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Bistatic electromagnetic scatteringfrom a three-dimensional perfectelectric conducting object above aGaussian rough surface based on theKirchhoff–Helmholtz and electric fieldintegral equationXi-Min Li a , Chuang-Ming Tong a b , Shu-Hong Fu a & Jing-Jing Li aa Missile Institute , Air Force Engineering University , P.O. Box 25,Sanyuan, Shaanxi, Chinab Southeast University , State Key Laboratory of Millimeter Waves,Nanjing, ChinaPublished online: 14 Apr 2011.

To cite this article: Xi-Min Li , Chuang-Ming Tong , Shu-Hong Fu & Jing-Jing Li (2011) Bistaticelectromagnetic scattering from a three-dimensional perfect electric conducting object abovea Gaussian rough surface based on the Kirchhoff–Helmholtz and electric field integral equation,Waves in Random and Complex Media, 21:3, 389-404, DOI: 10.1080/17455030.2011.571725

To link to this article: http://dx.doi.org/10.1080/17455030.2011.571725

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Waves in Random and Complex MediaVol. 21, No. 3, August 2011, 389–404

Bistatic electromagnetic scattering from a three-dimensional perfect

electric conducting object above a Gaussian rough surface based on

the Kirchhoff–Helmholtz and electric field integral equation

Xi-Min Lia*, Chuang-Ming Tonga,b, Shu-Hong Fua and Jing-Jing Lia

aMissile Institute, Air Force Engineering University, P.O. Box 25, Sanyuan, Shaanxi,China; bSoutheast University, State Key Laboratory of Millimeter Waves, Nanjing, China

(Received 27 September 2010; final version received 25 February 2011)

A hybrid integral equation is developed to solve the problem ofelectromagnetic (EM) scattering from a three-dimensional (3D) perfectelectric conducting (PEC) object above a two-dimensional (2D) PEC ordielectric Gaussian rough surface. Firstly, the Kirchhoff–Helmholtz (KH)equation is adopted to describe the wave reflection on the rough surface;only one integral operation on the rough surface is needed, and thescattering from the object can be described by solving the electric fieldintegral equation (EFIE) on the surface of the object. Moreover, accordingto scattering theory, the KH equation and the EFIE are coupled together(KH-EFIE) to describe wave propagation between the object and the roughsurface. Then method of moments (MoM) is adopted to solve theKH-EFIE, and the current is obtained to calculate the scattering field.Finally, compared with other methods, the accuracy of the proposedapproach is validated, and its efficiency is proved to be much higher thannumerical solutions. Furthermore, by calculating the statistic compositeradar cross-section (RCS) of the object/surface and the difference radarcross-section (DRCS) of the object, the influence of the rough surface rootmean square (rms) height, the correlation length, the medium permittivity,the shape of the object, and the altitude of the object on the scatteringcharacteristic is investigated.

1. Introduction

The problem of electromagnetic (EM) wave scattering from an object above a roughsurface has been widely studied because of its military and academic applications inremote sensing and target identification. The conventional methods for solvingthis problem can be classified into numerical methods and analytical approximatemethods. However, each method has its disadvantages in solving scatteringproblems for both a rough surface and an object. Numerical methods are accurateenough but require huge memory and computational effort [1–5]; analytical approxi-mate methods greatly save CPU time but sometimes give inaccurate results [6,7].To overcome these disadvantages, some authors have tried to develop hybrid

*Corresponding author. Email: bigboy11272000@126.com

ISSN 1745–5030 print/ISSN 1745–5049 online

� 2011 Taylor & Francis

DOI: 10.1080/17455030.2011.571725

http://www.informaworld.com

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methods to investigate scattering from an object and a rough surface [8,9].Burkholder et al. [10] adopted the generalized forward–backward (GFB) methodto study composite EM scattering of two-dimensional ship-like targets on randomrough sea surfaces. Johnson [11,12] proposed a numerical model for scattering froman object above a half-space/rough-surface. Ye and Jin [8] proposed a hybrid methodcombining an analytic Kirchhoff approximation (KA) and the numerical method ofmoments (MoM), and the final solution of scattering from a two-dimensional (2D)perfect electric conducting (PEC) object above a one-dimensional (1D) randomlyrough PEC surface was obtained by adopting an iterative approach. A half-spaceGreen’s function with a rough surface interface was first derived from the Kirchhoffapproximation by Guan et al. [9], and MoM was applied to analyze the scatteringproblem of a 3D arbitrarily shaped PEC object above a 2D PEC rough surface.A finite-difference time-domain approach for EM scattering characteristic from a 2Dinfinitely long target with arbitrary cross-section above a 1D randomly rough seasurface is presented by Li et al. [13]. Kubicke et al. extended the PILE (Propagation-Inside-Layer Expansion) method [14] to the more general case of two illuminatedsurfaces, and applied this extended PILE method to an object located above a roughsurface [15]. Finally, a 3D multilevel UV method was used to calculate vector wavescattering from a target and a rough surface composite model by Deng et al. [16].

In this paper, a hybrid Kirchhoff–Helmholtz electric field integral equation(KH-EFIE) is proposed to study the bistatic scattering characteristic of a 3D PECobject above a 2D Gaussian rough surface. The KH equation in [17,18] is adopted todescribe the reflection of EM waves on the rough surface, and the EFIE, in which theilluminating wave contains the incident wave and the multiple scattering wave fromthe underlying rough surface, is built on the surface of the object. On the other hand,the incident wave and the multiple scattering wave induced by the current on thesurface of the object illuminate the rough surface, and are treated together as theilluminating wave in the KH equation. Finally, KH and EFIE are combined togetherand solved by employing MoM.

2. Hybrid integral equations: KH-EFIE

2.1. Description of composite EM scattering

As shown in Figure 1, the incident EM wave with an incident angle of �i,’iilluminates a system composed of a PEC object and an underlying rough surface.The altitude of the object is denoted by d, and R is the enclosed radius. The incidentwave (tinc) illuminates the rough surface and results in the scattering wave (tr).tr and tinc illuminate the object and induce a current (J) on the surface of the object.The scattering wave (ts) induced by J causes a new scattering wave (tsr) on the roughsurface. tsr is then incident on the object and makes an additional contribution to J.Then the new J induces a new ts, and the new ts causes a new tsr. In this way,the interactions between the object and surface are considered order by order, i.e.first solve the surface, then illuminate the target, then solve the target, thenilluminate the surface, and then solve the surface. J and tsr mutually affect and

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update each other, order by order. However, during the propagation of scattering

wave, the induced current reaches convergence as soon as the incident wave

illuminates on the object and surface. So the final J and tsr are invariable but

unknown. In order to accurately describe the interactions between J and tsr, and

obtain their final values, tsr is introduced into the EFIE in which J is unknown. Let

us denote the electric field as E, when a plane EM wave illuminates the system. Then

the EFIE is built on the surface (St) of the object by

�jk�A� r� ¼ E incðrÞ þ E rðrÞ þ EsrðrÞ ð1Þ

where A and � are defined by

A ¼

ZSt

Jðr0ÞGðr, r0ÞdSt � ¼j

k

ZSt

r0 � Jðr0ÞGðr, r0ÞdSt ð2Þ

where r, r0 2St, respectively, denote the field point and the source point, k, � are the

wavenumber and wave impedance in free space, respectively, and G(r, r0)¼ exp(�

jkjr� r0j)/jr� r0j is the 3D Green’s function of infinite free space. As J and Esr are all

unknowns, we can write (1) as

�jk�A� r�� EsrðrÞ ¼ E incðrÞ þ E rðrÞ: ð3Þ

Figure 1. (Color online) Model of the 3D object above the underlying rough surface.

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2.2. Formulation of hybrid equations

The excitation terms on the right-hand side of formulation (3) must be known before

solving the EFIE by MoM. E inc(r) is the electric field of the incident plane wave and

can be written as

E incðrÞ ¼ eiE0 expð�jki � rÞ ð4Þ

where ki ¼ kki ¼ kðx sin �i cos ’i þ y sin �i sin ’i � z cos �iÞ denotes the incident wave-vector and ei is the polarization vector of the electric field.

Furthermore, E r(r), as the electric field of the scattering wave after the plane

wave has impinged upon the rough surface, can be written in terms of the KH

equation

E rðrÞ ¼

ZSr

f jk�Gðr, r00Þ � ½n�Hinctotalðr

00Þ� þ r �Gðr, r00Þ � ½n� E inctotalðr

00Þ�gdSr ð5Þ

where Sr is denoted as the rough surface, r00 2Sr, n is the unit normal vector on the

rough surface, and E inctotalðr

00Þ is defined as the total electric field on the rough surface

because of the excitation of E inc(r00). The dyadic Green’s function G(r, r00) in free

space is defined by

Gðr, r00Þ ¼ Iþrr

k2

� �expð�jkjr� r00jÞ

4�jr� r00jð6Þ

where I is the unit tensor. Let an orthonormal system ðpi, qi, kiÞ be defined at the

point r00, with

qi ¼ki � n

jki � njpi ¼ qi � ki: ð7Þ

The unit vectors pi and qi are the local perpendicular and parallel polarization

vectors at the point r00, respectively. Then, we have

n� E inctotalðr

00Þ ¼ E0fðei � qiÞðn� qiÞð1þ RhÞ þ ðei � piÞðn � kiÞqið1� RvÞg expð�jki � r00Þ

ð8Þ

n�Hinctotalðr

00Þ ¼E0

�f�ðei � qiÞðn � kiÞð1� RhÞ þ ðei � piÞðn� qiÞð1þ RvÞg expð�jki � r

00Þ

ð9Þ

where Rh and Rv are the local Fresnel reflection coefficients [19] for perpendicular

and parallel polarization, respectively, and their definitions are given by

Rh ¼cos � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"r�r � sin2 �

pcos � þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"r�r � sin2 �

p , Rv ¼"r cos � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"r�r � sin2 �

p"r cos � þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"r�r � sin2 �

p ð10Þ

where "r and �r are the relative dielectric permittivity and permeability of the surface

medium, respectively.

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According to formulations (5), (8) and (9), E r(r) can be expressed in terms of

E inc(r00). Similarly, Esr(r) can be expressed in terms of the scattering field E s(r00), and

E s(r00) induced by J can be written as

E sðr00Þ ¼ jk�

ZSt

Jðr0ÞGðr00, r0Þ þ1

k2r0 � Jðr0Þr00Gðr00, r0Þ

� �dSt: ð11Þ

Replacing eiE0 expð�jki � r00Þ in (8) and (9) with Es(r00) in (11), then substituting (8)

and (9) in (5), Esr(r) can be obtained as

EsrðrÞ ¼jk�

ZSr

ZSt

fkGðr, r00Þ � ½h�ðJðr0ÞGðr00, r0Þ þ1

k2r0 � Jðr0Þr00Gðr00, r0ÞÞ � qiiðn � k

0iÞ

� ð1� RhÞqi þ hðJðr0ÞGðr00, r0Þ þ

1

k2r0 � Jðr0Þr00Gðr00, r0ÞÞ � piið1þ RvÞðn� qiÞ�

� r �Gðr, r00Þ � ½hðJðr0ÞGðr00, r0Þ þ1

k2r0 � Jðr0Þr00Gðr00, r0ÞÞ � qiið1þ RhÞðn� qiÞ

� hðJðr0ÞGðr00, r0Þ þ1

k2r0 � Jðr0Þr00Gðr00, r0ÞÞ � piið1� RvÞðn � k

0iÞqi�gdStdSr ð12Þ

where k0i ¼ ðr00� r0Þ=jr00 � r0j denotes the unit vector of the scattering wave ts from

point r0 to point r00 with the assumption that ts is treated as a plane wave, and

qi ¼ ðk0i � nÞ=jk0i � nj, pi ¼ qi � k0i.

Changing the integral order of the double integral in (12), then substituting (12)

in (3), the KH-EFIE is finally obtained as

� jk�

ZSt

f½Jðr0ÞGðr, r0Þ þ1

k2r0 � Jðr0ÞrGðr, r0Þ� �

ZSr

½�kGðr, r00Þ � h�ðJðr0ÞGðr00, r0Þ

þ1

k2r0 � Jðr0Þr00Gðr00, r0ÞÞ � qiðn � k

0iÞð1� RhÞqi þ ðJðr

0ÞGðr00, r0Þ þ1

k2r0 � Jðr0Þr00Gðr00, r0ÞÞ

� pið1þ RvÞðn� qiÞi þ r �Gðr, r00Þ � hðJðr0ÞGðr00, r0Þ þ1

k2r0 � Jðr0Þr00Gðr00, r0ÞÞ

� qið1þ RhÞðn� qiÞ þ ðJðr0ÞGðr00, r0Þ þ

1

k2r0 � Jðr0Þr00Gðr00, r0ÞÞ

� piðn � k0iÞð1� RvÞqii�dSrgdSt ¼ E incðrÞ þ E rðrÞ: ð13Þ

3. MoM solution of the KH-EFIE

According to the standard MoM procedure, we first divide the object into Mf planar

triangular patches with M edges. Assume that the current J is approximated by

RWG basis functions [20]

Jðr0Þ ¼XMm¼1

Jm fmðr0Þ: ð14Þ

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Then, MoM can be used to solve Equation (13). After substituting (14) into (13), thematrix equation is finally obtained as

ZI ¼ V ð15Þ

where Z is the impedance matrix, and I and V are the unknown vector and excitationvector, respectively, and the elements of Z can be expressed by

Zmn ¼ Zttmn þ Ztr

mn ð16Þ

where Zttmn represents the self impedance of the object, and Ztr

mn represents themutually-coupled impedance of the object/rough surface,

Zttmn ¼ �lm jðAþmn �

�þm2þ A�mn �

��m2Þ þ�þmn ���mn

� �ð17Þ

Ztrmn ¼ �

Zsr

Fnðrcþm , r00ÞdSr

� ��lm�þm

2�

Zsr

Fnðrc�m , r00ÞdSr

� ��lm��m

2ð18Þ

where

A�mn ¼

ZT�n

fnðr0ÞGðr�m, r

0ÞdSt ��mn ¼ �1

j!"

ZT�n

r0 � fnðr0ÞGðr�m, r

0ÞdSt ð19Þ

Fnðrc�m , r00Þ ¼ �kGðrc�m , r00Þ � ½�ðEn � qiÞðn � k

0iÞð1� RhÞqi þ ðEn � piÞð1þ RvÞðn� qiÞ�

þ r �Gðrc�m , r00Þ � ½ðEn � qiÞð1þ RhÞðn� qiÞ þ ðEn � piÞðn � k0iÞð1� RvÞqi� ð20Þ

En ¼ln�þn

2Gðr00, rcþn Þ þ

lnk2r00Gðr00, rcþn Þ þ

ln��n

2Gðr00, rc�n Þ �

lnk2r00Gðr00, rc�n Þ ð21Þ

where Fnðrc�m , r00Þ denotes the reflected electric field on patch T�m , when the point r00 of

the rough surface is illuminated by the scattering field En induced by the current overthe patches Tþn and T�n . The elements of V can be written as

Vm ¼ lm½Eincðrcþm Þ þ E rðrcþm Þ� �

�þm2þ lm½E

incðrc�m Þ þ E rðrc�m Þ� ���m2: ð22Þ

4. Calculation of the far field

After solving the KH-EFIE by using MoM, the current over the surface of the objectis known, and the scattering field at any point above the rough surface can beobtained by formulation (5), (11) and (12). Considering the approximation of theGreen’s functions and the dyadic Green’s functions in the far field regions,

Gðr, r0Þ ffiexpð�jkrÞ

4rexp½ jkðks � r

0Þ� ð23Þ

Gðr, r00Þ ffiexpð�jkrÞ

4rðvsvs þ hshsÞ exp½ jkðks � r

00Þ� ð24Þ

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r �Gðr, r00Þ ffi jkexpð�jkrÞ

4rðhsvs � vshsÞ exp½ jkðks � r

00Þ� ð25Þ

where

ks ¼ x sin �s cos ’s þ y sin �s sin’s þ z cos �s

hs ¼ �x sin ’s þ y cos ’s

vs ¼ x cos �s cos ’s þ y cos �s sin ’s � z sin �s:

8><>:

ð26Þ

E r(r) and E sr(r) can be calculated by the KH equations, and E s(r) can beobtained by (11). Finally, the difference radar cross-section �d (DRCS [12])contributed by the presence of the object above the rough surface is obtained by

�d ¼ limr�!1

2�r2jE s þ Esrj2

jE incj2ð27Þ

and the composite RCS � of the object/rough surface can be calculated by

� ¼ limr�!1

2�r2jE s þ Esr þ E rj2

jE incj2: ð28Þ

5. Numerical results and analysis

5.1. Accuracy and efficiency of the computations

To validate the accuracy of the MoM solution of the KH-EFIE, all the numericalresults of the different methods below are obtained for the same rough surfacesample created by the Monte Carlo (MC) method [21]; six points per wavelengthwere used in discretizing the surface profile.

In the following examples, the incident plane wave can be classified into fourkinds: plane wave 1 with �i¼ 30�, ’i¼ 90�, plane wave 2 with �i¼ 35�, ’i¼ 90�, planewave 3 with �i¼ 20�, ’i¼ 90�, plane wave 4 with �i¼ 25�, ’i¼ 90�, and the observingplane can be classified into three kinds: observing plane 1 (’s¼ 90�), observing plane2 (’s¼ 270�), observing plane 3 (’s¼ 0�).

Case 1: consider a PEC cube with side length of a¼ 0.2� (� is the wavelength infree space) at an altitude of d¼ 5� above a PEC flat surface. For incident plane wave1, surface length Lx¼Ly¼ 40�, correlation length lx¼ ly¼ 4.0�, rms height h¼ 0,and the CPU employed is a Pentium (R) Dual-Core 2.5 GHz processor with2GBytes of RAM. Figure 2 shows the element, In, of the unknown vector, I, versusthe discretizing number, n, by using the half-space Green’s function method,FBM-CG (forward–backward method and conjugate gradient) in [4] and the MoMsolution of the KH-EFIE, respectively. The order of the discretizing number on thecube surface is also shown in Figure 2. The three methods are almost perfectlymatched, but the CPU time consumed is 10, 923 s for FBM-CG with 28, 872unknowns, and 1012 s for the proposed approach with 72 unknowns. The taperedwave parameter g used in the FBM-CG can be decided by consulting reference [19],and gx¼Lx/3, gy¼Ly/3 in the numerical examples of this work.

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Case 2: using the same parameters as in case 1 except for h¼ 0.5�, Figure 3presents the hh-polarized DRCS in observing plane 1 with the MoM solution of theKH-EFIE and FBM-CG. It demonstrates that the proposed approach and FBM-CG have almost the same accuracy for a PEC rough surface. The CPU timeconsumed is 18, 762 s for FBM-CG and 1114 s for the proposed approach.

Case 3: considering a sphere with a radius of R¼ 0.5� at an altitude of d¼ 4.5�above the dielectric rough surface, given incident plane wave 2, then lx¼ ly¼ 1.2�,h¼ 0.1�, Lx¼Ly¼ 40�, "r¼ 2.0, and �r¼ 1.0. Figures 4 and 5 present the hh-polarized RCS of the sphere and the composite RCS of sphere/surface in observing

–90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90

–35

–30

–25

–20

–15

–10 MoM solution of KH_EFIE

FBM-CG

DR

CS

-hh

(dB

)

qs(degree)

Figure 3. DRCS of the cube above a PEC rough surface.

,

Figure 2. Unknowns versus the discretizing number on the cube surface.

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–90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90

–70

–60

–50

–40

–30

–20

–10

0

10

RCS of sphere in free space

MoM solution of KH_EFIE

FBM_CG

RC

S-h

h (d

B)

qs(degree)

Figure 4. RCS of the sphere above the rough surface.

–90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90

–10

–5

0

5

10

15

20

25

30

35

40

45

50 MoM solution of KH-EFIE

FBM-CG

Numerical model in reference [12]

RC

S-h

h (d

B)

qs(degree)

Figure 5. Composite RCS of the sphere/rough surface.

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plane 2, by adopting the proposed approach, FBM-CG and the numerical model in[12], respectively. The agreement of the RCS results by the three solutions validatesthe accuracy of the proposed approach for a dielectric rough surface. In Figure 4, theRCS enhancement demonstrates that object/rough surface interaction effects canmake important contributions to the object scattering. The CPU time consumed is124, 512 s for FBM-CG with 58, 764 unknowns and 11, 763 s for the proposedapproach with 1164 unknowns.

Case 4: since DRCS does not contain the surface returns by the incident wave, itcan be used to denote the interaction of the sphere/surface by changing the length anddiscretization of the rough surface in case 3. The DRCS varies slightly with differentsurface length and discretization in Figure 6. In fact, when the distance between theobject and surface patches is too large, the coupling scattering of the object/surfacebecomes infinitesimal. In order to confirm this conclusion, the scattering electric fieldon the surface (Lx¼Ly¼ 80�) by the current of the object is presented in Figure 7.The scattering electric field becomes very weak on patches far away (more than 20�)from the center of the surface. Therefore, to reduce computational work and ensureaccuracy, the surface length can be truncated to Lx¼Ly 40�.

Compared with the results of the different methods, the MoM solution of theKH-EFIE is proved to have almost the same accuracy as the others, but is moreefficient than the numerical methods such as FBM-CG.

5.2. Statistic composite EM scattering

Making 100 MC simulations of the rough surface, given an incident plane wave, wecan obtain the mean composite RCS of the objects/rough surface by the MoMsolution of the KH-EFIE.

–90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90

–5–4–3–2–10123456789

101112

Lx=Ly=80l,6 points per l

Lx=Ly=40l,6 points per l

Lx=Ly=20l,6 points per l

Lx=Ly=40l,10 points per l

DR

CS

-hh

(dB

)

qs(degree)

Figure 6. DRCS of sphere/rough surfaces with different surface lengths and discretizations.

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Case 5: let a PEC cylinder with a radius of R¼ 0.25� and a length of H¼ � lie at

an altitude of d¼ 4.5� above a Gaussian dielectric rough surface. Given incident

plane wave 3, and Lx¼Ly¼ 40�, the relative dielectric permittivity and permeability

are "r¼ 10.0 and �r¼ 1.0, respectively. Choosing observing plane 2, Figure 8

presents the composite vv-polarized bistatic RCS of the cylinder and underlying

rough surface with different correlation length (lx¼ ly¼ l) and rms height, h.In Figure 8, it is obvious that the composite bistatic RCS is closely correlated

with the rms height and correlation length, and the composite RCS appears as a peak

near �s¼�20� (forward scattering), which is more significant for the smoother

surface with a lower value of h/l. The lower h/l is, the more remarkable the forward

scattering (coherent scattering component) becomes, and incoherent scattering

increases and coherent scattering decreases as the roughness increases. So the

forward scattering peak disappears for h/l¼ 0.1.Case 6: let a PEC plane with a side length of a¼ � lie at an altitude of d¼ 5.0�

above a Gaussian dielectric rough surface, with the PEC plane parallel to the xOy

plane. Given incident plane wave 1, Lx¼Ly¼ 40�, h¼ 0.2�, and lx¼ ly¼ 4.0�.Figure 9 presents the composite hh-polarized bistatic RCS results in observing plane

2 for different permittivities. The permittivity has an important influence on the

composite scattering characteristic. The surface with higher permittivity has higher

reflectance, and the interaction between object and surface is more intense. So the

composite RCS is larger for rough surface with higher permittivity. The solid line in

Figure 7. Scattering electric field on the surface by the current of the object.

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RC

S-h

h (d

B)

–80 –60 –40 –20 0 20 40 60 80

0

10

20

30

40

50er=2.0

er=10.0

er=50.0

er=PEC surface

qs(degree)

Figure 9. Composite bistatic RCS of the plane/rough surface with different mediumpermittivities.

–90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90

0

10

20

30

40

50

x

y

O

zH

R

h/l=0.1h/l=0.05h/l=0.005

RC

S-v

v (d

B)

qs(degree)

Figure 8. Composite bistatic RCS of the cylinder/rough surface with different correlationlengths and rms heights.

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Figure 9 denotes the composite RCS result for a PEC rough surface with infinitepermittivity ("r¼1).

Case 7: let a cube, sphere, and tetrahedron with the same surface area of S¼��2

lie at an altitude of d¼ 5� above the Gaussian dielectric rough surface, respectively,

–90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90

–25

–20

–15

–10

–5

0

5

10

15

20

25

Cube and rough surface

Sphere and rough surface

Tetrahedron and rough surface

Rough surface without objects

RC

S-h

h (d

B)

qs(degree)

Figure 11. Composite bistatic RCS of different objects/rough surfaces.

–90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90–15

–10

–5

0

5

O

x

x

y

y

z

O

x

x

y

y

z

d

O

x

x

y

y

z

Cube

Sphere

Tetrahedron

DR

CS

-hh

(dB

)

qs(degree)

Figure 10. DRCS of different objects with the same surface area.

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as shown in Figure 10. Given incident plane wave 4, lx¼ ly¼ 4.0�, Lx¼Ly¼ 40�,"r¼ 2.0 and �r¼ 1.0.

Comparing the results in Figure 10, although the three objects have the samesurface area, their DRCS results in observing plane 2 appear to differ greatly because

–14

–12

–10

–8

–6

–4

–2

0

2

4(a)

(b)

x

y

O

z

xy d

120o

ab

d=5ld=50ld=100l

DR

CS

-hh

(dB

)

qs(degree)

–90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90

–90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90–55

–50

–45

–40

–35

–30

–25

–20

–15

–10

–5

0

5

d=5ld=50ld=100lD

RC

S-h

v (d

B)

qs(degree)

Figure 12. DRCS of the dihedral above the rough surface for different altitude: (a) hhpolarization, (b) hv polarization.

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of their different geometrical shapes. However, in Figure 11, the composite RCS ofthe different objects/rough surface in observing plane 2 presents a slight variance,although it is higher than the RCS of the rough surface only. It is difficult to identifythe object by composite RCS results, because the scattering component of theunderlying surface is much larger than that of the object. So the DRCS data mayhave greater value than the composite RCS data in target identification.

Case 8: as the last example, let a PEC dihedral with side length of a¼ �, b¼ 0.5� lieabove a Gaussian dielectric rough surface. Given incident plane wave 1, h¼ 0.2�,lx¼ ly¼ 4.0�, Lx¼Ly¼ 40�, "r¼ 10.0 and �r¼ 1.0. Figure 12 presents the hh/hv-polarized DRCS results in observing plane 3 for different altitudes of d¼ 5�, d¼ 50�and d¼ 100�. Comparing the DRCS results for the different altitudes in Figure 12, theDRCS for d¼ 5� is much larger than that for d¼ 50� and d¼ 100�, because the objectat the lower altitude has more intense interaction with the underlying rough surface,and when the altitude is high enough (i.e. d¼ 50� and d¼ 100�), the interaction of theobject/rough surface is so weak that DRCS varies slowly with variance of the altitude.

Since the surface is still of finite size and the incident wave is the plane wave,truncation effects appear in the numerical results in terms of the oscillations versusangle that are observed.

6. Conclusion

The hybrid integral equation KH-EFIE is proposed to describe the scattering from aPEC object above a randomly rough surface. MoM is used to numerically solve theKH-EFIE. Compared with the methods for solving the EFIE built on both a roughsurface and an object [1,3–5], the proposed approach avoids the calculation andinversion of the impedancematrix of the rough surface, and the number of unknowns isonly related to the object discretization density. Moreover, the accuracy of the MoMsolution of the KH-EFIE is validated by adopting the Green’s function method, theFBM-CG and the four-path model in the same numerical example. In the last fournumerical examples, the rms height, correlation length, the medium permittivity, andthe object altitude are found to have a great influence on the composite RCS of theobject/rough surface, and the DRCS appears to vary greatly for different objects withthe same surface area. However, since the scattering component of the rough surface isanalytically calculated based on the KH equations similar to the KA approximation,the proposed approach has its valid range restricted to small surface roughness, lowincident angle (�i), and so on. In order to extend its valid range, we are trying tocombine the proposed approach with other numerical or analytical methods.

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