a study of electromagnetic scattering from conducting targets above and below the dielectric rough...

A study of electromagnetic scattering from conducting targets above and below the

dielectric rough surface

Lixin Guo,* Yu Liang, and Zhensen Wu

School of Science, Xidian University, Shaanxi 710071, China *[email protected]

Abstract: The composite scattering from the conducting targets above and below the dielectric rough surface using the extended Propagation-Inside-Layer Expansion (EPILE) combined with the Forward-Backward method (FBM) is studied. The established integral equations are approved by comparing with the related theory. The accuracy and efficiency of the EPILE + FBM are compared with the method of moments (MOM). The influences of target size, target height/depth, target position, and the rms height, the correlation length, as well as the incident angle on the bistatic scattering coefficient (BSC) for different polarizations are also investigated. The presented algorithm is of generality for the target-rough surface composite scattering problems.

©2011 Optical Society of America

OCIS codes: (290.5880) Scattering, rough surfaces; (2990.0290) Scattering.

References and links

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#140332 - $15.00 USD Received 4 Jan 2011; revised 3 Mar 2011; accepted 7 Mar 2011; published 14 Mar 2011(C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5785

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1. Introduction

The scattering from randomly rough surface and the composite scattering from targets and rough surface have been important research subjects over the past several decades because of their important applications in many domains, such as electromagnetics, applied optics, remote sensing, oceanography, material science, target recognition, electronic countermeasure (ECM), etc. Both the approximate, analytical and numerical methods have been developed to study single rough surface scattering problem, of which analytical methods mainly include: the small-perturbation method (SPM) [1], the Kirchhoff or tangent plane approximation (KA) [2], the physical optics (PO) method, the extended boundary condition method [3], the two-scale method (TSM) [4], the phase perturbation method (PPM) [5], the small-slope approximation (SSA) [6], etc. The numerical methods mainly include: the method of moments (MOM) [7], the Monte Carlo method [8], the finite difference time domain (FDTD) method [9], the finite element method (FEM) [10], the method of multiple interactions (MOMI) [11], the fast multipole method (FMM) [12], the Banded-Matrix-Iterative-Approach/Canonical Grid (BMIA-CAG) [13], the forward-backward method (FBM) [14], etc. And methods in solving the rough surface-target composite scattering problem can be also categorized into the analytical and numerical methods, and even the analytical-numerical combined methods, such as the MOM [15], the FEM [16], the FDTD [17], the hybrid SPM/MOM technique [18], the hybrid KA/MOM technique [19], the hybrid PO/MOM technique [20], the reciprocity theorem [21,22], the Generalized FBM (GFBM) [23], the FBM/SAA [24], etc.

Although the aforementioned numerical methods are of respective advantages, however, some of them are efficient in computation time but not accurate enough, some of them show good accuracy but are a little time- consuming, and also some of them are only valid for studying the composite scattering between one target and the rough surface. Hence, it is natural and interesting to develop new methods, both exact and fast, to investigate composite scattering from more targets and the rough surface below or above them. In 2006, a fast numerical method, Propagation-Inside-Layer Expansion (PILE) was presented by N.


Déchamps et al. [25]. This method, which shows high efficiency and high accuracy, is able to handle problems configured with a huge number of unknowns. In the beginning, the PILE was devoted to the scattering by layered rough surface [25]. Later, the Extended PILE combined with the Forward-Backward method (FBM), used to study the scattering by an object above or below a randomly rough surface, was presented by G. Kubické et al. [26] and C. Bourlier et al. [27], respectively, which can be abbreviated as the EPILE + FBM.

In this paper, based on the previous work [25–27], we extend EPILE + FBM to scattering by two conducting targets located each in different dielectric media separated by a rough one-dimensional boundary (particularly, one of them being the free space). In EPILE + FBM, the new integral equations about the composite scattering from the targets both above and below the dielectric rough surface are established, and the new divided elements of the matrices are obtained, where EPILE is used to analyze the local and coupling interactions, while FBM is employed to calculate the local interactions on the middle rough surface to accelerate the EPILE. Differing from the presented in references [26,27], as for the reconstructed characteristic matrix of the EPILE is concerned, the dual coupling interactions from the targets to the rough surface or from the rough surface to the targets and the dual local interactions from the targets are considered, in other words, it should be an improvement or extension of the traditional EPILE. The FBM has a computational complexity of o(N

2), where

N is the number of samples of the discretized rough surface.

This paper is organized as follows. In Section 2, the geometry of the problem is defined, and the basic formulas of the composite scattering problem are given. In Section 3, the EPILE + FBM for the composite scattering problem is presented. In Section 4, numerical results are exhibited and detailed discussions are given. Finally, concluding remarks are addressed in Section 5.

2. Formulation of the composite problem

Figure 1 illustrates the geometry considered: a perfect electrically conducting (PEC) target (configuration arbitrary) is located above and a PEC target (configuration arbitrary) is located

below the dielectric rough surface with the rms height and the correlation length l . The

problem is assumed to be one-dimensional (variant in the x-z plane). ik represents the

propagation vector of the incident wave. i is the incident angle. Xu , Xd is the horizontal

distance from the center of the target above and the target below, respectively, to the center of

the rough surface (z-axis). Hu , Dd is the height of the target above and the depth of the

target below, Ru , Rd denotes the maximum radius of the targets above and below circled by

the dashed line, respectively. The regions 0 0 0( , ) , 1 1 1( , ) denotes the space above and

below the rough surface. 0 0 = 1,

1 is the relative permittivity of the dielectric rough

surface. The rough surface is assumed as nonmagnetic, i.e., 1 0 . The randomly rough

surface is generated by the Monte-Carlo spectral method [28]. The exponential spectrum 2 2 2 2 3/2( ) 2 (1 )i iK KW l l is applied to model the rough surface.

iK is the space

wavenumber [28]. L is the length of the rough surface. Each point on the rough surface, the target above, and target below is denoted by the two-dimensional position vector

ˆ ˆr r rx z r x z , ˆ ˆ

ou ou oux z r x z , and ˆ ˆod od odx z r x z , respectively, where

rx , oux ,

odx is

the discretized abscissa and, rz ,

ouz , odz is the discretized height.


Dd

iik

L

z

Target2

x

0 0 0( , )

Xd1 1 1( , )

Rough surfaceIntensity distribution of the

Thorsos’tapered waveStrong WeakWeak Strong

Rd

RuXuHu

Target1

Fig. 1. Geometric model of targets located both above and below the dielectric rough surface.

To avoid the edge limitations, the following Thorsos‟ tapered plane wave [29] is chosen as the incident field

2 2 2 2 2( ) exp( (1 [2( tan ) / 1] / ( cos ) ))exp( ( tan ) / ),inc i i i ij x z g kg x z g kr r (1)

in which g is tapered parameter. ik is the incident wave vector [28]. The time dependence of

exp( )j t is assumed and suppressed throughout this paper. j denotes unit imaginary

number. The tapered parameter g and the length of the rough surface L should be chosen

properly to satisfy the requirements of the wave equation, energy truncation [23].

Suppose a transverse electric (TE) wave ˆ ( )incE y r or transverse magnetic (TM) wave

ˆ ( )inc rH y impinges on the rough surface, as shown in Fig. 1. According to the Ewald-

Oseen‟ extinction theorem and the boundary conditions on the rough surface and the targets [15], for TE case (HH polarization), the following electric field integral equations (EFIE) can be obtained as

10 0 0 0 0 0 0

1( ) [ ( ') ' ( , ') ( , ') ' ( ')] ' ( , ') ' ( ') ' ( ) ,

2 r ti r

S SG G ds G ds S r r r r r r r r r rE r E r n n E n E E (2)

1

0 0 0 0 0 0 1[ ( ') ' ( , ') ( , ') ' ( ')] ' ( , ') ' ( ') ' ( ) ,r t

i tS S

G G ds G ds S E r n r r r r n E r r r n E r E r r (3)

2

1 1 1 1 1 1 1

1( ) [ ( ') ' ( , ') ( , ') ' ( ') ] ' ( , ') ' ( ') ' 0 ,

2 r tr

S SG G ds G ds S r r r r r r r r r r rE E n n E n E (4)

2

1 1 1 1 1 1 2[ ( ') ' ( , ') ( , ') ' ( ') ] ' ( , ') ' ( ') ' 0 ,r t

tS S

G G ds G ds S r r r r r r r r r rE n n E n E (5)

and for TM case (VV polarization), the magnetic field integral equations(MFIE) are given below

1

0 0 0 0 0 0 0 1

1( ) [ ( ') ' ( , ') ( , ') ' ( ')] ' ( ') ' ( , ') ' ( ) / ,

2 r ti r t

S SG G ds G ds S S H r H r n r r r r n H r H r n r r H r r (6)

2

1 1 1 1 1 1 1 1 1

1( ) [ ( ') ' ( , ') ( , ') ' ( ') ] ' [ ( ') ' ( , ') ( , ') ' ( ')] '

2

0 ,

r tS S

r

G G ds G G ds

S

r r r r r r r r r r r r r

r

H H n n H H n n H (7)

2

1 1 1 1 1 1 1

2

1( ) [ ( ') ' ( , ') ( , ') ' ( ') ] ' ( ') ' ( , ') '

2

0 ,

r tS S

t

G G ds G ds

S

r r r r r r r r r r

r

H H n n H H n (8)


where for HH polarization, 1 0/ and for VV polarization, 1 0/ . 'n is the unit

normal vector of the surface or targets, is the gradient operator. iE , 0E ,

1E denote the

incident electric field, electric fields in region 0 and 1 , respectively.

iH , 0H , 1H denote

the incident magnetic field, magnetic fields in region 0 and

1 , respectively. The use of the

method of moments (MOM) [15] with point matching and pulse basis functions leads to the following linear system

1 2 1 2 1 2 1 2( 2 ) ( 2 ) ( 2 ) ( 2 ) ,t t r t t r t t r t t rN N N N N N N N N N N N Z X S (9)

in which 1 2 1 2( 2 ) ( 2 )t t r t t rN N N N N N Z denotes the impedance matrix,

1 2( 2 )t t rN N N X is the induced

unknown vector, 1 2( 2 )t t rN N N S is the incident source item.

1tN ,2tN are the numbers of

sampling points on the target above and target below, respectively. rN represents the number

of sampling points on the rough surface. The impedance matrix is expressed as follows

1 1 1 2 1 1

2 1 2 2 2 2

1 2

1 2 1 2

1 2

( 2 )( 2 )

.

t t t t t r t r

t t t t t r t r

t t r

t t r r t r t r r r r

r t r t r r r r

N N N N N N N N

N N N N N N N N

N N NN N N N N N N N N N N

N N N N N N N N

A 0 B C

0 D E FZ

G 0 H I

0 J K L

(10)

1 2 1 2(2 ) (2 )r t t r t tN N N N N N Z can be divided into four blocks and expressed as

1 2 1 2 1 2

1 2

1 21 2

1, 2 1, 2

( ) ( ) ( ) (2 )

( 2 ) 1, 2( 2 ) (2 ) ( ) (2 ) (2 )

,t t t t t t r

t t r

t t rr t t r r

t t r t t

N N N N N N N

N N N t t r rN N N N N N N N

A AZ

A A (11)

1 1 2 2 1 1

1 2 1 2

1 2 2 1 2 1 2 2

1, 2 1, 2

( ) ( )( ) (2 )

,t t t t t r t r

t t t t

t t rt t t t t r t r

N N N N N N N Nt t r t t

N N N NN N NN N N N N N N N

A 0 B CA A

0 D E F (12)

1 2

1 2 1 2

1, 2

(2 ) (2 )( ) (2 )

.r t r t r r r r

r r

t t rr t r t r r r r

N N N N N N N Nt t r r

N NN N NN N N N N N N N

G 0 H IA A

0 J K L (13)

(2 ) (2 )r r

r

N NA corresponds exactly to the impedance matrices of the rough surface.

1 2

1, 2(2 ) ( )r t t

t t rN N N A and

1 2

1, 2

( ) (2 )t t r

r t t

N N N

A can be interpreted as coupling matrices between the targets

and the rough surface. 1 2 1 2

1, 2

( ) ( )t t t t

t t

N N N N A corresponds exactly to the impedance matrices of the

targets. The unknown vector and the source item are defined as

1 2 1 2 1 2 1 2( 2 ) 1, 2( ) (2 ) ( 2 ) 1, 2( ) (2 ) ,

t t r t t r t t r t t r

T T T T T T

N N N t t N N r N N N N t t N N r N X X X S S S (14)

with

1 2

1 2

1

1 1 1 0 0

1, 2( ) 0 1 0 1 1 2 1 2 (2 ) 0 0

( ) ( )( ) ( ), ( ) ( ) ( ) ( ), ,

r

t t r

t t r

N

N N NT T r r

t t N N t t t t r N r r

r r

r rX r r r r X r r

n n (15)

1 2

1 2

1 1 1 1

1, 2( ) 1 1 (2 )( ) ( ),0 0 ( ) ( ),0 0 ,t t r r

t t r

N N N NT T

t t N N i t i t r N i r i r S r r S r r (16)


in which the superscript T stands for the transpose operator, and / n stands for the normal

derivative operator. i ,

0 , 1 corresponds to

iE , 0E ,

1E for TE case and iH ,

0H ,1H for

TM case, respectively. The elements of the impedance matrix are shown as below:

(1)0(1)1 00 0

1(1)( ) ( )

20 0

ˆ( ) ( )( ) 44

, ,( )1

[ / (2 )] 2 44

q pq p qq p q

pq pqt pHH VV

q q

p

jkj x n R H k r r p qx H k r r p q

A Az x xj

p qx H k x e p q

(1) (1)0

1 0 0 0ˆ( ) ( ), ( ),

4 4pn n pn p n pn n p n

jk jB x n R H k r r C x H k r r

(1)1(1)1 10 1

( ) ( ) 2(1)

20 1

ˆ( ) ( )( )44

, ,( )1

[ / (2 )] 2 44

w vw v ww v w

vw vwHH VV t v

w w

v

jkjx n R H k r r v wx H k r r v w

D Dz x xj

v wx H k x e v w

(1) (1)1

1 1 0 1ˆ( ) ( ), ( ),

4 4vn n vn v n vn n v n

jk jE x n R H k r r F x H k r r

(1) (1)0

0 0 1 0

( ) ( )

ˆ( - ), ( ) ( - ),4 4

mq q m q mq q mq m q

HH VV

jkjG x H k r r G x n R H k r r

(1)0 (1)1 0 0 0

(1)

2 0 0

ˆ( ) ( ) ( ) 4 4

, ,( )1

[ / (2 )] 2 4 4

n mn m n n m n

mn mnr m

m m

m

jk jx n R H k r r m n x H k r r m n

H Iz x x j

m n x H k x e m n

(1) (1)1

0 1 1 1( ) ( )

ˆ( ), ( ) ( ),4 4

mv v m v mv v mv m vHH VV

jkjJ x H k r r J x n R H k r r

(1)1 (1)1 1 0 1

(1)

2 0 1

ˆ( ) ( ) ( ) 4 4

, ,( )1

[ / (2 )] 2 4 4

n mn m n n m n

mn mnr m

m m

m

jk jx n R H k r r m n x H k r r m n

K Lz x x j

m n x H k x e m n

where ( , , )n

n

n

m p v

r rR

r r, 21 [ ( )]m r mz x , 21 [ ( )]n r nz x ,

2

11 [ ( )]p t pz x , 2

21 [ ( )]q t qz x , 2

21 [ ( )]v t vz x , 2

21 [ ( )]w t wz x ,

2

ˆ ˆ( )ˆ

1 [ ( )]

r n

r n

z x

z x

x zn , ( ,m n ) refer to the sampling points on the rough surface, ( ,p q ) refer to

the sampling points on the target above, and ( ,v w ) refer to the sampling points on the target

below. 0 2 /k is the wavenumber in the free space, where is the incident wavelength

and 1 0 1k k is the wavenumber in the transmitted medium 1 . x is the sampling step.

ˆ ˆm m mx z r x z and ˆ ˆ

n n nx z r x z denote arbitrary two points on the rough surface.


ˆ ˆp p px z r x z and ˆ ˆ

q q qx z r x z denote arbitrary two points of the target above.

ˆ ˆv v vx z r x z and ˆ ˆw w wx z r x z denote arbitrary two points of the target below.

1tz , 2tz

and rz is the first-order differential of height function of the target above, target below, and

the rough surface, respectively, and 1tz , 2tz , rz corresponds to the second- order differential

of height function. e = 2.718214, = 1.781072, (1)0H is the zeroth-order Hankel function of

the first kind. (1)

1H is the first-order Hankel function of the first kind.

Upon solving the matrix Eq. (4) using the conjugate gradient method (CGM) [30] or bi-conjugate gradient method (BCGM) [31] or the direct LU inversion, the scattering field in the free space is given by [15]

0

( ) ( , ).jk r

N

s s s i

e

r r (17)

For TE and TM case, ( , )Ns s i is expressed as

1

1~ 1~24

(TE,HH) 0

2 2

1 1

2ˆ( , ) exp( ) [ ( ) ( ) ( )]

4

1 [ ( )] ( ) exp( ) 1 [ ( )]

{

},

r r r

r

t

jN N NN

s s i s s r rS

r t s tS

je j j x x

k

z x dx x j z x dx

k r n k X X

X k r

(18)

1

1~ 1~2 24

(TM,VV) 0

2

1 1 1

2ˆ( , ) [ ( ) ( ) ( )] exp( ) 1 [ ( )]

4

ˆ( ) ( ) exp( ) 1 [ ( )]

{

},

r r r

r

t

jN N NN

s s i s r r s rS

t s t s tS

je j x x j z x dx

k

j x j z x dx

n k X X k r

n k X k r

(19)

in which 2

1 1 1ˆ ˆ ˆ( ( ) ) 1 [ ( )]t t tz x z x n x z . The expression for the normalized far-field bistatic

scattering coefficient (BSC) with the Thorsos‟ tapered plane wave incidence is given as

2

2 2 2 2

0

( , )( , ) .

/ 2 cos (1 (1 2 tan ) / (2 cos ))

N

s s i

s s i

i i ig k g

(20)

Usually, when the number of samples for the target, and the length of the rough surface increases, the computational cost of solving the matrix equation using the CGM [30], BCGM [31] or the direct LU inversion becomes prohibitive, in Section 3 and Section 4, a fast and accurate numerical method, i.e., the EPILE + FBM is presented to speed up the scattering calculation.


3. The EPILE combined with the FBM for conducting targets both above and below the dielectric rough surface

0p

1p

2p

(0)X

(1)X

(2)X

Target1

Local

interactions

Surface-

Target1

coupling

Target2-

Surface

coupling

Surface

Local

interactions1

(2 ) (2 )( )r r

r

N N

A

1 2

1, 2

(2 ) ( )r t t

t t r

N N N

A

1 2

1, 2

( ) (2 )t t r

r t t

N N N

A1 2 1 2

1, 2 1

( ) ( )( )t t t t

t t

N N N N

A

Target2

Target1 Target1

Target2

Target2

Target1 Target1

Target2

Target2

Local

interactions

Target1-

Surface

coupling

Surface-

Target2

coupling

Surface

Local

interactions

Fig. 2. Physical interpretation of the EPILE for targets both above and below the dielectric rough surface.

N. Déchamps et al. have presented the PILE to investigate the layer rough surface scattering [25], and then the Extended Propagation Inside Layer Expansion (EPILE) to analyze the scattering from a single target above or below the rough surface was presented by G. Kubické et al. [26] and C. Bourlier et al. [27]. Here, based on their work [25–27], we apply the EPILE to study the composite electromagnetic scattering from targets both above and

below the rough surface. The inverse matrix of 1 2 1 2( 2 ) ( 2 )t t r t t rN N N N N N Z can be partitioned into

four blocks (it should be noted that, some of the following formulations are of the similar form with those in [25–27], but the influence of dual targets both above and below are included).

1 2 1 2

1

(2 ) (2 ) ,r t t r t tN N N N N N

T UZ

V W (21)

where

1 2 1 2 1 2 1 2

1, 2 1, 2 1 1, 2 1

( ) ( ) ( ) (2 ) (2 ) (2 ) (2 ) ( )[ ( ) ] ,t t t t t t r r r r t t

t t r t t r t t r

N N N N N N N N N N N N

T A A A A (22)

1 2 1 2 1 2 1 2

1 2

1, 2 1, 2 1 1, 2 1

( ) ( ) ( ) (2 ) (2 ) (2 ) (2 ) ( )

1, 2 1

( ) (2 ) (2 ) (2 )

[ ( ) ]

( ) ,

t t t t t t r r r r t t

t t r r r

t t r t t r t t r

N N N N N N N N N N N N

r t t r

N N N N N

U A A A A

A A (23)

1 2 1 2 1 2 1 2

1 2

1 1, 2 1, 2 1, 2(2 ) (2 ) (2 ) ( ) ( ) ( ) ( ) (2 )

1 1, 2 1(2 ) (2 ) (2 ) ( )

( ) [

( ) ] ,

r r r t t t t t t t t r

r r r t t

r t t r t t r t tN N N N N N N N N N N N

r t t rN N N N N

V A A A A

A A (24)

1 2 1 2 1 2 1 2

1 2 1 2

1 1 1, 2 1, 2 1, 2

(2 ) (2 ) (2 ) (2 ) (2 ) ( ) ( ) ( ) ( ) (2 )

1 1, 2 1 1, 2 1

(2 ) (2 ) (2 ) ( ) ( ) (2 ) (2 ) (2 )

( ) ( ) [

( ) ] ( ) .

r r r r r t t t t t t t t r

r r r t t t t r r r

r r t t r t t r t t

N N N N N N N N N N N N N N

r t t r r t t r

N N N N N N N N N N

W A A A A A

A A A A (25)

The total induced unknown vector on the rough surface can be expressed as follows

1 2 1 2 1 2 1 2 1 2

1, 2 1, 2 1 1, 2 1 1, 2 1

( ) ( ) ( ) (2 ) (2 ) (2 ) (2 ) ( ) ( ) (2 ) (2 ) (2 )[ ( ) ] ( ) ,t t t t t t r r r r t t t t r r r

t t r t t r t t r r t t r

r N N N N N N N N N N N N N N N N N r

X A A A A A A S (26)

and the total induced unknown vector on the targets above and below can be given by

1 2 1 2 1 2 1 2

1, 2 1, 2 1 1, 2 11, 2 1, 2 (2 ) (2 ) 1, 2( ) ( ) ( ) (2 ) (2 ) ( )[ ( ) ] .

r rt t t t t t r r t t

t t r t t r t t rt t t t N N t tN N N N N N N N N NT

X S A A A A S (27)


Introducing the characteristic matrix ( )c rM on the rough surface, i.e.,

1 2 1 2 1 2 1 2

1 1, 2 1, 2 1 1, 2

( ) (2 ) (2 ) (2 ) ( ) ( ) ( ) ( ) (2 )( ) ( )r r r t t t t t t t t r

r t t r t t r t t

c r N N N N N N N N N N N N

M A A A A (28)

From Eq. (26) and Eq. (28), the total induced vector on the rough surface can be expressed as

1 2 1 2 1 2

( ) 1 1, 2 1, 2 1

(2 ) ( ) (2 ) (2 ) (2 ) ( ) ( ) ( ) 1, 20( ) ( ( ) ),

r r r r t t t t t t

Pp p r t t r t t

r N c r N N r N N N N N N N t tp

X M A S A A S

(29)

where1 2 1 2 1 2 1 2

1, 2 1 1, 2 1 1, 2

( ) ( ) ( ) (2 ) (2 ) (2 ) (2 ) ( )0( ) ( )

t t t t t t r r r r t t

P p t t r t t r t t r

c N N N N N N N N N N N Np

M I A A A A , I is the

identity matrix.

Similarly, introducing the characteristic matrix ( 1, 2)c t tM on the targets

1 1 2 1 2 1 2

1, 2 1 1, 2 1 1, 2

( 1, 2) (2 ) ( ) ( ) (2 ) (2 ) (2 ) (2 ) ( )( ) ( ) .t t t t t r r r r t t

t t r t t r t t r

c t t N N N N N N N N N N N

M A A A A (30)

It should be addressed that both ( )c rM and ( 1, 2)c t tM are differed from the expressions

presented in [26,27] due to consider the dual local and coupling interactions. The total induced vector on the targets will yield by

1 2 1 2 1 2 1 2

( ) 1, 2 1 1, 2 1

1, 2( ) ( 1, 2) ( ) ( ) 1, 2 ( ) (2 ) (2 ) (2 )0( ) ( ( ) ).

t t t t t t t t r r r

Pp p t t r t t r

t t N N c t t N N N N t t N N N N N rp

X M A S A A S (31)

Therefore, ( )

(2 )r

p

r NX and 1 2

( )

1, 2( )t t

p

t t N NX can be expressed as follows

1 2

( ) ( ) ( ) ( )

(2 ) ( ) 1, 2( ) ( 1, 2)0 0, .

r t t

P Pp p p p

r N r t t N N t tp p X Y X Y (32)

Each item in expressions above is given below

1 2 1 2 1 2

(0) 1 1, 2 1, 2 1 ( ) ( 1)

(2 ) (2 ) (2 ) ( ) ( ) ( ) 1, 2 ( )( ) ( ( ) ), ,r r r t t t t t t

r t t r t t p p

r N N r N N N N N N N t t r c r r

Y A S A A S Y M Y (33)

1 2 1 2 1 2

(0) 1, 2 1 1, 2 1 ( ) ( 1)

( 1, 2) ( ) ( ) 1, 2 ( ) (2 ) (2 ) (2 ) ( 1, 2) ( 1, 2) ( 1, 2)( ) ( ( ) ), .t t t t t t r r r

t t r t t r p p

t t N N N N t t N N N N N r t t c t t t t

Y A S A A S Y M Y (34)

p denotes the iteration order. In Eq. (28) and Eq. (30), the 1 2 1 2

1, 2 1

( ) ( )( )t t t t

t t

N N N N

A accounts

for the local interactions on the targets, and 1

(2 ) (2 )( )r r

r

N N

A accounts for the local interactions

on the rough surface. 1 2

1, 2

( ) (2 )t t r

r t t

N N N

A propagates the resulting field on the rough surface toward

the targets (surface-targets coupling), and 1 2

1, 2

(2 ) ( )r t t

t t r

N N N

A propagates the resulting field on the

targets toward the rough surface (targets-surface coupling), and so on for the subsequent terms ( )prY , just as shown in Fig. 2. Usually, at each iteration step, the number

1 2t tN N of samples

for the targets is less than the numberrN of samples for the rough surface. The term

1 2 1 2

1, 2 1( ) ( )( )

t t t t

t tN N N N

A ς ( ς denotes the unknown column vector of length

1 2t tN N ) will be

solved by the MOM(CGM / BCGM / LU), while, to decrease the computing cost and speed

up the calculation of the term 1(2 ) (2 ) (2 )( )

r r r

rN N r N

A ζ (i.e., single rough surface, (2 )rr Nζ denotes

the unknown column vector of length 2 rN ), the FBM by A. Iodice [14] can be applied.

Assume 1(2 ) (2 ) (2 )( )

r r r

rN N r N

A ζ equals to (2 )rNξ ( ξ denotes the unknown column vector of

length 2 rN ). The impedance matrix and the induced unknown vector can be decomposed as

follow

((2 ) (2 )) ((2 ) (2 )) ((2 ) (2 )) (2 ) (2 )((2 ) (2 )) (2 ), ,r r r r r r r rr r r

f fd b br N N r N N r N N r N r Nr N N r N A A A A ξ ξ ξ (35)


in which (2 2 )r r

f

r N NA denotes the lower triangle part of (2 2 )r rr N NA , ((2 ) (2 ))r r

dr N NA denotes the

diagonal part of ((2 ) (2 ))r rr N NA , and ((2 ) (2 ))r r

br N NA denotes the upper triangle part of

((2 ) (2 ))r rr N NA . (2 )r

f

r Nξ and (2 )r

b

r Nξ are the forward, backward induced unknown vector on the

rough surface, respectively.

Assume that (2 ) 1 ( ) 2 ( )[ ]r r r

T Tr N r N r Nξ ξ ξ and (2 ) 1 ( ) 2 ( )[ ]

r r r

T T

r N r N r Nζ ζ ζ , therefore,

1 ( ) 1 ( )

2 ( ) 2 ( )

,r r r r r r

r r r r r r

T TN N N N r N r N

T TN N N N r N r N

H I ξ ζ

K L ξ ζ (36)

, ,r r r r r r r r r r r r r r r r

f s b f s b

N N N N N N N N N N N N N N N N H H H H I I I I (37)

, .r r r r r r r r r r r r r r r r

f s b f s b

N N N N N N N N N N N N N N N N K K K K L L L L (38)

The further decomposition and iteration operation of these matrixes applying the FBM can be found in [14], hence, these equations are not listed here, the iteration number is denoted as i . Therefore, the EPILE combined with FBM (EPILE + FBM) can be applied to investigate

composite scattering from targets above and below the rough surface. To validate the accuracy and the efficiency of the EPILE + FBM, the Relative Residual

Error and the computational complexity () are necessarily investigated. The Relative

Residual Error of the scattering coefficient obtained by using the EPILE + FBM is defined

as the norm of the following form

90 2

(EPILE+FBM) (MOM(CGM/LU/BCGM))90

90 2

MOM(CGM/LU/BCGM)90

| |RRE .

| |

(39)

The complexity per iteration of the terms ( )1, 2p

t tY , ( )prY is given below, respectively

1 2 1 2 1 2 1 2

1 2

2 31 2 1 2

( 1) 1 1, 2 1, 2 1 1, 2 ( 1)

( ) (2 ) (2 ) (2 ) ( ) ( ) ( ) ( ) (2 )

(( ) (2 )) ( )

( 12( ) ) (4( /3 /3) ) ( )

( ) ( )r r r t t t t t t t t r

t t r

iter t t t t

p r t t r t t r t t p

c r r N N N N N N N N N N N N r

N N N a

M N N or N N b

M Y A A A A Y

1 2

2

((2 ) ( )) ( )

((2 ) ) ( )

,

r t t

r

N N N c

N d

(40)

1 2 1 2 1 2 1 2

1 2

2

1 2

( 1) 1, 2 1 1, 2 1 1, 2 ( 1)

( 1, 2) ( 1, 2) ( ) ( ) ( ) (2 ) (2 ) (2 ) (2 ) ( ) ( 1, 2)

((2 ) ( )) ( )

((2 ) ) ( )

((

( ) ( )t t t t t t r r r r t t

r t t

r

t t

p t t r t t r t t r p

c t t t t N N N N N N N N N N N N t t

N N N e

N f

N N

M Y A A A A Y

2 31 2 1 2

) (2 )) ( )

( 12( ) ) (4( /3 /3) ) ( )

.

r

iter t t t t

N g

M N N or N N h

(41)

Operations (a), (c), (e), and (g) are matrix-vector multiplications: their computational complexities are 1 2(( ) (2 ))t t rN N N . Operations (d), (f) are the fast FBM iterative inversions,

their complexities are 2((2 ) )rN . Operations (b) and operations (h) are the MOM (CGM or

LU scheme), whose complexities are 2

1 2( 12( ) )iter t tM N N or 31 2((4 / 3 4 / 3) )t tN N , where

iterM is

the number of iterations in the CGM iteration scheme. Therefore, the total computational

complexity is 2 2

1 2 1 2 1 2( (( ) (2 ) 12( ) (2 ) ( ) (2 ) ))t t r iter t t r t t rp N N N M N N N N N N / 3 2

1 2 1 2 1 2( (( ) (2 ) (4 / 3 4 / 3) (2 ) ( ) (2 ) ))t t r t t r t t rp N N N N N N N N N for both the

calculation of the terms ( )1, 2p

t tY and ( )p

rY , where p is the number of iterations in the EPILE

scheme, for 1 2( )r t tN N N , i.e., the number of samples for the rough surface is much more


than those of the targets, or the size of the rough surface is far larger than that of the targets,

the complexity is about 2( (2 ( ) ))r rp N N , p is generally less than 10, so this method is

much faster than the direct LU inversion, of order 3((4 / 3) )rN and the CGM, of order 2( 12( ) )iter rM N .

4. Numerical results and discussions

In this Section, above all, the established integral equations will be validated, then, compared with MOM (CGM), the Relative Residual Error and the average computational time of the ordered EPILE + FBM are discussed, where the targets are two cylinders with infinite length. Subsequently, using the ordered PILE + FBM scheme, the BSC are investigated with changes of the target radius, the target height/depth, the rms height, the correlation length and the incident angle. The aforementioned numerical algorithms are tested on the computer with a 2.33GHz processor (Intel Core 2 Quad Q8200), 4GB Memory, ASUSTekP5Mainboard, Microsoft Windows XP operation system, and Fortran PowerStation 4.0 compiler. The

incident frequency is 3GHz (i.e., the wavelength is 0.1m) and the relative permittivity of

the dielectric rough surface 1 = (6.91,0.63) in the following numerical simulations. Both the

length L of the rough surface and the tapered parameter g (L/4) satisfy the aforementioned requirement in Section 2. 50 surface realizations are averaged in all the numerical examples except Fig. 4, Fig. 5, Table 1 and Table 2. The number of sampling points on the cylinder

above the surface and below the surface 1tN = 2tN = 100, and sampling points on the rough

surface rN = 1024. It is also noted the numerical simulation of the BSC are plotted in decibel

(dB) scale. To validate the integral Eqs. (2)-(8), our scheme has been compared with the results of X.

Wang‟s method [15], which is applied for composite scattering from a PEC target situated above or below a dielectric rough surface for TM case (VV). To explore the „PEC target above + dielectric rough surface‟ or the „PEC target below + dielectric rough surface‟ composite scattering using our theory that is applied to the case of the „PEC target above +

dielectric rough surface + PEC target below‟ composite scattering, the uR , or dR is assumed

as infinitesimal, respectively. Other parameters are stated in the figures. As are illustrated in Fig. 3, the curves of our scheme show very good agreements with those of X. Wang‟s method over all scattering angles, which not only guarantees the validity of the Eqs. (2)-(8), but also suggests the applicability of our scheme in solving „target above/below + rough surface‟ composite scattering.

-80 -60 -40 -20 0 20 40 60 80

-25

-20

-15

-10

-5

0

5

VV

Hu=Dd=3.3

Xu=Xd=0

Ru=2.0, Rd=/

=0.1 , l=1.0

i0, L=100

1= (6.91,0.63)

BS

C(d

B)

scattering angle()

X.Wang's method(target above+rough surface)

Our scheme(target above+rough surface

+target below)

-80 -60 -40 -20 0 20 40 60 80

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

i0, L=100

1= (6.91,0.63)

VV

Hu=3.3, Dd=4.3

Xu=Xd=0

Ru=/, Rd=4

=0.1 , l=1.0

BS

C(d

B)

scattering angle()

X.Wang's method(target below+rough surface)

Our scheme(target above+rough surface

+target below)

Fig. 3. Comparison of the angular distribution of BSC by our scheme and X. Wang‟s method.


-80 -60 -40 -20 0 20 40 60 80-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3

Ru=Rd=1, Xu=Xd=0

L=100

HH

EPILE+FBM

MOM(CGM)

(a)

-80 -60 -40 -20 0 20 40 60 80-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

(b)

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3

Ru=Rd=1, Xu=Xd=0

L=200

HH

EPILE+FBM

MOM(CGM)

Fig. 4. BSC versus the scattering angle (HH polarization).

Figure 4(a) shows the comparison of the EPILE + FBM and the MOM (CGM) for BSC of a cylinder target located above and a cylinder target located below the rough surface versus

the scattering angle for HH polarization with the incident angle i 0°. The radius, depth of

both the cylinders above and below are u dR R = 1 , d dH D = 3.3 , = 0.1 , l =

1.0 , L = 100 , rN = 1024. 10 Monte-Carlo surface realizations are averaged. In the

EPILE scheme, the number of iterations is 2. In the FBM scheme, the number of iterations is

6. The Relative Residual Error is 3.879 × 106

, the average computational time by the EPPILE(2) + FBM(6) is about 50 seconds, by the MOM (CGM) is 333 seconds. Figure 4(b) is

the case that L increases to 200 (rN = 2048), while other parameters are fixed. The EPILE

(2) + FBM (6) scheme is also applied. The average computational time by EPILE + FBM is about 165 seconds, by the MOM is about 1484 seconds, the Relative Residual Error is 4.207 ×

106

.

Figure 5 gives the case for VV polarization. In Fig. 5(a), rN = 1024, the Relative Residual

Error is 2.249 × 106

, the average computational time by the EPILE(2) + FBM(6) is about 48

seconds, by the MOM (CGM) is 201 seconds. In Fig. 5(b), rN = 2048, the Relative Residual

Error is 3.559 × 106

, the average computational time by the EPILE(2) + FBM(6) is about 168 seconds, by the MOM (CGM) is about 852 seconds. Comparing the scattering pattern of BSC by the EPILE(2) + FBM(6) and the MOM (CGM) in Fig. 4 and Fig. 5, it is found that the scattering curves match well with each other for both HH and VV polarization, which indicates the ordered EPILE + FBM is exact and timesaving. With increasing the rough surface length, the advantage of the EPILE + FBM in the computational time is more evident.

-80 -60 -40 -20 0 20 40 60 80

-25

-20

-15

-10

-5

0

5

(a)

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3

Ru=Rd=1

Xu=Xd=0

L=100

VV

EPILE+FBM

MOM(CGM)

-80 -60 -40 -20 0 20 40 60 80-35

-30

-25

-20

-15

-10

-5

0

5

10

(b)

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3

Ru=Rd=1

Xu=Xd=0

L=200

VV

EPILE+FBM

MOM(CGM)

Fig. 5. BSC versus the scattering angle (VV polarization).


We have also calculated the 2, 3 order EPILE combined with 2, 3, 4, 5, 6 iteration number of FBM for both HH and VV polarizations, the Relative Residual Error and the computational time are listed in Table 1, Table 2, respectively. The comparison of the data in row suggests that, for a given EPILE iteration number, the Relative Residual Error decreases as the FBM iteration number increases, the FBM increases one iteration step, the computational time increases 4~5 seconds. The comparison of the data in column shows that, for a given FBM order, the Relative Residual Error decreases as the EPILE order increases, the EPILE increases one order, the computational time increases 4~8 seconds. In addition, the computational time of the EPILE(2) + FBM(3) almost equals to that of the EPILE(3) + FBM(2).

Table 1. Comparison of Different Order EPILE Combined with Different Iteration Number of FBM in Relative Residual Error and Computational Time for One Dielectric

Rough Surface Realization (HH Polarization)

EPILE() + FBM()

(2) + (2) (2) + (3) (2) + (4) (2) + (5) (2) + (6)

Relative Residual Error

8.307 × 103 7.220 × 104 6.721 × 105 1.746 × 105 1.661 × 105

computational time (second)

37 41 45 50 55

EPILE() + FBM()

(3) + (2) (3) + (3) (3) + (4) (3) + (5) (3) + (6)


7.960 × 103 6.906 × 104 6.376 × 105 1.566 × 105 1.191 × 105


42 47 53 58 64

Table 2. Comparison of Different Order EPILE Combined with Different Iteration Number of FBM in Relative Residual Error and Computational Time for One Dielectric

Rough Surface Realization (VV Polarization)

EPILE() + FBM()

(2) + (2) (2) + (3) (2) + (4) (2) + (5) (2) + (6)


1.492 × 104 2.045 × 105 1.268 × 105 1.231 × 105 1.223 × 105


34 38 43 47 52

EPILE() + FBM()

(3) + (2) (3) + (3) (3) + (4) (3) + (5) (3) + (6)


1.436 × 104 1.938 × 105 1.156 × 105 1.102 × 105 1.096 × 105


38 44 49 55 60

-80 -60 -40 -20 0 20 40 60 80

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=4.3

Rd=1, Xu=Xd=0

i=0, L=100

HH

Ru=0.1

Ru=1.0

Ru=3.9(a)

-80 -60 -40 -20 0 20 40 60 80

-25

-20

-15

-10

-5

0

5

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=4.3

Rd=1, Xu=Xd=0

i=0, L=100

VV

Ru=0.5

Ru=1.5

Ru=3.5 (b)

Fig. 6. BSC versus the scattering angle (VV polarization).


-80 -60 -40 -20 0 20 40 60 80

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3

Rd=Ru=1, Xd=0

i=0, L=100

HH

Xu=0.4L

Xu=0.3L

Xu=0.2L (a)

-80 -60 -40 -20 0 20 40 60 80-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3

Rd=Ru=1 , Xd=0

i=0, L=100

VV

Xu=0.4L

Xu=0.3L

Xu=0.2L(b)

Fig. 7. BSC versus the scattering angle (different Xu).

-80 -60 -40 -20 0 20 40 60 80

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Dd=3.3, Xu= Xd=0

Ru=Rd=1

i=45, L=100

HH

Hu=40.3

Hu=27.3

Hu=10.3 (a)

-80 -60 -40 -20 0 20 40 60 80

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Dd=3.3, Xu= Xd=0

Ru=Rd=1

i=45, L=100

VV

Hu=40.3

Hu=27.3

Hu=10.3 (b)

Fig. 8. BSC versus the scattering angle (different Hu).

Figure 6 exhibits the BSC versus the scattering angle for the „target above + rough surface

+ target below‟ with different radius uR of the „target above‟ using the EPILE(2) + FBM(6)

for both HH and VV polarization. It is shown that, with the increasing of uR , the specular

coherent scattering changes slightly, while the incoherent scattering at non-specular region increases evidently, as the coupling scattering between the „target above‟ and the „rough

surface‟ increases simultaneously. The dependency of the BSC on the horizontal distance uX

of the „target above‟ versus the scattering angle for both HH and VV polarization is shown in

Fig. 7. It is readily found that, with the increasing of uX , the BSC decreases at non-specular

region due to the fact that the intensity of the Thorsos‟ tapered wave decreases gradually from the center to the edge of the rough surface(shown in Fig. 1), the coupling scattering from the

„target above‟ and the rough surface is strongest when uX adjoins the center of the rough

surface, and decreases gradually when uX is close to the truncation point of it. The

dependency of the BSC on the height uH of the „target above‟ versus the scattering angle for

both HH and VV polarization is depicted in Fig. 8. It is observed that, with the increasing of

uH , the specular coherent scattering changes slightly, while the incoherent scattering at non-

specular region decreases evidently, especially at the backward direction, as the coupling scattering between the „target above‟ and the rough surface decreases simultaneously.


-80 -60 -40 -20 0 20 40 60 80

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3, Xd=0

Xu=0.45L, Ru=2

i=45, L=100

HH

Rd=2.5

Rd=2.8

Rd=3.0 (a)

-80 -60 -40 -20 0 20 40 60 80-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3, Xd=0

Xu=0.45L, Ru=2

i=45, L=100

VV

Rd=2.5

Rd=2.8

Rd=3.0 (b)

Fig. 9. BSC versus the scattering angle (different Rd).

-80 -60 -40 -20 0 20 40 60 80

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3, Xu=0.45L

Ru=2, Rd=3

i=45, L=100

HH

Xd=0.3L

Xd=0.2L

Xd=0.0L (a)

-80 -60 -40 -20 0 20 40 60 80-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3, Xu=0.45L

Ru=2, Rd=3

i=45, L=100

VV

Xd=0.3L

Xd=0.2L

Xd=0.0L (b)

Fig. 10. BSC versus the scattering angle (different Xd).

-80 -60 -40 -20 0 20 40 60 80

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=3.3

Xu=0.45L, Xd=0.0L

Ru=2, Rd=3

i=45, L=100

HH

Dd=4.3

Dd=3.6

Dd=3.3 (a)

-80 -60 -40 -20 0 20 40 60 80-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=3.3

Xu=0.45L, Xd=0.0L

Ru=2, Rd=3

i=45, L=100

VV

Dd=4.3

Dd=3.6

Dd=3.3(b)

Fig. 11. BSC versus the scattering angle (different Dd).

Figure 9–11 present the BSC versus the scattering angle for the „target above + rough

surface + target below‟ with different radius dR , horizontal distance

dX , and depth dD of

the „target below‟ using the EPILE(2) + FBM(6) for both HH and VV polarization. It is

shown from Fig. 9 that, as dR increases, the specular coherent scattering changes slightly, due

to increasing of the coupling scattering between the „target below‟ and the „rough surface‟, while the incoherent scattering at backward non-specular region increases evidently.

Figure 10 gives that, with the increasing of dX , the scattering curve decreases at non-specular


region. Figure 11 indicates that, as dD increases, the specular coherent scattering changes

slightly, while the incoherent scattering at non-specular region decreases evidently, especially at the backward direction. All of the phenomena above are similar with the effects of

uR ,uX ,

uH for the composite model in Figs. 6–8, and the inducements can also be obtained

on the analogy of those discussed above for Figs. 6–8.

-80 -60 -40 -20 0 20 40 60 80

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

Hu=Dd=3.3

Xu=0.45L, Xd=0.0L

Ru=Rd=1, l=1.0

i=45, L=100

=0.1

=0.2

=0.4

HH

(a)

-80 -60 -40 -20 0 20 40 60 80

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

BS

C(d

B)

scattering angle()

Hu=Dd=3.3

Xu=0.45L, Xd=0.0L

Ru=Rd=1, l=1.0

i=45, L=100

=0.1

=0.2

=0.4

VV

(b)

Fig. 12. BSC versus the scattering angle (different ).

-80 -60 -40 -20 0 20 40 60 80

-50

-40

-30

-20

-10

0

10

BS

C(d

B)

scattering angle()

Hu=Dd=3.3

Xu=0.45L, Xd=0.0L

Ru= Rd=1, =0.1

i=45, L=100

HH

l=1.0

l=2.0

l=4.0 (a)

-80 -60 -40 -20 0 20 40 60 80

-50

-40

-30

-20

-10

0

10

B

SC

(dB

)

scattering angle()

Hu=Dd=3.3

Xu=0.45L, Xd=0.0L

Ru= Rd=1, =0.1

i=45, L=100

VV

l=1.0

l=2.0

l=4.0 (b)

Fig. 13. BSC versus the scattering angle (different l ).

In Fig. 12, using the EPILE(2) + FBM(6), the influence of rms height of the rough

surface on the BSC of the composite model for both HH and VV polarization is examined.

Obviously, with the increasing of the rms height , the specular coherent scattering

decreases, while the incoherent scattering at non-specular direction increases evidently, especially at the backward direction. We attribute this behavior to the fact that, the bigger the rms height is, the rougher the surface is, hence, the stronger the intensity of coupling scattering between the target and the rough surface is. Figure 13 gives the dependence of BSC

on correlation length l of the composite model using the EPILE(2) + FBM(6)for both HH

and VV polarization. Evidently, with the increasing of the correlation length, the specular coherent scattering decreases, but the specular peak becomes wider, the incoherent scattering at non- specular direction decreases. The reason for this is that the bigger the correlation length is, the smoother the rough surface is, hence, the weaker the intensity of coupling scattering between the target and the rough surface is. To further explore the important scattering characteristics of the composite model, in Fig. 14, using the EPILE(2) + FBM(6), for HH and VV polarization, the BSC of the „target above + rough surface + target below‟ composite model is examined for different incident angles 30°, 45°, 60°, respectively. It can be observed that, with the increasing of the incident angle, for HH polarization, the specular


coherent scattering and the forward incoherent scattering increases, while for VV polarization, the specular coherent scattering and the forward incoherent scattering decreases, for both HH and VV polarization, and also the width of the specular peak becomes wider.

-80 -60 -40 -20 0 20 40 60 80

-50

-40

-30

-20

-10

0

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3

Xu=0.45L, Xd=0.0L

Ru= Rd=1

L=100

HH

i=30

i=45

i=60 (a)

-80 -60 -40 -20 0 20 40 60 80

-50

-40

-30

-20

-10

0

10

BS

C(d

B)

scattering angle()

=0.1, l=1.0

Hu=Dd=3.3

Xu=0.45L, Xd=0.0L

Ru= Rd=1

L=100

VV

i=30

i=45

i=60 (b)

Fig. 14. BSC versus the scattering angle (different θi).

5. Conclusions

In this paper, the fast method EPILE + FBM, i.e., the extended Propagation Inside Layer Expansion combined with the Forward-Backward method is applied to study composite scattering from targets both above and below the dielectric rough surface. Although this method is based on the rigorous PILE method and the Forward- Backward method, but different from the previous relevant works, the dual targets are considered in our algorithms, the integral equations are reestablished and the EPILE is improved for taking the dual local and coupling interactions into account. The Extended PILE method was applied to the case of targets above and below the rough surface, and the Forward-Backward method was applied to the rough surface. For the large size rough surface, the EPILE + FBM and schemes can reduce the computational complexity. Using the EPILE + FBM schemes, scattering from the cylinder targets above and below the exponential spectrum rough surface is investigated. Generally speaking, an accurate result can be obtained by this method only through a few iterations, while the computing efficiency (i.e., time) is improved more evidently when the surface size increases, compared with the MOM (CGM). Especially speaking, the presented scheme is generalized, that is to say, all the composite scattering problems including „PEC target above + rough surface‟, „PEC target below + rough surface‟, and „PEC target above + rough surface + PEC target below‟ can be efficiently solved by it for both HH and VV polarization. It needs to be pointed out that the future investigation on this topic will include the composite scattering from the 3-D arbitrary target and the 2-D randomly rough surface by this algorithm.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 60971067), by the Fundamental Research Funds for the Central Universities (Grant No. 20100203110016), and the Fundamental Research Funds for the Central Universities. The authors would like to thank the reviewers for their helpful and constructive suggestions.


a study of electromagnetic scattering from conducting targets above and below the dielectric rough...

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