The Factorization Method for Reconstructing a Penetrable Obstacle with Unknown Buried Objects

Abstract

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous penetrable obstacle with buried objects inside. We prove under certain conditions that the factorization method can be applied to reconstruct the penetrable obstacle from far-field data without knowing the buried objects inside. Numerical examples are also provided illustrating the inversion algorithm. Introduction.In this paper, we study the problem of scattering of timeharmonic acoustic plane waves by an inhomogeneous penetrable obstacle with buried objects inside. This type of problem occurs in various applications such as radar, remote sensing, geophysics, medical imaging, and nondestructive testing. Let D 0 denote an impenetrable obstacle which is embedded in a bounded, penetrable obstacle D with C 2 -boundary ∂D, that is, D 0 ⊂ D. Assume that D 1 := D \ D 0 is filled with an inhomogeneous material characterized by the refractive index n ∈ L ∞ (D 1 ) with Re[n(x)] > 0 and Im[n(x)] ≥ 0 for almost all x ∈ D 1 and D 2 := R 3 \ D is filled with a homogeneous material with the constant refractive index 1. Then the scattering of time-harmonic acoustic waves by D and D 0 can be modeled by the Helmholtz equation with boundary conditions on the interface ∂D and boundary ∂D 0 :

The factorization method for inverse acoustic scattering by a penetrable anisotropic obstacle

“…The research of X. Liu was supported in part by the NNSF of China under grants 11101412 and 11071244 and the Alexander von Humboldt Foundation. The authors would also like to thank Professor Bo Zhang for sending us their preprint that hints us to introduce the operator .…”

“…where u solves the interior Neumann boundary value problems (18)- (19) with boundary data g 2 H 1=2 . @D/ Properties of G are collected in the following lemma.…”

“…Assume that k 2 is neither a Dirchlet eigenvalue of the boundary value problems (16)-(17) nor a Neumann eigenvalue of the boundary value problems (18)- (19). Then, the far field operator F : L 2 .S 2 / !…”

A modification of the factorization method for scatterers with different physical properties

“…For recent works discussing this case, we refer to previous studies. [5][6][7][8][9] In this paper, we study the factorization method for a scatterer consisting of two objects with different physical properties. Especially, we consider the following two cases: One is the case when each object has the different boundary condition, and the other one is when different penetrability.…”

Locating a complex inhomogeneous medium with an approximate factorization method

“…For the case when Re[n(x)] > 1 or Re[n(x)] < 1 in D, based on a Lippmann-Schwinger integral equation method, [13] proved the validity of the factorization method for recovering the inhomogeneous obstacle D. Recently, a factorization method has been developed in [26] in determining a penetrable obstacle D with unknown buried objects inside in the case when the solution is discontinuous across the interface ∂D, that is, u| + = u| − , ∂ ν u| + = λ∂ ν u| − on ∂D for λ = 1. However, the method used in [26] can not be applied to the case when the solution is continuous across the interface ∂D, that is, λ = 1 (see [26,Remark 2.5]). To overcome this difficulty, in [22] an approximate factorization method was proposed to solve the same inverse problem as that in [26] for the case when the solution is continuous across the interface ∂D.…”