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 OSA

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 [2527], 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 δ and the correlation length l. The problem is assumed to be one-dimensional (variant in the x-z plane). ki represents the propagation vector of the incident wave. θi 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 Ω0(ε0,μ0), Ω1(ε1,μ1) denotes the space above and below the rough surface. ε0=μ0 = 1, ε1 is the relative permittivity of the dielectric rough surface. The rough surface is assumed as nonmagnetic, i.e., μ1=μ0. The randomly rough surface is generated by the Monte-Carlo spectral method [28]. The exponential spectrum W(Ki)=2πδ2l2(1+Ki2l2)3/2 is applied to model the rough surface. Ki 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 rr=xrx^+zrz^, rou=xoux^+zouz^, and rod=xodx^+zodz^, respectively, where xr, xou, xod is the discretized abscissa and, zr, zou, zod is the discretized height.

 

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

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

φinc(r)=exp(jkir(1+[2(x+ztanθi)2/g21]/(kgcosθi)2))exp((x+ztanθi)2/g2),
in which g is tapered parameter. ki is the incident wave vector [28]. The time dependence of exp(jωt) 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 E=y^φinc(r) or transverse magnetic (TM) wave H=y^φinc(r) 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

12E0(r)Sr[E0(r')n'G0(r,r')G0(r,r')n'E0(r')]ds'+   St1G0(r,r')n'E0(r')ds'     =Ei(r)       rSr,
Sr[E0(r')n'G0(r,r')G0(r,r')n'E0(r')]ds'   St1G0(r,r')n'E0(r')ds'     =Ei(r)         rSt1,
12E1(r)+Sr[E1(r')n'G1(r,r')G1(r,r')n'E1(r')ρ]ds'St2G1(r,r')n'E1(r')ds'   =0       rSr,
Sr[E1(r')n'G1(r,r')G1(r,r')n'E1(r')ρ]ds'St2G1(r,r')n'E1(r')ds'=0         rSt2,

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

12H0(r)Sr[H0(r')n'G0(r,r')G0(r,r')n'H0(r')]ds'St1H0(r')n'G0(r,r')ds'     =Hi(r)           rSr/St1,
12H1(r)+Sr[H1(r')n'G1(r,r')G1(r,r')n'H1(r')ρ]ds'+St2[H1(r')n'G1(r,r')G1(r,r')n'H1(r')]ds'   =0       rSr,
12H1(r)+Sr[H1(r')n'G1(r,r')G1(r,r')n'H1(r')ρ]ds'+St2H1(r')n'G1(r,r')ds'=0       rSt2,

where for HH polarization, ρ=μ1/μ0 and for VV polarization, ρ=ε1/ε0. n' is the unit normal vector of the surface or targets, ∇ is the gradient operator. Ei, E0, E1 denote the incident electric field, electric fields in region Ω0 and Ω1, respectively. Hi, H0, H1 denote the incident magnetic field, magnetic fields in region Ω0 and Ω1, respectively. The use of the method of moments (MOM) [15] with point matching and pulse basis functions leads to the following linear system

Z¯(Nt1+Nt2+2Nr)×(Nt1+Nt2+2Nr)X(Nt1+Nt2+2Nr)=S(Nt1+Nt2+2Nr),

in which Z¯(Nt1+Nt2+2Nr)×(Nt1+Nt2+2Nr) denotes the impedance matrix, X(Nt1+Nt2+2Nr) is the induced unknown vector, S(Nt1+Nt2+2Nr) is the incident source item. Nt1,Nt2 are the numbers of sampling points on the target above and target below, respectively. Nr represents the number of sampling points on the rough surface. The impedance matrix is expressed as follows

Z¯   (Nt1+Nt2+2Nr)×(Nt1+Nt2+2Nr)=[ANt1×Nt10Nt1×Nt2BNt1×NrCNt1×Nr0Nt2×Nt1DNt2×Nt2ENt2×NrρFNt2×NrGNr×Nt10Nr×Nt2HNr×NrINr×Nr0Nr×Nt1JNr×Nt2KNr×NrρLNr×Nr].

can be divided into four blocks and expressed as

Z¯   (Nt1+Nt2+2Nr)×(Nt1+Nt2+2Nr)=[A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2A¯(Nt1+Nt2)×(2Nr)rt1,t2A¯(2Nr)×(Nt1+Nt2)t1,t2rA¯(2Nr)×(2Nr)r],
 A¯   (Nt1+Nt2)×(Nt1+Nt2)t1,t2=[ANt1×Nt10Nt2×Nt20Nt2×Nt1DNt2×Nt1]      A¯(Nt1+Nt2)×(2Nr)rt1,t2=[BNt1×NrCNt1×NrENt2×NrρFNt2×Nr],
A¯   (2Nr)×(Nt1+Nt2)t1,t2r=[GNr×Nt1       0Nr×Nt20Nr×Nt1     JNr×Nt2]      A¯(2Nr)×(2Nr)r=[HNr×NrINr×NrKNr×NrρLNr×Nr].

A¯(2Nr)×(2Nr)r corresponds exactly to the impedance matrices of the rough surface. A¯(2Nr)×(Nt1+Nt2)t1,t2r and A¯(Nt1+Nt2)×(2Nr)rt1,t2 can be interpreted as coupling matrices between the targets and the rough surface. A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2 corresponds exactly to the impedance matrices of the targets. The unknown vector and the source item are defined as

X(Nt1+Nt2+2Nr)T=[Xt1,t2(Nt1+Nt2)TXr(2Nr)T]     S(Nt1+Nt2+2Nr)T=[St1,t2(Nt1+Nt2)TSr(2Nr)T],
with
Xt1,t2(Nt1+Nt2)T=[ψ0(rt11)ψ0(rt1Nt1),ψ1(rt21)ψ1(rt2Nt2)]    Xr(2Nr)T=[ψ0(rr1)ψ0(rrNr),ψ0(rr1)nrψ0(rrNr)nr],
St1,t2(Nt1+Nt2)T=[ψi(rt11)ψi(rt1Nt1),010Nt2]    Sr(2Nr)T=[ψi(rr1)ψi(rrNr),010Nr],
in which the superscript T stands for the transpose operator, and /n stands for the normal derivative operator. ψi, ψ0, ψ1 corresponds to Ei, E0, E1 for TE case and Hi,H0,H1 for TM case, respectively.

The elements of the impedance matrix are shown as below:

Apq(HH)={γqΔxj4H0(1)(k0|rprq|)                    pqγqΔxj4H0(1)[k0Δxγq/(2e)]     p=q,   Apq(VV)={γqΔxjk04(n^Rpq)H1(1)(k0|rprq|)       pq                   12zt1(xp)Δx4πγp2                                                                                                                     p=q,
Bpn=γnΔxjk04(n^Rpn)H1(1)(k0|rprn|),   Cpn=γnΔxj4H0(1)(k0|rprn|),
Dvw(HH)={γwΔxj4H0(1)(k1|rvrw|)                       vwγwΔxj4H0(1)[k1Δxγw/(2e)]     v=w,   Dvw(VV)={γwΔxjk14(n^Rvw)H1(1)(k1|rvrw|)       vw                 12+zt2(xv)Δx4πγv2                                                                                                         v=w,
Evn=γnΔxjk14(n^Rvn)H1(1)(k1|rvrn|),   Fvn=γnΔxj4H0(1)(k1|rvrn|),
Gmq(HH)=γqΔxj4H0(1)(k0|rm-rq|),   Gmq(VV)=γqΔxjk04(n^×Rmq)H1(1)(k0|rm-rq|),
Hmn={γnΔxjk04(n^Rmn)H1(1)(k0|rmrn|)       mn                   12zr(xm)Δx4πγm2                                                                                                                       m=n,   Imn={γnΔxj4H0(1)(k0|rmrn|)                            mnγmΔxj4H0(1)[k0Δxγm/(2e)]     m=n,
Jmv(HH)=γvΔxj4H0(1)(k1|rmrv|),   Jmv(VV)=γvΔxjk14(n^Rmv)H1(1)(k1|rmrv|),
Kmn={γnΔxjk14(n^Rmn)H1(1)(k1|rmrn|)     mn                 12zr(xm)Δx4πγm2                                                                                                       m=n,    Lmn={γnΔxj4H0(1)(k1|rmrn|)                            mnγmΔxj4H0(1)[k1Δxγm/(2e)]       m=n,
whereRαn=rαrn|rαrn|             (α=m,p,v),γm=1+[zr(xm)]2,γn=1+[zr(xn)]2,γp=1+[zt1(xp)]2, γq=1+[zt2(xq)]2,γv=1+[zt2(xv)]2, γw=1+[zt2(xw)]2, n^=zr(xn)x^+z^1+[zr(xn)]2, (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. k0=2π/λ is the wavenumber in the free space, where λ is the incident wavelength and k1=k0ε1 is the wavenumber in the transmitted medium Ω1. Δx is the sampling step. rm=xmx^+zmz^ and rn=xnx^+znz^ denote arbitrary two points on the rough surface. rp=xpx^+zpz^ and rq=xqx^+zqz^ denote arbitrary two points of the target above. rv=xvx^+zvz^ and rw=xwx^+zwz^ denote arbitrary two points of the target below. zt1, zt2 and zr is the first-order differential of height function of the target above, target below, and the rough surface, respectively, and zt1, zt1, zr corresponds to the second- order differential of height function. e = 2.718214, γ = 1.781072, H0(1) is the zeroth-order Hankel function of the first kind. H1(1) 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]

ψs(r)=ejk0rrψsN(θs,θi).

For TE and TM case, ψsN(θs,θi) is expressed as

ψsN(θs,θi)(TE,HH)=j42πk0ejπ4{Srexp(jksr)[j(n^ks)Xr1~Nr(x)XrNr+1~2Nr(x)]                                   1+[zr(x)]2dxSt1Xt1(x)exp(jksr)1+[zt1(x)]2dx},
ψsN(θs,θi)(TM,VV)=j42πk0ejπ4{Sr[j(n^ks)Xr1~Nr(x)XrNr+1~2Nr(x)]exp(jksr)1+[zr(x)]2dx                                   St1j(n^t1ks)Xt1(x)exp(jksr)1+[zt1(x)]2dx},
in which n^t1=(zt1(x)x^+z^)1+[zt1(x)]2. The expression for the normalized far-field bistatic scattering coefficient (BSC) with the Thorsos’ tapered plane wave incidence is given as

σs(θs,θi)=|ψsN(θs,θi)|2gπ/2cosθi(1(1+2tan2θi)/(2k02g2cos2θi)).

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

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 [2527], we apply the EPILE to study the composite electromagnetic scattering from targets both above and below the rough surface. The inverse matrix of Z¯(Nt1+Nt2+2Nr)×(Nt1+Nt2+2Nr) can be partitioned into four blocks (it should be noted that, some of the following formulations are of the similar form with those in [2527], but the influence of dual targets both above and below are included).

Z¯(2Nr+Nt1+Nt2)×(2Nr+Nt1+Nt2)1=[T¯U¯V¯W¯],
where

T¯=[A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r]1,
U¯=[A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r]1A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1,
V¯=(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r[A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r]1,
W¯=(A¯(2Nr)×(2Nr)r)1(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r[A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2A¯(Nt1+Nt2)×(2Nr)rt1,t2           (A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r]1A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1.

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

Xr=[A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r]1A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1Sr,
and the total induced unknown vector on the targets above and below can be given by

Xt1,t2=TSt1,t2=[A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r]1St1,t2.

Introducing the characteristic matrix Mc(r) on the rough surface, i.e.,

Mc(r)=(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r(A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2)1A¯(Nt1+Nt2)×(2Nr)rt1,t2

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

Xr(2Nr)(p)=(p=0PMc(r)p)(A¯(2Nr)×(2Nr)r)1(SrA¯(2Nr)×(Nt1+Nt2)t1,t2r(A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2)1St1,t2),
wherep=0PMcp=I(A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2)1A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r, I¯ is the identity matrix.

Similarly, introducing the characteristic matrix Mc(t1,t2) on the targets

Mc(t1,t2)=(A¯(2Nt1)×(Nt1+Nt2)t1,t2)1A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r.

It should be addressed that both Mc(r) and Mc(t1,t2) 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

Xt1,t2(Nt1+Nt2)(p)=(p=0PMc(t1,t2)p)(A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2)1(St1,t2A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1Sr).

Therefore, Xr(2Nr)(p) and Xt1,t2(Nt1+Nt2)(p) can be expressed as follows

Xr(2Nr)(p)=p=0PY(r)(p),   Xt1,t2(Nt1+Nt2)(p)=p=0PY(t1,t2)(p).

Each item in expressions above is given below

Yr(0)=(A¯(2Nr)×(2Nr)r)1(SrA¯(2Nr)×(Nt1+Nt2)t1,t2r(A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2)1St1,t2),   Yr(p)=Mc(r)Yr(p1),
Y(t1,t2)(0)=(A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2)1(St1,t2A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1Sr),   Y(t1,t2)(p)=Mc(t1,t2)Y(t1,t2)(p1).

p denotes the iteration order. In Eq. (28) and Eq. (30), the (A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2)1 accounts for the local interactions on the targets, and (A¯(2Nr)×(2Nr)r)1 accounts for the local interactions on the rough surface. A¯(Nt1+Nt2)×(2Nr)rt1,t2 propagates the resulting field on the rough surface toward the targets (surface-targets coupling), and A¯(2Nr)×(Nt1+Nt2)t1,t2r propagates the resulting field on the targets toward the rough surface (targets-surface coupling), and so on for the subsequent terms Yr(p), just as shown in Fig. 2 . Usually, at each iteration step, the number Nt1+Nt2 of samples for the targets is less than the numberNrof samples for the rough surface. The term (A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2)1ς (ς denotes the unknown column vector of length Nt1+Nt2) will be solved by the MOM(CGM / BCGM / LU), while, to decrease the computing cost and speed up the calculation of the term (A¯(2Nr)×(2Nr)r)1ζr(2Nr) (i.e., single rough surface, ζr(2Nr) denotes the unknown column vector of length 2Nr), the FBM by A. Iodice [14] can be applied. Assume(A¯(2Nr)×(2Nr)r)1ζr(2Nr) equals to ξ(2Nr) (ξ denotes the unknown column vector of length 2Nr). The impedance matrix and the induced unknown vector can be decomposed as follow

A¯r((2Nr)×(2Nr))=A¯r((2Nr)×(2Nr))f+A¯r((2Nr)×(2Nr))d+A¯r((2Nr)×(2Nr))b,   ξr(2Nr)=ξr(2Nr)f+ξr(2Nr)b,
in which A¯r(2Nr×2Nr)f denotes the lower triangle part of A¯r(2Nr×2Nr), A¯r((2Nr)×(2Nr))d denotes the diagonal part of A¯r((2Nr)×(2Nr)), and A¯r((2Nr)×(2Nr))b denotes the upper triangle part of A¯r((2Nr)×(2Nr)). ξr(2Nr)f and ξr(2Nr)b are the forward, backward induced unknown vector on the rough surface, respectively.

 

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

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Assume that ξr(2Nr)=[ξ1r(Nr)Tξ2r(Nr)T] and ζr(2Nr)=[ζ1r(Nr)Tζ2r(Nr)T], therefore,

[HNr×NrINr×NrKNr×NrρLNr×Nr][ξ1r(Nr)Tξ2r(Nr)T]=[ζ1r(Nr)Tζ2r(Nr)T],
HNr×Nr=HNr×Nrf+HNr×Nrs+HNr×Nrb,   INr×Nr=INr×Nrf+INr×Nrs+INr×Nrb,
KNr×Nr=KNr×Nrf+KNr×Nrs+KNr×Nrb,   LNr×Nr=LNr×Nrf+LNr×Nrs+LNr×Nrb.

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 ο () are necessarily investigated. The Relative Residual Error of the scattering coefficient σ obtained by using the EPILE + FBM is defined as the norm of the following form

RRE=9090|σ(EPILE+FBM)σ(MOM(CGM/LU/BCGM))|29090|σMOM(CGM/LU/BCGM)|2.

The complexity per iteration of the terms Yt1,t2(p), Yr(p) is given below, respectively

Mc(r)Yr(p1)=(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2r(A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2)1A¯(Nt1+Nt2)×(2Nr)rt1,t2Yr(p1)ο((Nt1+Nt2)×(2Nr))   (a)ο(Miter12(Nt1+Nt2)2)orο(4(Nt1/3+Nt2/3)3)   (b)ο((2Nr)×(Nt1+Nt2))   (c)ο((2Nr)2)   (d),
Mc(t1,t2)Y(t1,t2)(p1)=(A¯(Nt1+Nt2)×(Nt1+Nt2)t1,t2)1A¯(Nt1+Nt2)×(2Nr)rt1,t2(A¯(2Nr)×(2Nr)r)1A¯(2Nr)×(Nt1+Nt2)t1,t2rY(t1,t2)(p1)ο((2Nr)×(Nt1+Nt2))   (e)ο((2Nr)2)   (f)ο((Nt1+Nt2)×(2Nr))   (g)ο(Miter12(Nt1+Nt2)2)orο(4(Nt1/3+Nt2/3)3)   (h).

Operations (a), (c), (e), and (g) are matrix-vector multiplications: their computational complexities areο((Nt1+Nt2)×(2Nr)). Operations (d), (f) are the fast FBM iterative inversions, their complexities areο((2Nr)2). Operations (b) and operations (h) are the MOM (CGM or LU scheme), whose complexities are ο(Miter12(Nt1+Nt2)2) or ο((4Nt1/3+4Nt2/3)3), where Miter is the number of iterations in the CGM iteration scheme. Therefore, the total computational complexity is ο(p((Nt1+Nt2)×(2Nr)+Miter12(Nt1+Nt2)2+(2Nr)×(Nt1+Nt2)+(2Nr)2))/ ο(p((Nt1+Nt2)×(2Nr)+(4Nt1/3+4Nt2/3)3+(2Nr)×(Nt1+Nt2)+(2Nr)2)) for both the calculation of the terms Yt1,t2(p) and Yr(p), where p is the number of iterations in the EPILE scheme, for Nr(Nt1+Nt2), 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 ο(p(2Nr+(Nr)2)), p is generally less than 10, so this method is much faster than the direct LU inversion, of order ο((4Nr/3)3) and the CGM, of order ο(Miter12(Nr)2).

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 λ is 0.1m) and the relative permittivity of the dielectric rough surface ε1 = (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 Nt1 = Nt2 = 100, and sampling points on the rough surface Nr = 1024. It is also noted the numerical simulation of the BSC are plotted in decibel (dB) scale.

 

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

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

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Tables Icon

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)

Tables Icon

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)

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 Ru, or Rd 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.

 

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

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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 θi=0°. The radius, depth of both the cylinders above and below are Ru=Rd = 1λ, Hd=Dd = 3.3λ, δ = 0.1λ, l = 1.0λ, L = 100λ, Nr = 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 × 10−6, 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 200λ (Nr = 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 × 10−6.

Figure 5 gives the case for VV polarization. In Fig. 5(a), Nr = 1024, the Relative Residual Error is 2.249 × 10−6, 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), Nr = 2048, the Relative Residual Error is 3.559 × 10−6, 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.

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).

Figure 6 exhibits the BSC versus the scattering angle for the ‘target above + rough surface + target below’ with different radius Ru of the ‘target above’ using the EPILE(2) + FBM(6) for both HH and VV polarization. It is shown that, with the increasing of Ru, 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 Xu 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 Xu, 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 Xu adjoins the center of the rough surface, and decreases gradually when Xu is close to the truncation point of it. The dependency of the BSC on the height Hu 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 Hu, 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.

 

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

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

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

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Figure 911 present the BSC versus the scattering angle for the ‘target above + rough surface + target below’ with different radius Rd, horizontal distance Xd, 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 Rd 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 Xd, 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 Ru , Xu,Hu for the composite model in Figs. 68, and the inducements can also be obtained on the analogy of those discussed above for Figs. 68.

 

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

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

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

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

 

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

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

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

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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.

References and links

1. S. O. Rice, “Reflection of Electromagnetic Waves from Slightly Rough Surfaces,” in Theory of Electromagnetic Waves, M. Kline, ed. (Wiley, 1951).

2. D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987). [CrossRef]  

3. L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004). [CrossRef]  

4. S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990). [CrossRef]  

5. D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985). [CrossRef]  

6. A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half- spaces,” Waves Random Media 4(3), 337–367 (1994). [CrossRef]  

7. R. R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surfaces,” Radio Sci. 9(12), 1139–1146 (1974). [CrossRef]  

8. R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978). [CrossRef]  

9. C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991). [CrossRef]  

10. S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991). [CrossRef]  

11. D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996). [CrossRef]  

12. V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998). [CrossRef]  

13. L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993). [CrossRef]  

14. A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002). [CrossRef]  

15. X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003). [CrossRef]  

16. Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000). [CrossRef]  

17. J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).

18. Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999). [CrossRef]  

19. H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008). [CrossRef]  

20. S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007). [CrossRef]  

21. T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999). [CrossRef]  

22. G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002). [CrossRef]  

23. M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

24. Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001). [CrossRef]  

25. N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006). [CrossRef]  

26. G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008). [CrossRef]  

27. C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008). [CrossRef]  

28. L. Tsang, and J. A. Kong, Scattering of Electromagnetic Waves- Numerical Simulations, (Wiley, 2000).

29. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988). [CrossRef]  

30. M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).

31. G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

References

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  • |

  1. S. O. Rice, “Reflection of Electromagnetic Waves from Slightly Rough Surfaces,” in Theory of Electromagnetic Waves, M. Kline, ed. (Wiley, 1951).
  2. D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987).
    [CrossRef]
  3. L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004).
    [CrossRef]
  4. S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990).
    [CrossRef]
  5. D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985).
    [CrossRef]
  6. A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half- spaces,” Waves Random Media 4(3), 337–367 (1994).
    [CrossRef]
  7. R. R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surfaces,” Radio Sci. 9(12), 1139–1146 (1974).
    [CrossRef]
  8. R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978).
    [CrossRef]
  9. C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
    [CrossRef]
  10. S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991).
    [CrossRef]
  11. D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996).
    [CrossRef]
  12. V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
    [CrossRef]
  13. L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
    [CrossRef]
  14. A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002).
    [CrossRef]
  15. X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
    [CrossRef]
  16. Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000).
    [CrossRef]
  17. J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).
  18. Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999).
    [CrossRef]
  19. H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008).
    [CrossRef]
  20. S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007).
    [CrossRef]
  21. T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999).
    [CrossRef]
  22. G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002).
    [CrossRef]
  23. M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).
  24. Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001).
    [CrossRef]
  25. N. Déchamps, N. de Beaucoudrey, C. Bourlier, and S. Toutain, “Fast numerical method for electromagnetic scattering by rough layered interfaces: propagation-inside-layer expansion method,” J. Opt. Soc. Am. A 23(2), 359–369 (2006).
    [CrossRef]
  26. G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008).
    [CrossRef]
  27. C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008).
    [CrossRef]
  28. L. Tsang, and J. A. Kong, Scattering of Electromagnetic Waves- Numerical Simulations, (Wiley, 2000).
  29. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
    [CrossRef]
  30. M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).
  31. G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

2008 (4)

J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).

H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008).
[CrossRef]

G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008).
[CrossRef]

C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008).
[CrossRef]

2007 (1)

S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007).
[CrossRef]

2006 (1)

2004 (1)

L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004).
[CrossRef]

2003 (1)

X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
[CrossRef]

2002 (2)

A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002).
[CrossRef]

G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002).
[CrossRef]

2001 (1)

Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001).
[CrossRef]

2000 (1)

Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000).
[CrossRef]

1999 (2)

Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999).
[CrossRef]

T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999).
[CrossRef]

1998 (2)

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
[CrossRef]

1996 (1)

D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996).
[CrossRef]

1994 (1)

A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half- spaces,” Waves Random Media 4(3), 337–367 (1994).
[CrossRef]

1993 (2)

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

1991 (2)

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
[CrossRef]

S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991).
[CrossRef]

1990 (1)

S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990).
[CrossRef]

1988 (1)

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
[CrossRef]

1987 (1)

D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987).
[CrossRef]

1985 (1)

D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985).
[CrossRef]

1978 (1)

R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978).
[CrossRef]

1974 (1)

R. R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surfaces,” Radio Sci. 9(12), 1139–1146 (1974).
[CrossRef]

1952 (1)

M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).

Axline, R. M.

R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978).
[CrossRef]

Balasubramaniam, S.

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
[CrossRef]

Bourlier, C.

Braunisch, H.

Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999).
[CrossRef]

Brown, G. S.

D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996).
[CrossRef]

Burkholder, R. J.

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

Chan, C. H.

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
[CrossRef]

S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991).
[CrossRef]

Chew, W. C.

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
[CrossRef]

Cheyoung, K.

G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002).
[CrossRef]

Chiu, T.

T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999).
[CrossRef]

de Beaucoudrey, N.

Déchamps, N.

Durden, S. L.

S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990).
[CrossRef]

Fokkema, D. R.

G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

Fung, A. K.

R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978).
[CrossRef]

Gan, Y. B.

X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
[CrossRef]

Guo, L. X.

J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).

L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004).
[CrossRef]

He, S. Y.

S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007).
[CrossRef]

Hestenes, M. R.

M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).

Holliday, D.

D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987).
[CrossRef]

Iodice, A.

A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002).
[CrossRef]

Ishimaru, A.

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985).
[CrossRef]

Jandhyala, V.

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
[CrossRef]

Jin, Y. Q.

H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008).
[CrossRef]

Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001).
[CrossRef]

Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000).
[CrossRef]

Kapp, D. A.

D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996).
[CrossRef]

Kong, J. A.

Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
[CrossRef]

Kubické, G.

G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008).
[CrossRef]

C. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25(4), 891–902 (2008).
[CrossRef]

Landesa, L.

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

Lentz, R. R.

R. R. Lentz, “A numerical study of electromagnetic scattering from ocean-like surfaces,” Radio Sci. 9(12), 1139–1146 (1974).
[CrossRef]

Li, G.

Y. Q. Jin and G. Li, “Detection of a scatter target over a randomly rough surface by using the angular correlation function in a finite-element approach,” Waves Random Media 10(2), 273–280 (2000).
[CrossRef]

Li, J.

J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).

Li, L. W.

X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
[CrossRef]

Li, Z.

Z. Li and Y. Q. Jin, “Bistatic scattering from a fractal dynamic rough sea surface with a ship presence at low grazing-angle incidence using the FBM/SAA,” Microw. Opt. Technol. Lett. 31(2), 146–151 (2001).
[CrossRef]

Lixin, G.

G. Lixin and K. Cheyoung, “Light scattering models for a spherical particle above a slightly dielectric rough surface,” Microw. Opt. Technol. Lett. 33(2), 142–146 (2002).
[CrossRef]

Lou, S. H.

S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
[CrossRef]

Michielssen, E.

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
[CrossRef]

Obelleiro, F.

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

Phu, P.

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

Pino, M. R.

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

Rodriguez, J. L.

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

Saillard, J.

G. Kubické, C. Bourlier, and J. Saillard, “Scattering by an object above a randomly rough surface from a fast numerical method: Extended PILE method combined with FB-SA,” Waves Random Complex Media 18(3), 495–519 (2008).
[CrossRef]

Sangani, H.

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

Sarabandi, K.

T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999).
[CrossRef]

Sleijpeny, G. L. G.

G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

Stiefel, E.

M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. 49, 409–436 (1952).

Thorsos, E. I.

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
[CrossRef]

Toutain, S.

Tsang, L.

L. Tsang, C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, “A Banded Matrix Iterative Approach to Monte Carlo simulations of large scale random rough surface scattering: TE case,” J. Electromagn. Waves Appl. 7(9), 1185–1200 (1993).
[CrossRef]

C. H. Chan, S. H. Lou, L. Tsang, and J. A. Kong, “Electromagnetic scattering of waves by random rough surface: A finite- difference time-domain approach,” Microw. Opt. Technol. Lett. 4(9), 355–359 (1991).
[CrossRef]

S. H. Lou, L. Tsang, and C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Media 1(4), 287–307 (1991).
[CrossRef]

Vesecky, J. F.

S. L. Durden and J. F. Vesecky, “A numerical study of the separation wavenumber in the two-scale scattering approximation,” IEEE Trans. Geosci. Rem. Sens. 28(2), 271–272 (1990).
[CrossRef]

Voronovich, A.

A. Voronovich, “Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half- spaces,” Waves Random Media 4(3), 337–367 (1994).
[CrossRef]

Wang, C. F.

X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
[CrossRef]

Wang, X.

X. Wang, C. F. Wang, Y. B. Gan, and L. W. Li, “Electromagnetic scattering from a circular target above or below rough surface,” Prog. Electromagn. Res. 40, 207–227 (2003).
[CrossRef]

Winebrenner, D.

D. Winebrenner and A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20(2), 161–170 (1985).
[CrossRef]

Wu, Z. S.

L. X. Guo and Z. S. Wu, “Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface,” J. Electromagn. Waves Appl. 18(9), 1219–1234 (2004).
[CrossRef]

Yang, Y. E.

Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999).
[CrossRef]

Ye, H.

H. Ye and Y. Q. Jin, “A hybrid KA-MOM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface,” Radio Sci. 43(3), RS3005 (2008).
[CrossRef]

Zeng, H.

J. Li, L. X. Guo, and H. Zeng, “FDTD investigation on the electromagnetic scattering from a target above a randomly rough sea surface,” Waves Random Media 10, 273–280 (2008).

Zhang, Y.

Y. Zhang, Y. E. Yang, H. Braunisch, and J. A. Kong, “Electromagnetic wave interaction of conducting object with rough surface by hybrid SPM/MOM technique,” Prog. Electromagn. Res. 22, 315–335 (1999).
[CrossRef]

Zhu, G. Q.

S. Y. He and G. Q. Zhu, “A hybrid MM-PO method combining UV technique for scattering from two-dimensional target above a rough surface,” Microw. Opt. Technol. Lett. 49(12), 2957–2960 (2007).
[CrossRef]

Electron. Trans. Numer. Anal. (1)

G. L. G. Sleijpeny and D. R. Fokkema, “Bicgstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

IEEE Trans. Antenn. Propag. (6)

T. Chiu and K. Sarabandi, “Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface,” IEEE Trans. Antenn. Propag. 47(5), 902–913 (1999).
[CrossRef]

M. R. Pino, L. Landesa, J. L. Rodriguez, F. Obelleiro, and R. J. Burkholder, “The generalized forward-backward method for analyzing the scattering from targets on ocean-like rough surfaces,” IEEE Trans. Antenn. Propag. 3, 961–968 (1998).

D. Holliday, “Resolution of a controversy surrounding the Kirchhoff approach and the small perturbation method in rough surface scattering theory,” IEEE Trans. Antenn. Propag. 35(1), 120–122 (1987).
[CrossRef]

R. M. Axline and A. K. Fung, “Numerical computations of scattering from a perfectly conducting random surface,” IEEE Trans. Antenn. Propag. 26(3), 482–488 (1978).
[CrossRef]

A. Iodice, “Forward–backward method for scattering from dielectric rough surfaces,” IEEE Trans. Antenn. Propag. 50(7), 901–911 (2002).
[CrossRef]

D. A. Kapp and G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antenn. Propag. 44(5), 711–721 (1996).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens. (2)

V. Jandhyala, E. Michielssen, S. Balasubramaniam, and W. C. Chew, “A combined steepest descent-fast multipole algorithm for the fast analysis of three-dimensional scattering by rough surfaces,” IEEE Trans. Geosci. Rem. Sens. 36(3), 738–748 (1998).
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Figures (14)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

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

Fig. 4
Fig. 4

BSC versus the scattering angle (HH polarization).

Fig. 5
Fig. 5

BSC versus the scattering angle (VV polarization).

Fig. 3
Fig. 3

Comparison of the angular distribution of BSC by our scheme and X. Wang’s method.

Fig. 6
Fig. 6

BSC versus the scattering angle (VV polarization).

Fig. 7
Fig. 7

BSC versus the scattering angle (different Xu).

Fig. 8
Fig. 8

BSC versus the scattering angle (different Hu).

Fig. 9
Fig. 9

BSC versus the scattering angle (different Rd).

Fig. 11
Fig. 11

BSC versus the scattering angle (different Dd).

Fig. 10
Fig. 10

BSC versus the scattering angle (different Xd).

Fig. 12
Fig. 12

BSC versus the scattering angle (different δ).

Fig. 13
Fig. 13

BSC versus the scattering angle (different l).

Fig. 14
Fig. 14

BSC versus the scattering angle (different θi ) .

Tables (2)

Tables Icon

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)

Tables Icon

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)

Equations (49)

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φ i n c ( r ) = exp ( j k i r ( 1 + [ 2 ( x + z tan θ i ) 2 / g 2 1 ] / ( k g cos θ i ) 2 ) ) exp ( ( x + z tan θ i ) 2 / g 2 ) ,
1 2 E 0 ( r ) S r [ E 0 ( r ' ) n ' G 0 ( r , r ' ) G 0 ( r , r ' ) n ' E 0 ( r ' ) ] d s ' +     S t 1 G 0 ( r , r ' ) n ' E 0 ( r ' ) d s '       = E i ( r )         r S r ,
S r [ E 0 ( r ' ) n ' G 0 ( r , r ' ) G 0 ( r , r ' ) n ' E 0 ( r ' ) ] d s '     S t 1 G 0 ( r , r ' ) n ' E 0 ( r ' ) d s '       = E i ( r )           r S t 1 ,
1 2 E 1 ( r ) + S r [ E 1 ( r ' ) n ' G 1 ( r , r ' ) G 1 ( r , r ' ) n ' E 1 ( r ' ) ρ ] d s ' S t 2 G 1 ( r , r ' ) n ' E 1 ( r ' ) d s '     = 0         r S r ,
S r [ E 1 ( r ' ) n ' G 1 ( r , r ' ) G 1 ( r , r ' ) n ' E 1 ( r ' ) ρ ] d s ' S t 2 G 1 ( r , r ' ) n ' E 1 ( r ' ) d s ' = 0           r S t 2 ,
1 2 H 0 ( r ) S r [ H 0 ( r ' ) n ' G 0 ( r , r ' ) G 0 ( r , r ' ) n ' H 0 ( r ' ) ] d s ' S t 1 H 0 ( r ' ) n ' G 0 ( r , r ' ) d s '       = H i ( r )             r S r / S t 1 ,
1 2 H 1 ( r ) + S r [ H 1 ( r ' ) n ' G 1 ( r , r ' ) G 1 ( r , r ' ) n ' H 1 ( r ' ) ρ ] d s ' + S t 2 [ H 1 ( r ' ) n ' G 1 ( r , r ' ) G 1 ( r , r ' ) n ' H 1 ( r ' ) ] d s '     = 0         r S r ,
1 2 H 1 ( r ) + S r [ H 1 ( r ' ) n ' G 1 ( r , r ' ) G 1 ( r , r ' ) n ' H 1 ( r ' ) ρ ] d s ' + S t 2 H 1 ( r ' ) n ' G 1 ( r , r ' ) d s ' = 0         r S t 2 ,
Z ¯ ( N t 1 + N t 2 + 2 N r ) × ( N t 1 + N t 2 + 2 N r ) X ( N t 1 + N t 2 + 2 N r ) = S ( N t 1 + N t 2 + 2 N r ) ,
Z ¯     ( N t 1 + N t 2 + 2 N r ) × ( N t 1 + N t 2 + 2 N r ) = [ A N t 1 × N t 1 0 N t 1 × N t 2 B N t 1 × N r C N t 1 × N r 0 N t 2 × N t 1 D N t 2 × N t 2 E N t 2 × N r ρ F N t 2 × N r G N r × N t 1 0 N r × N t 2 H N r × N r I N r × N r 0 N r × N t 1 J N r × N t 2 K N r × N r ρ L N r × N r ] .
Z ¯     ( N t 1 + N t 2 + 2 N r ) × ( N t 1 + N t 2 + 2 N r ) = [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r A ¯ ( 2 N r ) × ( 2 N r ) r ] ,
  A ¯     ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 = [ A N t 1 × N t 1 0 N t 2 × N t 2 0 N t 2 × N t 1 D N t 2 × N t 1 ]        A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 = [ B N t 1 × N r C N t 1 × N r E N t 2 × N r ρ F N t 2 × N r ] ,
A ¯     ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r = [ G N r × N t 1         0 N r × N t 2 0 N r × N t 1       J N r × N t 2 ]        A ¯ ( 2 N r ) × ( 2 N r ) r = [ H N r × N r I N r × N r K N r × N r ρ L N r × N r ] .
X ( N t 1 + N t 2 + 2 N r ) T = [ X t 1 , t 2 ( N t 1 + N t 2 ) T X r ( 2 N r ) T ]       S ( N t 1 + N t 2 + 2 N r ) T = [ S t 1 , t 2 ( N t 1 + N t 2 ) T S r ( 2 N r ) T ] ,
X t 1 , t 2 ( N t 1 + N t 2 ) T = [ ψ 0 ( r t 1 1 ) ψ 0 ( r t 1 N t 1 ) , ψ 1 ( r t 2 1 ) ψ 1 ( r t 2 N t 2 ) ]      X r ( 2 N r ) T = [ ψ 0 ( r r 1 ) ψ 0 ( r r N r ) , ψ 0 ( r r 1 ) n r ψ 0 ( r r N r ) n r ] ,
S t 1 , t 2 ( N t 1 + N t 2 ) T = [ ψ i ( r t 1 1 ) ψ i ( r t 1 N t 1 ) , 0 1 0 N t 2 ]      S r ( 2 N r ) T = [ ψ i ( r r 1 ) ψ i ( r r N r ) , 0 1 0 N r ] ,
A p q ( H H ) = { γ q Δ x j 4 H 0 ( 1 ) ( k 0 | r p r q | )                      p q γ q Δ x j 4 H 0 ( 1 ) [ k 0 Δ x γ q / ( 2 e ) ]       p = q ,     A p q ( V V ) = { γ q Δ x j k 0 4 ( n ^ R p q ) H 1 ( 1 ) ( k 0 | r p r q | )         p q                     1 2 z t 1 ( x p ) Δ x 4 π γ p 2                                                                                                                       p = q ,
B p n = γ n Δ x j k 0 4 ( n ^ R p n ) H 1 ( 1 ) ( k 0 | r p r n | ) ,     C p n = γ n Δ x j 4 H 0 ( 1 ) ( k 0 | r p r n | ) ,
D v w ( H H ) = { γ w Δ x j 4 H 0 ( 1 ) ( k 1 | r v r w | )                         v w γ w Δ x j 4 H 0 ( 1 ) [ k 1 Δ x γ w / ( 2 e ) ]       v = w ,     D v w ( V V ) = { γ w Δ x j k 1 4 ( n ^ R v w ) H 1 ( 1 ) ( k 1 | r v r w | )         v w                   1 2 + z t 2 ( x v ) Δ x 4 π γ v 2                                                                                                           v = w ,
E v n = γ n Δ x j k 1 4 ( n ^ R v n ) H 1 ( 1 ) ( k 1 | r v r n | ) ,     F v n = γ n Δ x j 4 H 0 ( 1 ) ( k 1 | r v r n | ) ,
G m q ( H H ) = γ q Δ x j 4 H 0 ( 1 ) ( k 0 | r m - r q | ) ,     G m q ( V V ) = γ q Δ x j k 0 4 ( n ^ × R m q ) H 1 ( 1 ) ( k 0 | r m - r q | ) ,
H m n = { γ n Δ x j k 0 4 ( n ^ R m n ) H 1 ( 1 ) ( k 0 | r m r n | )         m n                     1 2 z r ( x m ) Δ x 4 π γ m 2                                                                                                                         m = n ,     I m n = { γ n Δ x j 4 H 0 ( 1 ) ( k 0 | r m r n | )                              m n γ m Δ x j 4 H 0 ( 1 ) [ k 0 Δ x γ m / ( 2 e ) ]       m = n ,
J m v ( H H ) = γ v Δ x j 4 H 0 ( 1 ) ( k 1 | r m r v | ) ,     J m v ( V V ) = γ v Δ x j k 1 4 ( n ^ R m v ) H 1 ( 1 ) ( k 1 | r m r v | ) ,
K m n = { γ n Δ x j k 1 4 ( n ^ R m n ) H 1 ( 1 ) ( k 1 | r m r n | )       m n                   1 2 z r ( x m ) Δ x 4 π γ m 2                                                                                                         m = n ,      L m n = { γ n Δ x j 4 H 0 ( 1 ) ( k 1 | r m r n | )                              m n γ m Δ x j 4 H 0 ( 1 ) [ k 1 Δ x γ m / ( 2 e ) ]         m = n ,
ψ s ( r ) = e j k 0 r r ψ s N ( θ s , θ i ) .
ψ s N ( θ s , θ i ) ( TE , HH ) = j 4 2 π k 0 e j π 4 { S r exp ( j k s r ) [ j ( n ^ k s ) X r 1 ~ N r ( x ) X r N r + 1 ~ 2 N r ( x ) ]                                     1 + [ z r ( x ) ] 2 d x S t 1 X t 1 ( x ) exp ( j k s r ) 1 + [ z t 1 ( x ) ] 2 d x },
ψ s N ( θ s , θ i ) ( TM , VV ) = j 4 2 π k 0 e j π 4 { S r [ j ( n ^ k s ) X r 1 ~ N r ( x ) X r N r + 1 ~ 2 N r ( x ) ] exp ( j k s r ) 1 + [ z r ( x ) ] 2 d x                                     S t 1 j ( n ^ t 1 k s ) X t 1 ( x ) exp ( j k s r ) 1 + [ z t 1 ( x ) ] 2 d x },
σ s ( θ s , θ i ) = | ψ s N ( θ s , θ i ) | 2 g π / 2 cos θ i ( 1 ( 1 + 2 tan 2 θ i ) / ( 2 k 0 2 g 2 cos 2 θ i ) ) .
Z ¯ ( 2 N r + N t 1 + N t 2 ) × ( 2 N r + N t 1 + N t 2 ) 1 = [ T ¯ U ¯ V ¯ W ¯ ] ,
T ¯ = [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 ,
U ¯ = [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 ,
V ¯ = ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 ,
W ¯ = ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2             ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 .
X r = [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 S r ,
X t 1 , t 2 = T S t 1 , t 2 = [ A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ] 1 S t 1 , t 2 .
M c ( r ) = ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2
X r ( 2 N r ) ( p ) = ( p = 0 P M c ( r ) p ) ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 ( S r A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 S t 1 , t 2 ) ,
M c ( t 1 , t 2 ) = ( A ¯ ( 2 N t 1 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r .
X t 1 , t 2 ( N t 1 + N t 2 ) ( p ) = ( p = 0 P M c ( t 1 , t 2 ) p ) ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 ( S t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 S r ) .
X r ( 2 N r ) ( p ) = p = 0 P Y ( r ) ( p ) ,     X t 1 , t 2 ( N t 1 + N t 2 ) ( p ) = p = 0 P Y ( t 1 , t 2 ) ( p ) .
Y r ( 0 ) = ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 ( S r A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 S t 1 , t 2 ) ,     Y r ( p ) = M c ( r ) Y r ( p 1 ) ,
Y ( t 1 , t 2 ) ( 0 ) = ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 ( S t 1 , t 2 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 S r ) ,     Y ( t 1 , t 2 ) ( p ) = M c ( t 1 , t 2 ) Y ( t 1 , t 2 ) ( p 1 ) .
A ¯ r ( ( 2 N r ) × ( 2 N r ) ) = A ¯ r ( ( 2 N r ) × ( 2 N r ) ) f + A ¯ r ( ( 2 N r ) × ( 2 N r ) ) d + A ¯ r ( ( 2 N r ) × ( 2 N r ) ) b ,     ξ r ( 2 N r ) = ξ r ( 2 N r ) f + ξ r ( 2 N r ) b ,
[ H N r × N r I N r × N r K N r × N r ρ L N r × N r ] [ ξ 1 r ( N r ) T ξ 2 r ( N r ) T ] = [ ζ 1 r ( N r ) T ζ 2 r ( N r ) T ] ,
H N r × N r = H N r × N r f + H N r × N r s + H N r × N r b ,     I N r × N r = I N r × N r f + I N r × N r s + I N r × N r b ,
K N r × N r = K N r × N r f + K N r × N r s + K N r × N r b ,     L N r × N r = L N r × N r f + L N r × N r s + L N r × N r b .
RRE = 90 90 | σ ( EPILE + FBM ) σ ( MOM ( CGM / LU / BCGM ) ) | 2 90 90 | σ MOM ( CGM / LU / BCGM ) | 2 .
M c ( r ) Y r ( p 1 ) = ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 Y r ( p 1 ) ο ( ( N t 1 + N t 2 ) × ( 2 N r ) )     ( a ) ο ( M i t e r 12 ( N t 1 + N t 2 ) 2 ) o r ο ( 4 ( N t 1 / 3 + N t 2 / 3 ) 3 )     ( b ) ο ( ( 2 N r ) × ( N t 1 + N t 2 ) )     ( c ) ο ( ( 2 N r ) 2 )     ( d ) ,
M c ( t 1 , t 2 ) Y ( t 1 , t 2 ) ( p 1 ) = ( A ¯ ( N t 1 + N t 2 ) × ( N t 1 + N t 2 ) t 1 , t 2 ) 1 A ¯ ( N t 1 + N t 2 ) × ( 2 N r ) r t 1 , t 2 ( A ¯ ( 2 N r ) × ( 2 N r ) r ) 1 A ¯ ( 2 N r ) × ( N t 1 + N t 2 ) t 1 , t 2 r Y ( t 1 , t 2 ) ( p 1 ) ο ( ( 2 N r ) × ( N t 1 + N t 2 ) )     ( e ) ο ( ( 2 N r ) 2 )     ( f ) ο ( ( N t 1 + N t 2 ) × ( 2 N r ) )     ( g ) ο ( M i t e r 12 ( N t 1 + N t 2 ) 2 ) o r ο ( 4 ( N t 1 / 3 + N t 2 / 3 ) 3 )     ( h ) .

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