# An Efficient Hybrid Method for 3D Scattering from Inhomogeneous Object Buried beneath a Dielectric Randomly Rough Surface

Authors: Hong-jie He; Li-xin Guo; Wei Liu
Journal: International Journal of Antennas and Propagation
 (2017)

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## Abstract

An efficient iterative analytical-numerical method is proposed for three-dimensional (3D) electromagnetic scattering from an inhomogeneous object buried beneath a two-dimensional (2D) randomly dielectric rough surface. In the hybrid method, the electric and magnetic currents on the dielectric rough surface are obtained by current-based Kirchhoff approximation (KA), while the scattering from the inhomogeneous object is rigorously studied by finite element method (FEM) combined with the boundary integral method (BIM). The multiple interactions between the buried object and rough surface are taken into account by updating the electric and magnetic current densities on them. Several numerical simulations are considered to demonstrate the algorithm’s ability to deal with the scattering from the inhomogeneous target buried beneath a dielectric rough surface, and the effectiveness of our proposed method is also illustrated.


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## Body

## 1. Introduction

The scattering from an object buried beneath a randomly dielectric rough surface which can be modelled as sea surface and sands is a subject of great interest in many applications, for example, geophysical exploration, buried target detection, and remote sensing. Many researchers have paid much attention to this problem and many numerical methods have been proposed in the past decades. The method of moments (MoM) was applied to discuss the scattering from a rough surface with a two-dimensional (2D) target in [1, 2]. The MoM with the dyadic Green’s function of the background medium is applied in [3] to study the forward-scattering problem of three-dimensional (3D) objects buried under a 2D locally rough surface. To improve the computational efficiency and reduce the memory requirement, some asymptotic methods and hybrid methods are developed to study the scattering from a target buried beneath the rough surface. The cylindrical wave approach combined with the first-order small perturbation method (SPM) [4] was applied in [5] to study the scattering from a cylinder buried below a slightly rough surface. The SPM was applied to study the electromagnetic scattering from a 2D dielectric cylinder buried beneath a slightly rough surface [6]. The second-order small slope approximation was used in [7] to study the scattering from one dimensional (1D) sea surface. The fields scattered by an object below a rough surface were computed by the efficient propagation-inside-layer-expansion (PILE) method combined with the physical optics approximation [8]. Hybrid KA-BIM was proposed in [9] to study the scattering from a 2D target buried beneath a rough surface. An efficient hybrid KA-MoM was proposed to analyse the electromagnetic scattering from a 3D perfectly electric conducting (PEC) object buried beneath a dielectric rough surface [10]. However, these numerical methods and hybrid methods mentioned above, for both 2D and 3D scattering problems, are applicable to PEC or dielectric objects. They all suffer the inconvenience of dealing with the scattering from inhomogeneous objects buried beneath dielectric rough surface.For general inhomogeneous objects, the volume integral equation method is suitable. A fast volume integral equation algorithm with dyadic Green’s function [11] was developed to simulate the electromagnetic scattering from large inhomogeneous objects embedded in a planarly layered medium. The other appropriate choice for inhomogeneous objects is the finite element method (FEM) combined with the boundary integral method (BIM). The hybrid FEM/BIM method with the layered medium dyadic Green’s function [12] was proposed to predict the scattering from inhomogeneous dielectric objects embedded in multilayered medium. But these methods cannot be applied to deal with the scattering from inhomogeneous object buried beneath randomly rough surface, since the analytical expression of the dyadic Green’s function for the half space with rough surface can hardly be obtained. A hybrid FEM/MoM was applied in [13] to study the scattering from a 3D dielectric object above a 2D conductive rough surface. The FEM/BIM combined with KA was used in [14] to model the scattering from a 3D coated target above 2D PEC rough surface. The efficient and accurate modelling of inhomogeneous object buried beneath the randomly dielectric rough surface is still a challenging problem.In this paper, a hybrid method is firstly designed for the fast and accurate solution of the 3D scattering problem related to a 3D inhomogeneous target buried beneath a 2D randomly dielectric rough surface. In order to achieve a significant reduction in CPU time and memory requirements, the asymptotic current-based KA is applied in the hybrid method to obtain the electric and magnetic current densities on the dielectric rough surface. To model the scattering from the buried inhomogeneous target, the powerful FEM/BIM is used in this paper. The multiple interactions between the target and rough surface are taken into account by updating the electric and magnetic fields on them. The accuracy of our hybrid method has been proven by comparing the results with that of the numerical method. And the effectiveness of our hybrid method is also discussed.


## 2. Hybrid Theory and Formulations

The geometry of the electromagnetic scattering problem is depicted in Figure1. The whole space is separated into two half-spaces by the rough surface ΓKA, which can be defined by a function z=fx,y with mean fx,y=0. The rough surface generated by using the Monte-Carlo method based on the corresponding power spectrum function is confined to a square region L=Lx=Ly on x-y plane of Cartesian coordinate system. The 2D Gaussian roughness spectrum used in this paper is given by (1)Skx,ky=δ2lxly4πexp⁡-kx2lx2+ky2ly24,where δ is the root-mean-square height of the rough surface, lx and ly are the correlation lengths in the x^ and y^ directions, respectively, and kx and ky are the spatial frequencies in the x^ and y^ directions, respectively.Figure 1
Geometry of the scattering problem of a 3D complex dielectric object buried under a 2D randomly dielectric rough surface.The half-spaces above and below the rough surface are marked as region 1 and region 2, respectively. The permittivity and permeability of regioni are εi and μi, respectively. The inhomogeneous target buried in region 2 is bounded by ΓBIM. The interior region ΩFEM of the target can be composed of inhomogeneous dielectric objects with complex relative permittivity εrr and permeability μrr, and PEC. To reduce the edge effects caused by the truncation of the finite surface length, the rough surface is illuminated by a TE polarized tapered wave [15] in this paper. The source of the tapered wave is in region 1 and the tapered wave can be expressed as(2)Eincr=exp-jk1·r1+wexp-tx-ty,where(3)w=1k122tx-1g2cos2θinc+2ty-1g2tx=cosθinccosφincx+cosθincsinφincy+sinθincz2g2cos2θincty=-sinφincx+cosφincy2g2,where θinc and φinc are the incident angles, g is the tapered factor, and k1 is the wavenumber in the half space above the rough surface.Hybrid FEM/BIM is a powerful technique to deal with the inhomogeneous media problem due to the differential equation nature. As shown in Figure1, region 2 can be decomposed into two parts by boundary ΓBIM, the interior region ΩFEM, and the outer region. In FEM/BIM, the interior region ΩFEM is analysed by FEM and the outer region is modelled by BIM, which naturally satisfies the radiation condition. The interior field and the outer field are coupled together via the field continuity conditions on the boundary ΓBIM.In the FEM regionΩFEM, the electric field satisfies the following Helmholtz’s equation:(4)∇×1μrr∇×Er-k02εrrEr=-jk0η0Jrr∈ΩFEM,where k0 is the free-space wavenumber, Jr is the electric current in the space, η0 is the free-space intrinsic impedance, and Er is the total electric field.Based on the theory of functional analysis, the equivalent variational problem can be formulated as [16](5)δFE=0n^×Er=0r∈Γ0,where (6)FE=12∫∫∫ΩFEM1μrr∇×E·∇×E-k02εrrE·EdV-jk0η0∯ΓBIME·n^×HtscatdS,where n^ is the out-ward unit vector normal to boundary ΓBIM and Htscat is the magnetic field scattered by the target on the boundary ΓBIM. Discretizing the FEM region with tetrahedral elements, the fields can be expanded by the vector basis function [16] Nmr:(7)Er=∑m=1MEmNmrHr=∑m=1MHmNmr,where M is the number of the vector bases defined in FEM region. Then the following weak form of the matrix equation can be obtained:(8)∫∫∫ΩFEM1μrr∇×Ni·∇×E-k02εrrNi·EdV-jk0η0∫∫ΓBIMNi·n^×HtscatdS=0.The outer region can be modelled by the integral equation method. The total electric fieldEr in the outer region satisfies the following integral equation:(9)Er-Etscatr=Esscatr,where Esscatr is the field excited by the equivalent surface electric and magnetic currents (Jseq and Mseq) on the rough surface, which act as the source illuminating on the target. Etscatr is the field excited by the equivalent surface electric and magnetic currents (Jteq and Mteq) on the boundary ΓBIM. The fields Etscatr and Esscatr can be obtained from the following expressions, respectively:(10)Etscatr=η2L2Jteq-K2Mteq(11)Esscatr=-η2L2Jseq+K2Mseq.The operators Li and Ki are defined as(12)LiX=-jki∫∫ΓXr′Gir,r′+1ki2∇′·Xr′∇Gir,r′dΓ′KiX=-∫∫ΓXr′×∇Gir,r′dΓ′.The symbols ηi, ki, and Gi are the complex impedance, the wavenumber, and Green’s function associated with region i, respectively. The expression of the Green’s function Gi is given by(13)Gir,r′=e-jkir-r′4πr-r′,where r′ and r are the position vectors of the source and observation points, respectively.On the boundaryΓBIM, the equivalent electric and magnetic currents (Jteq and Mteq) have the relation with surface fields as Jteq=n^×Htscat and Mteq=-n^×Etscat, respectively. Then the electric and magnetic currents on the boundary can be discretized as(14)Jteqr=∑n=1NtHnfnrMteqr=∑n=1NtEnfnr,where fnr is the Rao-Wilton-Glisson (RWG) basis functions which have relation with Nnr as fnr=-n^×Nnr. When MoM is applied, the following matrix equations can be obtained by discretizing (9):(15)PE+QH=b,where P and Q are the impedance matrices and b is the vector related to the fields scattered from the rough surface. And they can be obtained by the following expressions:(16)Pmn=fmr,Nnr+∫Snfnr′×∇′G2r,r′dS′Qmn=fmr,jk2η2∫Snfnr′G2r,r′dS′+fmr,jη2k2∫Sn∇′·fnr′∇G2r,r′dS′bm=fmr,Esscatr′.Then we can obtain the final matrix equation for the scattering problem of the inhomogeneous target buried under the dielectric rough surface as(17)KIIKIS0KSIKSSC0PQEIESHS=00b,where KII, KIS, KSI, and KSS are the FEM matrices. The subscript I is denoting in the interior region ΩFEM and subscript S is denoting on the boundary ΓBIM. C is the coupling matrix between FEM and BIM. K and C can be represented by the following expressions, respectively:(18)Kmn=∭Ω1μr∇×Nm·∇×Nn-k02εrNm·NndVCmn=jk0η0∬ΓmNm·n^×NndS.Note that the effects of the rough surface are considered in (17) as terms b. To get the field Esscat scattered from the rough surface, we should firstly compute the equivalent surface electric and magnetic currents (Jseq and Mseq) on the rough surface. They can be rigorously achieved by solving the following integral equations with MoM: (19)-η2L2Jseq-η1L1Jseq+K2Mseq+K1Mseq=Einc-Etscat-K2Jseq-K1Jseq-1η1L1Mseq-1η2L2Mseq=Hinc-Htscat.However, solving these integral equations with MoM is very time-consuming and requires a great amount of storage memory, especially for 3D electrically large rough surface. To reduce the computation time and memory requirement, analytical KA is a good choice.Based on the superposition principle, the electric and magnetic currents on the rough surface can be represented by the combination of two independent parts as(20)Jseq=Jsinc+Jsobj(21)Mseq=Msinc+Msobj, where Jsinc and Msinc are the electric and magnetic currents induced by the incident wave in the absence of the buried target. Jsobj and Msobj are the electric and magnetic currents generated by the fields scattered by the buried target in the absence of the incident wave source.According to the KA theory and equivalent principle, the electric and magnetic currents (Jsinc and Msinc) at any point r can be expressed as(22)Jsinc=n^s×1+RUTMp^i·Eincη1q^i-q^i·Eincη1p^i-RUTEk^r×q^i,(23)Msinc=-n^s×1+RUTEq^i·Eincq^i+p^i·Eincp^i-RUTMk^r×q^i,where (p^i, q^i, k^inc) is a local orthonormal basis established at any point r on the rough surface. The symbol k^inc is the unit incident wave vector. The unit vectors p^i and q^i are the local perpendicular and parallel polarization vectors, respectively, which can be defined as(24)q^ir=k^inc×n^srk^inc×n^srp^ir=q^ir×k^inc.The symbol k^r is the unit local reflection direction vector. Based on the tangent plane approximation, k^r=k^inc-2n^sn^s·k^inc. RUTE and RUTM are the local Fresnel reflection coefficients of TE polarization and TM polarization, respectively, which can be expressed as(25)RUTE=cosθli-ε2r-sin2θlicosθli+ε2r-sin2θliRUTM=ε2rcosθli-ε2r-sin2θliε2rcosθli+ε2r-sin2θli,where θli is the local incident angle on the rough surface which satisfies cosθli=-n^s·k^i and ε2r is the relative permittivity of region 2.Similarly, the electric and magnetic currents (Jsobj and Msobj) generated by the fields scattered by the buried target can be expressed as(26)Jsobj=n^s×1+RDTMp^s·Etscatη2q^s-q^s·Etscatη2p^s-RDTEk^rs×q^s,(27)Msobj=-n^s×1+RDTEq^s·Etscatq^s+p^s·Etscatp^s-RDTMk^rs×q^s,where (p^s, q^s, r^s) is a local orthonormal basis established at any point r on the rough surface. r^s is the unit vector from the source point r′ on the target to the field point r on the rough surface, which satisfies r^s=r-r′/r-r′. The unit vectors p^s and q^s are the local perpendicular and parallel polarization vectors, respectively, which can be defined as(28)q^sr=r^s×n^srr^s×n^srp^sr=q^sr×r^s,where k^rs is the unit local reflection direction of the wave induced by the electric and magnetic currents on the target, which act as another incident source illuminating on the rough surface from the bottom. RDTE and RDTM are the local Fresnel reflection coefficients of TE polarization and TM polarization, respectively, which can be expressed as(29)RDTE=ε2rcosθls-1-ε2rsin2θlsε2rcosθls+1-ε2rsin2θlsRDTM=cosθls-ε2r1-ε2rsin2θlscosθls+ε2r1-ε2rsin2θls,where θls is the local incident angle and satisfies cosθls=n^sr·r^s.Then the scattering information from the rough surface can be obtained by (20), (22), and (26), if the scattering field Etscat from the target is calculated beforehand. It should be pointed out that, since the scattering from the rough surface is considered by KA, the limitation of KA [17] should be taken into account in our hybrid method; namely, the curvature radius ρ of the rough surface should be very large compared to the wavelength of the incident wave, ρ≫λ.The multiple interaction between the rough surface and the buried target can be taken into account by an iterative approach. To give a clearer depiction, the iterative process in our method is shown in Figure2. The iterative error at the ith iteration is defined as(30)γi=Jti-Jti-1Jti.Figure 2
The flowchart of the iterative method.Once the scattering information from the rough surface is obtained, the bistatic normalized radar cross-section (NRCS) can be calculated by(31)NRCS=limr→∞r2Escat2Pinc,where Escat is the field scattered by the rough surface at any point r in the region 1, which can be obtained with the following expression:(32)Escat=η1L1Jseq-K1Mseq.The symbol Pinc is the incident beam power which can be given by (33)Pinc=π2g2cosθinc1-1+cos2θinc+2tan2θinc2k12g2cos2θinc.


## 3. Numerical Results

In this part, several numerical simulations are considered to demonstrate the algorithm’s ability to deal with the scattering from the inhomogeneous target buried beneath a dielectric rough surface. The effectiveness of our proposed method is also illustrated. In all examples, the region 1 is considered as free space. In addition, all sizes are given in terms of free-space wavelengthλ which is equal to 1 m in this paper unless otherwise specified. Finally, the relative permittivity of region 2 is ε2r=(2.0,-0.01).Our numerical code is firstly applied to calculate the bistatic NRCS of a 3D PEC sphere buried under a Gaussian dielectric rough surface. The radius of the sphere is 0.5 m and the burial depth is 4.0 m. The incident angles areθinc=0° and φinc=0°. The length of the truncated rough surface is L= 9.0 m. The tapering parameter of the tapered incident wave is g=L/4. The root-mean-square height and the correlation length of the rough surface are δ= 2.0 m and lx=ly=l= 0.1 m, respectively.Figures3(a) and 3(b) show the bistatic NRCS as a function of scattering angle θs in X-Z and Y-Z planes, respectively. And the results are obtained with single rough surface sample. In the legends, “Rs + Object” denotes the model of an object buried under the rough surface. “Rs” denotes the model of the rough surface only. Figure 3 shows that the results obtained by our method agree with solutions of FEM/MoM very well. It can also be observed that the total NRCS is mainly contributed by the rough surface within the scattering angles θs∈-15°,15°, while in other scattering angles, the total NRCS is mainly contributed by the buried target.Figure 3
Bistatic NRCS of a 3D PEC sphere buried under dielectric rough surface: (a)X-Z Plane and (b) Y-Z Plane.
(a)
(b)Table1 shows the comparisons of the computing resource for a single rough surface sample for different methods. The matrix of FEM/BIM is stored with compress sparse row format, and the parallel LU decomposition is used to solve the matrix equation. Table 1 indicates that the number of unknowns in our method is reduced to 14.5% of that in FEM/MoM, and the memory requirement and solving time are reduced to 1.3% and 3.3%, respectively. The result shows that our method is exact and efficient in dealing with the scattering from the target buried beneath rough surface.Table 1
Comparisons of the computing resource for different methods.
Method
Number of unknowns
Memory requirement (G)
Solvingtime (s)
FEM/MoM
83836
107.34
3807
New method
12186
1.43
127
Calculated by a computer with a 2.8 GHz processor (AMD Opteron (tm) Processor 4280), RAM 128 GB.To further demonstrate the ability of our method to deal with the composite scattering from the inhomogeneous target buried under rough surface, the scattering from a 3D inhomogeneous sphere buried under a dielectric rough surface, as shown in Figure4, is calculated. The default parameters are set as follows: the incident angles are θinc=15° and φinc=0°, the truncated length of the rough surface is L= 10.5 m, and the root-mean-square height and the correlation length of the rough surface are δ= 1.5 m and lx=ly=l= 0.1 m, respectively.Figure 4
Cross-section of an inhomogeneous sphere.Figures5(a) and 5(b) show the bistatic NRCS as a function of scattering angle θs in X-Z and Y-Z planes, respectively. And the results are obtained with single rough surface sample. It can be seen from Figure 5 that the results obtained by our hybrid method agree with solutions of FEM/MoM very well, demonstrating the ability of our method to deal with the scattering from inhomogeneous target.Figure 5
Bistatic NRCS of a 3D inhomogeneous sphere buried under dielectric rough surface: (a)X-Z Plane and (b) Y-Z Plane.
(a)
(b)In the third example, the influence of the surface roughness on the scattering properties of the inhomogeneous sphere buried under dielectric rough surface is studied. The parameters of the inhomogeneous sphere are shown in Figure4. The other parameters are set as follows: the length of the rough surface is L= 50 m, the incident angles are θinc=30° and φinc=0°. The parameters of the root-mean-square height δ and the correlation length lx=ly=l of the rough surface can be found in the Figure 6.Figure 6
Scattering results of a 3D inhomogeneous sphere buried under dielectric rough surface with differentδ and l: (a) X-Z Plane and (b) Y-Z Plane.
(a)
(b)The curves in Figure6 present the influence of the surface roughness on the scattering properties of the inhomogeneous sphere buried under dielectric rough surface. The results are obtained by taking the average of 50 rough surface realizations. It can be seen from Figure 6(a) that the scattering coefficients have a small change with the change of the roughness in the scattering directions defined by θs∈30°,90∘. Within the scattering angles θs∈-90°,30∘, the scattering coefficient decreases in the specular direction with the increase in the root-mean-square height of the rough surface but increases in other scattering directions in the X-Z Plane. In the Y-Z Plane, the NRCS increase evidently with the increase in the root-mean-square height of the rough surface in the scattering directions defined by θs∈-45°,45∘. The increase in the correlation length weakens the NRCS in the scattering directions θs∈-30°,30° in the Y-Z Plane.


## 4. Conclusions

In this paper, an efficient hybrid method has been presented to study the scattering from a 3D inhomogeneous object buried beneath a 2D dielectric rough surface. The electric and magnetic currents on the rough surface are analytically obtained by KA, which can greatly reduce the computing resource. The hybrid vector FEM/BIM, a powerful technique to model complex dielectric problem, is chosen to calculate the electric and magnetic currents on the inhomogeneous object. The matrix of FEM/BIM is stored with compress sparse row format, and the parallel LU decomposition is used to solve the matrix. The validity of our hybrid method is verified by numerical method. The effectiveness of our method is also illustrated. The results show that the memory requirement in our hybrid method is greatly reduced compared with the numerical method, and the computational time is also greatly reduced. The ability of our method to deal with the scattering inhomogeneous object buried beneath rough surface is also demonstrated.



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