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

Lixin Guo; Yu Liang; Zhensen Wu

doi:10.1364/OE.19.005785

1. Introduction

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

Although the aforementioned numerical methods are of respective advantages, however, some of them are efficient in computation time but not accurate enough, some of them show good accuracy but are a little time- consuming, and also some of them are only valid for studying the composite scattering between one target and the rough surface. Hence, it is natural and interesting to develop new methods, both exact and fast, to investigate composite scattering from more targets and the rough surface below or above them. In 2006, a fast numerical method, Propagation-Inside-Layer Expansion (PILE) was presented by N. Déchamps et al. [25]. This method, which shows high efficiency and high accuracy, is able to handle problems configured with a huge number of unknowns. In the beginning, the PILE was devoted to the scattering by layered rough surface [25]. Later, the Extended PILE combined with the Forward-Backward method (FBM), used to study the scattering by an object above or below a randomly rough surface, was presented by G. Kubické et al. [26] and C. Bourlier et al. [27], respectively, which can be abbreviated as the EPILE + FBM.

In this paper, based on the previous work [25–27], we extend EPILE + FBM to scattering by two conducting targets located each in different dielectric media separated by a rough one-dimensional boundary (particularly, one of them being the free space). In EPILE + FBM, the new integral equations about the composite scattering from the targets both above and below the dielectric rough surface are established, and the new divided elements of the matrices are obtained, where EPILE is used to analyze the local and coupling interactions, while FBM is employed to calculate the local interactions on the middle rough surface to accelerate the EPILE. Differing from the presented in references [26,27], as for the reconstructed characteristic matrix of the EPILE is concerned, the dual coupling interactions from the targets to the rough surface or from the rough surface to the targets and the dual local interactions from the targets are considered, in other words, it should be an improvement or extension of the traditional EPILE. The FBM has a computational complexity of o(N ²), where N is the number of samples of the discretized rough surface.

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

2. Formulation of the composite problem

Figure 1 illustrates the geometry considered: a perfect electrically conducting (PEC) target (configuration arbitrary) is located above and a PEC target (configuration arbitrary) is located below the dielectric rough surface with the rms height δ and the correlation length l. The problem is assumed to be one-dimensional (variant in the x-z plane). $k_{i}$ represents the propagation vector of the incident wave. $θ_{i}$ is the incident angle. $X u$ , $X d$ 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). $H u$ , $D d$ is the height of the target above and the depth of the target below, $R u$ , $R d$ denotes the maximum radius of the targets above and below circled by the dashed line, respectively. The regions $Ω_{0} (ε_{0}, μ_{0})$ , $Ω_{1} (ε_{1}, μ_{1})$ denotes the space above and below the rough surface. $ε_{0} = μ_{0}$ = 1, $ε_{1}$ is the relative permittivity of the dielectric rough surface. The rough surface is assumed as nonmagnetic, i.e., $μ_{1} = μ_{0}$ . The randomly rough surface is generated by the Monte-Carlo spectral method [28]. The exponential spectrum $W (K_{i}) = \sqrt{2 π} δ^{2} l^{2} {(1 + K_{i}^{2} l^{2})}^{- 3 / 2}$ is applied to model the rough surface. $K_{i}$ 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 $r_{r} = x_{r} \hat{x} + z_{r} \hat{z}$ , $r_{o u} = x_{o u} \hat{x} + z_{o u} \hat{z}$ , and $r_{o d} = x_{o d} \hat{x} + z_{o d} \hat{z}$ , respectively, where $x_{r}$ , $x_{o u}$ , $x_{o d}$ is the discretized abscissa and, $z_{r}$ , $z_{o u}$ , $z_{o d}$ 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

φ_{i n c} (r) = \exp (j k_{i} \cdot 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}),

in which g is tapered parameter.

k_{i}

is the incident wave vector [28]. The time dependence of

\exp (- j ω t)

is assumed and suppressed throughout this paper. j denotes unit imaginary number. The tapered parameter g and the length of the rough surface L should be chosen properly to satisfy the requirements of the wave equation, energy truncation [23].

Suppose a transverse electric (TE) wave $E = \hat{y} φ_{i n c} (r)$ or transverse magnetic (TM) wave $H = \hat{y} φ_{i n c} (r)$ 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

\frac{1}{2} E_{0} (r) - \int_{S_{r}} [E_{0} (r') n' \cdot \nabla G_{0} (r, r') - G_{0} (r, r') n' \cdot \nabla E_{0} (r')] d s' + \int_{S_{t 1}} G_{0} (r, r') n' \cdot \nabla E_{0} (r') d s' = E_{i} (r) r \in S_{r},

\int_{S_{r}} [E_{0} (r') n' \cdot \nabla G_{0} (r, r') - G_{0} (r, r') n' \cdot \nabla E_{0} (r')] d s' - \int_{S_{t 1}} G_{0} (r, r') n' \cdot \nabla E_{0} (r') d s' = - E_{i} (r) r \in S_{t 1},

\frac{1}{2} E_{1} (r) + \int_{S_{r}} [E_{1} (r') n' \cdot \nabla G_{1} (r, r') - G_{1} (r, r') n' \cdot \nabla E_{1} (r') ρ] d s' - \int_{S_{t 2}} G_{1} (r, r') n' \cdot \nabla E_{1} (r') d s' = 0 r \in S_{r},

\int_{S_{r}} [E_{1} (r') n' \cdot \nabla G_{1} (r, r') - G_{1} (r, r') n' \cdot \nabla E_{1} (r') ρ] d s' - \int_{S_{t 2}} G_{1} (r, r') n' \cdot \nabla E_{1} (r') d s' = 0 r \in S_{t 2},

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

\frac{1}{2} H_{0} (r) - \int_{S_{r}} [H_{0} (r') n' \cdot \nabla G_{0} (r, r') - G_{0} (r, r') n' \cdot \nabla H_{0} (r')] d s' - \int_{S_{t 1}} H_{0} (r') n' \cdot \nabla G_{0} (r, r') d s' = H_{i} (r) r \in S_{r} / S_{t 1},

\begin{array}{r} \frac{1}{2} H_{1} (r) + \int_{S_{r}} [H_{1} (r') n' \cdot \nabla G_{1} (r, r') - G_{1} (r, r') n' \cdot \nabla H_{1} (r') ρ] d s' + \int_{S_{t 2}} [H_{1} (r') n' \cdot \nabla G_{1} (r, r') - G_{1} (r, r') n' \cdot \nabla H_{1} (r')] d s' \\ = 0 r \in S_{r}, \end{array}

\begin{array}{r} \frac{1}{2} H_{1} (r) + \int_{S_{r}} [H_{1} (r') n' \cdot \nabla G_{1} (r, r') - G_{1} (r, r') n' \cdot \nabla H_{1} (r') ρ] d s' + \int_{S_{t 2}} H_{1} (r') n' \cdot \nabla G_{1} (r, r') d s' \\ = 0 r \in S_{t 2}, \end{array}

where for HH polarization, $ρ = μ_{1} / μ_{0}$ and for VV polarization, $ρ = ε_{1} / ε_{0}$ . $n'$ is the unit normal vector of the surface or targets, ∇ is the gradient operator. $E_{i}$ , $E_{0}$ , $E_{1}$ denote the incident electric field, electric fields in region $Ω_{0}$ and $Ω_{1}$ , respectively. $H_{i}$ , $H_{0}$ , $H_{1}$ denote the incident magnetic field, magnetic fields in region $Ω_{0}$ and $Ω_{1}$ , respectively. The use of the method of moments (MOM) [15] with point matching and pulse basis functions leads to the following linear system

{\bar{Z}}_{(N_{t 1} + N_{t 2} + 2 N_{r}) \times (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})},

in which ${\bar{Z}}_{(N_{t 1} + N_{t 2} + 2 N_{r}) \times (N_{t 1} + N_{t 2} + 2 N_{r})}$ denotes the impedance matrix, $X_{(N_{t 1} + N_{t 2} + 2 N_{r})}$ is the induced unknown vector, $S_{(N_{t 1} + N_{t 2} + 2 N_{r})}$ is the incident source item. $N_{t 1}$ , $N_{t 2}$ are the numbers of sampling points on the target above and target below, respectively. $N_{r}$ represents the number of sampling points on the rough surface. The impedance matrix is expressed as follows

{\bar{Z}}_{\begin{array}{l} (N_{t 1} + N_{t 2} + 2 N_{r}) \\ \times (N_{t 1} + N_{t 2} + 2 N_{r}) \end{array}} = [\begin{matrix} A_{N_{t 1} \times N_{t 1}} & 0_{N_{t 1} \times N_{t 2}} & B_{N_{t 1} \times N_{r}} & C_{N_{t 1} \times N_{r}} \\ 0_{N_{t 2} \times N_{t 1}} & D_{N_{t 2} \times N_{t 2}} & E_{N_{t 2} \times N_{r}} & - ρ F_{N_{t 2} \times N_{r}} \\ G_{N_{r} \times N_{t 1}} & 0_{N_{r} \times N_{t 2}} & H_{N_{r} \times N_{r}} & I_{N_{r} \times N_{r}} \\ 0_{N_{r} \times N_{t 1}} & J_{N_{r} \times N_{t 2}} & K_{N_{r} \times N_{r}} & - ρ L_{N_{r} \times N_{r}} \end{matrix}] .

can be divided into four blocks and expressed as

{\bar{Z}}_{\begin{array}{l} (N_{t 1} + N_{t 2} + 2 N_{r}) \\ \times (N_{t 1} + N_{t 2} + 2 N_{r}) \end{array}} = [\begin{matrix} {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2} & {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \\ {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r} & {\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r} \end{matrix}],

{\bar{A}}_{\begin{array}{l} (N_{t 1} + N_{t 2}) \\ \times (N_{t 1} + N_{t 2}) \end{array}}^{t 1, t 2} = [\begin{matrix} A_{N_{t 1} \times N_{t 1}} & 0_{N_{t 2} \times N_{t 2}} \\ 0_{N_{t 2} \times N_{t 1}} & D_{N_{t 2} \times N_{t 1}} \end{matrix}] {\bar{A}}_{\begin{array}{l} (N_{t 1} + N_{t 2}) \\ \times (2 N_{r}) \end{array}}^{r \to t 1, t 2} = [\begin{matrix} B_{N_{t 1} \times N_{r}} & C_{N_{t 1} \times N_{r}} \\ E_{N_{t 2} \times N_{r}} & - ρ F_{N_{t 2} \times N_{r}} \end{matrix}],

{\bar{A}}_{\begin{array}{l} (2 N_{r}) \\ \times (N_{t 1} + N_{t 2}) \end{array}}^{t 1, t 2 \to r} = [\begin{matrix} G_{N_{r} \times N_{t 1}} & 0_{N_{r} \times N_{t 2}} \\ 0_{N_{r} \times N_{t 1}} & J_{N_{r} \times N_{t 2}} \end{matrix}] {\bar{A}}_{\begin{array}{l} (2 N_{r}) \\ \times (2 N_{r}) \end{array}}^{r} = [\begin{matrix} H_{N_{r} \times N_{r}} & I_{N_{r} \times N_{r}} \\ K_{N_{r} \times N_{r}} & - ρ L_{N_{r} \times N_{r}} \end{matrix}] .

${\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r}$ corresponds exactly to the impedance matrices of the rough surface. ${\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r}$ and ${\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2}$ can be interpreted as coupling matrices between the targets and the rough surface. ${\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2}$ corresponds exactly to the impedance matrices of the targets. The unknown vector and the source item are defined as

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}],

with

X_{t 1, t 2 (N_{t 1} + N_{t 2})}^{T} = [ψ_{0} (r_{t 1}^{1}) \cdot \cdot \cdot ψ_{0} (r_{t 1}^{N_{t 1}}), ψ_{1} (r_{t 2}^{1}) \cdot \cdot \cdot ψ_{1} (r_{t 2}^{N_{t 2}})] X_{r (2 N_{r})}^{T} = [ψ_{0} (r_{r}^{1}) \cdot \cdot \cdot ψ_{0} (r_{r}^{N_{r}}), \frac{\partial ψ_{0} (r_{r}^{1})}{\partial n_{r}} \cdot \cdot \cdot \frac{\partial ψ_{0} (r_{r}^{N_{r}})}{\partial n_{r}}],

S_{t 1, t 2 (N_{t 1} + N_{t 2})}^{T} = [ψ_{i} (r_{t 1}^{1}) \cdot \cdot \cdot ψ_{i} (r_{t 1}^{N_{t 1}}), 0^{1} \cdot \cdot \cdot 0^{N_{t 2}}] S_{r (2 N_{r})}^{T} = [ψ_{i} (r_{r}^{1}) \cdot \cdot \cdot ψ_{i} (r_{r}^{N_{r}}), 0^{1} \cdot \cdot \cdot 0^{N_{r}}],

in which the superscript T stands for the transpose operator, and

\partial / \partial n

stands for the normal derivative operator.

ψ_{i}

,

ψ_{0}

,

ψ_{1}

corresponds to

E_{i}

,

E_{0}

,

E_{1}

for TE case and

H_{i}

,

H_{0}

,

H_{1}

for TM case, respectively.

The elements of the impedance matrix are shown as below:

\underset{(H H)}{A_{p q}} = {\begin{matrix} γ_{q} Δ x \frac{j}{4} H_{0}^{(1)} (k_{0} | r_{p} - r_{q} |) p \neq q \\ γ_{q} Δ x \frac{j}{4} H_{0}^{(1)} [k_{0} Δ x γ_{q} / (2 e)] p = q \end{matrix}, \underset{(V V)}{A_{p q}} = {\begin{matrix} - γ_{q} Δ x \frac{j k_{0}}{4} (\hat{n} \cdot R_{p q}) H_{1}^{(1)} (k_{0} | r_{p} - r_{q} |) p \neq q \\ \frac{1}{2} - \frac{{z^{″}}_{t 1} (x_{p}) Δ x}{4 π γ_{p}^{2}} p = q \end{matrix},

B_{p n} = - γ_{n} Δ x \frac{j k_{0}}{4} (\hat{n} \cdot R_{p n}) H_{1}^{(1)} (k_{0} | r_{p} - r_{n} |), C_{p n} = γ_{n} Δ x \frac{j}{4} H_{0}^{(1)} (k_{0} | r_{p} - r_{n} |),

\underset{(H H)}{D_{v w}} = {\begin{matrix} γ_{w} Δ x \frac{j}{4} H_{0}^{(1)} (k_{1} | r_{v} - r_{w} |) v \neq w \\ γ_{w} Δ x \frac{j}{4} H_{0}^{(1)} [k_{1} Δ x γ_{w} / (2 e)] v = w \end{matrix}, \underset{(V V)}{D_{v w}} = {\begin{matrix} γ_{w} Δ x \frac{j k_{1}}{4} (\hat{n} \cdot R_{v w}) H_{1}^{(1)} (k_{1} | r_{v} - r_{w} |) v \neq w \\ \frac{1}{2} + \frac{{z^{″}}_{t 2} (x_{v}) Δ x}{4 π γ_{v}^{2}} v = w \end{matrix},

E_{v n} = γ_{n} Δ x \frac{j k_{1}}{4} (\hat{n} \cdot R_{v n}) H_{1}^{(1)} (k_{1} | r_{v} - r_{n} |), F_{v n} = γ_{n} Δ x \frac{j}{4} H_{0}^{(1)} (k_{1} | r_{v} - r_{n} |),

\underset{(H H)}{G_{m q}} = γ_{q} Δ x \frac{j}{4} H_{0}^{(1)} (k_{0} | r_{m} - r_{q} |), \underset{(V V)}{G_{m q}} = γ_{q} Δ x \frac{j k_{0}}{4} (\hat{n} \times R_{m q}) H_{1}^{(1)} (k_{0} | r_{m} - r_{q} |),

H_{m n} = {\begin{matrix} - γ_{n} Δ x \frac{j k_{0}}{4} (\hat{n} \cdot R_{m n}) H_{1}^{(1)} (k_{0} | r_{m} - r_{n} |) m \neq n \\ \frac{1}{2} - \frac{{z^{″}}_{r} (x_{m}) Δ x}{4 π γ_{m}^{2}} m = n \end{matrix}, I_{m n} = {\begin{matrix} γ_{n} Δ x \frac{j}{4} H_{0}^{(1)} (k_{0} | r_{m} - r_{n} |) m \neq n \\ γ_{m} Δ x \frac{j}{4} H_{0}^{(1)} [k_{0} Δ x γ_{m} / (2 e)] m = n \end{matrix},

\underset{(H H)}{J_{m v}} = - γ_{v} Δ x \frac{j}{4} H_{0}^{(1)} (k_{1} | r_{m} - r_{v} |), \underset{(V V)}{J_{m v}} = γ_{v} Δ x \frac{j k_{1}}{4} (\hat{n} \cdot R_{m v}) H_{1}^{(1)} (k_{1} | r_{m} - r_{v} |),

K_{m n} = {\begin{matrix} γ_{n} Δ x \frac{j k_{1}}{4} (\hat{n} \cdot R_{m n}) H_{1}^{(1)} (k_{1} | r_{m} - r_{n} |) m \neq n \\ \frac{1}{2} - \frac{{z^{″}}_{r} (x_{m}) Δ x}{4 π γ_{m}^{2}} m = n \end{matrix}, L_{m n} = {\begin{matrix} γ_{n} Δ x \frac{j}{4} H_{0}^{(1)} (k_{1} | r_{m} - r_{n} |) m \neq n \\ γ_{m} Δ x \frac{j}{4} H_{0}^{(1)} [k_{1} Δ x γ_{m} / (2 e)] m = n \end{matrix},

where

R_{α n} = \frac{r_{α} - r_{n}}{| r_{α} - r_{n} |} (α = m, p, v)

,

γ_{m} = \sqrt{1 + {[z_{r}^{'} (x_{m})]}^{2}}

,

γ_{n} = \sqrt{1 + {[z_{r}^{'} (x_{n})]}^{2}}

,

γ_{p} = \sqrt{1 + {[z_{t 1}^{'} (x_{p})]}^{2}}

,

γ_{q} = \sqrt{1 + {[z_{t 2}^{'} (x_{q})]}^{2}}

,

γ_{v} = \sqrt{1 + {[z_{t 2}^{'} (x_{v})]}^{2}}

,

γ_{w} = \sqrt{1 + {[z_{t 2}^{'} (x_{w})]}^{2}}

,

\hat{n} = \frac{- z_{r}^{'} (x_{n}) \hat{x} + \hat{z}}{\sqrt{1 + {[z_{r}^{'} (x_{n})]}^{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.

k_{0} = 2 π / λ

is the wavenumber in the free space, where λ is the incident wavelength and

k_{1} = k_{0} \sqrt{ε_{1}}

is the wavenumber in the transmitted medium

Ω_{1}

.

Δ x

is the sampling step.

r_{m} = x_{m} \hat{x} + z_{m} \hat{z}

and

r_{n} = x_{n} \hat{x} + z_{n} \hat{z}

denote arbitrary two points on the rough surface.

r_{p} = x_{p} \hat{x} + z_{p} \hat{z}

and

r_{q} = x_{q} \hat{x} + z_{q} \hat{z}

denote arbitrary two points of the target above.

r_{v} = x_{v} \hat{x} + z_{v} \hat{z}

and

r_{w} = x_{w} \hat{x} + z_{w} \hat{z}

denote arbitrary two points of the target below.

z_{t 1}^{'}

,

z_{t 2}^{'}

and

z_{r}^{'}

is the first-order differential of height function of the target above, target below, and the rough surface, respectively, and

z_{t 1}^{″}

,

z_{t 1}^{″}

,

{z^{″}}_{r}

corresponds to the second- order differential of height function. e = 2.718214, γ = 1.781072,

H_{0}^{(1)}

is the zeroth-order Hankel function of the first kind.

H_{1}^{(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]

ψ_{s} (r) = \frac{e^{j k_{0} r}}{\sqrt{r}} ψ_{s}^{N} (θ_{s}, θ_{i}) .

For TE and TM case, $ψ_{s}^{N} (θ_{s}, θ_{i})$ is expressed as

\begin{array}{l} \underset{(TE, HH)}{ψ_{s}^{N} (θ_{s}, θ_{i})} = \frac{j}{4} \sqrt{\frac{2}{π k_{0}}} e^{- j \frac{π}{4}} {\int_{S_{r}} \exp (- j k_{s} \cdot r) \cdot [- j (\hat{n} \cdot k_{s}) X_{r}^{1 ~ N_{r}} (x) - X_{r}^{N_{r} + 1 ~ 2 N_{r}} (x)] \\ \cdot \sqrt{1 + {[z_{r}^{'} (x)]}^{2}} d x - \int_{S_{t 1}} X_{t 1}^{} (x) \cdot \exp (- j k_{s} \cdot r) \sqrt{1 + {[z_{t 1}^{'} (x)]}^{2}} d x}, \end{array}

\begin{array}{l} \underset{(TM, VV)}{ψ_{s}^{N} (θ_{s}, θ_{i})} = \frac{j}{4} \sqrt{\frac{2}{π k_{0}}} e^{- j \frac{π}{4}} {\int_{S_{r}} [- j (\hat{n} \cdot k_{s}) X_{r}^{1 ~ N_{r}} (x) - X_{r}^{N_{r} + 1 ~ 2 N_{r}} (x)] \cdot \exp (- j k_{s} \cdot r) \sqrt{1 + {[z_{r}^{'} (x)]}^{2}} d x \\ - \int_{S_{t 1}} j ({\hat{n}}_{t 1} \cdot k_{s}) X_{t 1}^{} (x) \cdot \exp (- j k_{s} \cdot r) \sqrt{1 + {[z_{t 1}^{'} (x)]}^{2}} d x}, \end{array}

in which

{\hat{n}}_{t 1} = (- z_{t 1}^{'} (x) \hat{x} + \hat{z}) \sqrt{1 + {[z_{t 1}^{'} (x)]}^{2}}

. The expression for the normalized far-field bistatic scattering coefficient (BSC) with the Thorsos’ tapered plane wave incidence is given as

σ_{s} (θ_{s}, θ_{i}) = \frac{{| ψ_{s}^{N} (θ_{s}, θ_{i}) |}^{2}}{g \sqrt{π / 2} \cos θ_{i} (1 - (1 + 2 \tan^{2} θ_{i}) / (2 k_{0}^{2} g^{2} \cos^{2} θ_{i}))} .

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 [25–27], we apply the EPILE to study the composite electromagnetic scattering from targets both above and below the rough surface. The inverse matrix of ${\bar{Z}}_{(N_{t 1} + N_{t 2} + 2 N_{r}) \times (N_{t 1} + N_{t 2} + 2 N_{r})}$ can be partitioned into four blocks (it should be noted that, some of the following formulations are of the similar form with those in [25–27], but the influence of dual targets both above and below are included).

{\bar{Z}}_{(2 N_{r} + N_{t 1} + N_{t 2}) \times (2 N_{r} + N_{t 1} + N_{t 2})}^{- 1} = [\begin{matrix} \bar{T} & \bar{U} \\ \bar{V} & \bar{W} \end{matrix}],

where

\bar{T} = {[{\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2} - {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \cdot {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r}]}^{- 1},

\begin{matrix} \bar{U} = - [{\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2} - {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \cdot ({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})^{- 1} \cdot {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r}]^{- 1} \\ \cdot {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \cdot ({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})^{- 1}, \end{matrix}

\begin{matrix} \bar{V} = - ({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})^{- 1} \cdot {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r} \cdot [{\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2} - {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \\ \cdot ({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})^{- 1} \cdot {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r}]^{- 1}, \end{matrix}

\begin{array}{l} \bar{W} = ({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})^{- 1} - ({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})^{- 1} \cdot {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r} \cdot [{\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2} - {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \\ \cdot ({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})^{- 1} \cdot {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r}]^{- 1} \cdot {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \cdot ({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})^{- 1} . \end{array}

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

X_{r} = - {[{\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2} - {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \cdot {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r}]}^{- 1} \cdot {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \cdot {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot S_{r},

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

X_{t 1, t 2} = T \cdot S_{t 1, t 2} = {[{\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2} - {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \cdot {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r}]}^{- 1} \cdot S_{t 1, t 2} .

Introducing the characteristic matrix $M_{c (r)}$ on the rough surface, i.e.,

M_{c (r)} = {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r} \cdot {({\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1} \cdot {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2}

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

X_{r (2 N_{r})}^{(p)} = (\sum_{p = 0}^{P} M_{c (r)}^{p}) \cdot {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot (S_{r} - {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r} {({\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1} \cdot S_{t 1, t 2}),

where

\sum_{p = 0}^{P} M_{c}^{p} = I - {({\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1} {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r}

,

\bar{I}

is the identity matrix.

Similarly, introducing the characteristic matrix $M_{c (t 1, t 2)}$ on the targets

M_{c (t 1, t 2)} = {({\bar{A}}_{(2 N_{t 1}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1} \cdot {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \cdot {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r} .

It should be addressed that both $M_{c (r)}$ and $M_{c (t 1, t 2)}$ 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

X_{t 1, t 2 (N_{t 1} + N_{t 2})}^{(p)} = (\sum_{p = 0}^{P} M_{c (t 1, t 2)}^{p}) \cdot {({\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1} (S_{t 1, t 2} - {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot S_{r}) .

Therefore, $X_{r (2 N_{r})}^{(p)}$ and $X_{t 1, t 2 (N_{t 1} + N_{t 2})}^{(p)}$ can be expressed as follows

X_{r (2 N_{r})}^{(p)} = \sum_{p = 0}^{P} Y_{(r)}^{(p)}, X_{t 1, t 2 (N_{t 1} + N_{t 2})}^{(p)} = \sum_{p = 0}^{P} Y_{(t 1, t 2)}^{(p)} .

Each item in expressions above is given below

Y_{r}^{(0)} = {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot (S_{r} - {\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r} {({\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1} \cdot S_{t 1, t 2}), Y_{r}^{(p)} = M_{c (r)} \cdot Y_{r}^{(p - 1)},

Y_{(t 1, t 2)}^{(0)} = {({\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1} (S_{t 1, t 2} - {\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} {({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot S_{r}), Y_{(t 1, t 2)}^{(p)} = M_{c (t 1, t 2)} \cdot Y_{(t 1, t 2)}^{(p - 1)} .

p denotes the iteration order. In Eq. (28) and Eq. (30), the ${({\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1}$ accounts for the local interactions on the targets, and ${({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1}$ accounts for the local interactions on the rough surface. ${\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2}$ propagates the resulting field on the rough surface toward the targets (surface-targets coupling), and ${\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r}$ propagates the resulting field on the targets toward the rough surface (targets-surface coupling), and so on for the subsequent terms $Y_{r}^{(p)}$ , just as shown in Fig. 2 . Usually, at each iteration step, the number $N_{t 1} + N_{t 2}$ of samples for the targets is less than the number $N_{r}$ of samples for the rough surface. The term ${({\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1} ς$ (ς denotes the unknown column vector of length $N_{t 1} + N_{t 2}$ ) will be solved by the MOM(CGM / BCGM / LU), while, to decrease the computing cost and speed up the calculation of the term ${({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} ζ_{r (2 N_{r})}$ (i.e., single rough surface, $ζ_{r (2 N_{r})}$ denotes the unknown column vector of length $2 N_{r}$ ), the FBM by A. Iodice [14] can be applied. Assume ${({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} ζ_{r (2 N_{r})}$ equals to $ξ_{(2 N_{r})}$ (ξ denotes the unknown column vector of length $2 N_{r}$ ). The impedance matrix and the induced unknown vector can be decomposed as follow

{\bar{A}}_{r ((2 N_{r}) \times (2 N_{r}))} = {\bar{A}}_{r ((2 N_{r}) \times (2 N_{r}))}^{f} + {\bar{A}}_{r ((2 N_{r}) \times (2 N_{r}))}^{d} + {\bar{A}}_{r ((2 N_{r}) \times (2 N_{r}))}^{b}, ξ_{r (2 N_{r})} = ξ_{r (2 N_{r})}^{f} + ξ_{r (2 N_{r})}^{b},

in which

{\bar{A}}_{r (2 N_{r} \times 2 N_{r})}^{f}

denotes the lower triangle part of

{\bar{A}}_{r (2 N_{r} \times 2 N_{r})}

,

{\bar{A}}_{r ((2 N_{r}) \times (2 N_{r}))}^{d}

denotes the diagonal part of

{\bar{A}}_{r ((2 N_{r}) \times (2 N_{r}))}

, and

{\bar{A}}_{r ((2 N_{r}) \times (2 N_{r}))}^{b}

denotes the upper triangle part of

{\bar{A}}_{r ((2 N_{r}) \times (2 N_{r}))}

.

ξ_{r (2 N_{r})}^{f}

and

ξ_{r (2 N_{r})}^{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 $ξ_{r (2 N_{r})} = [ξ_{1}_{r (N_{r})}^{T} ξ_{2}_{r (N_{r})}^{T}]$ and $ζ_{r (2 N_{r})} = [ζ_{1}_{r (N_{r})}^{T} ζ_{2}_{r (N_{r})}^{T}]$ , therefore,

[\begin{matrix} H_{N_{r} \times N_{r}} & I_{N_{r} \times N_{r}} \\ K_{N_{r} \times N_{r}} & - ρ L_{N_{r} \times N_{r}} \end{matrix}] \cdot [\begin{matrix} ξ_{1}_{r (N_{r})}^{T} \\ ξ_{2}_{r (N_{r})}^{T} \end{matrix}] = [\begin{matrix} ζ_{1}_{r (N_{r})}^{T} \\ ζ_{2}_{r (N_{r})}^{T} \end{matrix}],

H_{N_{r} \times N_{r}} = H_{N_{r} \times N_{r}}^{f} + H_{N_{r} \times N_{r}}^{s} + H_{N_{r} \times N_{r}}^{b}, I_{N_{r} \times N_{r}} = I_{N_{r} \times N_{r}}^{f} + I_{N_{r} \times N_{r}}^{s} + I_{N_{r} \times N_{r}}^{b},

K_{N_{r} \times N_{r}} = K_{N_{r} \times N_{r}}^{f} + K_{N_{r} \times N_{r}}^{s} + K_{N_{r} \times N_{r}}^{b}, L_{N_{r} \times N_{r}} = L_{N_{r} \times N_{r}}^{f} + L_{N_{r} \times N_{r}}^{s} + L_{N_{r} \times N_{r}}^{b} .

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 = \frac{\sum_{- 90}^{90} | σ_{(EPILE + FBM)} - σ_{(MOM (CGM / LU / BCGM))} |^{2}}{\sum_{- 90}^{90} | σ_{MOM (CGM / LU / BCGM)} |^{2}} .

The complexity per iteration of the terms $Y_{t 1, t 2}^{(p)}$ , $Y_{r}^{(p)}$ is given below, respectively

M_{c (r)} \cdot Y_{r}^{(p - 1)} = \underset{ο ({(2 N_{r})}^{2}) (d)}{\underset{︸}{{({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot \underset{ο ((2 N_{r}) \times (N_{t 1} + N_{t 2})) (c)}{\underset{︸}{{\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r} \cdot \underset{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)}{\underset{︸}{{({\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1} \cdot \underset{ο ((N_{t 1} + N_{t 2}) \times (2 N_{r})) (a)}{\underset{︸}{{\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \cdot Y_{r}^{(p - 1)}}}}}}}}},

M_{c (t 1, t 2)} \cdot Y_{(t 1, t 2)}^{(p - 1)} = \underset{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)}{\underset{︸}{{({\bar{A}}_{(N_{t 1} + N_{t 2}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2})}^{- 1} \cdot \underset{ο ((N_{t 1} + N_{t 2}) \times (2 N_{r})) (g)}{\underset{︸}{{\bar{A}}_{(N_{t 1} + N_{t 2}) \times (2 N_{r})}^{r \to t 1, t 2} \cdot \underset{ο ({(2 N_{r})}^{2}) (f)}{\underset{︸}{{({\bar{A}}_{(2 N_{r}) \times (2 N_{r})}^{r})}^{- 1} \cdot \underset{ο ((2 N_{r}) \times (N_{t 1} + N_{t 2})) (e)}{\underset{︸}{{\bar{A}}_{(2 N_{r}) \times (N_{t 1} + N_{t 2})}^{t 1, t 2 \to r} \cdot Y_{(t 1, t 2)}^{(p - 1)}}}}}}}}} .

Operations (a), (c), (e), and (g) are matrix-vector multiplications: their computational complexities are $ο ((N_{t 1} + N_{t 2}) \times (2 N_{r}))$ . Operations (d), (f) are the fast FBM iterative inversions, their complexities are $ο ({(2 N_{r})}^{2})$ . Operations (b) and operations (h) are the MOM (CGM or LU scheme), whose complexities are $ο (M_{i t e r} 12 {(N_{t 1} + N_{t 2})}^{2})$ or $ο ({(4 N_{t 1} / 3 + 4 N_{t 2} / 3)}^{3})$ , where $M_{i t e r}$ is the number of iterations in the CGM iteration scheme. Therefore, the total computational complexity is $ο (p ((N_{t 1} + N_{t 2}) \times (2 N_{r}) + M_{i t e r} 12 {(N_{t 1} + N_{t 2})}^{2} + (2 N_{r}) \times (N_{t 1} + N_{t 2}) + {(2 N_{r})}^{2}))$ / $ο (p ((N_{t 1} + N_{t 2}) \times (2 N_{r}) + {(4 N_{t 1} / 3 + 4 N_{t 2} / 3)}^{3} + (2 N_{r}) \times (N_{t 1} + N_{t 2}) + {(2 N_{r})}^{2}))$ for both the calculation of the terms $Y_{t 1, t 2}^{(p)}$ and $Y_{r}^{(p)}$ , where p is the number of iterations in the EPILE scheme, for $N_{r} ≫ (N_{t 1} + N_{t 2})$ , 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 $ο (p (2 N_{r} + {(N_{r})}^{2}))$ , p is generally less than 10, so this method is much faster than the direct LU inversion, of order $ο ({(4 N_{r} / 3)}^{3})$ and the CGM, of order $ο (M_{i t e r} 12 {(N_{r})}^{2})$ .

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 $ε_{1}$ = (6.91,0.63) in the following numerical simulations. Both the length L of the rough surface and the tapered parameter g (L/4) satisfy the aforementioned requirement in Section 2. 50 surface realizations are averaged in all the numerical examples except Fig. 4 , Fig. 5 , Table 1 and Table 2 . The number of sampling points on the cylinder above the surface and below the surface $N_{t 1}$ = $N_{t 2}$ = 100, and sampling points on the rough surface $N_{r}$ = 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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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)

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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)

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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 $R_{u}$ , or $R_{d}$ 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 $θ_{i} =$ 0°. The radius, depth of both the cylinders above and below are $R_{u} = R_{d}$ = 1λ, $H_{d} = D_{d}$ = 3.3λ, δ = 0.1λ, l = $1.0 λ$ , L = 100λ, $N_{r}$ = 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 ( $N_{r}$ = 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), $N_{r}$ = 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), $N_{r}$ = 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 $R_{u}$ of the ‘target above’ using the EPILE(2) + FBM(6) for both HH and VV polarization. It is shown that, with the increasing of $R_{u}$ , 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 $X_{u}$ 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 $X_{u}$ , 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 $X_{u}$ adjoins the center of the rough surface, and decreases gradually when $X_{u}$ is close to the truncation point of it. The dependency of the BSC on the height $H_{u}$ 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 $H_{u}$ , 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 9 –11 present the BSC versus the scattering angle for the ‘target above + rough surface + target below’ with different radius $R_{d}$ , horizontal distance $X_{d}$ , and depth $D_{d}$ of the ‘target below’ using the EPILE(2) + FBM(6) for both HH and VV polarization. It is shown from Fig. 9 that, as $R_{d}$ 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 $X_{d}$ , the scattering curve decreases at non-specular region. Figure 11 indicates that, as $D_{d}$ 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 $R_{u}$ , $X_{u}$ , $H_{u}$ for the composite model in Figs. 6–8, and the inducements can also be obtained on the analogy of those discussed above for Figs. 6–8.

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 θ_i)_.

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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.

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EPILE() + FBM()	(2) + (2)	(2) + (3)	(2) + (4)	(2) + (5)	(2) + (6)
Relative Residual Error	8.307 × 10⁻³	7.220 × 10⁻⁴	6.721 × 10⁻⁵	1.746 × 10⁻⁵	1.661 × 10⁻⁵
computational time (second)	≈37	≈41	≈45	≈50	≈55
EPILE() + FBM()	(3) + (2)	(3) + (3)	(3) + (4)	(3) + (5)	(3) + (6)
Relative Residual Error	7.960 × 10⁻³	6.906 × 10⁻⁴	6.376 × 10⁻⁵	1.566 × 10⁻⁵	1.191 × 10⁻⁵
computational time (second)	≈42	≈47	≈53	≈58	≈64

EPILE() + FBM()	(2) + (2)	(2) + (3)	(2) + (4)	(2) + (5)	(2) + (6)
Relative Residual Error	1.492 × 10⁻⁴	2.045 × 10⁻⁵	1.268 × 10⁻⁵	1.231 × 10⁻⁵	1.223 × 10⁻⁵
computational time (second)	≈34	≈38	≈43	≈47	≈52
EPILE() + FBM()	(3) + (2)	(3) + (3)	(3) + (4)	(3) + (5)	(3) + (6)
Relative Residual Error	1.436 × 10⁻⁴	1.938 × 10⁻⁵	1.156 × 10⁻⁵	1.102 × 10⁻⁵	1.096 × 10⁻⁵
computational time (second)	≈38	≈44	≈49	≈55	≈60

EPILE() + FBM()	(2) + (2)	(2) + (3)	(2) + (4)	(2) + (5)	(2) + (6)
Relative Residual Error	8.307 × 10⁻³	7.220 × 10⁻⁴	6.721 × 10⁻⁵	1.746 × 10⁻⁵	1.661 × 10⁻⁵
computational time (second)	≈37	≈41	≈45	≈50	≈55
EPILE() + FBM()	(3) + (2)	(3) + (3)	(3) + (4)	(3) + (5)	(3) + (6)
Relative Residual Error	7.960 × 10⁻³	6.906 × 10⁻⁴	6.376 × 10⁻⁵	1.566 × 10⁻⁵	1.191 × 10⁻⁵
computational time (second)	≈42	≈47	≈53	≈58	≈64

EPILE() + FBM()	(2) + (2)	(2) + (3)	(2) + (4)	(2) + (5)	(2) + (6)
Relative Residual Error	1.492 × 10⁻⁴	2.045 × 10⁻⁵	1.268 × 10⁻⁵	1.231 × 10⁻⁵	1.223 × 10⁻⁵
computational time (second)	≈34	≈38	≈43	≈47	≈52
EPILE() + FBM()	(3) + (2)	(3) + (3)	(3) + (4)	(3) + (5)	(3) + (6)
Relative Residual Error	1.436 × 10⁻⁴	1.938 × 10⁻⁵	1.156 × 10⁻⁵	1.102 × 10⁻⁵	1.096 × 10⁻⁵
computational time (second)	≈38	≈44	≈49	≈55	≈60

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

Abstract

1. Introduction

2. Formulation of the composite problem

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

4. Numerical results and discussions

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (14)

Tables (2)

Equations (49)

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