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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 5, SEPTEMBER 1999 2597

Fast Algorithm for Electromagnetic Scattering byBuried 3-D Dielectric Objects of Large Size

Tie Jun Cui, Member, IEEE, and Weng Cho Chew, Fellow, IEEE

Abstract A fast algorithm for electromagnetic scattering byburied three dimensional (3-D) dielectric objects of large sizeis presented by using the conjugate gradient (CG) method andfast Fourier transform (FFT). In this algorithm, the Galerkinmethod is utilized to discretize the electric field integral equations,where rooftop functions are chosen as both basis and testingfunctions. Different from the 3-D objects in homogeneous space,the resulting matrix equation for the buried objects contains bothcyclic convolution and correlation terms, either of which canbe solved rapidly by the CG-FFT method. The near-scatteredfield on the observation plane in the upper space has beenexpressed by two-dimensional (2-D) discrete Fourier transforms(DFTs), which also can be rapidly computed. Because of the

use of FFTs to handle the Toeplitz matrix, the Sommerfeldintegrals evaluation which is time consuming yet essential forthe buried object problem, has been reduced to a minimum. Thememory required in this algorithm is of order

NNN

(the number ofunknowns), and the computational complexity is of order NNN

i t e r

NNN

l o g NNN

, in whichNNN

i t e r

is the iteration number, andNNN

i t e r

NNN

isusually true for a large problem.

Index TermsBuried objects, CG-FFT, fast algorithm, Som-merfeld integrals.

I. INTRODUCTION

ELECTROMAGNETIC (EM) scattering by dielectric and

conducting objects buried in a half space or layered media

is very important in modeling geophysical prospection, remotesensing, and wave propagation. Hence, it has been investigated

intensively in the past few decades using various methods

[1][17]. However, most of these methods are only efficient

for small objects. When the buried objects become large,

the number of Sommerfeld integrals to be evaluated and the

memory requirement for the resulting matrix increase rapidly,

and the matrix inversion becomes very CPU intensive, making

it impossible to solve using a small computer.

In this paper, a fast algorithm for EM scattering by buried

three-dimensional (3-D) large dielectric objects of arbitrary

shape is presented using the CG-FFT method. The CG-FFT

method is one of the most efficient techniques to analyze large-

scale problems, and it has been used widely and investigated

Manuscript received August 7, 1998; revised February 16, 1999. Thiswork was supported by the Department of Energy under Grant DEFG07-97ER 14835, by the Air Force Office of Scientific Research under MURIGrant F49620-96-1-0025, by the Office of Naval Research under GrantN00014-95-1-0872, and by the National Science Foundation under Grant NSFECS93-02145.

The authors are with the Center for Computational Electromagnetics,Department of Electrical and Computer Engineering, University of Illinois,Urbana, IL 61801-2991 USA (e-mail: [email protected]).

Publisher Item Identifier S 0196-2892(99)06282-8.

in EM scattering and radiation in a homogeneous space as well

as in microstrip antennas [18][34].

In comparison to the objects in a homogeneous space,

the main difference arising in the buried object problem is

that the integral equations contain a reflected field term from

the ground, which is expressed by the Sommerfeld integrals

(besides the primary field term in homogeneous space). In this

paper, the Galerkin method is utilized to discretize the electric-

field integral equations (EFIE), in which rooftop functions are

used as both basis and testing functions. After discretization,

the primary field term yields a cyclic convolution similar

to that in a homogeneous space [24], while the reflectedfield term yields a cyclic correlation. Both of these can be

evaluated by FFT. Meanwhile, the near-scattered field on an

observation plane in the upper space, determined by the other

type of Sommerfeld integrals, also can be expressed by two-

dimensional (2-D) DFT forms.

Due to the use of FFT to handle the cyclic convolutions

and correlation, the Sommerfeld integrals evaluation has been

reduced to a minimum. In the meantime, the memory required

in this algorithm is only of order , and the computational

complexity is of order . Therefore, it is possible

to solve large buried object problems on a small computer

by using this algorithm. Several results are presented, some

of which have been compared to those from the method ofmoments (MoM). The good agreement shows the validity of

this algorithm.

II. EFIE AND SCATTERED ELECTRIC FIELD

Consider a 3-D dielectric object of arbitrary shape that is

buried in the lower region of a half space and is characterized

by relative permittivities and , as shown in Fig. 1. Both

and can be complex to represent the lossy case, but usually

the upper region is free space . The arbitrarily shaped

dielectric object with complex permittivity is assumed

to be inscribed in a cuboid that is parallel

to the interface of the half space. The bottom of the cuboid

is separated from the interface by . In this paper, the time

dependence of is assumed and suppressed.

Under the Cartesian coordinate system shown in Fig. 1, the

dyadic Greens functions in Region and Region can be

formulated as follows when the source is located in the lower

region [15], [16]:

(1)

01962892/99$10.00 1999 IEEE


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2598 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 5, SEPTEMBER 1999

Fig. 1. Three-dimensional dielectric object buried in a half space.

(2)

(3)

in which , , and represent the primary, reflected, and

transmitted fields, respectively, and

(4)

is the scalar Greens function in a homogeneous space

(5)

and

(6)

are Sommerfeld integrals. Here, , and ,, , and are reflection and transmission coeffi-

cients of TE wave and TM wave from Region to Region ,

respectively. The mixed reflection coefficient is defined

as

in which ; ;

; ; ; ;

; ; ; ;

.

Hence, the scattered electric fields in lower and upper spaces

by buried dielectric objects can be formulated from the dyadic

Greens functions

(7)

(8)

where is the induced electric current density

inside the dielectric object.

Substituting (1)(3) for (7) and (8) and using Greens

theorem, we have

(9)

(10)

in which the induced electric current is related to the total

electric field inside the dielectric object by

Considering the relationship of incident, scattered, and total

fields inside the dielectric object , one easily

obtains the electric field integral equations of the induced

current density inside the buried dielectric object

(11)

(12)


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CUI AND CHEW: FAST ALGORITHM FOR ELECTROMAGNETIC SCATTERING 2599

Fig. 2. Dielectric object is inscribed in a cuboid and the partition.

where

(13)

(14)

(15)

and

III. DISCRETIZATION OF THE EFIE

In this section, we use the MoM to discretize the above

integral equations. As shown in Fig. 2, we divide the bounded

box into cuboidal

cells of volume , where

and is the division number in the -direction. Here and

after, we let To simplify the expressions in

this paper, we define the following numbers:

and.

From (11) and (12), both the volumetric currents and

their derivatives are included in the EFIEs. To ensure

the existence of the derivatives, the basis function of mustbe continuous in the -direction. A simple but efficient basis

function is a triangle in -direction and pulses in two other

directions. For example, the basis function for is written as

in which

else

TABLE ITOTAL NUMBER OF SOMMERFELD INTEGRALS IN VARIOUS METHODS

Fig. 3. Comparison of copolar electric currents on the bottom slice of ahomogeneous cuboid, using CG-FFT and plain MoM.

and

else

are triangle and pulse functions. Using the basis functions, the

electric currents can be expressed as

(16)

Notice that the discrete functions should be zero

when the basis function is located outside the actual dielectric

object. From (16), one easily obtains the derivatives of

(17)

which consists of two adjacent 3-D pulses with opposite

amplitude, representing two opposite electric charges. Here,

the 3-D pulse function is defined as

It can be shown that the moments of the basis functions

is the same as that of and is equal to .

In this paper, the Galerkin method is used. Because three

basis functions are applied for different components of the

electric current, the testing functions also should be different

(18)


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and act on the three components of the EFIE. Similarly, the

derivatives of the testing functions are also two adjacent 3-D

pulses

(19)

Using the definition of inner product of functions and

(in which hereafter represents the complex conjugate), we

can obtain the discrete forms of (11) and (12) by applying

their interactions with the corresponding testing functions

(20)

in which and are operator expressions of the second

to fifth terms in (11) and (12), and

(21)

(22)

(23)

(24)

in which when , respectively, and

In the inner products (22) and (23), ,when and ,

. Here, the superscript indicates the summation of

primary field part and reflected field part

where and are related to the Greens func-

tions

(25)

(26)

in which . Clearly, the inner products in

(20) are all 3-D summations of the product of Greens func-

tions and discrete electric currents, which are time consuming.

IV. CYCLIC CONVOLUTION AND CORRELATION

From the theory of Fourier transform and discrete Fourier

transform (DFT) [35], the cyclic convolution of discrete sig-

nals and is defined as

(27)

which can be fast calculated using FFT

(28)

in which and are the DFT of and .

Similarly, from the continuous correlation, we can define a

cyclic correlation of discrete signals and

(29)

which can be easily shown to satisfy

(30)

Note that in (27)(30), both the discrete signals and their DFT

have a cyclic property

Using the above definition and property, we can calculate

rapidly all the terms in the discrete EFIEs by FFT because thesummations in primary field parts resemble a 3-D cyclic con-

volution, and the summations in reflected-field parts resemble a

cyclic convolution in plane and correlation in direction.

However, the computational domain of these discrete signals

must be extended to from

since

for


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which does not satisfy the cyclic properties. Hence, we define

new discrete Greens functions and current distributions in the

extended domain

(31)

else

(32)

in which

, and

else,

else,

else.

With the new definitions, the terms in (20) can be computed

rapidly and exactly by using FFT. Substituting all the terms

into (20), we obtain

(33)

where

in which and are the DFTs of

and , respectively, and

Here, , , and are given by

From (33), we notice that the discrete integral equation (20)

has been expressed by the DFT and inverse DFT. Combined

with the CG method, this equation can be solved quickly.

V. CG METHOD

The CG algorithm is an efficient method for solving linear

system equations [18], [19]. In this algorithm, an adjoint

operation defined by

is required. Hence, we must evaluate the adjoint operations

in (20) to use the CG algorithm. After many derivations, one

obtains the adjoint operator of (33), which is expressed as

(34)

where

For a given initial guess of the current distribution ,

which is usually set to zero, the CG algorithm for (33) is

described as follows:

(35)

(36)

(37)


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in which ; ;

; . For

(39)

(40)

(41)

Here, and have a similar meaning to that of .

The error of this algorithm can be controlled by

tolerance

(42)

in which the norm is defined as .

VI. FAST COMPUTATION OF NEAR SCATTERED FIELD

From (10), the components of the scattered electric field in

the upper space can be written as

(43)

(44)

in which and

Usually, we compute the scattered electric field on a plane

of constant in Region . Let the computational domain be

Then, by substituting (16) into (43) and (44), we obtain

(45)

(46)

in which

(47)

where when , respectively.Similar to (31), here we extend the computational domain

of the transmitted Greens functions. Then (45) and (46)

can be written as 2-D cyclic convolutions. According to the

convolution theorem, we have

(48)

(49)

where is the 2-D DFT of .

From (48) and (49), we see clearly that all the electric fields

in the computational domain can

be computed rapidly by using the 2-D FFT. This scheme is

valid for both near field and far field.

VII. EFFICIENT EVALUATION OF SOMMERFELD INTEGRALS

From the above analysis, the integrals ,

, and should be

evaluated in the CG-FFT algorithm and near field computation.However, it is very time consuming to evaluate these integrals

exactly. Considering the fact that the moment of rooftop

function is the same as that of the pulse function, we can

obtain

else

(50)

(51)

(52)

in which

Now we evaluate the Sommerfeld integrals in (51) and (52).

Considering the reflection and transmission coefficients of half


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(a) (b)

Fig. 4. (a) Copolar electric currents on the bottom slice of the inhomogeneous cuboid by CG-FFT and plain MoM. (b) Copolar scattered electricfields on the observation plane.

(a) (b)

Fig. 5. (a) Copolar electric currents on the bottom slice of the inhomogeneous cuboid by CG-FFT and plain MoM. (b) Current distribution on thewhole bottom slice.

space [16], [33], these Sommerfeld integrals can be simplified

further

(53)

(54)

(55)

(56)

(57)

(58)

(59)

in which is the scalar Greens function in homogeneousspace shown in (4), ,

and

(60)

(61)

(62)


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(a) (b)

Fig. 6. (a) Copolar electric currents on the three central axis of the dielectric cube. (b) Current distribution on the horizontal center slice.

(a) (b)

Fig. 7. (a) Copolar electric currents on the three central axes of the weak dielectric cube. (b) Current distribution on the horizontal center slice.

(63)

where

The simplified Sommerfeld integrals (60)(63) can be evalu-

ated efficiently because the integrands have simple forms and

benign behavior.

On the other hand, the number of the Sommerfeld integrals

to be computed in this algorithm also can be reduced. As

we know, one has to calculate the Sommerfeld integrals

times to fill in the dense matrix

in the plain MoM. In the CG-FFT algorithm, however, only

times are required.

Furthermore, from the definition of these integrals, they are

only the functions of and . Thus, we evaluate just the

Sommerfeld integrals in a 2-D region

and then obtain by linear interpolation.

This can improve the efficiency greatly.

For example, we consider a dielectric

cube. If and sample points

per wavelength are used, the total numbers of Sommerfeld

integrals to be computed in various methods are listed in

Table I.

VIII. NUMERICAL RESULTS

In the following numerical results, the upper region is

assumed to be free space and the lower region is

assumed to be dry sand where . A plane wave with

polarization and normalized electric field normally is incident

from the free space. The scattered fields are measured on the

plane of in the free space. For reason of space, only

the amplitudes of current distributions and scattered fields are

shown in this paper.


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(a) (b)

Fig. 8. (a) Copolar electric currents on the three central axes of the dielectric sphere. (b) Current distibution on the horizontal center slice.

(a) (b)

Fig. 9. (a) Copolar electric currents on the bottom slice of the large-sized dielectric cuboid. (b) Copolar scattered electric fields on the observation plane.

To test the validity of the fast algorithm, we consider a small

dielectric cuboid

buried at , which has been divided

into cells. Fig. 3 shows the comparison

of copolar electric currents on the bottom slice of the cuboid,

using the CG-FFT and plain MoM. In the plain MoM results,Galerkins method has been used, which leads to the same

discrete integral equation as (20). This is solved by matrix

inversion. From Fig. 3, we can see that the results from CG-

FFT and plain MoM are nearly the same. This is because the

cyclic convolution, correlation, and FFT are exact.

The above example only shows the correctness of the FFT

and CG procedures, since the two methods solve the same

matrix equation. To test the fast algorithm further, we consider

the other MoM results, in which pulse and delta functions

have been chosen as the basis and testing functions, and

the differential operations have been put to the Sommerfeld

integrals. This leads to different integral equations [16]. Fig. 4

gives the comparison of copolar electric currents on the bottom

slice and the scattered electric fields on the observation plane

by using the CG-FFT and the MoM. Clearly, the two results

fit very well, showing the validity of the fast algorithm.

When the buried dielectric cuboid is inhomogeneous (when and when ), the

copolar electric currents on the bottom slice computed by CG-

FFT and MoM are shown in Fig. 5(a). Again, the good agree-

ment of the two results shows the robustness of the algorithm.

Fig. 5(b) shows the current distribution on the whole slice.

It is interesting to compare the CPU time used in the

CG-FFT and MoM. Despite the difference in evaluating the

Sommerfeld integrals, it takes 72.47 s to solve the matrix

equation in MoM, while the CPU time is only 0.059 s/iteration

for the CG-FFT. After 15 iterations, the error reaches

0.000 97.


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(a)

(b)

Fig. 10. (a) Current distribution on the bottom slice. (b) Scattered-field distribution on the observation plane.

In the next example, we consider a buried dielectric cube

with

and . When the cube is partitioned by using

and grids, part of the numerical results

from the CG-FFT are shown in Fig. 6, in which Fig. 6(a)depicts the copolar electric currents on the three central axes

of the cube, and Fig. 6(b) displays the current distributions on

the horizontal central slice. In this example, the CPU time is

1.12 s for grids. After 47 iterations, the error

is 0.000 9 5. For grids, the CPU time is 11.63

s/iteration. After 49 iterations, the error is 0.000 92.

To investigate the relation of the convergence rate to the

dielectric properties, we consider the same cube with low

dielectric contrast . The numerical results

from the CG-FFT are illustated in Fig. 7, in which the CPU

time is the same as that in Fig. 6 per iteration, while the

convergence is much faster. For both and

grids, the error becomes 0.000005 after three

iterations.

If the dielectric cube in Fig. 6 is replaced by a dielectric

sphere with diameter , the same partitioningis needed. Corresponding to above examples, the numerical

results for the sphere are shown in Fig. 8.

Again, for the grids, the CPU time is 1.12

s/iteration. But after 35 iterations, the error reaches 0.000 90.

For the grids, the CPU time is 11.63 s/iteration.

After 35 iterations, the error becomes 0.000 83.

Finally, we consider a large dielectric cuboid

, buried at

. When the cuboid is partitioned by grids,

the CG-FFT results are illustrated in Figs. 9 and 10. Fig. 9

shows the copolar electric currents on the bottom slice and


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Fig. 11. Actual memory requirement of the algorithm.

the copolar scattered fields on the observation plane. Fig. 10

displays the current and scattered field distributions on the

whole planes.In this example, the total cell number is 131 072. Consid-

ering that the current in each cell has three components, the

total number of unknowns is 393 216. For so many unknowns,

it is impossible to use the plain MoM, in which the CPU time

is estimated to be s on a Dec Alpha workstation.

However, the CPU time is only 130.01 s/iteration by usingthe fast algorithm. After 63 iterations, the error becomes

0.0059. The memory requirement for this problem is 185 MB.

Generally, the actual memory requirement of this algorithm

is proportional to the number of unknowns at the following

factor

MBytes

which is obtained by the least-square method. The comparison

of actual memory requirement and the least-square estimation

is shown in Fig. 11.

IX. CONCLUSION

This paper presents a fast algorithm for EM scattering by

buried 3-D large dielectric objects of arbitrary shape, using

the CG-FFT method. In this algorithm, both the integral

equation and near-scattered field can be handled rapidly, and

the Sommerfeld integrals evaluation, which is essential for

buried object problems, has been reduced to a minimum. Thememory required for this algorithm is only of order , the

total cell number, and the computational complexity is of order

in each iteration.

We have noticed that the convergence rate may be increased

if a RazorBlade function is used as the testing function

instead of the Galerkin procedure. In addition, the evaluation

of Sommerfeld integrals in the spatial domain could be avoided

if the convolution and correlation were performed in the

spectral domain. However, the singularity of the spectral

domain Greens function must be resolved in this case. This

will be our future work.

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Tie Jun Cui (M98), for photograph and biography, see p. 900 of the Mar.1999 issue of this TRANSACTIONS.

Weng Cho Chew (S79M80SM86F93), for photograph and biography,see p. 900 of the Mar. 1999 issue of this T RANSACTIONS.

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