Estimating the effective number of looks in interferometric SAR data

The probability density function (pdf) of the multi-look interferometric phase between two complex synthetic aperture radar (SAR) images is parameterized by the number of looks and the complex correlation coefficient. In practice, adjacent pixels in a real SAR interferogram, are statistically dependent due to filtering, and hence, the number of looks is usually smaller than the number of samples averaged. It has been shown that compensation with an effective number of looks, rather than an intractable rederivation of the pdf, can account for the statistical dependence. This paper addresses the challenge of how to determine a suitable value for the effective number of looks. It is shown that an optimum value can be found via a maximum-likelihood estimator (MLE) based on the interferometric phase pdf. However, since such an MLE is computationally intensive and numerically unstable, an estimator based on the method of moments (MoM) possessing similar fidelity is proposed. MoM is fast and robust and can be used in operational applications, such as determining constant false alarm rate (CFAR) detection thresholds for moving-target detection in SAR along-track interferometry.     

Human exposure assessment in the near field of GSM base-station antennas using a hybrid finite element/method of moments technique

A hybrid finite-element method (FEM)/method of moments (MoM) technique is employed for specific absorption rate (SAR) calculations in a human phantom in the near field of a typical group special mobile (GSM) base-station antenna. The MoM is used to model the metallic surfaces and wires of the base-station antenna, and the FEM is used to model the heterogeneous human phantom. The advantages of each of these frequency domain techniques are, thus, exploited, leading to a highly efficient and robust numerical method for addressing this type of bioelectromagnetic problem. The basic mathematical formulation of the hybrid technique is presented. This is followed by a discussion of important implementation details-in particular, the linear algebra routines for sparse, complex FEM matrices combined with dense MoM matrices. The implementation is validated by comparing results to MoM (surface equivalence principle implementation) and finite-difference time-domain (FDTD) solutions of human exposure problems. A comparison of the computational efficiency of the different techniques is presented. The FEM/MoM implementation is then used for whole-body and critical-organ SAR calculations in a phantom at different positions in the near field of a base-station antenna. This problem cannot, in general, be solved using the MoM or FDTD due to computational limitations. This paper shows that the specific hybrid FEM/MoM implementation is an efficient numerical tool for accurate assessment of human exposure in the near field of base-station antennas.     

基于矩量法的手机天线辐射的研究

基于矩量法,分析了三维各向同性介质的散射问题。并以此为基础,研究了反映各向同性介质在电磁场下热效应的参数-比吸收率（SAR）。针对实际手机使用中的辐射问题,给出了一个简单模型下的SAR数值仿真结果,并与IEEE标准作了比较,提出了具体的安全距离建议。最后,比较了矩量法,时域有限差分法,有限元法在手机天线辐射研究中的优缺点。     

Ultrawide-band synthetic aperture radar for detection of unexplodedordnance: modeling and measurements

Electromagnetic (EM) scattering from subsurface unexploded ordnance (UXO) is investigated both theoretically and experimentally. Three EM models are considered: the multilevel fast multipole algorithm (MLFMA), the method of moments (MoM), and physical optics (PO). The relative accuracy of these models is compared for several scattering scenarios. Moreover, the model results are compared to data measured by an experimental synthetic aperture radar (SAR) system, SAR images have been generated for subsurface UXO targets, in particular 155-mm shells. We compare SAR images from the measured data with theoretical images produced by the MoM and PO simulations, using a standard back-projection imaging technique. In addition to such comparisons with measurement, we consider additional buried-UXO scattering scenarios to better understand the underlying wave phenomenology     

On the convergence of MoM and mode matching solutions for infinite array and waveguide problems

It is shown that the method of moment (MoM) fails to analyze an infinite array of strip discontinuities if the number of basis functions and the number of expansion modes are set equal, however large they individually may be. Using spectral analysis and linear vector space theory it is found that for a given number of basis functions, there exists a minimum number of expansion modes required for a reasonable accuracy. Subsequently, for a given number of expansion modes, there is an upper limit for the number of basis functions. The mode matching (MM) analysis is viewed as a special case of the MoM analysis and the accuracy of the analysis is discussed with respect to the number of modes in the waveguides.     

An efficient method for electromagnetic characterization of 2-Dgeometries in stratified media

A numerically efficient technique, based on the spectral-domain method of moments (MoM) in conjunction with the generalized pencil-of-functions (GPOF) method, is developed for the characterization of two-dimensional geometries in multilayer planar media. This approach provides an analytic expression for all the entries of the MoM matrix, explicitly including the indexes of the basis and testing functions provided that the Galerkin's MoM is employed. This feature facilitates an efficient modification of the geometry without the necessity of recalculating the additional elements in the MoM matrix. To assess the efficiency of the approach, the results and the matrix fill times are compared to those obtained with two other efficient methods, namely, the spatial-domain MoM in conjunction with the closed-form Green's functions, and a fast Fourier transform algorithm to evaluate the MoM matrix entries. Among these, the spectral-domain MoM using the GPOF algorithm is the most efficient approach for printed multilayer geometries     

Use of the Finite-Difference Time-Domain Method in Calculating EM Absorption in Human Tissues

Although there are acceptable methods for calculating whole body electromagnetic absorption, no completely acceptable method for calculating the local specific absorption rate (SAR) at points within the body has been developed. Frequency domain methods, such as the method of moments (MoM) have achieved some success; however, MoM requires computer storage on the order of (3N) 2 and computation time on the order of (3N) 3 where N is the number of cells. The finite-difference time-domain (FDTD) method has been employed extensively in calculating the scattering of metallic objects, and recently is seeing some use in calculating the interaction of EM fields with complex, lossy dielectric bodies. Since the FDTD method has storage and time requirements proportional to N, it presents an attractive alternative to calculating SAR distribution in large bodies. This paper describes the FDTD method and evaluates it by comparing its results to analytic solutions in two and three dimensions. The utility of the FDTD method is demonstrated by a 3D scan of the human torso. The results obtained demonstrate that the FDTD method is capable of calculating internal SAR distribution with acceptable accuracy. With the availability of supercomputers, such as the CRAY II, the calculation of SAR distribution in a man model of 50 000 cells (1.27 cm per cell) appears to be feasible.     

基于Mellin变换的K分布参数估计新方法

 K分布是目前SAR图像建模领域应用最广泛、最著名的统计模型之一.当前普遍采用的是基于矩估计的参数估计方法,但其存在等效视数值需要经验获取、容易出现错误估计以及造成K分布失效等问题.为此,本文提出了一种基于Mellin变换的K分布参数估计新方法.该方法以Mellin变换为基础,详细推导了K分布对应的第一个、第二个第二类型的特征函数和它们各自对应的对数矩和对数累积量,最终获得了K分布参数估计简洁的迭代表达式.所提方法不但有效克服了K分布失效的问题,更为重要的是,其把视数同形状参数、尺度参数一样视为待估计的参数,且能够快速准确地迭代出它们的估计值,保证了K分布的拟合精度.实验结果证明了所提方法的有效性.     

Higher order hybrid method of moments-physical optics modeling technique for radiation and scattering from large perfectly conducting surfaces

An efficient and accurate higher order, large-domain hybrid computational technique based on the method of moments (MoM) and physical optics (PO) is proposed for analysis of large antennas and scatterers composed of perfectly conducting surfaces of arbitrary shapes. The technique utilizes large generalized curvilinear quadrilaterals of arbitrary geometrical orders in both the MoM and PO regions. It employs higher order divergence-conforming hierarchical polynomial basis functions in the context of the Galerkin method in the MoM region and higher order divergence-conforming interpolatory Chebyshev-type polynomial basis functions in conjunction with a point-matching method in the PO region. The results obtained by the higher order MoM-PO are validated against the results of the full MoM analysis in three characteristic realistic examples. The truly higher order and large-domain nature of the technique in both MoM and PO regions enables a very substantial reduction in the number of unknowns and increase in accuracy and efficiency when compared to the low-order, small-domain MoM-PO solutions. The PO part of the proposed technique, on the other hand, allows for a dramatic reduction in the computation time and memory with respect to the pure MoM higher order technique, which greatly extends the practicality of the higher order MoM with a smooth transition between low- and high-frequency applications.     

Analysis of Large Complex Structures With the Synthetic-Functions Approach

An innovative procedure is presented that allows the method of moments (MoM) analysis of large and complex antenna and scattering problems at a reduced memory and CPU cost, bounded within the resources provided by a standard (32 bit) personal computer. The method is based on the separation of the overall geometry into smaller portions, called blocks, and on the degrees of freedom of the field. The blocks need not be electrically unconnected. On each block, basis functions are generated with support on the entire block, that are subsequently used as basis functions for the analysis of the complete structure. Only a small number of these functions is required to obtain an accurate solution; therefore, the overall number of unknowns is drastically reduced with consequent impact on storage and solution time. These entire-domain basis functions are called synthetic functions; they are generated from the solution of the electromagnetic problem for the block in isolation, under excitation by suitably defined sources. The synthetic functions are obtained from the responses to all sources via a procedure based on the singular-value decomposition. Because of the strong reduction of the global number of unknowns, one can store the MoM matrix and afford a direct solution. The method is kernel-free, and can be implemented on existing MoM codes.     

Analysis of low frequency scattering from penetrable scatterers

A method is presented for solving the surface integral equation using the method of moments (MoM) at very low frequencies, which finds applications in geoscience. The nature of the Helmholtz decomposition leads the authors to choose loop-tree basis functions to represent the surface current. Careful analysis of the frequency scaling property of each operator allows them to introduce a frequency normalization scheme to reduce the condition number of the MoM matrix. After frequency normalization, the MoM matrix can be solved using LU decomposition. The poor spectral properties of the matrix, however, makes it ill-suited for an iterative solver. A basis rearrangement is used to improve this property of the MoM matrix. The basis function rearrangement (BFR), which involves inverting the connection matrix, can be viewed as a pre-conditioner. The complexity of BFR is reduced to O(N), allowing this method to be combined with iterative solvers. Both rectilinear and curvilinear patches have been used in the simulations. The use of curvilinear patches reduces the number of unknowns significantly, thereby making the algorithm more efficient. This method is capable of solving Maxwell's equations from quasistatic to electrodynamic frequency range. This capability is of great importance in geophysical applications because the sizes of the simulated objects can range from a small fraction of a wavelength to several wavelengths     

Higher Order Hybrid FEM-MoM Technique for Analysis of Antennas and Scatterers

A novel higher order large-domain hybrid computational electromagnetic technique based on the finite element method (FEM) and method of moments (MoM) is proposed for three-dimensional analysis of antennas and scatterers in the frequency domain. The geometry of the structure is modeled using generalized curved parametric hexahedral and quadrilateral elements of arbitrary geometrical orders. The fields and currents on elements are modeled using curl- and divergence-conforming hierarchical polynomial vector basis functions of arbitrary approximation orders, and the Galerkin method is used for testing. The elements can be as large as about two wavelengths in each dimension. As multiple MoM objects are possible in a global exterior region, the MoM part provides much greater modeling versatility and potential for applications, especially in antenna problems, than just as a boundary-integral closure to the FEM part. The examples demonstrate excellent accuracy, convergence, efficiency, and versatility of the new FEM-MoM technique, and very effective large-domain meshes that consist of a very small number of large flat and curved FEM and MoM elements, with $p$-refined field and current distributions of high approximation orders. The reduction in the number of unknowns is by two orders of magnitude when compared to available data for low-order FEM-MoM modeling.      

Threshold Power of Canonical Antennas for Inducing SAR at Compliance Limits in the 300–3000 MHz Frequency Range

A study of the specific absorption rate (SAR) in an exposed body induced by canonical antennas is presented, with the aim of determining an upper bound for the antenna transmit power that demonstrates that a product is inherently compliant with internationally accepted radio frequency (RF) exposure limits. Starting from the fundamental limits in antenna quality factor (Q) and the corresponding bandwidth, several antenna sizes are selected, and their SAR distributions are computed using the method of moments (MoM) and finite-difference time domain (FDTD) method in the frequency range 300-3000 MHz. The threshold powers are then determined, below which the peak 1-g and 10-g averaged SAR would not exceed the limits specified in international exposure standards. From the data, simple expressions are derived to estimate the threshold power over a wide range of antenna sizes, frequencies, and distances from the body. It is demonstrated that the results presented in this paper are conservative in comparison with the measured SAR data of real products as well as other published data     

Higher order hierarchical discretization scheme for surface integral equations for layered media

This paper presents an efficient technique for the analysis of electromagnetic scattering by arbitrarily shaped perfectly conducting objects in layered media. The technique is based on a higher order method of moments (MoM) solution of the electric field, magnetic field, or combined-field integral equation. This higher order MoM solution comprises higher order curved patches for the geometry modeling and higher order hierarchical basis functions for expansion of the electric surface current density. Due to the hierarchical property of the basis functions, the order of the expansion can be selected separately on each patch depending on the wavelength in the layer in which the patch is located and the size of the patch. In this way, a significant reduction of the number of unknowns is achieved and the same surface mesh can be reused in a wide frequency band. It is shown that even for fairly large problems, the higher order hierarchical MoM requires less memory than existing fast multipole method (FMM) or multilevel FMM implementations.     

A novel grid-robust higher order vector basis function for themethod of moments

A set of novel, grid-robust, higher order vector basis functions is proposed for the method-of-moments (MoM) solution of integral equations for three-dimensional (3-D) electromagnetic (EM) problems. These basis functions are defined over curvilinear triangular patches and represent the unknown electric current density within each patch using the Lagrange interpolation polynomials. The highlight of these basis functions is that the Lagrange interpolation points are chosen to be the same as the nodes of the well-developed Gaussian quadratures. As a result, the evaluation of the integrals in the MoM is greatly simplified. Additionally, the surface of an object to be analyzed can be easily meshed because the new basis functions do not require the side of a triangular patch to be entirely shared by another triangular patch, which is a very stringent requirement for traditional vector basis functions. The proposed basis functions are implemented with point matching for the MoM solution of the electric-field integral equation, the magnetic-field integral equation, and the combined-field integral equation. Numerical examples are presented to demonstrate the higher order convergence and the grid robustness for defective meshes using the new basis functions     

Computation of specific absorption rate in the human body due to base-station antennas using a hybrid formulation

A procedure for computational dosimetry to verify safety standards compliance of mobile communications base stations is presented. Compared with the traditional power density method, a procedure based on more rigorous physics was devised, requiring computation or measurement of the specific absorption rate (SAR) within the biological tissue of a person at an arbitrary distance. This uses a hybrid method of moments/finite difference time domain (MoM/FDTD) numerical method in order to determine the field or SAR distribution in complex penetrable media, without the computational penalties that would result from a wholly FDTD simulation. It is shown that the transmitted power allowed by the more precise SAR method is, in many cases, between two and five times greater than that allowed by standards implementing the power flux density method.     

Limitations of the Cubical Block Model of Man in Calculating SAR Distributions

Block models of man which consist of a limited number of cubical cells are commonly used to predict the internal electromagnetic (EM) fields and specific absorption rate (SAR) distributions inside the human body. Numerical results, for these models, are obtained based on moment-method solutions of the electric-field integral equation (EFIE) with a pulse function being used as the basis for expanding the unknown internal field. In this paper, we first examine the adequacy of the moment-method procedure, with pulse basis functions, to determine SAR distributions in homogeneous models. Calculated results for the SAR distributions in some block models are presented, and the stability of the solutions is discussed. It is shown that, while the moment-method, using pulse basis functions, gives good values for whole-body average SAR, the convergence of the solutions for SAR distributions is questionable. A new technique for improving the spatial resolution of SAR distribution calculations using a different EFIE and Galerkin's method with linear basis functions and polyhedral mathematical cells is also described.     

An improved MoM model for line-fed patch antennas and printed circuits

In this paper, an improved model is proposed to analyze the edge-connected line-fed patch antennas and printed circuits based on the method of moments (MoM), where the number of unknowns can be significantly reduced using simplified meshes. In the presented model, three types of basis functions are used to describe the currents on the antenna patch and circuit, the feedline and the feedline-patch junction. A new feedline-patch junction basis function is proposed based on the conventional wire-surface junction basis function. Numerical results are given to illustrate the accuracy and efficiency of the improved MoM model.     

On solving first-kind integral equations using wavelets on abounded interval

The conventional method of moments (MoM), when applied directly to integral equations, leads to a dense matrix which often becomes computationally intractable. To overcome the difficulties, wavelet-bases have been used previously which lead to a sparse matrix. The authors refer to “MoM with wavelet bases” as “wavelet MoM”. There have been three different ways of applying the wavelet techniques to boundary integral equations: 1) wavelets on the entire real line which requires the boundary conditions to be enforced explicitly, 2) wavelet bases for the bounded interval obtained by periodizing the wavelets on the real line, and 3) “wavelet-like” basis functions. Furthermore, only orthonormal (ON) bases have been considered. The present authors propose the use of compactly supported semi-orthogonal (SO) spline wavelets specially constructed for the bounded interval in solving first-kind integral equations. They apply this technique to analyze a problem involving 2D EM scattering from metallic cylinders. It is shown that the number of unknowns in the case of wavelet MoM increases by m-1 as compared to conventional MoM, where m is the order of the spline function. Results for linear (m=2) and cubic (m=4) splines are presented along with their comparisons to conventional MoM results. It is observed that the use of cubic spline wavelets almost “diagonalizes” the matrix while maintaining less than 1.5% of relative normed error. The authors also present the explicit closed-form polynomial representation of the scaling functions and wavelets     

Efficient method of moments formulation for large PEC scatteringproblems using asymptotic phasefront extraction (APE)

A method is introduced for reducing the exorbitant dependence on computer storage and solution time in the method of moments (MoM) for electrically large electromagnetic (EM) scattering problems. The unknown surface currents on large, smooth parts of a perfect electrical conductor (PEC) scatterer are expressed by an efficient set of linearly phased surface current basis functions. The phasefront characteristics of the surface currents are numerically extracted from known current samples obtained from a lower-frequency solution of the same configuration. The use of such basis functions for efficiently representing the surface currents that are constructed in terms of linearly phased currents at higher frequencies is justified by considering the form of the surface currents predicted by high-frequency asymptotic ray methods. The procedure for extracting the current phasefronts is purely numerical, obviating computationally expensive and nonrobust operations such as ray-tracing, and thus, is amenable to general purpose scattering codes. The new MoM with linearly phased basis functions is shown to greatly relieve the storage and solution time of the conventional MoM while accurately reproducing the induced surface currents and scattered fields of some chosen targets