Sparse-matrix Canonical-grid Method Research Articles

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.

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.

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.

Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals

By using the sparse-matrix canonical-grid method, we performed large-scale multiple-scattering calculations to study band-edge states in large-sized two-dimensional photonic quasicrystals. We find that the band-edge states evolve in an abrupt and irregular way when the sample size is increased. New states with reduced symmetries can emerge at the band edge in large samples. Strong multifractal behaviors in the wave functions are also observed. Our findings unveil important differences between quasiperiodic and periodic systems at band edges.

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.

Multiple Scattering Among Vias in Planar Waveguides Using Preconditioned SMCG Method

Full-wave modeling for cylindrical vias in planar waveguides is formulated using Foldy-Lax multiple scattering equations. Recently, a sparse-matrix canonical-grid method based on fast Fourier transform and an iterative algorithm was proposed to solve a large-scale via problem. In this paper, we further improve computational efficiency by a preconditioning scheme based on the dominant information contained in the near field. We also discuss two methods-the sparse-matrix LU decomposition and sparse matrix iterative methods-for constructing the preconditioner, providing the tradeoffs between CPU time and memory. Results show an order of magnitude improvement over the nonpreconditioned case on the convergence rate. As an example, for 20 000 vias simulation, the solution time (per excitation and per frequency) is approximately 13 s per iteration, and 75 min to reach convergence in 320 iterations with memory requirements of 560 MB on a single Pentium 2.4-GHz processor machine. Numerical simulations are illustrated for physical problems such as ball-grid array and large-scale randomly distributed vias, where we have observed the shielding effects of the terminated vias that reduce the coupling between vias.

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

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.

Parallel analysis of electromagnetic scattering from random rough surfaces

An efficient approach for simulation of random rough surface scattering is developed based on using a single integral equation formulation and a multilevel sparse-matrix canonical-grid method. Merits of the scheme are demonstrated using two wind-driven ocean surfaces, one which is very rough and the other large in size.

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.

Multiple scattering among vias in lossy planar waveguides using SMCG method

Large-scale full-wave multiple scattering among cylindrical vias in planar waveguides is modeled using the Foldy-Lax equation. The formulation includes the skin effects of the conducting power/ground plane. Numerical solution of the Foldy-Lax equation with large number of unknowns is computed efficiently using the sparse-matrix canonical-grid method. In this method, interactions among vias are decomposed into the strong interactions part and the weak interactions part. The calculation of the weak part is carried out using two-dimensional (2-D)-fast Fourier transform (FFT) by translating the locations of the vias onto the uniform grids. The final solution of the Foldy-Lax equations is calculated by an iterative method. The results show O (N log N) CPU efficiency and O (N) memory efficiency. This makes large scale via problems possible for computer simulation.

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.

Multilevel expansion of the sparse-matrix canonical grid method for two-dimensional random rough surfaces

Wave scattering from two-dimensional (2-D) random rough surfaces up to several thousand square wavelengths has been previously analyzed using the sparse-matrix canonical grid (SMCG) method. The success of the SMCG method highly depends on the roughness of the random surface for a given surface area. We present a multilevel expansion algorithm to overcome this limitation. The proposed algorithm entails the use of a three-dimensional (3-D) canonical grid. This grid is generated by a uniform discretization of the vertical displacement along the height (z-axis) of the rough surface in addition to the uniform sampling of the rough surface along the x-y plane. The Green's function is expanded about the 3-D canonical grid for the far interactions. The trade-off in computer memory requirements and CPU time between the neighborhood distance and the number of discretization levels along the x-axis are discussed for both perfectly electric conducting (PEC) and lossy dielectric random rough surfaces. Ocean surfaces of the Durden-Vesecky (1985) spectrum with various bandlimits are also studied.

Application of physics-based two-grid method and sparse matrix canonical grid method for numerical simulations of emissivities of soils with rough surfaces at microwave frequencies

The simulations of emissivities from a two-dimensional (2D) wet soil with random rough surfaces are studied with numerical solutions of three-dimensional (3D) Maxwell equations. The wet soils have large permittivity. For media with large permittivities, the surface fields can have large spatial variations on the surface. Thus, a dense discretization of the surface is required to implement the method of moment (MoM) for the surface integral equations. Such a dense discretization is also required to ensure that the emissivity can be calculated to the required accuracy of 0.01 for passive remote sensing applications. It has been shown that the physics-based two-grid method (PBTG) can efficiently compute the accurate surface fields on the dense grid. In this paper, the numerical results are calculated by using the PBTG in conjunction with the sparse-matrix canonical grid method (SMCG). The emissivities are illustrated for random rough surfaces with Gaussian spectrum for different soil moisture conditions. The results are calculated for L- and C-bands using the same physical roughness parameters. The numerical solutions of Maxwell's equations are also compared with the popular H and Q empirical model.

Bistatic scattering and emissivities of random rough dielectric lossy surfaces with the physics-based two-grid method in conjunction with the sparse-matrix canonical grid method

Bistatic EM wave scattering from 2-D lossy dielectric random rough surfaces (3-D scattering problem) with large permittivity is studied. For media with large permittivities, the fields can vary rapidly on the surface. Thus, a dense discretization of the surface is required to implement the method of moments (MoM) for the surface integral equations. Such a dense discretization is also required to ensure that the emissivity can be calculated to the required accuracy of 0.01 for passive remote sensing applications. We have developed a physics-based two-grid method (PBTG) that can give the accurate results of the surface fields on the dense grid and also the emissivities. The PBTG consists of using two grids on the surface, the coarse grid and the required dense grid. The PBTG only requires moderate increase in central processing unit (CPU) and memory. In this paper, the numerical results are calculated by using the PBTG in conjunction with the sparse-matrix canonical grid (SMCG) method. The computational complexity and memory requirement for the present algorithm are O(N/sub scg/log(N/sub scg/)) and O(N/sub scg/), respectively, where N/sub scg/ is the number of grid points on the coarse grid. Numerical simulations are illustrated for root mean square (rms) height of 0.3 wavelengths and correlation length of 1.0 wavelength. The relative permittivity used is as high as (17+2i). The numerical results are compared with that of the second-order small perturbation method (SPM). The comparisons show that a large difference in brightness temperature exists between the SPM and numerical simulation results for cases with moderate rms slope.