A hybrid KA-MoM algorithm for computation of scattering from a 3-D PEC target above a dielectric rough surface

Authors


Abstract

[1] A new hybrid analytical-numerical iterative algorithm, which combines the Kirchhoff approximation (KA) and the Method of Moment (MoM), is developed for computing the electromagnetic scattering from a three-dimensional (3-D) perfect electric conducting (PEC) target above a 2-D infinite randomly rough dielectric surface. The equations of difference scattering due to the target presence above the rough surface are derived. The induced difference fields on the rough surface due to the interactions between the target and the rough surface are calculated by using the KA method. The excitation term on the right-hand-side (RHS) of target's surface integral equation (SIE), which contains the difference scattering of the rough surface, is then updated for calculating new target currents with the Conjugate Gradient (CG) procedure. Then the target currents are used to compute the difference field induced on the rough surface with the KA method. Multiple iterations take account of the multiorder interactions between the target and the underlying rough surface. Numerical quadrature upon the rough surface is performed only once to compute the coupling scattering field from the rough surface to the target, and it takes N steps (N is the discretized mesh number of rough surface). By using this hybrid KA-MoM algorithm, the requirements of memory and CPU time can be reduced significantly. Moreover, the validity conditions and the convergence performance of this hybrid algorithm are also discussed. With Monte-Carlo generations of randomly rough surfaces, bistatic scattering from different-shaped targets above a Gaussian rough surface is numerically simulated. Finally, dependence of bistatic scattering pattern on the surface dielectric property and the target geometry is discussed.

1. Introduction

[2] In recent years, electromagnetic scattering problems of targets above or beneath a rough surface have drawn more and more attentions, because of its extensive applications on radar surveillance, target identification, GPR (ground-penetration radar) probing, etc. Numerical simulation of the composite target/surface model should take into account the complicated interactions between target and rough surface background. Some numerical approaches of target-surface modeling have emerged recently, such as the generalized forward backward method with spectral acceleration algorithm (GFBM/SAA) [Pino et al., 1999, 2002], the finite element method and domain decomposition method (FEM-DDM) [Liu and Jin, 2004], the steepest descent path method and the fast multipole method (SDP/FMM) [El-Shenawee et al., 2001], the coupled canonical grid/discrete dipole approach (CCG/DDA) [Johnson and Burkholder, 2001], etc. However, most of them are restricted to 2-D model because of large computation complexity of rough surface scattering. Very few exceptive examples of 3-D models can be found, e.g., the steepest descent path (SDP) method and the fast multipole method (FMM) for analyzing the scattering from a target beneath a rough surface. But it still requires tremendous memory and CPU time as in the order of O(NlgN) [El-Shenawee, 2002; El-Shenawee et al., 2001].

[3] If the underlying surface is flat, the image method with Green's function of half-space is usually employed, e.g., complex image method and multilevel fast multipole algorithm (MLFMA) [Li et al., 2003]. As the underlying surface becomes randomly rough, the Green's function becomes the sum of infinite series due to the multiple scattering of rough surface. It causes a large difficulty for numerical simulation.

[4] In a 2-D target-surface model [Ye and Jin, 2006], it has been found that scattering computation of rough surface costs most of CPU time. This yields a hybrid analytic-numerical method applying the analytic Kirchhoff approximation (KA) method to coarsely compute the induced currents on rough surface [Ye and Jin, 2007]. Then the computation cost is significantly reduced, and good efficiency and precision are preserved.

[5] In this paper, the hybrid KA-MoM algorithm is generalized to the 3-D model of a PEC target above a 2-D dielectric randomly rough surface. The difference electric and magnetic fields induced on the dielectric rough surface are introduced in coupling interactions. The KA expressions for the coupling scattering fields between the target and the underlying rough surface are presented, based on the tangential plane approximation and the local orthogonal decomposition of the radiation fields of target's currents. Induced currents on the target are obtained in an iterative procedure: 1) compute the radiation field of the target's induced currents (solved in the previous iteration step) over the rough surface; 2) use the surface equivalence theorem and the KA method to derive the induced fields on the rough surface; 3) calculate the radiation fields of these equivalent induced surface fields to generate the difference scattering field of the rough surface; 4) update the RHS excitation of the discrete MoM equation with the difference scattering fields and solve it to obtain the new induced currents on the target.

[6] In numerical process, the target and the randomly rough surface samples (generated by Monte-Carlo method) are discretized into small patches, and the interaction is numerically calculated for each patch. In order to derive the KA expression for coupling scattering fields of the rough surface, it is presumed that the radiation electric and magnetic fields from the source current unit on the target to the discrete patches of the rough surface satisfy the right-handed helix relationship, just the same as the general TEM plane wave, which leads the validity condition of kR > 20. Moreover, the incident and scattering angles should satisfy tan θi < s−1 and θs < s−1 to avoid the shadowing effects.

[7] As a code validation, the hybrid algorithm is first examined for the composite model of a sphere above a PEC flat surface, and is compared with the image Green's function method. Then, it is applied to numerically study the bistatic difference scattering from different-shaped targets above a dielectric randomly rough surface.

2. Coupling Surface Integral Equations

[8] As shown in Figure 1, a 3-D PEC target is located at the altitude d above a 2-D randomly rough surface, which is described by the random height function z = ς(x, y), with the mean height 〈ς(x, y)〉 = 0. The underlying medium is homogeneous with dielectric constants ɛ1 = ɛ0ɛr and μ1 = μ0. The incident plane electromagnetic fields are written as

equation image

where equation imagei = kequation imagei, and k and η are respectively the wave number and impedance of free space. Note that an harmonic time convention eiωt is assumed and suppressed throughout this paper. The incident polarized coordinate system is defined as

equation image

where equation imagei and equation imagei are the polarization bases of incident wave, and equation imagei is the incident wave vector.

Figure 1.

A 3-D composite model of a PEC target above a dielectric rough surface.

[9] In this paper, the difference scattering field, i.e., the difference value of scattering fields from two corresponding models of with and without the target presence above the rough surface, is introduced to expose the target scattering characteristics from the diffuse scattering of rough surface background. Similar to the 2-D model [Ye and Jin, 2006, 2007], the composite scattering can be described by a coupling set of surface integral equations (SIEs) for the target and the underlying rough surface, respectively. On the PEC target surface, the Dirichlet boundary condition is forced and the SIE can be expressed as

equation image

where equation image is the normal vector on the target surface, equation imagest(equation image) is the direct scattering from the target (the subscript t denotes ‘target’ and the subscript s denotes ‘scattering’), equation imagesfd(equation image) is the difference scattering field of the rough surface due to the target presence above it (the subscripts f, d denote the additional or difference (d) scattering of the rough surface (f)).

[10] On the rough surface, the surface equivalence principle can also be applied to obtain a series of SIEs, which describe the equivalent upper and down problems of the rough surface for two situations: with and without the target presence. However, in the following KA method, the induced surface fields on the rough surface are directly calculated with the excitation fields, instead of solving the rough surface's SIEs, so these SIEs of rough surface are suppressed here.

[11] The difference scattering d-RCS due to target presence above the rough surface is defined as

equation image

where R is the distance between the origin point and the observation point in far-field region. It is noticeable that the d-RCS contains the target's self-scattering and its coupling scattering with the rough surface, while the direct scattering of the rough surface excited by the incident plane wave is subtracted, see References [Johnson, 2002; Ye and Jin, 2006]. Different from the incident coordinate system, the scattering polarized orthogonal coordinate system is defined as

equation image

[12] In equation (3), equation image0(equation image) is the total electric field illuminating on the target, which includes the incident field equation imagei(equation image) and the direct scattering field from the rough surface equation imagesf0(equation image) (the subscripts f0 denote the scattering from the rough surface (f) without the target presence (0)), i.e.

equation image

[13] The scattering field equation imagest(equation image) of the target is produced by the induced currents on the target equation imaget, written as

equation image
equation image

where g(equation image, equation image′) = equation image is the scalar Green's function of free space, R = ∣equation imageequation image′∣ is the distance between the field point equation image and the source point equation image′. Using the RWG (Rao-Wilton-Glisson) vector base functions [Wilton et al., 1984] to discretize the induced current on the target surface as equation imaget(equation image) = equation imageJnequation imagen(equation image) (Here equation imagen is the RWG base function on mesh Tn, and Jn is the expansion coefficient to be solved with the CG (Conjugate Gradient) procedure, M is the discrete number), and using the Galerkin's method to test equation (3), the following MoM equations can be obtained.

equation image

On the RHS (right hand side) of equation (9), the difference scattering field equation imagesfd(equation image) denotes the coupling interaction between the target and the rough surface, which is included as an excitation term changing the target induced currents. Then equation (9) can be solved with CG procedure, so as to obtain the induced currents on the target. Moreover the FMM or MLFMA methods can be employed to accelerate the matrix-vector multiplication operations.

[14] If a 2-D rough surface is quadrilateralized, the discrete mesh number is N = LxLy/(ΔxΔy) (Lx and Ly are respectively the rough surface size in x and y directions, Δx and Δy are the discrete intervals). It should be noticed that the rough surface size Lx × Ly is related to the target size and its altitude [Ye and Jin, 2006, 2007], and it is usually very large, so it requires a large memory and CPU time for the solution of the rough surface's SIE, which is regarded as the bottleneck of numerical simulation of the composite target/surface scattering model. In next section, with the aid of analytical KA method, the computation expressions of scattering fields of the rough surface, equation imagesf0(equation image) and equation imagesfd(equation image), will be derived, and they need only once numerical quadrature along the rough surface. Therefore, the requirements of memory and CPU time can be significantly reduced.

3. KA Computation of Direct Scattering Field equation imagesf0

[15] Following from the Huygens equivalence principle, which expresses the field at an observation point in terms of fields at the boundary surface, the following expressions are obtained for the direct scattering field equation imagesf0(equation image) of the rough surface [Tsang and Kong, 2001a, 2001b].

equation image

where equation image is the normal vector of the rough surface, and equation image = equation image × equation image and equation image = equation image × equation image are the equivalent electric and magnetic currents on the rough surface. The 3-D dyadic Green's functions are written as

equation image
equation image

where equation image is an unit dyadic. Equations (11) and (12) are further written as

equation image
equation image

where A = (1 + equation imageequation image) and B = (equation imageequation image−1)equation image. Note that the far-field approximation cannot be directly used for the dyadic Green's function in scattering computation of equation imagesf0(equation image), because the target is located in the near zone of the rough surface.

[16] Based on the tangential plane approximation, the local incident angle on the rough surface θℓi satisfies cosθℓi = −equation image · equation imagei. Consider an orthogonal coordinate system (equation image, equation image, equation imagei) on the rough surface, where the polarization bases are defined as

equation image

It can be seen that the polarization vectors change in different positions of the rough surface. Because the incident electric field equation imagei and magnetic field equation imagei exhibit the TEM right-handed helix relationship, the following polarized decomposition can be obtained [Tsang and Kong, 2001a, 2001b].

equation image
equation image

[17] According to the tangential plane approximation (the curvature radius of the rough surface should satisfy ρλ), the local reflection direction is equation imager = equation imagei − 2equation image(equation image · equation imagei), and the reflected fields equation imager and equation imager also satisfy the TEM right-handed helix relationship. So the polarized components of equation imager and equation imager can be expressed as

equation image
equation image

where RTE and RTM are the local Fresnel reflection coefficients of TE and TM polarization, respectively as follows [Tsang and Kong, 2001a, 2001b]

equation image

[18] The total electromagnetic fields on the rough surface are the summation of incidence and reflection of two polarized electromagnetic fields, written as

equation image
equation image

Substituting equations (16) and (17) into equations (21) and (22), it yields

equation image
equation image

With the induced fields on the rough surface expressed in equations (23) and (24), the direct scattering field equation imagesf0(equation image) of the rough surface can be calculated using equation (10), which is only a single quadrature on the rough surface. Then, the total illuminating field on the target can be calculated as equation image0(equation image) = equation imagei(equation image) + equation imagesf0(equation image), which is kept invariable during the coupling iteration of the target and the rough surface.

[19] When the permittivity ɛr → ∞, the Fresnel reflection coefficients become RTE = −1 and RTM = 1, which yields equation image × equation image = 2equation image × equation imagei and equation image × equation image = 0. Then equation (10) becomes

equation image

4. KA Computation of the Difference Scattering Field equation imagesfd

[20] equation imagesfd is the coupling scattering between the target and the rough surface denoting their multiple interactions. If we regard the radiation field of the target induced currents as an illuminating field on the rough surface, analogically to equation (10), the following expression can be obtained.

equation image

where equation image(equation image′) and equation image(equation image′) are the induced electric and magnetic fields on the rough surface caused by the radiation field of the target currents.

[21] Consider the radiation field of each discrete current unit on the target surface, e.g., equation imagestn and equation imagestn of current unit Jnequation imagen, which is presumed satisfying the local right-handed helix relationship (This presumption will be demonstrated in section 5, and it yields a corresponding validity condition). The following polarized decomposition expressions can be obtained.

equation image
equation image

where equation imagestn(equation image′) and equation imagestn(equation image′) are expressed as

equation image
equation image

Here ds″ is operated upon the target, while ds′ in equation (26) is operated on the rough surface.

[22] In equation (26), the induced fields equation image(equation image′) and equation image(equation image′) on the rough surface are caused by the radiation fields of the target currents, similarly to equations (23) and (24), they can be expressed as the series summation of the KA solutions caused by all current units as follows.

equation image
equation image

[23] In this paper the rough surface is discretized into quadrilaterals, while the target discretized into triangles, and the dimensions of discrete patches are restricted much less than a wavelength (as λ/10). So the local incident direction equation imageti can be expressed as the orientation vector from the triangle center equation image′ on the target to the tetragon center equation image on the rough surface, e.g., equation imageti = equation image = equation image, which changes with different patches of the target and the rough surface. Note that the polarized vectors equation image and equation image, the Fresnel reflection coefficients RTE and RTM, and the local reflection vector equation imagetr are defined in the same way as in section 3, but with equation imagei replaced by equation imageti. They are all variational for different source and field patches.

[24] Therefore, the KA computation for the difference scattering field of the dielectric rough surface can be summarized as follows: first compute the radiation field equation imagestn from the target current unit with equation (29); then compute the induced fields equation image and equation image on the rough surface with equations (31) and (32); and finally compute the difference scattering field equation imagesfd with equation (26).

[25] For PEC rough surface, equation (26) is simplified as

equation image

[26] It can be seen that the computation of equation imagesfd needs only once numerical integral on the rough surface, and the CPU time is N × M (M and N are the discrete numbers of target and rough surface). From equation (26), it can be seen that only the induced electric and magnetic fields over the rough surface need to be stored, and the memory requirement is 2N. As a comparison, some other methods, such as SDFMM and UV/SMCG, usually require memory and CPU time in the order of O(N lg N) [El-Shenawee et al., 2001].

5. Validity Analysis

[27] Consider the interaction between a triangle patch (the dark-gray one) on the target and a quadrilateral patch on the rough surface (the light-gray one), as shown in Figure 2. Since the dimensions of the discrete patches on the target and the rough surface are both restricted enough small (generally much less than a wavelength λ, and the patches are smooth), the local incident vector can be approximated as the radiation direction from the target patch center equation imagenc to the rough surface patch center equation imagemc, written as equation imageti = equation image = equation image. It can be seen that equation imageti approximately holds its direction for given patches of the target and the rough surface, but changes for different source and field patches.

Figure 2.

Interaction of discretized patches of target and rough surface.

[28] According to vector operation relationships, the following expressions can be obtained.

equation image
equation image

In equation (34), ∇g = equation image is used to remove the second term on the RHS. Comparing equation (34) with equation (35), it yields an approximation condition kR ≫ 1 for the right-handed helix relationship equation image. In the following numerical simulation kR > 20 (R > 3.2λ) is used.

[29] As a summary for the model in Figure 1, the discrete unit dimension of the target and the rough surface should be much less than a wavelength, and drh > 3.2λ is required (r and d are, respectively, the radius of the target and its altitude, h is the root mean square (rms) height of the rough surface) for KA validity.

[30] According to Ye and Jin [2006], the coupling field equation imagesfd decreases quickly toward the rough surface edges, and the surface length is truncated as L = 2(d + r)equation image with ϑ = 10−2, because the coupling interaction between the target and the rough surface outside this scale gradually decreases small enough to be neglected, and the computation and memory requirements are significantly reduced.

[31] Consider a Gaussian rough surface with the surface spectrum as [Pak et al., 1995]

equation image

where h is the root mean square (rms) height, and lx and ly are the correlation lengths along the x and y directions, respectively. Here h = 0.2λ and l = lx = ly = 4.0λ satisfy the KA condition of curvature radius ρ ≈ 23.3λλ [Tsang and Kong, 2001a, 2001b].

[32] To avoid the KA restriction in the model under low grazing angle and the shadowing effect, the incident and scattering angles should satisfy

equation image

which means that the angles are limited by large surface slope. As for the aforementioned Gaussian rough surface with the mean surface slope s = equation image = 0.071, the incident and scattering angles are restricted as ∣θi,s∣ < 85.9°.

6. Convergence Analysis of the KA-MoM Iteration

[33] To carry out the multiple interactions of the target and the rough surface, the coupling iteration is performed, and the process at the (i)-step is as follows: first using the target current equation image(i − 1) solved at the (i − 1)-step to calculate equation (29) to obtain equation imagestn(i− 1) on the rough surface; then calculate the difference scattering field equation imagesfd(i) of the rough surface with the KA method using equations (26), (31), and (32); finally using CG and FMM methods to solve equation (9) with the updated RHS to obtain the new current equation image(i). Iteration begins with equation image(0) = 0, and terminates when equation image(i) reaches convergence.

[34] Define an error function δ of the target current solution in two successive iterative steps as

equation image

where the notation of ∣equation image∣ indicates the Euclidean 2-norm of a vector, as ∣equation image∣ = equation image, equation image and equation image(i) are respectively the discretized MoM impedance matrix and the RHS term of equation (9). equation image(i) and equation image(i − 1) are the current coefficients at the (i)-th and (i − 1)-th steps, which have been solved by the CG process, i.e., equation image · equation image(i) = equation image(i) and equation image · equation image(i − 1) = equation image(i − 1). The RHS term of equation (9), equation image(i) = −〈equation imagem, equation image0〉 − 〈equation imagem, equation imagesfd(i)〉, contains the contribution of the rough surface scattering, which changes in each coupling iterative step. So in coupling iterations, the change of equation imagesfd(i − 1) causes the change of current coefficients equation image(i − 1), and the change of equation imagesfd(i). It means that Δequation imagesfd causes Δequation image, and Δequation image then induces Δequation imagesfd. High-order iteration computes high-order coupling scattering. Because high-order scattering decreases as the iteration keeps going, it causes the decrease of the current variation Δequation image. So this coupling iteration can always converge after several steps. Here the convergence condition is set as ɛ = 10−3.

[35] Consider a PEC target above a dielectric Gaussian rough surface as (1) a cubic with side length a = 3λ, (2) a sphere with radius r = 1.5λ, and (3) an ellipsoid with the semi-axes a = b = 1.5λ, c = 3.0λ and the major axis in x-direction. Gaussian rough surface parameters are h = 0.2λ and l = lx = ly = 4.0λ. The target altitude d = 5λ satisfies the aforementioned KA approximation condition drh > 3.2λ. The relative permittivity of the underlying dielectric rough surface is ɛr = 10.0. The TE-polarized plane wave (equation imagei = equation imagei = −equation image) is incident with incident angles θi = 30° and ϕi = 90°. The truncated surface length is L = 277.6λ, and the discretized unit dimension is Δx = Δy = λ/3 because of the large scale fluctuation of the rough surface.

[36] Figure 3 shows that the error function δ decreases exponentially as iteration steps i increases. It presents that the high-order interaction decreases quickly.

Figure 3.

The error function versus iteration steps.

[37] Figure 4 shows that the loop number for CG computation also decreases quickly as the hybrid KA-MoM iteration continues (here it is assumed that the CG loop terminates with ɛ′ = 10−3, and the solution in the previous iterative step is set as the initial guess for CG process). It means that high-order interaction generates very weak influence on the target currents, which yields good convergence. However, if the discretized dimension is large, or drh > 3.2λ is not satisfied, the hybrid iteration will not converge.

Figure 4.

The loops of CG versus the hybrid KA-MoM iteration steps.

[38] Total scattering field equation images includes the target's direct scattering equation imagest and the target-surface coupling (difference) scattering equation imagesfd, which can be computed with equation imaget. In order to compute the d-RCS, the far-field approximation condition can be used to simplify the dyadic Green's functions as follows. Note that in the above KA computation, the far-field approximation condition cannot be used, because the near field interaction must be computed to account the coupling scattering between target and rough surface.

equation image
equation image

where the scattering wave vector equation images and polarization vectors equation images, equation images correspond to equation image and equation image, equation image in the spherical coordinate system, respectively. So the scattering field can be decomposed as v-polarized and h-polarized components, and the bistatic d-RCS in equation (4) includes the v- and h- polarization scattering power, written as

equation image

In the nature of things, two polarized scattering RCS can be computed individually according to computation practical demands.

7. Numerical Results of d-RCS

[39] As an example of code validation, consider a PEC sphere with radius r = 3λ at altitude d = 10λ above a PEC flat surface. The incident angles are θi = 30° and ϕi = 90°. Since the underlying surface is now flat, it can be solved by the specular image of Green's function. The MoM equation is written as

equation image

Now equation image0 = equation imageiequation imager is produced to illuminate the target, where the subscript r indicates the reflection from the underlying flat surface, the scattering field of the target equation imagest(equation image) is expressed similarly to equation (7), with g(equation image, equation image′) replaced by g(equation image, equation image′) − g(equation image, equation image″).

[40] It can also be solved by the KA-MoM iterative method described above. Comparing equation (42) and equation (9), it can be seen that the specular image g(equation image, equation image″) in equation (42) takes account of the effect of the flat surface, which is equivalent to the term equation imagesfd in equation (9). Let the rough surface truncated with L = 100λ and discretized with Δx = Δy = λ/3, the unknown number is N = 90000. The target is discretized with RWG basis function, and the unknown number is M = 1920.

[41] The hybrid KA-MoM iterative method is compared with the specular image Green's function method. Figure 5 compares the target current coefficients solved by two methods. They are well matched.

Figure 5.

The coefficients Jn of induced current on the target (Figure 5b is the enlarged part of Figure 5a).

[42] Two surface lengths are chosen for comparison, with discretized unit dimension Δx = Δy = λ/3, and the unknown numbers are respectively N(L = 100λ) = 90000 and N(L = 300λ) = 810000. In numerical simulation, the loop termination conditions for CG and KA-MoM iteration are set as ɛ = ɛ′ = 10−3, and four coupling iterative steps of target and rough surface, which are corresponding to the 1-order to 4-order coupling scattering, are computed to reach convergence. It means that the 5-order and higher order coupling scatterings are small enough to be neglected.

[43] For two surface lengths, the total CPU time for CG computation both approximates 62.54s, but for KA computation the CPU time are different, respectively consuming approximately 1427.6s for L = 100λ and 12695.5s for L = 300λ, respectively. It can be seen that the KA computation consumes the leading CPU time because the discretized unit number of the rough surface is much larger than the target. The total CPU time for two surface lengths are respectively 1489.1s and 12757.2s. It should be noted that for different Monte-Carlo realization the CPU time may be different slightly.

[44] Figure 6 shows the bistatic d-RCS in the y-z plane (incident plane). The results are coincident for most scattering angles θs, except trivial deviation at low grazing scattering directions. It can also be seen that large surface length is necessary for scattering computation at low grazing scattering angles. But it causes much more memory and computation time.

Figure 6.

Bistatic d-RCS σd of a sphere above a PEC flat surface.

[45] Suppose a Gaussian rough surface with RMS height h = 0.2λ, correlation length lx = ly = 4.0λ, and the surface length L = 300λ. A PEC sphere with radius r = 1.86λ is located at altitude d = 5λ above the rough surface. The permittivity of the rough surface takes ɛr = 2.0, 10.0 and a PEC rough surface, respectively. The TE-polarized plane wave (equation imagei = equation imagei = −equation image) is incident with incident angle θi = 30° and ϕi = 90°.

[46] Using Monte-Carlo method to generate 50 randomly Gaussian rough surface samples, the mean bistatic d-RCS is shown in Figures 7a7c for different dielectric or PEC underlying rough surfaces. It can be seen that in the (Figures 7a and 7b) cases of dielectric rough surface, the major peaks appear in backscattering direction (where θs = 30°, ϕs = 270°), and some secondary peaks appear in specular forward direction (where θs = 30°, ϕs = 90°), which are caused by coupling scattering of the underlying rough surface, especially when the permittivity becomes large. While in the (Figure 7c) case of PEC rough surface, the forward scattering peaks become stronger while the backscattering peak is reduced. The reason is that when the surface permittivity increases, more energy is scattered or rebounded into the upper half-space, especially in the specular direction.

Figure 7.

Bistatic d-RCS of a PEC sphere above dielectric rough surface with different permittivity.

[47] Figure 8 shows the induced difference fields on the dielectric rough surface sample (one Monte-Carlo realization) as shown in equations (31) and (32). They are locally concentrated on a finite surface region under the target, which makes reasons to the finite truncation of the rough surface for interaction calculation. Moreover, the difference induced fields on the rough surface become stronger as the surface permittivity increases, which makes more electromagnetic energy rebounded into the upper space.

Figure 8.

The induced difference field over rough surface due to a PEC sphere presence: (a) electric field ∣E(equation image)∣ and (b) magnetic field ∣H(equation image)∣.

[48] Figure 9 specifically compares the mean bistatic d-RCS on the incident plane for 50 rough surface realizations with a spherical target presence. It can be seen that d-RCS increases as the permittivity ɛr increases, especially in forward direction, because of the strong coupling interaction between the target and the underlying rough surface.

Figure 9.

Bistatic d-RCS of sphere on the incident plane for ɛr = 2, 10 and PEC cases.

[49] To make a complicated target, some hexahedrons are shown in Figure 10: (1) a regular cubic with side length 2λ, (2) a cutoff cubic with the cut size P1P5 = 0.2λ, P2P6 = 1.4λ, P3P7 = 2λ and P4P8 = 0.8λ, (3) a transformed hexahedron with linear mapping mode of (xi, yi, zi) → (Xi, Yi, Zi), i.e.

equation image

The vertexes of Figure 10 (3) can be obtained as

equation image
equation image

It is shown that the linearly transformed hexahedron is a parallelepiped.

Figure 10.

Section and transformation of a regular cubic.

[50] Figures 11a and 11b show the bistatic d-RCS of the three hexahedrons (shown in Figure 10) above a dielectric Gaussian rough surface with ɛr = 10.0, and the comparison with the case of no underlying rough surface, respectively. The incident angle is θi = 30° and ϕi = 90°. Comparing with Figure 11b, multiple peaks in Figure 11a are caused by the coupling interaction between the target and the underlying rough surface. It can also be seen that the angular scattering from target-surface becomes more concentrated around the incident plane.

Figure 11.

Bistatic d-RCS of three hexahedrons: (a) with underlying surface and (b) no underlying surface.

[51] The small side face P1P4P8P5 of the cutoff cubic (Figure 10 (2)) contributes less to the backscattering field. However, the scattering contribution of the cutting face P5P6P7P8 makes the backscattering field still very strong.

[52] The side faces of the transformed hexahedron (Figure 10 (3)) change their orientations, which makes the incident direction unparallel or unvertical to any side face of the hexahedron. So the scattering peaks deviate from the incident plane, and the bistatic scattering pattern shows asymmetric, which can also be seen from Figure 13b.

[53] Figure 12 shows the induced difference fields on the underlying dielectric rough surface due to the hexahedral targets presence, which are concentrated on a finite local surface region under the target. Figures 12a and 12b show apparent interference fringes with the centers shifted in y direction, while Figure 12c shows a strong peak with unsymmetry, which is caused by the side face P1P4P8P5 of the transformed hexahedron in Figure 10 (3).

Figure 12.

Difference electromagnetic fields on rough surface induced by different targets: (a) a regular cubic, (b) a cutoff cubic, and (c) a transformed hexahedron.

[54] Figure 13 compares the bi-static d-RCS of three hexahedral targets on the planes, parallel and vertical to the incident plane, respectively. The d-RCS of a regular cubic shows forward and backward peaks on parallel plane, and another peak of θs = 0° on vertical plane. As comparison, the d-RCS of the cutoff cubic shows additional peaks at low grazing scattering angle on the vertical plane, and the d-RCS of the transformed hexahedron shows more additional peaks on the parallel and vertical planes, which are caused by reflections of the self face of the hexahedron.

Figure 13.

Comparison of bistatic d-RCS on the parallel and vertical planes.

8. Conclusions

[55] A hybrid analytic KA and numerical MoM algorithm for computation of the difference scattering from a 3-D PEC target above a dielectric rough surface is developed. Numerical MoM is used to solve the target's SIE, and the KA method is used for rough surface scattering computation. Iterations take account of the multiple target-surface interactions. This hybrid method significantly reduces computation cost, shows physical insight in complicated target-surface interaction, and preserves the solution precision by using the iterative scheme. High order interactions quickly become convergent. The right-handed helix relationship of radiation fields of target's currents is validated, and the requirement for KA computation is discussed.

[56] Using the Monte-Carlo method to generate rough surface samples, bistatic difference scattering from a PEC target above a dielectric Gaussian rough surface is numerically simulated. Induced fields on the rough surface sample show the limited region of interactions between the target and the rough surface. As the permittivity of the dielectric rough surface increases, the penetrated electromagnetic power though the interface becomes weaker, and the coupling scattering of the target and rough surface makes the d-RCS stronger in the upper half-space. Nonregular target, e.g., transformed cubic, makes d-RCS with multipeaks away from the incident plane.

Acknowledgments

[57] This work was supported by the National Science Foundation of China 40637033 and 60571050.

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