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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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CUI AND CHEW: FAST ALGORITHM FOR ELECTROMAGNETIC SCATTERING 2601
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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2602 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 5, SEPTEMBER 1999
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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CUI AND CHEW: FAST ALGORITHM FOR ELECTROMAGNETIC SCATTERING 2603
(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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CUI AND CHEW: FAST ALGORITHM FOR ELECTROMAGNETIC SCATTERING 2605
(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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2606 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 5, SEPTEMBER 1999
(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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CUI AND CHEW: FAST ALGORITHM FOR ELECTROMAGNETIC SCATTERING 2607
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.