The Far-Field Equations in Linear Elasticity for Disconnected Rigid Bodies and Cavities

In this paper the far-field equations in linear elasticity for scattering from disjoint rigid bodies and cavities are considered. The direct scattering problem is formulated in differential and integral form. The boundary integral equations are constructed using a combination of single- and double-layer potentials. Using a Fredholm type theory it is proved that these boundary integral equations are uniquely solvable. Assuming that the incident field is produced by a superposition of plane incident waves in all directions of propagation and polarization it is established that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed for the far-field region. The properties of the Herglotz functions are used to derive solvability conditions for the far-field equations. It is also proved that the far-field operators, in terms of which we can express the far-field equations, are injective and have dense range. An analytical example for spheres illuminates the theoretical results.     

The use of polarization effects in electromagnetic inverse scattering problems

We consider the scattering of plane, time-harmonic electromagnetic waves by a perfect conductor D. We first show that the set ?_λ consisting of the span of a fixed linear combination of the electric and magnetic far-field patterns is dense in the space of square-integrable tangential vector fields defined on the unit sphere for all values of the wave number k ≠ 0 provided that Im k ≥0. We next consider the affine hull ??_λ of ?_λ and characterize the set ? in terms of electric Herglotz fields. This result is then used to derive an optimization scheme for solving the inverse scattering problem of determining D from a knowledge of ?_λ and it is shown that this (unconstrained) optimization scheme has zero as its greatest lower bound.     

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

Consider the reconstruction of Dirichlet-to-Neumann map(D-to-N map) from the far-field patterns of the scattered waves in inverse scattering problems, which is the first step in detecting the obstacle boundary by the probe method using far-field measurements corresponding to all incident plane waves. In principle, this problem can be reduced to solving an integral equation of the second kind with the kernels involving the derivatives of the scattered waves for point sources. Based on the mixed reciprocity principle, we propose two simple and feasible numerical schemes for reconstructing D-to-N map. Compared with the well-known obstacle boundary recovering schemes using the simulation of D-to-N map directly, the proposed schemes give the possible ways to realizing the probe methods using practical far-field data, with the advantage of no numerical differentiation for scattered wave in their implementations. We present some numerical examples for the D-to-N map, showing the validity and stability of our schemes.     

A spectrally accurate algorithm for electromagnetic scattering in three dimensions

In this work we develop, implement and analyze a high-order spectrally accurate algorithm for computation of the echo area, and monostatic and bistatic radar cross-section (RCS) of a three dimensional perfectly conducting obstacle through simulation of the time-harmonic electromagnetic waves scattered by the conductor. Our scheme is based on a modified boundary integral formulation (of the Maxwell equations) that is tolerant to basis functions that are not tangential on the conductor surface. We test our algorithm with extensive computational experiments using a variety of three dimensional perfect conductors described in spherical coordinates, including benchmark radar targets such as the metallic NASA almond and ogive. The monostatic RCS measurements for non-convex conductors require hundreds of incident waves (boundary conditions). We demonstrate that the monostatic RCS of small (to medium) sized conductors can be computed using over one thousand incident waves within a few minutes (to a few hours) of CPU time. We compare our results with those obtained using method of moments based industrial standard three dimensional electromagnetic codes CARLOS, CICERO, FE-IE, FERM, and FISC. Finally, we prove the spectrally accurate convergence of our algorithm for computing the surface current, far-field, and RCS values of a class of conductors described globally in spherical coordinates.     

The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence

In this work, we consider the method of non-linear boundary integral equation for solving numerically the inverse scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder in three dimensions. We consider the indirect method and simple representations for the electric and the magnetic fields in order to derive a system of five integral equations, four on the boundary of the cylinder and one on the unit circle where we measure the far-field pattern of the scattered wave. We solve the system iteratively by linearizing only the far-field equation. Numerical results illustrate the feasibility of the proposed scheme.     

Reciprocity relations and completeness of far-field pattern vectors for obstacle scattering of acoustic wave in a stratified medium

Under a generalized Sommerfeld radiation condition, we proved the uniqueness and existence of the direct obstacle scattering problem of time-harmonic acoustic waves in a stratified medium [8]. In this paper, we study the asymptotic behaviour of the scattered waves and prove three reciprocity relations among the free-wave far-field patterns and the guided-wave far-field pattern vectors corresponding to incident distorted plane waves and normal mode waves. Then we prove conditions under which a set of far-field patterns is complete in a Hilbert space based on the reciprocity relation. These properties are important in investigating the inverse obstacle scattering problems.     

Inverse diffraction by a doubly periodic structure

Let us consider the scattering of electromagnetic waves by a doubly periodic structure. Above the structure, the medium is assumed to be homogeneous with a fixed real dielectric coefficient. The medium is a perfect conductor below the structure. For a given incident plane wave, the tangential electric field is measured away from the structure. An inverse problem arises: To what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor? In this paper, results on uniqueness and stability are established for the inverse problem. A crucial step in our proof is to obtain a lower bound for the first eigenvalue of an eigenvalue problem.     

An approximation problem in inverse scattering theory

In [3] a new method was introduced for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium. This method is based on the solution of a new class of boundary value problems for the reduced wave equation called interior transmission problems. In this paper it is shown that if there is absorption there exists at most one solution to the interior transmission problem and an approximate solution can be found such that the metaharmonic part is a Herglotz wave function. These results provide the necessary theoretical basis for the inverse scattering method introduced in [3]     

Inverse scattering from an open arc

A Newton method is presented for the approximate solution of the inverse problem to determine the shape of a sound-soft or perfectly conducting arc from a knowledge of the far-field pattern for the scattering of time-harmonic plane waves. Fréchet differentiability with respect to the boundary is shown for the far-field operator, which for a fixed incident wave maps the boundary arc onto the far-field pattern of the scattered wave. For the sake of completeness, the first part of the paper gives a short outline on the corresponding direct problem via an integral equation method including the numerical solution.     

Radiation condition and scattering problem for time-harmonic acoustic waves in a stratified medium with a nonstratified inhomogeneity

In this paper, a generalized Sommerfeld radiation conditionis presented for the scattering waves in a stratified mediumwith a nonstratified inhomogeneity. Using integral equationmethods, the uniqueness and existence of the direct scatteringproblem are proved. Relations between the scattered acousticwaves in the far field and the sound profile of the inhomogeneityare obtained. Using these relations, the author proves threereciprocity relations between the free-wave far-field patternsand the guided-wave far-field pattern vectors correspondingto incident distorted plane waves and normal mode waves. Thenconditions under which a set of far-field patterns is completein a Hilbert space are determined using the reciprocity relation.These properties are important in investigating inverse scatteringproblems.     

Note on a Uniqueness Theorem of Schiffer

AMS (MOS): 35R30, 76Q05

The uniqueness of the inverse scattering problem is examined. It is shown that the boundary conditions must be suitably restricted to secure uniqueness. For one class the far-field pattern for one incident field is sufficient for uniqueness and, for a wider class, information from a finite number of distinct incident waves suffices. In either case, the far-field also dictates the boundary condition on the scatterer.     

The inverse scattering problem by an elastic inclusion

In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti’s formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method.     

The singular sources method for cracks

The singular sources method is given to detect the shape of a thin infinitely cylindrical obstacle from a knowledge of the TM‐polarized scattered electromagnetic field in large distance. The basic idea is based on the singular behaviour of the scattered field of the incident point source on the cross‐section of the cylinder. We assume that the scatterer is a perfect conductor which is possibly coated by a material and investigate two models with different boundary conditions. Also we give a uniqueness proof for the shape reconstruction. Copyright © 2006 John Wiley & Sons, Ltd.     

A complementary theory of light scattering by homogeneous spheres

A theory of the scattering of electromagnetic waves by homogeneous spheres, the so-called Mie theory, is presented in a unique and coherent manner in this paper. We begin with Maxwell's equations, from which the vector wave equations are derived and solved by means of the two orthogonal solutions to the scalar wave equation. The transverse incident electric field is mapped in spherical coordinates and expanded in known mathematical functions satisfying the scalar wave equation. Determination of the unknown coefficients in the scattered and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of a sphere. Far-field solutions for the electric field are then given in terms of the scattering functions. Transformation of the electric field to the reference plane containing incident and scattered waves is carried out. Extinction parameters and the phase matrix are derived from the electric field perpendicular and parallel to the reference plane. On the basis of the independent-scattering assumption, the theory is extended to cases involving a sample of homogeneous spheres.     

The interior transmission problem for anisotropic Maxwell's equations and its applications to the inverse problem

The interior transmission problem appears naturally in the study of the inverse scattering problem of determining the shape of a penetrable medium from a knowledge of the time harmonic incident waves and the far field patterns of the scattered waves. We propose a variational study of this problem in the case of Maxwell's equations in an inhomogeneous anisotropic medium. Then we apply the obtained results to build an ‘extented far field’ operator and give a characterization of the medium from the knowledge of the range of this operator. We then show how the linear sampling method can be viewed as an approximation of this characterization. Copyright © 2004 John Wiley & Sons, Ltd.     

Mathematical simulation of acoustic wave scattering in fractured media

The simulation of acoustic waves in fractured media is considered. A self-consistent field model is proposed that describes the formation of a scattered field and the attenuation of the incident field. For the total field, a wave equation with a complex velocity is derived and the corresponding dispersion equation is studied. A frequency-dependent field damping law and an energy variation law are established. An initial and a boundary value problem for waves in a fractured medium is addressed. A finite-difference scheme for the initial value problem is constructed, and a condition for its stability is established. Numerical results are presented.     

Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern

The problem considered is that of determining the shape of aplane acoustically sound-soft obstacle from the knowledge ofthe far-field pattern for one time-harmonic incident field.An iterative procedure is proposed based on two boundary integralsrepresenting the incident field and the far-field pattern, respectively.Numerical examples are included which show that the proceduregives accurate numerical approximations in relatively few iterations.     

On the Two-dimensional Inverse Scattering Problems in Electromagnetics

In recent years, inverse scattering problems have received much attention because of their important applications. Given the incident and scattered waves, an inverse scattering problem in general is to determine the properties of the scatterer. In radar or sonar a known incident wave and observed scattered wave are used to detect the properties and the presence of aircraft or submarine objects; in MRI scanning, tomography X-rays and ultrasound, scattered waves are used to determine the presence or properties of tumors by detecting density variations, to name a few. In this article, we are concerned with the two-dimensional electromagnetic inverse scattering problem. An iterative algorithm for the transversal electric waves will be given based on a singular domain integral equation formulation. Basic features of a scattering object such as shape, location and index of refraction will be recovered from measurements of the field scattered by the object (when illuminated by electromagnetic waves with the magnetic vector polarized along the cylinder axis). Some numerical experiments are included to illustrate the efficiency of the algorithm.     

Recovery of interior eigenvalues from reduced near field data

We consider inverse obstacle and transmission scattering problems where the source of the incident waves is located on a smooth closed surface that is a boundary of a domain located outside of the obstacle/inhomogeneity of the media. The domain can be arbitrarily small but fixed.The scattered waves are measured on the same surface. An effective procedure is suggested for recovery of interior eigenvalues by these data.     

海洋波导中三维可穿透目标的快速声学成像

研究了用声传播远场分布信息来成像海洋波导环境中三维可穿透目标的反问题.建立了求解这类反问题的远场方程,基于内透射边界值问题的分析,讨论了远场方程解的唯一性和可解性,证明了总能找到远场方程的一个在最小平方意义下的近似解,其模在可穿透目标内部的取值是小的,而在外部的取值是大的,进而发展了一种快速成像可穿透目标的一种指示器样本方法.数值试验表明了这种方法是有效的,即使在有限孔径测量方式的情况,也能够得到未知目标的一个理想成像,而且不需要先验知道可穿透目标的任何几何与物理信息.