Sparse Matrix Canonical Grid Research Articles

SMCG research on electromagnetic scattering in arid area

Abstract This article is devoted to the study of electromagnetic scattering characteristics in arid areas, and proposes environmental monitoring and improvement methods. A four-component soil dielectric model was established to study the relationship of soil dielectric constant with soil moisture and frequency. As the Monte Carlo Method combined with Gaussian spectral function was used to simulate the actual dry ground, the Sparse Matrix/Canonical Grid (SMCG) algorithm model based on the surface current equation was established to calculate the electromagnetic scattering coefficient of arid areas. To verify the correctness of the proposed algorithm, the results obtained by SMCG was compared with those calculated by Method of Moments (MOM), which showed great consistency. Many results were obtained by using the algorithm in this paper, based on the measured soil data in the southeastern area of Ejin Banner, Inner Mongolia. It was found that soil moisture content, area roughness and incident electromagnetic wave segment had influence on the scattering echo and showed regular change. The results of this paper are of guiding significance for soil moisture monitoring, desertification control and agricultural planting in arid areas.

Modeling and electromagnetic analysis of typical environmental clutter based on SMCG

AbstractIn the research of the natural ecological environment, environmental scattering is an important research content whose results are widely used. In order to fully study the relevant characteristics of the natural grassland environment, this paper established a grassland broadband clutter model, summarized the dominant and recessive laws of the grassland environment broadband clutter, and provided a new perspective for related environmental monitoring which would be helpful to solve the problem of target detection in complex environments. The main work of this paper is as follows: The dielectric constant model of the grass was established, and the curve of the dielectric constant with frequency was obtained. The Monte Carlo method combined with the Gaussian spectrum function was used to generate a two-dimensional Gaussian surface to simulate the actual grass surface. A broadband clutter model was established, and, considering the calculation efficiency and accuracy, the Sparse Matrix/Canonical Grid (SMCG) was used to calculate the surface scattering coefficient. Then, the environmental clutter amplitude with different radar bandwidth (10, 40, 80 MHz), surface roughness (h = 0.1 m, h = 0.2 m, h = 0.4 m) and grazing angle (30°, 60°) were calculated and the probability density function (PDF) was obtained. The results show that the higher the radar resolution, the larger the incident angle, and the rougher the interface, all of which will cause the tail of the PDF to become longer which will not only reduce the detection probability of the radar, but also the tracking accuracy. The research results can be used for civilian and military field.

Accurate Calculations of Emissivities of Polar Ocean Surfaces Between 0.5 and 2 GHz Using an NIBC/Nystrom/SMCG Method

We use full-wave simulations of rough surfaces to calculate the emissivities of a polar ocean between 0.5 GHz and 2 GHz for applications in ocean salinity. High accuracy is required because the emissivity variations between flat and rough surfaces are at the order of magnitude of $10^{-3}$ . In addition, the changes of emissivity due to salinity change are only about $4.2\times 10^{-4}$ /psu to $2.6\times 10^{-3}$ /psu for 0.5–2 GHz over the polar region. An enhanced method is proposed by combining neighborhood impedance boundary condition (NIBC), Nystrom, and sparse matrix canonical grid (SMCG) to solve the dual surface integral equations (SIEs) using the method of moments (MoM). To simulate large-scale ocean waves using a complete ocean spectrum, the surface root-mean-square (RMS) heights are up to 22.5 cm corresponding to 1.5 times wavelength of 2 GHz. Thus, the surface lengths used are up to 333 wavelengths to account for the large RMS height. Simulation results are illustrated for 0.5–2 GHz. We show that the energy conservations are obeyed to $10^{-4}$ . The effects of roughness and salinity on broadband polar ocean emissivity are studied with the proposed approach. Numerical simulations show that the sensitivity to salinity is still preserved in spite of roughness.

Sea Surface Radar Scattering at L-Band Based on Numerical Solution of Maxwell’s Equations in 3-D (NMM3D)

Radar scattering from ocean surfaces is investigated by 3-D numerical solution of Maxwell’s equations [numerical Maxwell’s model in 3-D (NMM3D)] using the ocean surface profiles stochastically generated from a 3-D Durden–Vesecky ocean spectrum. The surface integral equations (SIEs) are formulated for dielectric surfaces using Green’s functions of the air and the ocean permittivities with the surface tangential electric and magnetic fields as the unknowns. In solving the SIEs using the method of moment, a fast matrix solver of the sparse matrix canonical grid is used in conjunction with Rao–Wilton–Glisson basis functions. The computation has been implemented on a high-performance parallel computing cluster for problems with up to six million surface unknowns. Unlike the two-scale model (TSM) approximation, NMM3D does not require division of the surface spectrum into large- and small-scale ocean waves. The results of backscattering simulations are compared to Aquarius satellite radar measurements for wind speeds of 5, 8, and 10 m/s and for incidence angles of 29°, 39°, and 46°. The results show that NMM3D ocean backscattering solutions at L-band are in good agreement with Aquarius satellite radar data for co-polarized VV, HH, and cross-polarized VH returns as well as for the VV/HH ratio. The azimuthal dependence of L-band backscatter is also assessed. Finally, NMM3D results are compared to TSM solutions and are shown to lie close to Aquarius data in observed VV/HH ratio, and their azimuthal dependencies.

Rough Surface and Volume Scattering of Soil Surfaces, Ocean Surfaces, Snow, and Vegetation Based on Numerical Maxwell Model of 3-D Simulations

In this paper, we give an overview and an update on the recent progress of our research group in numerical model of Maxwell equations in three dimensions (NMM3D) on random rough surfaces and discrete random media and their applications in active and passive microwave remote sensing. The random rough surface models were applied to soil surfaces and ocean surfaces. The discrete random media models were applied to snow and vegetation. For rough surface scattering, we use the surface integral equations of Poggio–Miller–Chang–Harrington–Wu–Tsai that are solved by the method of moments using the Rao–Wilton–Glisson basis functions. The sparse matrix canonical grid method is used to accelerate the matrix column multiplications. In modeling the rough surfaces, we use the exponential correlation functions for soil surfaces and the Durden–Vesecky ocean spectrum for ocean surfaces. In scattering by terrestrial snow and snow on sea ice, we use the volume integral equations formulated with the dyadic half-space Green's function. The microstructure of snow is modeled by the bicontinuous media. In scattering by vegetation, we use the discrete scatterers of cylinder. The NMM3D formulation is based on the Foldy–Lax multiple scattering equations in conjunction with the body of revolution for a single scatterer. For rough surface scattering, simulations results are compared with advanced integral equation model, small slope approximation, small perturbation method, and two scale model. For volume scattering by snow, results are compared with the bicontinuous dense media radiative transfer. For scattering by vegetation, results are compared with distorted Born approximation and radiative transfer equation. Comparisons are also made with experiments.

Adaptive Frequency-Sampling Method for Wideband Electromagnetic Scattering of Precipitation Particles

In this letter, an adaptive Stoer-Bulirsch frequency-sampling method (SB-AFS) is proposed in conjunction with Sparse-Matrix/Canonical Grid (SM/CG) method to analyze the broadband electromagnetic wave scattering from precipitation particles. Compared to other sampling methods, the Stoer-Bulirsch algorithm can obtain a rational interpolation function without suffering from singularity problems, as this algorithm is a recursive tabular method and no matrix inversion is required. Furthermore, since final results are obtained by interpolating with the adaptive frequency sampling method, no more information is needed except the sampled frequency and its true value. Hence, it can be appropriately used in the electromagnetic simulations of precipitation particles. Numerical results demonstrate that our proposed sampling strategy performs well in terms of both simulation time and computation accuracy.

Backscattering Coefficients, Coherent Reflectivities, and Emissivities of Randomly Rough Soil Surfaces at L-Band for SMAP Applications Based on Numerical Solutions of Maxwell Equations in Three-Dimensional Simulations

We used Numerical Maxwell Model in 3-D Simulations (NMM3D) to study the backscattering coefficients, coherent reflectivities, and emissivities of soil surfaces using Gaussian random rough surfaces with exponential correlation functions. The surface area used is 8 by 8 square wavelengths. A total of close to 200 cases are computed by varying rms height, correlation length, and soil permittivity. We consider a 40° incidence angle. For each case, 15 realizations of rough surface profiles are generated, and 30 solutions of Maxwell equations are computed because of two polarizations. The method for solving the Maxwell equations is based on the Method of Moments (MoM) with Rao-Wilton-Glisson (RWG) basis functions. The solutions are accelerated by the sparse matrix canonical grid method implemented on parallel computing. The rms height varies up to 0.126 wavelength. The results are compared with the Dubois formulation, Small Perturbation Method (SPM), Kirchhoff Approximation (KA), and Advanced Integral Equation Model (AIEM). The NMM3D results are also compared with VV and HH backscatter data of soil surfaces where ground truth rms heights and correlation lengths were both measured. Good agreement is found between the NMM3D results and experimental measurement data. Based on the computed cases, interpolation tables are derived that can be directly applied to L-band active and passive microwave remote sensing of soil moisture, such as for the upcoming Soil Moisture Active and Passive (SMAP) mission.

BISTATIC SCATTERING FROM TWO-DIMENSIONAL DIELECTRIC OCEAN ROUGH SURFACE WITH A PEC OBJECT PARTIALLY EMBEDDED BY USING THE G-SMCG METHOD

An e-cient approach called general sparse matrix canonical grid (G-SMCG) method is proposed to analyze the electromagnetic scattering from 2-D dielectric rough surface with a conducting object partially buried. In this paper, the scattering of 3-D arbitrarily shaped object is computed by using the traditional method of moments (MoM) with RWG basis function, and the scattering of rough surface is analyzed by using the SMCG method. The coupling interactions between an object and rough surface are calculated by iterative method. Combing the ocean rough surface with Pierson Moskowitz (PM) spectrum, the bistatic scattering coe-cients of typical objects buried in the ocean surface have been computed by using the proposed method. Then the accuracy and e-ciency of this method are discussed. Finally, the bistatic scattering coe-cients of a ship located on ocean surface are calculated, and the in∞uence of sea state and wind direction on the scattering coe-cients is indicated.

Numerical Analysis of Scattering by Dielectric Random Rough Surfaces Using Modified SMCG Scheme and Curvilinear RWG Basis Functions

An augmented sparse-matrix canonical-grid (SMCG) approach for numerical analysis of scattering by dielectric random rough surfaces is developed. It is an extension of the previous modified SMCG scheme for perfect-electric-conducting (PEC) surfaces to dielectric cases, and is enhanced by adopting curved triangulation modeling. By using the curved modeling with curvilinear Rao-Wilton-Glisson basis functions (CRWG bases), the number of unknowns can be reduced substantially to extract results of the same accuracy, compared with using the planar triangular discretization with planar RWG bases. Numerical results for Gaussian and ocean-like dielectric rough surfaces are provided to confirm the validity and efficacy of the proposed method.

Implementation and validation of the curvilinear coordinate method for the scattering of electromagnetic waves from two-dimensional dielectric random rough surfaces

The curvilinear coordinate method is applied for analysing 2-D dielectric random rough surfaces. The theory is based on Maxwell's equations written in a non-orthogonal coordinate system. For each medium, this method leads to an eigenvalue system. The scattered fields within two media are expanded as linear combinations of eigensolutions satisfying the outgoing wave condition. The boundary conditions allow the scattering amplitudes to be determined. The coherent and incoherent intensities are estimated by averaging the scattering amplitudes over several realizations. The theory is verified by comparison with results obtained by other exact method. A discussion on the C-method and the Sparse-Matrix CAnonical Grid method is proposed in terms of accuracy and computation time.

Electromagnetic Scattering From Randomly Rough Surfaces Using the Stochastic Second-Degree Method and the Sparse Matrix/Canonical Grid Algorithm

In applying the magnetic field integration equation for the numerical study of electromagnetic wave scattering from 2-D perfectly electric conducting randomly rough surfaces, we propose to apply the stochastic second-degree (SSD) method to improve the asymptotic convergence rate of the discretized system by the method of moment, in conjunction with the highly efficient method for calculating the product of the impedance matrix and a vector embedded in the sparse matrix/canonical grid (SMCG) algorithm. The proposed method, applied to a new matrix-splitting scheme, can also appreciably reduce the memory requirement for the same neighborhood distance parameter rd, or it can enlarge rd for the same memory requirement as that of SMCG. The preliminary results show that the proposed method has encouraging potential for moderate rough surfaces and for cases with a large number of unknowns. Moreover, in our approach, the structure of the new splitting in conjunction with SSD is quite general, and the involved key computation, namely, the product of the impedance matrix and a vector, is independent of the specifics of the approximation schemes. Hence, the formulation can be extended to a variety of approximation schemes, such as the recent multilevel expansion algorithm of the SMCG method.

A modified scheme of sparse‐matrix canonical‐grid method using Chebyshev interpolating Green′s function

AbstractA modified scheme of sparse‐matrix canonical‐grid (SMCG) method for simulation of rough surface scattering is developed. It employs an interpolating Green's function based on Chebyshev polynomial approximation to replace the Taylor series expansion in the previous SMCG methods, which becomes intractable for high‐order terms. This modified scheme is more efficient and flexible, and suitable for arbitrary roughness. © Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1844–1848, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23527

MICROWAVE EMISSION FROM TWO-DIMENSIONAL INHOMOGENEOUS DIELECTRIC ROUGH SURFACES BASED ON PHYSICS-BASED TWO-GRID METHOD

Numerical simulations of emission for two-dimensional randomly rough surfaces with an inhomogeneous layered medium are presented. The inhomogeneous layered medium is modeled by a generalized n-layered stratified media. The numerical method was adopted from the physics-based two-grid method (PBTG). To ensure the strict accuracy requirement while to relief the memory and CPU resources, the PBTG in conjunction with the sparse-matrix canonical grid method (SMCG) was used in this paper. In so doing the reflected terms of the dyadic Green's function that accounts for layered media were added into the integral equations governing the surface tangential fields. Since the reflected part of the dyadic Green's function does not

Scattering by rough surface using a hybrid technique combining the multilevel UV method with the sparse matrix canonical grid method

A procedure is developed for combining efficiently the multilevel UV method with the sparse matrix canonical grid (SMCG) method in the computation of three‐dimensional (3‐D) wave scattering from random rough surface. It handles near‐ and intermediate‐field interactions by the multilevel UV method and far‐field interactions by the SMCG method. This hybrid UV/SMCG method removes the large memory requirement of both the UV method in the far field and the SMCG method in the near field. The computational complexity for the hybrid method is O(NlogN). The tradeoffs in computer memory requirement and CPU time between the UV part and the SMCG part are examined for Gaussian random rough surfaces with Gaussian correlation function and with exponential correlation function. For a surface area of 44 × 44 square wavelengths with RMS of 1 wavelength and 123,904 surface unknowns, the UV/SMCG method requires CPU of 52.2 s per iteration with total CPU of 46.6 min for 22 iterations on a single processor of CPU speed of 2.66 GHz.

A Modification of the Canonical Grid Series Expansion in Order to Increase the Efficiency of the SMCG Method

In this letter, we present a Green's function approximation valid in the weak-interaction region that can be used with the sparse-matrix canonical grid (SMCG) method. It can be easily introduced into existing SMCG codes, allowing a reduction in size of the neighborhood region and, consequently, of the dynamic memory and the computation time requirements.

The application of the generalized product-type method based on Bi-CG to accelerate the sparse-matrix/canonical grid method

When the sparse-matrix/canonical grid (SMCG) method is applied to analyse scattering of randomly positioned dielectric spheroids, the impedance matrix is decomposed into a strong interaction matrix and a weak interaction matrix. The strong interaction portion of the matrix–vector multiplication is computed directly as the moment method (MOM). The far-interaction portion of matrix–vector multiplication is computed indirectly using fast Fourier transforms by a Taylor series expansion of impedance matrix elements about the canonical grid point. However, the condition number of the impedance matrix obtained from the SMCG method becomes large compared to the one from MOM. As a result, the conjugate gradient (CG) method converges slowly. To attack such a trouble, the generalized product-type method based on Bi-CG (GPBi-CG) is used as an iterative solver in this paper. The numerical results show that the GPBi-CG method can achieve good convergence improvement compared to the other iterative methods. Copyright © 2005 John Wiley & Sons, Ltd.

On the Analysis of Statistical Distributions of UWB Signal Scattering by Random Rough Surfaces Based on Monte Carlo Simulations of Maxwell Equations

The field strength level of received signal and its statistical distributions are important in the study of signal propagation and scattering for wireless communication and remote sensing. In this paper, the scattering by random rough surface is analyzed by solving the solution of Maxwell equation. The method of moments is used to discretize the integral equation into the matrix equation. The sparse-matrix canonical grid method, which is a fast matrix solver, is applied in the analysis. Conjugate gradient (CG) method is adopted to solve the solution of matrix equation. With the solution of Maxwell equations, the magnitude of the scattered field is then used to predict the field statistics. Both time harmonic scattering and ultrawide-band (UWB) scattering are considered. For the time domain response, the electromagnetic scattered field in the vicinity of center of rough surfaces is first calculated in the frequency domain. Then the time domain signal is obtained by mean of the Fourier transform. The fading statistics are compared with that of the Rayleigh and Nakagami distributions. Results reveal that UWB signal exhibits less fading than the narrow band signal.

Bistatic phase function and fast solution of scattering by 2D random distributed scatterers

AbstractWe present large‐scale Monte Carlo simulation results of the phase functions in multiple scattering by dense media of small 2D particles. Solution of the Foldy–Lax equations with large number of unknowns is done efficiently using the sparse‐matrix canonical‐grid (SMCG) method. The SMCG method facilitates the use of FFT and results in an N log N‐type efficiency for CPU and O(N) for memory. This dependence is demonstrated by the simulation of CPU time using up to 50000 particles that are randomly distributed through random walk in a large area of 400 square wavelengths. The bistatic phase functions for a random medium are computed. The phase function converges with the number of particles and the number of realizations. The simulation results indicate that the nonsticky particles, sticky particles, and independent scattering have similar angular distribution patterns of the phase functions. However, the dense sticky particles show stronger scattering than the independent scattering, while the dense nonsticky particles have smaller scattering than that of the independent scattering. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 38: 313–317, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11047

Sparse matrix/canonical grid method applied to 3-D dense medium simulations

The sparse matrix/canonical grid (SMCG) method, which has been shown to be an efficient method for calculating the scattering from one-dimensional and two-dimensional random rough surfaces, is extended to three-dimensional (3-D) dense media scattering. In particular, we study the scattering properties of media containing randomly positioned and oriented dielectric spheroids. Mutual interactions between scatterers are formulated using a method of moments solution of the volume integral equation. Iterative solvers for the resulting system matrix normally require O(N/sup 2/) operations for each matrix-vector multiply. The SMCG method reduces this complexity to O(NlogN) by defining a neighborhood distance, r/sub d/, by which particle interactions are decomposed into strong and weak. Strong interaction terms are calculated directly requiring O(N) operations for each iteration. Weak interaction terms are approximated by a multivariate Taylor series expansion of the 3-D background dyadic Green's function between any given pair of particles. Greater accuracy may be achieved by increasing r/sub d/, using a higher order Taylor expansion, and/or increasing mesh density at the cost of more interaction terms, more fast Fourier transforms (FFTs), and longer FFTs, respectively. Scattering results, computation times, and accuracy for large-scale problems with r/sub d/ up to 2 gridpoints, 14/spl times/14/spl times/14 canonical grid size, fifth-order Taylor expansion, and 15 000 discrete scatterers are presented and compared against full solutions.

Microwave emission of rough ocean surfaces with full spatial spectrum based on the multilevel expansion method

Microwave emission of ocean surfaces with full spatial spectrum is studied in this paper. For ocean surfaces with full spectrum, the rms height of roughness can be many wavelengths, and the surface size must be chosen to be larger than the longest scale wave in the spectrum. Due to computer resources, it is not straightforward to conduct numerical simulations of emission from rough surfaces with large rms height and size since a large number of unknowns will be involved. In this paper, the multilevel expansion of the sparse matrix canonical grid (SMCG) method, which is available for surfaces with large rms heights, is used to study the emission of one-dimensional (1-D) ocean surfaces. The computational complexity and the memory requirement are still on the order of O(N log (N)) and O (N), respectively, as in the SMCG method. Ocean surfaces with size 1024 wavelengths (21.9 m at 14 GHz) and spatial spectrum bandwidth between 0.858 rads/m (corresponding to the longest scale of 341.3 wavelengths) and 4691.5 rads/m (corresponding to the shortest scale of 1/16 wavelengths), which is rather wide to be regarded as a full spectrum, are studied. The maximum of the electromagnetic wavenumber-surface rms height product is up to 25.18. The surface is modeled as a lossy dielectric surface with large relative permittivity rather than as a perfectly conducting surface, which is often adopted as an approximation in the active remote sensing of ocean surfaces. A relatively high sampling density is used to ensure accuracy. The effects of the low and high portions of the spectrum on the emissivity are studied numerically. Monte Carlo simulation for ocean surfaces is also performed by exploiting the efficiency of the multilevel expansion method and the use of parallel computing techniques. The convergence of the results with respect to the sampling density is also illustrated.