An approximate factorization method for inverse medium scattering with unknown buried objects

Abstract

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous medium with different kinds of unknown buried objects inside. By constructing a sequence of operators which are small perturbations of the far-field operator in a suitable way, we prove that each operator in this sequence has a factorization satisfying the Range Identity. We then develop an approximate factorization method for recovering the support of the inhomogeneous medium from the far-field data. Finally, numerical examples are provided to illustrate the practicability of the inversion algorithm.

Inverse Obstacle Scattering in an Unbounded Structure

“…Zhang et al [43] developed a regularized conjugate gradient method with fast multipole acceleration for a fractal rough surface scattering problem. We refer to [7,8,12,13,32,34,40] for some mathematical and numerical studies on related inverse scattering problems.…”

Locating a complex inhomogeneous medium with an approximate factorization method

“…Accordingly, the total field, the scattered field and the far-field pattern are denoted as u(x; d), u s (x; d) and u ∞ ( x; d), respectively. By using a variational approach, it can be easily shown that the problem (1.1)-(1.2) has a unique solution (see, e.g., [7] or [22] for the case when D contains buried objects inside). In the current paper, we are interested in the inverse problem of reconstructing the shape and location of the inhomogeneous medium D from a knowledge of the far-field pattern u ∞ for incident plane waves.…”

“…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. However, the factorization method in [13,22,26] depends closely on the assumption that Re[n(x)] > 1 or Re[n(x)] < 1 in D. Therefore, the techniques developed in [13,22,26] can not be directly extended to deal with the case when D = K j=1 D j with Re[n(x)] > 1 in D l 1 and Re[n(x)] < 1 in D l 2 for some 1 ≤ l 1 = l 2 ≤ K which is the case of the inverse problem under consideration.…”

“…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. However, the factorization method in [13,22,26] depends closely on the assumption that Re[n(x)] > 1 or Re[n(x)] < 1 in D. Therefore, the techniques developed in [13,22,26] can not be directly extended to deal with the case when D = K j=1 D j with Re[n(x)] > 1 in D l 1 and Re[n(x)] < 1 in D l 2 for some 1 ≤ l 1 = l 2 ≤ K which is the case of the inverse problem under consideration. The reader is referred to [2,8,19] for applications of factorization method for the scattering by diffraction gratings, to [18] for the photonics and rough surfaces problems, to [3] and [27] for the cases of the conductive boundary condition and the generalized impedance boundary condition, and to [17,28,29] for the fluid-solid interaction problems.…”

“…In the present paper, we are motivated by [13,22] to solve the inverse problem of locating the inhomogeneous medium by developing an approximate factorization method in the case when the medium is filled with an inhomogeneous material characterizing by the complex refractive index. Due to the close dependence of the classical factorization method on the complex refractive index in D, we attempt to construct a sequence of perturbed operators F m of the far-field operator F in a suitable way such that F m satisfies the Range Identity in [15,Theorem 2.15] for each m ∈ N + .…”