when calculating the average shadowing function with double reflection the correlation between slopes and heights between points must be incorporated.     

Shadowing function with single reflection from anisotropic Gaussian rough surface. Application to Gaussian, Lorentzian and sea correlations

In this paper, the monostatic (transmitter and receiver are located at the same place) and bistatic (transmitter and receiver are distinct) statistical shadowing functions from an anisotropic two-dimensional randomly rough surface are presented. This parameter is especially important in the case of grazing angles for computing the bistatic scattering coefficient in optical and microwave frequencies. The objective of this paper is to extend the previous work (Bourlier C, Berginc G and Saillard J 2002 Waves Random Media 12 145-74), valid for a one-dimensional surface, to a two-dimensional anistropic surface by considering a joint Gaussian process of surface slopes and heights separating two points of the surface. The monostatic average (statistical shadowing function average over the statistical variables) shadowing function is then performed in polar coordinates with respect to the incidence angle, the azimuthal direction and the surface height two-dimensional autocorrelation function. In addition, for a bistatic configuration, it depends on the incidence angle and azimuthal direction of the receiver. For Gaussian and Lorentzian correlation profiles and practically important power-type spectra such as the Pierson-Moskowitz sea roughness spectrum, the numerical solution, obtained from generating the surface Gaussian elevations (Monte Carlo method), is compared with the uncorrelated and correlated models. The results show that the correlation underestimates the shadow slightly, whereas the uncorrelated results weakly overpredict the shadow and are close to the numerical solution.     

Bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase and scalar approximation with shadowing effect: comparisons with experiments and application to the sea surface

In this paper, the bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase method and scalar approximation with shadowing effect is investigated. Both of these approaches use the Kirchhoff integral. With the stationary phase, the bistatic cross section is formulated in terms of the surface height joint characteristic function where the shadowing effect is investigated. In the case of the scalar approximation, the scattering function is computed from the previous characteristic function and in terms of expected values for the integrations over the slopes, where the shadowing effect is analysed analytically. Both of these formulations are compared with experimental data obtained from a Gaussian one-dimensional randomly rough perfectly-conducting surface. With the stationary-phase method, the results are applied to a two-dimensional sea surface.     

Bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase and scalar approximation with shadowing effect: comparisons with experiments and application to the sea surface

Abstract

In this paper, the bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase method and scalar approximation with shadowing effect is investigated. Both of these approaches use the Kirchhoff integral. With the stationary phase, the bistatic cross section is formulated in terms of the surface height joint characteristic function where the shadowing effect is investigated. In the case of the scalar approximation, the scattering function is computed from the previous characteristic function and in terms of expected values for the integrations over the slopes, where the shadowing effect is analysed analytically. Both of these formulations are compared with experimental data obtained from a Gaussian one-dimensional randomly rough perfectly-conducting surface. With the stationary-phase method, the results are applied to a two-dimensional sea surface.     

Study of MPI based on parallel MOM on PC clusters for EM-beam scattering by 2-D PEC rough surfaces

This paper firstly applies the finite impulse response filter (FIR) theory combined with the fast Fourier transform (FFT) method to generate two-dimensional Gaussian rough surface. Using the electric field integral equation (EFIE), it introduces the method of moment (MOM) with RWG vector basis function and Galerkin's method to investigate the electromagnetic beam scattering by a two-dimensional PEC Gaussian rough surface on personal computer (PC) clusters. The details of the parallel conjugate gradient method (CGM) for solving the matrix equation are also presented and the numerical simulations are obtained through the message passing interface (MPI) platform on the PC clusters. It finds significantly that the parallel MOM supplies a novel technique for solving a two-dimensional rough surface electromagnetic-scattering problem. The influences of the root-mean-square height, the correlation length and the polarization on the beam scattering characteristics by two-dimensional PEC Gaussian rough surfaces are finally discussed.     

A monostatic illumination function with surface reflections from one-dimensional rough surfaces

When solving scattering or emissivity problems for rough surfaces, the shadowing effect is often taken into account. Furthermore, for rough surfaces with large root mean square slope, surface reflections of the incidence or emission ray should not be neglected, especially at large observation angles. In this paper, a model of the monostatic statistical illumination function for one-dimensional rough surfaces with single surface reflection is developed, which is based on the Smith illumination function. A Monte Carlo ray-tracing algorithm is used to evaluate the accuracy of the present model. It is shown that, when neglecting the correlation between heights and slopes of the surface, the present model agrees quite well with the Monte Carlo result. Moreover, the result is improved if the correlation between heights and slopes is taken into account. For practical purposes, an empirical factor is introduced to improve the performance of the uncorrelated first-order illumination function to avoid computing the correlated one, which takes a long computation time. Besides, the first-order illumination function is significant at large observation angles, which could be promising to overcome problems in models of surface infrared emissivity where underestimation occurs compared with experimental measurements.     

Theoretical study of the Kirchhoff integral from a two-dimensional randomly rough surface with shadowing effect: application to the backscattering coefficient for a perfectly-conducting surface

In this paper, the backscattering coefficient of a two-dimensional randomly rough perfectly-conducting surface is investigated using the Kirchhoff approach with a shadowing function. The rough surface height/slope correlations assumed to be Gaussian are accounted for in this analysis. The scattering coefficient is then formulated in terms of a characteristic function for the integrations over the surface heights, in terms of expected values for the integrations over the surface slopes. Numerical comparisons of Kirchhoff's approach (KA) with the stationary-phase (SP) approximation are made with respect to the choice of the one-dimensional surface height autocorrelation function and the shadowing effect. For an isotropic surface the results show that SP underestimated the incoherent backscattering coefficient compared with KA. Moreover, when the correlation between the slopes and the heights is neglected, the shadowing effect may be ignored.     

Theoretical study of the Kirchhoff integral from a two-dimensional randomly rough surface with shadowing effect: application to the backscattering coefficient for a perfectly-conducting surface

Abstract

In this paper, the backscattering coefficient of a two-dimensional randomly rough perfectly-conducting surface is investigated using the Kirchhoff approach with a shadowing function. The rough surface height/slope correlations assumed to be Gaussian are accounted for in this analysis. The scattering coefficient is then formulated in terms of a characteristic function for the integrations over the surface heights, in terms of expected values for the integrations over the surface slopes. Numerical comparisons of Kirchhoff's approach (KA) with the stationary-phase (SP) approximation are made with respect to the choice of the one-dimensional surface height autocorrelation function and the shadowing effect. For an isotropic surface the results show that SP underestimated the incoherent backscattering coefficient compared with KA. Moreover, when the correlation between the slopes and the heights is neglected, the shadowing effect may be ignored.     

一维粗糙面散射的射线追踪法研究

在考虑遮蔽效应的情况下,应用射线追踪法对一维导体和介质的高斯粗糙面的散射系数进行了研究,并分别计算了考虑一次反射和二次反射的散射系数。同时,利用蒙特卡罗法生成一维高斯粗糙面,计算了考虑一次、二次反射和遮蔽时不同均方根斜率粗糙面的平均散射系数。     

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.     

指数型粗糙地面与上方矩形截面柱宽带电磁散射的时域有限差分法研究

采用土壤介电常数的四成分模型表示实际地面的介电特性,应用指数型分布粗糙面模型和Monte Carlo方法模拟实际的粗糙地面,运用时域有限差分方法研究了微分高斯脉冲波照射时粗糙地面与上方目标的宽带电磁散射问题,得出了后向复合散射系数的频率响应曲线,计算了后向复合散射系数随粗糙地面高度起伏均方根、相关长度、土壤湿度、目标尺寸、高度、倾角等几何参数及介电参数等的变化情况,并做了详细分析与讨论,得到了指数型粗糙地面与上方目标宽带电磁散射特性.     

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)     

Directional hemispherical radiative properties of random dielectric rough surfaces

In this paper the directional hemispherical reflectivity and transmissivity of one-dimensional, randomly rough, dielectric surfaces are determined by the use of the integral method. This method is derived from electromagnetic theory without any restrictive hypotheses. Since this exact approach is computationally very intensive, a geometric optics approximation method is also developed. Curves displaying radiative properties versus the correlation length for a constant mean square deviation of the surface from flatness are presented. In this respect, the influence on the validity of the approximate method of multiple scattering, the shadowing effect and the real index of refraction of the dielectric have been investigated. Transverse electric and transverse magnetic polarized incident plane waves are considered. For the latter, our interest is focused on the influence of roughness on the reflected and transmitted intensities for an angle of incidence close to the Brewster angle.     

Directional hemispherical radiative properties of random dielectric rough surfaces

In this paper the directional hemispherical reflectivity and transmissivity of one-dimensional, randomly rough, dielectric surfaces are determined by the use of the integral method. This method is derived from electromagnetic theory without any restrictive hypotheses. Since this exact approach is computationally very intensive, a geometric optics approximation method is also developed. Curves displaying radiative properties versus the correlation length for a constant mean square deviation of the surface from flatness are presented. In this respect, the influence on the validity of the approximate method of multiple scattering, the shadowing effect and the real index of refraction of the dielectric have been investigated. Transverse electric and transverse magnetic polarized incident plane waves are considered. For the latter, our interest is focused on the influence of roughness on the reflected and transmitted intensities for an angle of incidence close to the Brewster angle.     

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)     

Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation

This paper presents a theory of scattering from very rough metallic and dielectric surfaces using the first- and second-order Kirchhoff approximations (KA) modified with the angular and propagation shadowing. The shadowing functions limit the single and double scattered waves which are illuminated and not shadowed by the surface. The theoretical results are compared with the Monte Carlo simulations showing the range of validity of the theory. The theory is applicable to the range where the RMS height is close to a wavelength and the RMS slope is close to unity, and the second medium is lossy. The second-order scattering includes two waves travelling in opposite directions on the surface, giving a physical explanation of the enhanced backscattering.     

雪层覆盖土壤表面与半埋柱体宽带复合散射FDTD方法

为满足半埋目标测量和检测的需要,应用指数型分布粗糙面模型和Monte Carlo方法模拟雪层、土壤表面,运用时域有限差分方法研究雪层覆盖的土壤表面与半埋矩形截面柱复合模型的宽带电磁散射,得出宽带复合散射系数的频率响应曲线,计算宽带复合散射系数随雪层表面、土壤表面高度起伏均方根、相关长度,雪层类型、雪层厚度,土壤层含水量,矩形截面柱几何参数、倾角、埋藏深度,电磁波入射角等参数的变化. 结果表明,雪层表面高度起伏均方根、相关长度对宽带复合散射系数影响较小,其它参数对宽带复合散射系数均有显著影响且较为复杂.     

粗糙体目标激光束散射场的互相关函数统计特征

依据高斯波束体目标散射特征,研究了粗糙体目标高斯波束散射场量的互相关函数统计特征。在激光波束入射下,通过数值方法讨论圆球类目标、不同半径、不同材料、不同入射波束极化等条件下,散射场量的互相关函数随散射角变化情况。数值计算结果表明:金属薄膜材料比非金属镀漆材料互相关函数量值要大,金属材料球体目标的后向散射互相关函数值较大,而非金属镀漆材料球体互相关函数值较小。材料表面的粗糙度和目标尺寸对互相关函数影响较大,而入射光的极化方式对相关函数有影响但影响较小。     

Unification of the Kirchhoff approximation and the method of moment for optical wave scattering from the lossy dielectric Gaussian random rough surface

The optical wave scattering from one-dimensional （1D） lossy dielectric Gaussian random rough surface is studied. The tapered incident wave is introduced into the classical Kirchhoff approximation （KA）, and the shadowing effect is also taken into account to make the KA results have a high accuracy. The definition of the bistatic scattering coefficient of the modified KA and the method of moment （MOM） are unified. The characteristics of the optical wave scattering from the lossy dielectric Gaussian random rough surface of different parameters are analyzed by implementing MOM.