A numerical simulation of scattering from one-dimensionalinhomogeneous dielectric random surfaces

An efficient numerical solution for the scattering problem of inhomogeneous dielectric rough surfaces is presented. The inhomogeneous dielectric random surface represents a bare soil surface and is considered to be comprised of a large number of randomly positioned dielectric humps of different sizes, shapes, and dielectric constants above an impedance surface. Clods with nonuniform moisture content and rocks are modeled by inhomogeneous dielectric humps and the underlying smooth wet soil surface is modeled by an impedance surface. In this technique, an efficient numerical solution for the constituent dielectric humps over an impedance surface is obtained using Green's function derived by the exact image theory in conjunction with the method of moments. The scattered field from a sample of the rough surface is obtained by summing the scattered fields from all the individual humps of the surface coherently ignoring the effect of multiple scattering between the humps. The statistical behavior of the scattering coefficient σ° is obtained from the calculation of scattered fields of many different realizations of the surface. Numerical results are presented for several different roughnesses and dielectric constants of the random surfaces. The numerical technique is verified by comparing the numerical solution with the solution based on the small perturbation method and the physical optics model for homogeneous rough surfaces. This technique can be used to study the behavior of scattering coefficient and phase difference statistics of rough soil surfaces for which no analytical solution exists     

Forward-backward method for scattering from dielectric rough surfaces

The iterative forward-backward (FB) method is a recently proposed efficient technique for numerical evaluation of scattering from perfectly conducting rough surfaces. Extension of the method to include scattering from imperfect conducting surfaces, with a high imaginary part of the complex dielectric constant, has also been proposed. The FB method is further generalized to analyze scattering from dielectric rough surfaces with arbitrary complex dielectric constant. Electric and magnetic equivalent surface currents are split into forward and backward components and equations governing these current components are obtained. As a solution, an iterative scheme is proposed and its convergence rate is analyzed. Finally, the effectiveness of the method is assessed by comparing the obtained scattering results with "exact" ones, computed by employing the usual method of moments (MoM).     

Electromagnetic Scattering From Multilayer Rough Surfaces With Arbitrary Dielectric Profiles for Remote Sensing of Subsurface Soil Moisture

Radar remote sensing of soil moisture content at low frequencies requires an accurate scattering model of realistic soils, which often involves multilayer rough surfaces and dielectric profiles. In this paper, a hybrid analytical/numerical solution to two-dimensional scattering from multilayer rough surfaces separated by arbitrary dielectric profiles based on the extended boundary condition method (EBCM) and scattering matrix technique is presented. The reflection and transmission matrices of rough interfaces are constructed using EBCM. The dielectric profiles are modeled as stacks of piecewise homogeneous dielectric thin layers, whose scattering matrices are computed by recursively cascading reflection and transmission matrices of individual dielectric interfaces. The interactions between the rough interfaces and stratified dielectric profiles are taken into account by applying the generalized scattering matrix technique. The scattering coefficients are obtained by combining the powers computed from the resulting Floquet modes of the overall system. The bistatic scattering coefficients are validated against existing analytical and numerical solutions. Field-collected soil moisture data are then used for numerical simulations to investigate the penetration capability at different frequencies and to address the potential of low-frequency radar systems in estimating deep soil moisture. In particular, soil moisture profiles during dry ground, wet ground, and wet subsurface layer conditions are examined. The results show that both backscattering coefficients and copolarized phase difference at low frequencies are sensitive to the roughness of subsurface interfaces and deep soil moisture. Also, much larger depth sensitivity can be achieved using copolarized phase difference than scattering coefficients     

一维粗糙介质分形海面电磁散射的扩展边界条件法

在采用经典扩展边界条件法处理导体分形粗糙面散射的基础上,将此方法推广到了一维粗糙介质分形海面的电磁散射.通过与传统的基尔霍夫近似法计算结果进行比较,验证了本文所给的方法,分析了不同入射角,不同分维和空间波数时海面双站散射振幅角分布的特点.     

Synthesis, construction, and validation of a fractal surface

Fractal geometry provides reliable models to describe geometrical properties of natural surfaces. Therefore, their use in the electromagnetic scattering methods deserves careful research. In order to have complete insight into the phenomenon, a measurement campaign on a fractal surface in a controlled environment is a key step. In this paper, we propose a technique for building a fractal surface that can be used for electromagnetic scattering evaluation purposes. The surface characteristics are imposed by computer synthesizing a bandlimited Weierstrass-Mandelbrot function, whose actual shape is constructed by means of a cheap innovative technique: the synthesized surface is made from cardboard covered with aluminum foil, which gives a conducting surface and creates the micro-scale conditions, useful to represent manufacturing errors. Statistics of the overall surface shape are then measured, analyzed and compared with the imposed ones, providing and verifying the rationale for a fully controlled surface to be applied in any kind of experiment on natural surfaces.     

Forward-backward method with spectral acceleration for scattering from Layered rough surfaces

A fast method of moments is presented to calculate electromagnetic wave scattering from layered one-dimensional rough surfaces. The formulation is provided for M stratified homogeneous regions, separated by M-1 rough surfaces, and solved using point matching and pulse basis functions. Compared to the single surface case, the solution of scattering from M-1 surfaces requires significantly more memory and computational time. To facilitate the solution, the forward-backward method with spectral acceleration is applied. As an example, a dielectric layer on a perfect electric conductor surface is studied. The numerical results are compared with the analytical solution for layered flat surfaces to partly validate the formulation. The accuracy, efficiency, and convergence of the method are then studied for various rough surfaces and layer permittivities.     

Extended boundary condition method for scattering and emission from natural surfaces modeled by fractals

The extended boundary condition method with the Weierstrass-Mandelbrot fractal function (WM-EBCM) has been recently employed to model and solve the problem of electromagnetic scattering from natural surfaces. In this paper we first of all show, on the basis of theoretical considerations and of numerical examples, that this method can be used also for the evaluation of electromagnetic emission from natural surfaces. In addition, a small roughness approximation of the WM-EBCM solution is presented to highlight the connection between EBCM and SPM, and to avoid matrix ill-conditioning in scattering problems. Achieved results show that the zero-order scattered field is the (deterministic) field reflected by the mean plane, and that the first-order (random) scattered field is directly proportional to surface roughness. Validity limits of the approximated method are discussed and verified by studying the scattered field behavior at different surface roughness conditions.     

Electromagnetic scattering from slightly rough surfaces withinhomogeneous dielectric profiles

Remote sensing of soil moisture using microwave sensors require accurate and realistic scattering models for rough soil surfaces. In the past, much effort has been devoted to the development of scattering models for either perfectly conducting or homogeneous rough surfaces. In practice, however, the permittivity of most soil surfaces is nonuniform, particularly in depth, for which analytical solution does not exist. The variations in the permittivity of a soil medium can easily be related to its soil moisture profile and soil type using the existing empirical models. In this paper, analytical expressions for the bistatic scattering coefficients of soil surfaces with slightly rough interface and stratified permittivity profile are derived. The scattering formulation is based on a new approach where the perturbation expansion of the volumetric polarization current instead of the tangential fields is used to obtain the scattered field. Basically, the top rough layer is replaced with an equivalent polarization current and, using the volumetric integral equation in conjunction with the dyadic Green's function of the remaining stratified half-space medium, the scattering problem is formulated. Closed-form analytical expressions for the induced polarization currents to any desired order are derived, which are then used to evaluate the bistatic scattered fields up to and including the third order. The analytical solutions for the scattered fields are used to derive the complete second-order expressions for the backscattering coefficients as well as the statistics of phase difference between the scattering matrix elements. The theoretical results are shown to agree well with the backscatter measurements of rough surfaces with known dielectric profiles and roughness statistics     

An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method

An efficient algorithm for wave scattering from two-dimensional lossy rough surfaces is proposed. It entails the use of a single magnetic field integral equation (SMFIE) in conjunction with a multilevel sparse-matrix canonical-grid (MSMCG) method. The Rao-Wilton-Glisson (RWG) triangular discretization is adopted to better model the rough surface than the pulse basis functions used in the well-established SMCG method. Using the SMFIE formulation, only one unknown per interior edge of the triangular mesh approximating the rough surface is required, and the iterative solution to the moment equation converges more rapidly than that of the conventional coupled equations for dielectric rough surfaces. The MSMCG method extends the applicability of the SMCG method to rougher surfaces. Parallel implementation of the proposed method enables us to model dielectric surfaces up to a few thousand square wavelengths. Simulation results are presented as bistatic scattering coefficients for Gaussian randomly rough surfaces.     

A stochastically optimized adaptive procedure for the location of MAS auxiliary monopoles: the case of electromagnetic scattering by dielectric cylinders

A genetically optimized technique that fully automates the potentially laborious allocation of the auxiliary monopoles for the method of auxiliary sources (MAS) is presented for the problem of electromagnetic (EM) scattering by isotropic dielectric cylinders with various cross sections. The proposed technique uses as input information not only the geometry of the scatterer but also the exciting field and the material properties of the cylinders are implicitly taking part in the optimization procedure. The resulting auxiliary surfaces, where the simulating monopoles are situated, are appropriately adapted to the original boundary surface and the MAS modeling is greatly facilitated. In addition, certain considerations are taken into account in order to avoid undesirable numerical dependencies between the fictitious monopoles. Finally, the accuracy of the numerical method combined with overdetermined systems of equations is examined for isotropic cylinders of various geometries and dielectric characteristics.     

带限Weierstrass-Mandelbrot分形土壤表面与置于其上组合目标复合电磁散射特性仿真及分析

为了满足置于粗糙面之上组合目标雷达回波特性的数据采集、特征提取的需要,采用Topp模型和电介质复介电常数计算公式模拟土壤介电常数的实部和虚部,应用带限Weierstrass-Mandelbrot分形函数模拟土壤表面,运用时域有限差分方法研究了带限Weierstrass-Mandelbrot分形土壤表面与置于其上组合目标...     

分形粗糙面的电磁散射－扩展边界条件

用扩展边界条件方法对分形粗糙良导体面及介质面的电磁散射问题进行了分析。用推广的Floquet模式，在分界面处将场分量用Fourier级数展开，根据边界条件及扩展边界条件得到了水平极化和垂直极化散射场的幅度分量的表达式。用其它近似方法（Kirchhoff和Rayleigh方法）及能量守恒准则验证了此方法的有效性。     

A Study of Optical Backscattering Enhancement from One-Dimensional Rough Surface Using Monte Carlo Method

In this paper, according to Kirchhoff approximation, the optical backscattering enhancement of one-dimensional random rough surface, which includes fractal rough surfaces and random rough surfaces with Gaussian and exponential correlation simulated by Monte Carlo method, is obtained. It is shown that backscattering enhancement of random rough surfaces will increase with increasing the rms height of rough surface for a given correlation length. The angle width of backscattering enhancement is directly proportional to incident wavelength and inverse proportional to correlation length of rough surface. Complex phase of scattering field from superposed rough surface is uniformly distributed, none of the directions is of more overweight. The backscattering enhancement is also studied by wavelet analysis. The numerical results show good consistent with that of the relative references.     

Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects

The recent development and extension of the method of moments technique for analyzing electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects is presented based on the combined field integral equations. The surfaces of the homogeneous three-dimensional arbitrary geometrical shapes are modeled using surface triangular patches, similar to the case of arbitrary shaped conducting objects. Further, the development and extensions required to treat efficiently three-dimensional lossy dielectric objects are reported. Numerical results and their comparisons are also presented for two canonical dielectric scatterers-a sphere and a finite circular cylinder.     

Monte Carlo simulations of large-scale one-dimensional randomrough-surface scattering at near-grazing incidence: Penetrable case

Scattering from dielectric one-dimensional (1-D) random rough surfaces at near grazing incidence is studied for both TE and TM cases. To obtain accurate results at incidence angles of 80°-85°, we use long surface lengths of up to 1000 wavelengths. Numerical results are illustrated for dielectric surfaces corresponding to soil surfaces with various moisture contents. Results indicate that TM backscattering is much larger than that of TE backscattering. The ratio of TM to TE backscattering increases as a function of soil moisture and can be used as an indicator of soil moisture in remote sensing applications. However, the ratio of TM to TE backscattering is much lower than that predicted by the small perturbation method. To facilitate computation of scattering by such long surfaces, the previously developed banded-matrix iteration approach/canonical grid method (BMIA/CG) has been extended to dielectric surfaces. The numerical algorithm consists of translating the nonnear-field interaction to a flat surface and the interaction subsequently calculated by fast Fourier transform (FFT)     

Fractal surfaces and electromagnetic extended boundary conditions

In this paper, we employ the extended boundary condition method with the Weierstrass-Mandelbrot (WM) fractal function to model and solve a relevant electromagnetic scattering problem. The key point of the procedure is the property of the WM to be an almost periodic function. This allows to generalize techniques employed for periodic problems and to express the field by means of a superposition of Floquet modes. The procedure is devised for the general case of dielectric surfaces. Criteria for assessing the validity of the method are discussed and provided. Validity of the method is confirmed by numerical results     

Validity of the Kirchhoff approximation for electromagnetic wave scattering from fractal surfaces

Valid application of the Kirchhoff approximation (KA) for scattering from rough surfaces requires that the surface radius of curvature exceed approximately the electromagnetic wavelength /spl lambda/. Fractal surface models have characteristic features on arbitrarily small scales, thereby posing problems in application of the electromagnetic boundary conditions in general as well as in the evaluation of surface radius of curvature pertinent to KA. Experiments and numerical simulations show variations in scattering behavior that are consistent with scattering from progressively smoother surfaces with increasing wavelength, demonstrating surface smoothing effects in the wave-surface interaction. We hypothesize control of KA scattering from fractal surfaces by an effective average radius of curvature as a function of the smallest lateral scale /spl Delta/x contributing to scattering at /spl lambda/. Solution of =/spl lambda/ for /spl lambda/ is one possible method for approximating the limit of KA validity, assuming that /spl Delta/x[/spl lambda/] is known. Investigation of the validity of KA for the calculation of scattering from perfectly conducting Weierstrass-Mandelbrot and fractional Brownian process fractal surface models shows that for both models the region of applicability of KA grows with increases in /spl lambda/ and the Hurst exponent H controlling large-scale roughness. Numerical simulations using the method of moments demonstrate the dependence of /spl Delta/x on /spl lambda/ and the surface parameters.     

Regularized integral equations and curvilinear boundary elementsfor electromagnetic wave scattering in three dimensions

The boundary integral equations (BIEs), in their original forms, which govern the electromagnetic (EM) wave scattering in three-dimensional space contain at least a hypersingularity (1/R³ ) or a Cauchy-singularity (1/R²), usually both. Thus, obtaining reliable numerical solutions using such equations requires considerable care, especially when developing systematic numerical integration procedures for realistic problems. Regularized BIEs for the numerical computation of time-harmonic EM scattering fields due to arbitrarily-shaped scatterers are introduced. Two regularization approaches utilizing an isolation method plus a mapping are presented to remove all singularities prior to numerical integration. Both approaches differ from all existing approaches to EM scattering problems. Both work for integral equations initially containing either hypersingularities or Cauchy-singularities, without the need to introduce surface divergences or other derivatives of the EM fields on the boundary. Also, neither approach is limited to flat surfaces nor flat-element models of curved surfaces. The Muller linear combination of the electric- and magnetic-field integral equations (EFIE) and (MFIE) is used to avoid the resonance difficulty that is usually associated with integral equation-based formulations. Some preliminary numerical results for EM scattering due to single and multiple dielectric spheres are presented and compared with analytical solutions     

土壤表面与置于其上组合目标复合电磁散射特性研究

为了满足置于粗糙面之上组合目标测量和检测的需要,该文分别采用Dobson半经验模型和电介质复介电常数公式表示土壤介电常数的实部和虚部,应用指数型分布粗糙面和Monte Carlo方法模拟实际的土壤表面。通过与矩量法得到的计算结果比较,验证了时域有限差分(FDTD)方法计算粗糙面与目标复合散射问题的有效性,进而运用该方法研究了土壤表面与置于其上组合目标的复合散射,得出了复合散射系数的角分布曲线。结果表明:复合散射系数随散射角振荡地变化,在镜反射方向处发生散射增强效应;土壤表面高度起伏均方根越大,复合散射系数越大;相关长度越大,复合散射系数越小;湿度越大,复合散射系数越小;组合目标尺度、介电常数、入射角对复合散射系数影响比较复杂。该文结果可用于求解地、海粗糙面与置于其上任意目标的复合电磁散射问题,与其它数值计算方法相比较,采用时域有限差分方法既可获得较高的准确性,同时又可减少计算时间和内存占用量。     

Calculation of Differential Reflection Coefficient for Isolated Microscopic Well Structure

We have calculated differential reflection coefficient for isolated well structure of micro-scale, etched on dielectric surface. The differential reflection coefficient is computed using Green's second integral theorem. The purpose of our computation is to find a class of well profiles which give maximal diffusive scattering. To have such a maximal effect, we have concluded that the waist radius of Gaussian beam and its wavelength should be comparable to the well width and that well depth has to be larger than a wavelength. Exact calculation of differential reflection coefficients of dielectric surface with isolated structure on it may be used for the examination of dielectric surfaces and also in making simple but efficient diffuser.