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This paper is concerned with the scattering problem of time-harmonic acoustic plane waves by an impenetrable obstacle buried in a piecewise homogeneous medium. The so-called generalized impedance boundary condition is imposed on the boundary of the obstacle. Firstly, the well posedness of the solution to the direct scattering problem is established by using the boundary integral method. Then a uniqueness result for the inverse scattering problem is proved; that is, both of the obstacle’s shape and the impedances (

This work is concerned with the scattering problem of time-harmonic acoustic plane waves by an impenetrable obstacle buried in a piecewise homogeneous medium. We set the generalized impedance boundary condition (GIBC) on the boundary of the obstacle and the transmission boundary conditions on the surface of the layered medium. The GIBC is commonly used to model thin coatings or gratings as well as more accurate models for imperfectly conducting obstacles. Addressing this problem is motivated by applications in nondestructive testing, medical imaging, remote sensing or radar, and so on; at the same time the background may be modeled as a layered medium. For simplicity, we just consider that the unknown obstacle is embedded in a two-layered medium, and the space is

To be precise, let

Multilayered scattering problem.

The scattering of time-harmonic acoustic plane waves by an obstacle with GIBC in a piecewise homogeneous medium in

In the following discussion, we use “

The constant surface impedance on

The total field

The

The

As usual in most of the inverse problems, the first issue is the uniqueness, that is, in what conditions, the shape of the obstacle

We solve the above-mentioned inverse problem by using the linear sampling method which was discussed early in 1996 by Colton and Kirsch [

The remaining part of the paper is organized as follows. In the next section, we will use integral equation method to solve direct scattering problem (

In this section, we will establish the well posedness of the direct scattering problem by employing the integral equation method. Let us consider a more general direct scattering problem: Given the transmission boundary conditions

Direct scattering problem (

Suppose that

Clearly, it is sufficient to show that

In order to establish the existence of the solution to problem (

Based on the method proposed in [

For further consideration, we define the single- and double-layer operators

Now we try to establish an integral system by employing the boundary integral equation approach. According to the presentation of the solution in the form of (

Define bounded linear operators

Based on the following two lemmas, we show the solvability of (

The operator

From [

Let

Now, we decompose

The entries

The operator

Let

Using the same

At this time, the potential given by (

By Fredholm theory, the above two lemmas show that the matrix operator

Under

As usual in most of the inverse problems, the first question to ask is the identifiability, that is, whether the scatterer

Let us go back to scattering problem (

To the incident plane wave

To the incident point source

The uniqueness result is based on the following mixed reciprocity relation.

For the scattering of plane waves

The mixed reciprocity relation has been established in the case of obstacle scattering problem [

We consider the case

Next, we consider the case

From (

On the other hand, by Green’s representation formula and Green’s second theorem, we have that

For the transmitted wave

Note that for the incident plane wave

Assume that

We are now in the position to present the uniqueness result based on the idea in [

Given the interface

If the obstacles are not the same, that is,

Consider the solution

Considering

Next, we show that

For this purpose, let

In this part, we give a mathematical basis to reconstruct the shape of the obstacle

We do some preparation firstly. Consider the total wave

We define four operators in the following.

The data-to-pattern operator

The auxiliary operator

The far field operator

The far field operator

Note that

From the boundary conditions on

Let

To prove the existence of an approximate solution of (

The data-to-pattern operator

First, injectivity is a direct consequence of Rellich’s lemma and analytic continuation of the solution to (

To prove compactness, using Green’s representation formula for

To show denseness of the range of

For any

Letting

Let

Firstly, let

Now let

If

Now, we turn our attention to the operator

For

For all

If

The adjoint operator of

We just need to show that the operator

Due to the fact that

In fact, if

Finally, we give the main result in this paper, that is, recovering the obstacles

Under

If

If

If

Next, we assume that

(1) From Theorem

(2) In this paper, we just consider the case of two-layered background medium; in fact, our result can be extended to the case of multilayered piecewise homogeneous medium.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by NSFC Grant no. 11171127 and NSFC Grant no. 11571132. This research is also supported by the Fundamental Research Funds for the Central Universities, nos. CZQ15020 and CZQ12014.