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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 11, NOVEMBER 2001

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Multilevel Expansion of the Sparse-Matrix Canonical Grid Method for Two-Dimensional Random Rough Surfaces Shu-Qing Li, Member, IEEE, Chi Hou Chan, Senior Member, IEEE, Ming-Yao Xia, Bo Zhang, and Leung Tsang, Fellow, IEEE

Abstract—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. In this paper, 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 ( -axis) of the rough surface in addition to the uniform plane. The Green’s sampling of the rough surface along the 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 -axis are discussed for both perfectly electric conducting (PEC) and lossy dielectric random rough surfaces. Ocean surfaces of Durden–Vesecky spectrum with various bandlimits are also studied. Index Terms—Electromagnetic scattering from rough surfaces.

I. INTRODUCTION

W

ITH the advent of modern computers and the development of efficient numerical algorithms, electromagnetic (EM) scattering from two-dimensional (2-D) random rough surfaces, once an intractable three-dimensional (3-D) problem, can now be computed in a reasonable amount of CPU time. Unlike classical analytic methods [1], [2], which are restricted in some domain of validity, numerical methods can provide accurate solutions of Maxwell’s equations. The most common numerical method for rough surface scattering is the surface integral equation method and its solution by the method of moments (MoM). opConventional implementation of the MoM requires computer memory storage, where is erations and the number of the surface unknowns. Such memory and operation requirements restrict the numerical method mostly to one-dimensional (1-D) rough surface problems [3]–[7]. To overcome these limitations, several fast numerical algorithms have Manuscript received June 16, 2000; revised January 2, 2001.This work was supported by the Hong Kong Research Grant Council, Hong Kong SAR, China, under Grant 9040372. S.-Q. Li, C. H. Chan, M.-Y. Xia and B. Zhang are with the Wireless Communications Research Center, Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. M.-Y. Xia is with the Institute of Electronics, Chinese Academy of Sciences, Beijing 100080, China. L. Tsang is with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195-2500, USA. Publisher Item Identifier S 0018-926X(01)09194-3.

been developed. Two of the most extensively used approaches are the sparse-matrix canonical-grid (SMCG) iterative method [8]–[17] and the fast multiple method (FMM) [18], [19]. In these two methods, the computational complexity and the memory and , respecrequirements are reduced to tively. However, the efficacy of these approaches still largely depends on the roughness of the surfaces. Over the last few years, we have developed and made significant improvements to the SMCG method [8]–[17]. This method was first formulated for the 1-D rough surface scattering problem [8] and then extended to the 2-D problem [9]–[13]. Using this approach, we have done extensive simulations of 3-D scattering and emission problems [9]–[13], and have applied the simulations to microwave remote sensing of ocean surfaces and soil surfaces [14]–[16]. More recently, we have implemented the SMCG method on a Beowulf parallel computing platform [17]. In the SMCG method, the interaction between two points on the surface is distinguished as either near (strong) interaction or far (weak) interaction. The near interactions among the source points and field points are computed using the exact Green’s function. In contrast, the far interactions are computed by the fast Fourier transform (FFT) through a Taylor series expansion of the Green’s function ) when the source and field point are about a flat surface ( far apart. Depending on the computer memory available, the near interactions can be computed repeatedly in the iterative solution or computed once and stored, which will lead to lower computation efficiency or large memory storage requirement, respectively. When the root mean square (rms) height of the rough surface increases, the applicability of the SMCG method highly depends on the memory storage or the efficiency of computing the near interactions and the accuracy of the Taylor series expansion of the far interactions. Therefore, the SMCG method is only suitable for surfaces up to a moderate roughness. A similar conclusion is true for other fast numerical methods [18], [19]. To overcome the limitation of the SMCG method for surface roughness, we have demonstrated that a multilevel expansion can be employed for 1-D rough surface with large rms height [14]. That is to say, we are not restricted to expand the as used in the Green’s function about the flat surface at SMCG method but about multilevel flat surfaces with equally displaced values of . In this paper, we extend this multilevel expansion approach to 2-D perfectly electric conducting (PEC) and lossy dielectric surfaces. The advantages of the multilevel

0018–926X/01$10.00 © 2001 IEEE

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expansion approach are two fold. First, the separation range designated for near interactions can be substantially reduced. Second, the number of Taylor series terms can also be reduced. Consequently, the CPU time and the associated memory of the near interactions can be drastically reduced. Furthermore, the number of FFT operations will also be reduced at the expense of replacing the 2-D FFTs by 3-D FFTs. The additional memory required for the 3-D FFTs is relatively small when the surface size is large, although the computation of the 3-D FFTs is less efficient than the 2-D FFTs. Above all, we can always obtain fast convergent results through fine discretization along the dimension, even for rough surfaces with large rms height. In this paper, the formulations for the SMCG method and the multilevel expansion of the Green’s function are given in Section II. The implementation of the 3-D FFTs for 2-D surfaces is described in detail. In Section III, numerical results are given. The trade-off in computer memory requirements and CPU time between the number of discretization levels along the height of the rough surface and the number of iterations are discussed. Ocean surfaces with large surface size and of Durden–Vesecky spectrum with various bandlimits are analyzed using the proposed method. It is found that lowering the lower limit of the spatial frequency of the ocean spectrum can change the bistatic scattering coefficients while increasing the upper limit of the spatial frequency in the ocean spectrum changes the emissivity. A conclusion is given in Section IV. The proposed algorithm is implemented on the Beowulf parallel computing platform for both the PEC and lossy dielectric surfaces.

(3)

(4)

(5) where and are the scalar Green’s functions in the free space and lossy dielectric medium (6)

II. FORMULATION

(7)

Consider an EM wave, and , with time depen, impinging upon a 2-D PEC or dielectric dence . The rough surface with a random height profile incident wave is tapered so that the illuminated rough surface . The incident field can be confined to the surface area expressions are given in [13] and will not be repeated here. and Let denote a source point and a field point on the rough surface, respectively. Then the magnetic-field integral equation on the PEC rough surface is

(1) represents the principal-value integral. where the integral The unit normal vectors and refer, respectively, to the unprimed and primed coordinates. They point in the upward direction away from the random surface. For the dielectric surface, the integral equations are

(2)

(8) and (9) A. Sparse-Matrix Canonical Grid Method In the SMCG method, we solve manner. The matrix is rewritten as

in an iterative

(10) represents near-field strong interactions and repwhere resents nonnear-field weak interactions. The strong and weak interactions are distinguished by the neighborhood distance . Let (11) represent the horizontal separation between the source and field , the interaction between two points is depoints. When fined as strong interaction. Outside this distance, the interaction is weak. The weak matrix is written as a series by Taylor series expansion of the Green’s functions. The term corresponds to the zero-order term of the Taylor expansion and is referred to as the flat-surface contribution. The iterative matrix-solving procedure is used to solve the matrix equation. The

LI et al.: MULTILEVEL EXPANSION OF THE SPARSE-MATRIX CANONICAL GRID METHOD

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Fig. 1. Multilevel expansion scheme for the Green’s function.

iterative procedure is given, for the first-order and higher-order solutions, as (12) (13) (14) Expansions of the Green’s functions at the flat surface , establish the flat-surface plane as a canonical grid, and are given by

accordingly. As a result, Taylor series expansion about the flat surface alone will not be sufficient. Fig. 1 shows the scheme of the multilevel expansion. The is unimaximum displacement in the vertical direction multiple flat surfaces. Taylor series formly discretized into expansion is then performed about these flat surfaces. Such expansion will convert the 2-D canonical grid of the original SMCG method into 3-D. An arbitrarily positioned surface point ) with is associated with its nearest 3-D ( ), where , which lies canonical grid point ( is the separation between on the th level flat surface, and two adjacent flat surfaces. Taylor expansions of the Green’s functions about the multilevel flat surfaces are

(15) (17) (16) . where for which the Taylor expansion will There is a minimum be convergent for a given rms height. When the RMS height is compared with the surface dimensions small, a very small can be chosen, and the SMCG method is very efficient. In contrast, when the roughness is large, a larger should be chosen rendering a large number of nonzero elements of the near-interaction matrix and the advantage of the SMCG disappears. Thus, the efficiency of the SMCG is largely dependent on the rms height of the surfaces. B. Multilevel Expansion of the SMCG Method To overcome the disadvantage of the SMCG for increasing surface roughness, we extend the multilevel expansion approach for 1-D surfaces [14] to 2-D. As indicated in (15) and (16), the accuracy of the Taylor series expansion depends on the ratio of . For rough surfaces with large rms height, the ratio of is large if the neighborhood distance does not increase

(18) reprewhere sents the distance between two points in the 3-D canonical grid. Truncations of the above two series depends on the ratio of ; a smaller ratio requires fewer terms in the series. Accurate expansions of the Green’s functions for rougher surfaces remains unchanged, can always be achieved, even when by choosing a finer discretization along the dimension. In , which this paper, we keep the expansion terms at results in ten terms for the expansion. These terms are listed for reference as (19a) (19b) (19c)

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(19d)

and

(19e)

(22a)

(19f) (22b) (19g) (19h) (22b) (19i) (19j) (22d)

and (20a) (20b) (20c) (20d) (20e) (20f)

(20g) (20h)

One can see that the coefficients of the Taylor expansions are translationally invariant and that FFTs can be employed in the matrix-vector multiplication of the weak interactions. Multilevel expansion makes the translationally invariant kernels functions of the distance between two points in the 3-D canonical grid. However, the matrix-vector product of the near-interaction matrix and the surface fields still employ a 2-D indexing scheme as in the original SMCG method. The 3-D scheme is only used for the product of the weak matrix and the surface fields. To connect the 2-D indexed problem with the 3-D canonical grid, the following algorithm is used. First, to perform the weak-matrix-vector multiplication by using the 3-D FFTs, the 2-D indexed unknown array is transformed into the 3-D indexed by setting

(20i) (20j) where (21a)

otherwise

(23)

) are the indices along the and directions, repreHere ( senting the position of a point on the rough surface. Index corresponds to the nearest grid point associated with the surface ) and can be defined as point ( (24)

(21b) After transforming the double-indexed unknown array into a triple-indexed one, 3-D FFTs can then be used to calculate the weak-matrix-vector multiplication (25) (26) (21c)

(21d)

and denote 3-D arrays obtained from the where 3-D Taylor series expansions of the Green’s functions, corresponding to the zero-order term and higher-order term of the and represent Taylor series, respectively. The terms the resultant triple-indexed arrays. To perform the iterative solution together with the 2-D indexed strong matrix, they should be transformed to 2-D indexed arrays. We denote the correand . With the above sponding 2-D indexed arrays by


implementation, we rewrite the iterative procedure of (12)–(14) as (27) (28) (29) where

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[12], and [21]. In this paper, to demonstrate the validity and the efficiency of the proposed method, comparisons are given among the results obtained respectively by the multilevel expansion, the original SMCG method, and the exact solution via directly solving the matrix equation. The proposed method is then used to study the scattering properties of various rough surfaces with different spectra and roughness. Two types of spectra, namely the Gaussian power spectrum and the ocean spectrum of Durden–Vesecky, are used in this paper. They are expressed respectively as

(30) (31) can be obtained from according to (23). Here From (27) to (29), one can see that the iterative procedure and the strong matrix calculation remain the same as those in the original SMCG method. Equations (27) and (28) are then solved using the conjugate gradient (CG) method. In the multilevel expansion method, the computational complexity and the memory requirement will both increase times compared with those of the original approximately SMCG method if the neighborhood distance and the number is the of Taylor expansion terms remain unchanged. Here number of discretization levels in the direction. In general, the neighborhood distance and the number of the Taylor expansion terms are smaller than those used in the original for a surface SMCG method, and we often have is the number of surface with large surface size. Here is fixed. Therefore, unknowns. Also, for a given rms height, the computational complexity and the memory requirement and , and can will be less than and still be approximately regarded as respectively, as increases while remains fixed. C. Parallel Processing The original SMCG method has been efficiently implemented on a Beowulf parallel computing platform that consists of 16 PCs with each PC a Pentium II 450 MHz machine having 256 MB RAM [17]. The multilevel expansion of the SMCG method is also implemented on the same parallel platform. The system software requires Redhat Linux Version 6.0 and the message passing interface (MPI) Version 1.1.2. The MPI version of the fast Fourier transform code, the Fastest Fourier Transform in the West (FFTW) developed by the Laboratory of Computer Science, Massachusetts Institute of Technology [20], is adopted to calculate the FFTs for performing the convolution of the weak part of the matrix-vector multiplication. The use of FFTW requires less computer memory than that of distributing the multiple FFT calls among the processors because the array required by the FFT operation is distributed uniformly among the processors in FFTW. III. NUMERICAL RESULTS AND DISCUSSION The original SMCG method has been rigorously tested for perfectly conducting and lossy dielectric rough surfaces [11],

(32) for Gaussian surfaces, where , correlation lengths in the , spatial frequencies in the surface rms height. On the other hand we have

and and

directions; directions;

(33) for ocean surfaces [22], where is the spectrum and describes anisotropy. For all the simulations in this paper, we choose the radius of the incident tapered wave spot size to be 1/3 of the edge length to provide sufficient field tapering at surface edges. For PEC surfaces, 64 points per square wavelength are used, while for lossy dielectric surfaces with permittivity and ocean surfaces with , 256 points per square wavelength are employed instead. The CG iterations are stopped when the error is less than 1% [17]. A. Results for Gaussian Surfaces First, to verify the validity of the multilevel expansion scheme, comparisons among the multilevel expansion expansion (denoted by M-SMCG), the original SMCG method (denoted by S-SMCG), and the exact numerical solution (denoted by Exact) are given. Figs. 2 and 3 show the comparisons of the bistatic scattering coefficients between the multilevel expansion and the original SMCG method for PEC and lossy dielectric surfaces with small roughness, respectively. The incident angle is 30 from normal. Surface lengths in the and directions are , which gives a surface , area of 64 . The rms height of the rough surface is . The neighborhood with a correlation length of is chosen in the original SMCG method, distance and the number of discretizations along while the direction are used in the multilevel expansion method. Figs. 4 and 5 show the comparisons between the multilevel expansion and the exact matrix equation solution for PEC surfaces with large roughness. Fig. 4 is for the surface with , choosing and , while rms height choosing Fig. 5 is for the surface with rms height and . The other parameters are the same as those used for Fig. 2. When the rms height increases to 0.8 , the solution of the original SMCG does not converge even when the neighborhood distance exceeds 6.0 . One can see that all the comparisons given from Fig. 2 to Fig. 5 are very

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Fig. 2. Comparison of the bistatic scattering coefficients of the PEC surface by the original SMCG method and the multilevel expansion method. The surface : . (a) Co-polarization. (b) Cross-polarization. area is 64 , h

=02

Fig. 4. Comparison of the bistatic scattering coefficients of the PEC surface by the exact numerical solution and the multilevel expansion method. The : , " i . (a) Co-polarization. (b) surface area is 64 , h Cross-polarization.

= 08

= 45 + 30

Fig. 3. Comparison of the bistatic scattering coefficients of the lossy dielectric surface by the original SMCG method and the multilevel expansion method. : , " i . (a) Co-polarization. (b) The surface area is 64 , h Cross-polarization.

=02

= 45 + 30

good. In Fig. 5, a closer agreement would be obtained if a larger neighborhood distance or number of discretization levels were chosen. Table I to Table VII, show the trade off of computer memory requirements and CPU time between the neighborhood distance and . First in Table I, we list the CPU time for computing the 2-D and 3-D FFTs for 100 times with different discretizations is twice that of . It can be along the direction, where seen that the CPU time increases about 12 times when the 2-D

Fig. 5. Comparison of the bistatic scattering coefficients of the PEC surface by the exact numerical solution and the multilevel expansion method. The surface area is 64 , h : , " i . (a) Co-polarization. (b) Cross-polarization.

= 16

= 45 + 30

FFTs is replaced by 3-D FFTs and the CPU time for . 3-D FFTs increases approximately linearly with Table II, Table III, andTable IV are for the PEC surfaces. In the PEC cases, the computer memory required for the strong


TABLE I COMPARISON

OF THE CPU TIME FOR ONE HUNDRED TIMES OF CONVOLUTIONS BY THE 2-D AND 3-D FFTS WITH DIFFERENT DISCRETIZATION , IS TWICE .

N N

N

interactions is manageable and they are calculated and stored before starting the iteration process. Table II is for the surface . The with surface area of 64 and rms height up to convergence of the solution in each case, as measured by the number of iterations and the CG terms in each iteration, is given in the last column for reference. The CG terms in each iteration refer to the number of CG iterations required to solve (27) or (28). For example, the first number in the first row of the leftmost column of Table II corresponds to the CG terms needed to solve equation (27), and the second and third numbers correspond to the numbers of CG terms for solving (28) with and . When the rms height is small or moderate, for exor 0.4 , the solutions of all cases with difample listed in the table converge ferent neighborhood distance and quickly. Results for bistatic scattering coefficients with the same rms height agree with each other very well. When the rms height increases to 0.8 , the solution of the original SMCG does not converge even when the neighborhood distance exceeds 6.0 . and , The multilevel expansion converges with but converges to the wrong results when comparing the bistatic scattering coefficients with other more exact results. The results of other cases compare well with the exact results, which means accurate results have been achieved. From Table II, one can find that the CPU time of the multilevel expansion algorithm is large compared with the original SMCG method and increases almost . This is because the number of strong matrix linearly with elements is very small when the PEC surface area is 64 . For example, the memory requirement of the strong matrix will be only 5.9, 22.2, and 158.2 MB when the neighborhood distance is 0.5, 1.0, and 3.0 , respectively. The CPU time spent on the calculation of strong elements and the strong matrix-vector multiplication is relatively small and most of the CPU time is spent on the calculation of 3-D FFTs. Table III gives the memory of the strong matrix and the CPU time distribution for the surface when the neighwith a surface area of 1024 and borhood is chosen to be 3.0 , 1.0 , and 0.5 . The total CPU time and the number of the iterations for each case are shown in Table IV. In Table III, Smvm and Fmvm denote the strong matrix-vector multiplication and the flat-surface-vector multiplication, respectively, which are in the right-hand side of the iteration equation, as shown in (12) and (29). Wmvm denotes the weak matrix-vector multiplication, which is in the left-hand side of the iteration equation, as shown in (13) and (30). The memory requirement for storing the strong matrix is very large when the neighborhood distance is 3.0 . Consequently, the CPU time of Smvm in each CG iteration term increases from 1.97 to

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41.93 seconds when the neighborhood distance increases from 1.0 to 3.0 . This results in discrepancy in the total CPU time of the original SMCG method for these two cases, as shown in Table IV. In contrast, when the surface area reduces to 64 , the total CPU time of the original SMCG is similar, as shown in Table II. For the cases shown in Table III and Table IV, much of the total CPU time is spent on the calculation of the Fmvm and Wmvm in the multilevel expansion method; more than 19% of the total CPU time is spent on the Wmvm. Overall, from Table II, Table III, and Table IV, we find that the CPU time of the multilevel expansion method is larger than that of the original SMCG method when the surface roughness is not so large that the original SMCG method is not applicable. The main reason is because the strong matrix is calculated once and stored before starting the iteration. The value of the total CPU time in the multilevel expansion algorithm is mainly determined by the calculation of the 3-D FFTs. However, the original SMCG method is not applicable when the surface is very rough but where the multilevel expansion method continues to work well. Table V, Table VI, and Table VII are for the lossy dielectric surfaces. The surface area is 64 . The strong interactions for lossy dielectric surfaces are computed repeatedly in the iterative solution because the memory associated with the strong interactions is too large to handle. The large memory associated with the strong matrix results from the fine discretization (256 points per square wavelength) for a dielectric surface with large dielectric constant. Table V is for the surface with rms height . The emissivity is also given in the table. The maximum difference of the emissivity among all the cases with difis less than 0.000 77, which results in 0.22K ferent and brightness temperature when the surface physical temperature is chosen to be 283K . Also the bistatic scattering coefficients for all the cases compare well. This implies that all the cases examined give reasonable results. From Table V, one can see that the difference of the CPU time among the cases, including the original SMCG, is small. For the cases with the same discretization, small neighborhood distance will result in slow convergence. While for the cases with the same neighborhood distance, a small number of discretization levels will also result in slow convergence. Using small neighborhood distance will reduce the repeated calculation time of the strong matrix, while slow convergence will lengthen the time for iterative solution. The fine discretization along the direction can accelerate the convergence but enlarges the dimension of the 3-D FFTs and therefore leads to longer computation time for the weak interaction computation. Overall, the computation time varies slightly for the different cases shown in Table V. It is flexible enough for us to be able to select various combinations of the neighborhood distance and the number of the discretization levels, depending mainly on the roughness of the surface. This is different from the cases of the PEC surfaces shown in Table II, where the CPU time for different cases is quite different. This is because of the large amount of elements of the strong matrix that must be recalculated in the each CG iterative term. As shown in Table VI, the CPU time of the strong matrix, including the calculation of the strong matrix elements and the Smvm, is comparable to that of the Fmvm in each CG term. Table VII is for the surface with rms

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TABLE II TRADE-OFF IN THE CPU TIME AND CONVERGENCE BETWEEN THE NEIGHBORHOOD DISTANCE AND THE NUMBER OF DISCRETIZATION LEVELS PEC SURFACE WITH AREA 64

TABLE III THE MEMORY OF THE STRONG MATRIX AND THE CPU TIME DISTRIBUTION FOR THE PEC SURFACE WITH AREA 1024

AND h

N

= 0:2

TABLE IV TRADE-OFF IN THE CPU TIME AND CONVERGENCE BETWEEN THE NEIGHBORHOOD DISTANCE AND THE NUMBER OF DISCRETIZATION LEVELS N PEC SURFACE WITH AREA 1024

TABLE V TRADE-OFF IN THE CPU TIME AND CONVERGENCE BETWEEN THE NEIGHBORHOOD DISTANCE AND THE NUMBER OF DISCRETIZATION LEVELS N DIELECTRIC SURFACE WITH AREA 64 AND h : . i

=20

height . The emissivity and the bistatic scattering coefficients still agree well with those of the original SMCG method. The CPU time is still not sensitive to the choice of the neighborhood distance. The convergence of the multilevel expansion is fast compared with that of the original SMCG method. From

= 45 + 30

FOR THE

FOR THE

FOR THE LOSSY

the above lossy dielectric surface cases, where the strong matrix must be recalculated in the iteration, we can conclude that the multilevel expansion algorithm is not only applicable when the surface is very rough, but may be more efficient than the original SMCG for slightly or moderately rough surfaces.


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TABLE VI COMPARISON OF THE CPU TIME OF THE STRONG MATRIX, INCLUDING THE CALCULATION OF THE STRONG MATRIX ELEMENTS AND THE SMVM, AND THAT OF THE FMVM IN EACH CG TERM FOR THE LOSSY DIELECTRIC SURFACE WITH DIFFERENT NEIGHBORHOOD . THE SURFACE AREA IS 64 , h : , i DISTANCE AND

N

=02

= 45 + 30

TABLE VII TRADEOFF IN THE CPU TIME AND CONVERGENCE BETWEEN THE NEIGHBORHOOD DISTANCE AND THE NUMBER OF DISCRETIZATION LEVELS N DIELECTRIC SURFACE WITH AREA 64 AND h : . i

=04

= 45 + 30

FOR THE

LOSSY

B. Results for Ocean Surfaces For ocean surfaces with the spectrum of Durden–Vesecky, [17] and [23] have shown some results for the bistatic scattering coefficients and the brightness temperatures. In these two papers, the surface size is only 8 by 8 wavelengths and the spatial frequency band of the spectrum is between 150 rad/m and 587 rad/m. In the following, we present results for ocean surfaces with size of 16 by 16 wavelengths and different spatial frequency bands. We know that changing the low- and high-cutoff spatial frequency of the spectrum will result in surfaces of different roughness. A very low cutoff spatial frequency will make the surface very rough. It is meaningful to study the ocean scattering properties with a wide spatial frequency band. We use and wind speed the magnitude parameter m/sec in the simulation. The spectrum is band-limited beand . We consider the scattering results at 14 GHz tween and incident angle at 30 degrees with the horizontal polarization. Fig. 6 shows the copolarization bistatic scattering coefficients via the scattering angle in the incident plane for surfaces with different spatial frequency bands. The spectrum is first rads/m and rads/m, band-limited between then the low-cutoff wavenumber is changed to 18 rads/m and the high-cutoff wavenumber is pushed to 1000 rads/m. The result of the corresponding flat surface is also given in Fig. 6(a), while the result using a larger number of the discretization levels along the direction for the roughest surface is given in Fig. 6(b) for reference. The corresponding EM wavenumber-surface rms height products and the brightness temperatures for different surfaces are listed in Table VIII. From the figures and the table, one can find that the roughness of ocean surface is determined mainly by the low-cutoff spatial frequency of the spectrum, and therefore the bistatic scattering coefficients are sensitive to the variation of the low-cutoff spatial frequency. The change of the high-cutoff spatial frequency has little effect on the bistatic scattering coefficients. The influence of the low- and high-cutoff spatial frequency on the brightness temperature can be observed from Table VIII. There is very small change in the brightness temperature when the low-cutoff spatial frequency is pushed down,

Fig. 6. The bistatic scattering coefficients of ocean surfaces with different i ,U m/secs, spatial frequency band and area 256 . " a : .

= 0 006

= 40 + 40

= 20

however the variation is relatively large when the high-cutoff spatial frequency is increased. The good comparison between and indicates that using the results using can give sufficiently accurate results. IV. CONCLUSION In this paper, we have presented a multilevel expansion algorithm of the SMCG method, which can be used to analyze random rough surfaces with large rms height. Using the proposed method, the separation range, which designates near interactions, can be substantially reduced and the number of Taylor series terms of the Green’s function can also be reduced

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TABLE VIII BRIGHTNESS TEMPERATURES OF OCEAN SURFACES WITH DIFFERENT LOW- AND HIGH-CUTOFF SPATIAL FREQUENCY. THE SURFACE AREA IS 256 , i ,U m/s, a :

= 40 + 40

compared with those used in the original SMCG method. The trade off in computer memory requirements and CPU time between the neighborhood distance and the number of discretization levels along the -axis are discussed for both PEC and lossy dielectric random rough surfaces. The results show that the CPU time of the multilevel expansion method is larger than that of the original SMCG method for the PEC surfaces considered in this paper, while the CPU times of the two methods are similar for lossy dielectric surfaces. The main reason for this is that the strong matrix is calculated once and stored before starting the iteration for the PEC surfaces while it is recalculated in each iteration term for the lossy dielectric surfaces. However, using the multilevel expansion method we can always obtain fast convergent results through fine discretization along the direction, even for rough surfaces with large rms heights when the original SMCG method is not applicable. The study on ocean surfaces with different spatial frequency band of the spectrum of Durden–Vesecky shows that increasing the high-cutoff spatial frequency has small effect on the scattering coefficients, but it can increase the brightness temperature. However the scattering coefficients are more sensitive to the variation of the low-cutoff spatial frequency. REFERENCES [1] A. Ishimaru, Wave Propagation and Scattering in Random Media. New York: Academic, 1978. [2] L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing. New York: Wiley, 1985. [3] R. M. Axline and A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propagat., vol. 26, pp. 482–488, 1978. [4] D. Maystre, M. Saillard, and J. Ingers, “Scattering by one- or two-dimensional randomly rough surfaces,” Waves Random Media, vol. 1, pp. 143–155, 1991. [5] E. I. Thorsos and D. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media, vol. 1, pp. 165–190. [6] S. H. Lou, L. Tsang, and C.H. Chan, “Application of the finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: Penetrable case,” Waves Random Media, vol. 1, pp. 287–307, 1991. [7] C. H. Chan, S.H. Lou, L. Tsang, and J.A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite-difference time-domain approach,” Microwave Opt. Technol. Lett., vol. 4, pp. 355–359, Nov. 1991. [8] L. Tsang, C. H. Chan, K. Pak, and H. Sangani, “Monte Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propagat., vol. 43, pp. 851–859, Aug. 1995. [9] L. Tsang, C. H. Chan, and K. Pak, “Monte Carlo simulations of a twodimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett., vol. 29, pp. 1153–1154, 1993. [10] L. Tsang, J. A. Kang, F. H. Ding, and C. D. Ao, “Scattering of Electromagnetic Wave; Numerical Simulation,” , New York: Wiley, 2001.

= 20

= 0 006

[11] K. Pak, L. Tsang, C.H. Chan, and J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Amer. A, vol. 12, pp. 2491–2499, Nov. 1995. [12] J. T. Johnson, L. Tsang, R. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, “Backscattering enhancement of electromagnetic waves for twodimensional perfectly conducting random rough surfaces: A comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propagat., vol. 44, pp. 748–756, May 1996. [13] K. Pak, L. Tsang, and J. Johnson, “Numerical simulations and backscattering enhancement of electromagnetic waves from two-dimensional dielectric random rough surfaces with the sparse-matrix canonical grid method,” J. Opt. Soc. Amer. A, vol. 14, no. 7, pp. 1515–1529, Nov. 1997. [14] C. H. Chan, L. Tsang, and Q. Li, “Monte Carlo simulations of largescale one-dimensional random rough-surface scattering at near-grazing incidence: penetrable case,” IEEE Trans. Antennas Propagat., vol. 46, pp. 142–149, 1998. [15] L. Tsang and Q. Li, “Numerical solution of scattering of waves by lossy dielectric surfaces using a physics-based two-grid method,” Microwave Opt. Technol. Lett., vol. 6, pp. 356–364, 1997. [16] Q. Li, C. H. Chan, and L. Tsang, “Monte-Carlo simulations of wave scattering for lossy dielectric random rough surfaces using the physics-based two-grid method and canonical grid method,” IEEE Trans. Antennas Propagat., vol. 47, pp. 752–763, Apr. 1999. [17] S.-Q. Li, C.H. Chan, L. Tsang, Q. Li, and L. Zhou, “Parallel implementation of the sparse-matrix/canonical grid method for the analysis of two-dimensional random rough surfaces (three-dimensional scattering problem) on a Beowulf system,” IEEE Trans. Geosci. Remote Sensing, vol. 38, no. 4, pp. 1600–1608, July 2000. [18] R. L. Wagner, J. Song, and W.C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surface,” IEEE Trans. Antennas Propagat., vol. 45, pp. 235–245, Feb. 1997. [19] V. Jandhyala, B. Shanker, E. Michielssen, and W.C. Chew, “Fast algorithm for the analysis of scattering by dielectric rough surfaces,” J. Opt. Soc. Amer. A, pp. 1877–1885, 1998. [20] M. Frigo and S.G. Johnson, “The fastest Fourier transform in the west,”, Tech. Rep, MIT-LCS-TR-728, 1997. [21] Q. Li, L. Tsang, K. S. Pak, and C.H. Chan, “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,” IEEE Trans. Antennas Propagat., vol. 48, pp. 1–12, 2000. [22] S. P. Durden and J. F. Vesecky, “A physical radar cross section model for a wind driven sea with swell,” IEEE J. Oceanic Eng., vol. 10, pp. 445–451, 1985. [23] J. T. Johnson, R. T. Shin, J.A. Kong, L. Tsang, and K. Pak, “A numerical study of ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sensing, vol. 37, pp. 8–20, Jan. 1999.

Shu-Qing Li (M’00) received the B. S. and M. S. degrees in Radio Science from Shandong University, China, and the Ph.D. degree in electronic engineering from Xi’an Jiaotong University, China, in 1991, 1994, and 1998, respectively. In 1998, she joined the Department of Electronic Engineering with the City University of Hong Kong as a Senior Research Assistant in and promoted to a Research Fellow in September 2000. Her current research interests include Computational Electromagnetics, Microwave Remote Sensing, High Frequency Interconnects, and Parallel Computing.


Chi Hou Chan received the Ph.D. degree in electrical engineering from the University of Illinois, Urbana, in 1987. From 1987 to 1989, he was a Visiting Assistant Professor with the University of Illinois, associated with the Electromagnetic Communication Laboratory. In 1989, he joined the Department of Electrical Engineering at the University of Washington, Seattle, as an Assistant Professor and promoted to Associate Professor with tenure in 1993. Since 1996, he has been with the Department of Electronic Engineering, City University of Hong Kong, where he is a Chair Professor of Electronic Engineering and Associate Dean of Faculty of Science and Engineering. He is a Guest Professor of Xi’an Jiaotong University, and a Consulting Professor with Nanjing University of Science and Technology in China. His research interests have been in computational electromagnetics and its applications. He is author of 3 book chapters and over 100 journal papers. Dr. Chan is the recipient of the 1991 US National Science Foundation Presidential Young Investigator Award. He is a fellow of the Chinese Institute of Electronics.

Ming-Yao Xia received the Ph.D. degree from the Chinese Academy of Sciences in China, in 1999. In 1988, he joined the Institute of Electronics, Chinese Academy of Sciences, as a Research Assistant, he became a Research Associate in 1990, and was promoted to an Associate Professor in 1995. He was a Visiting Scholar at the University of Oxford, England from 1995 to 1996. He was on leave with the City University of Hong Kong from June 1999 to August 2000. His research interests include electromagnetic theory, numerical methods, microwave remote sensing, and wavelet applications.

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Bo Zhang received the B.Sc. degree in mathematics from Shandong University, China, the M.Sc. degree in mathematics from Xi’an Jiaotong University, China, and the Ph.D. degree in applied mathematics from University of Strathclyde, U.K,, in 1983, 1985, and 1992, respectively. From 1992 to 1994, he was a Postdoctoral Research Fellow in the Department of Mathematics at Keele University, U.K. From 1995 to 1997, he was a Research Fellow in the Department of Mathematical Sciences at Brunel University, U.K. In 1997, he joined the School of Mathematical and Information Sciences at Coventry University, U.K., as a Senior Lecturer, where he was promoted to a Reader in Applied Mathematics in 2000. His current research interests include direct and inverse scattering problems in wave propagation, computational electromagnetics, numerical analysis and partial differential equations. Dr. Zhang is a member of the American and London Mathematical Societies and the Society for Industral and Applied Mathematics (SIAM).

Leung Tsang (S’73–M’75–SM’85–F’90) was born in Hong Kong. He received the S.B. degree, the S.M,, the E.E. degree, and the Ph.D. degree, all from the Massachusetts Institute of Technology, Cambridge, MA, in 1971, 1973, and 1976, respectively. Since September 2001, he has been with the City University of Hong Kong, on leave from the University of Washington where he is a Professor of Electrical Engineering. From 1976 to 1978, he was a Research Engineer with Schlumberger-Doll Research Center . From 1980 to 1983, he was with the Department of Electrical Engineering and Remote Sensing Center with Texas A&M University, College Station, TX. He is coauthor of Theory of Microwave Remote Sensing (New York: Wiley Interscience, 1985) and Scattering of Electromagnetic Waves, Vol. 1: Theories and Applications “(New York: Wiley, 2000). His current research interests include wave propagation in random media and rough surfaces, remote sensing, computational electromagnetics, and opto-electronics. Dr. Tsang served as the Editor-in-Chief of the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING between 1996 and 2001. He is on the Editorial board of the Journal of Electromagnetic Waves and Applications. He was the Technical Program Chairman of the 1994 IEEE Antennas and Propagation International Symposium, the URSI Radio Science Meeting, and the Technical Program Chairman of the 1995 Progress in Electromagnetics Research Symposium. He was the General Chairman of the 1998 IEEE International Geoscience and Remote Sensing Symposium. He was the recipient of the IEEE Geoscience and Remote Sensing Society Outstanding Service Award in 2000 and a recipient of the IEEE Third Millennium Medal.