The Mathematical Basis of the Inverse Scattering Problem for Cracks from Near-Field Data

We consider the acoustic scattering problem from a crack which has Dirichlet boundary condition on one side and impedance boundary condition on the other side.The inverse scattering problem in this paper tries to determine the shape of the crack and the surface impedance coefficient from the near-field measurements of the scattered waves, while the source point is placed on a closed curve.We firstly establish a near-field operator and focus on the operator’s mathematical analysis. Secondly, we obtain a uniqueness theorem for the shape and surface impedance. Finally, by using the operator’s properties and modified linear sampling method, we reconstruct the shape and surface impedance.


Introduction
In this paper, we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open crack as cross section in  2 .We assume that the cylinder is coated on one side by a material with surface impedance .This corresponds to the situation when the boundary or more generally a portion of the boundary is coated with an unknown material in order to avoid detection.Assuming that the electric field is polarized in the TM mode (see [1][2][3]), this leads to a mixed boundary value problem for the Helmholtz equation defined in the exterior of an open arc in  2 .
Briefly speaking, let Γ ⊂  2 be an oriented piecewise smooth nonintersecting crack without cups; that is, Γ = () :  ∈ [ 0 ,  1 ], where  : [ 0 ,  1 ] →  2 is an injective piecewise  1 function and the crack Γ is contained in a closed curve Λ.Then the mixed boundary value problem for the Helmholtz equation in  2 can be formulated as follows: Δ +  2  = 0, in  2 \ Γ,  + = 0, on Γ,  ∈ Λ, where  > 0 is the wave number and  > 0 is the surface impedance. is the total wave of the scattered wave   and the incident wave   = Φ(, ); that is,  =   +   , and Φ(, ) is the fundamental solution to the Helmholtz equation defined by with  (1)   0 being a Hankel function of the first kind of order zero.The scattered field   is required to satisfy the Sommerfeld radiation condition lim uniformly in x = /|| with  = ||.
In the following discussion, (⋅) ± means the limit approaching the boundary from outside and inside the domain.
The inverse scattering problem in this paper is trying to determine the shape of the arc (or crack) and the surface 2 Mathematical Problems in Engineering impedance coefficient from the near-field measurements of the scattered waves, while the source point is placed on a closed curve, the results are as follows.
Inverse Problem (Ip).In this paper, the inverse problem we are concerned about is to determine the crack Γ and the surface impedance  from these measurements   (, ) for ,  ∈ Λ.
In 1995, Kress considered the inverse scattering problem for cracks with sound-soft boundary condition in [4].The case of a sound-hard crack was considered by Monch in 1997 in [5].Both of the authors used Newton's method to reconstruct the shape of the crack from a knowledge of the far-field pattern.In 2003, Cakoni and Colton considered an inverse scattering problem by cracks, and they reconstructed the cracks by using the linear sampling method in [1].In 2005, Colton and Haddar discussed similar inverse scattering problem by cracks, and they recovered the cracks by using the reciprocity gap functional method in [6].Zeev and Cakoni considered the inverse scattering problem for a crack embedded in a known inhomogeneous background and recovered the crack (with a point source as incident field) in 2009 in [3].More related research works can be found in [2,3,7] and the references therein.
This paper is arranged as follows.In the next section, we formulate the scattering problem mathematically and prove that the associated near-field operator is injective with dense range under appropriate assumptions.In Section 3, we show that the crack Γ and the surface impedance  are uniquely determined from the near-field measurements of the scattered waves, while the source point is placed on a closed curve.We modify the linear sampling method and reconstruct the shape of the crack (or the surface impedance coefficient) in Section 4.

The Formulation of the Problem
We suppose that the crack Γ can be extended to an arbitrary piecewise smooth, simply connected, and closed curve Ω enclosing a bounded domain Ω such that the normal vector ] on Γ coincides with outward normal vector on Ω which we again denote by ].Λ is a closed curve; we denote by  the domain surrounded by Λ.We suppose that Ω is completely contained in , and we assume the normal vector ] on ] and  is mapped to the exteriors of the domain Ω and the domain , respectively.
In order to formulate our scattering problem more precisely, we need to properly define the trace space on Ω and .Let  be a bounded domain and let Σ be an open subset of the boundary .If  2 (),  1/2 (), and  −1/2 () denote the usual Sobolev spaces, we define the following spaces [8]: the dual space of H1/2 (Σ) , and we have the chain Then problem (1) can be rewritten as and   is required to satisfy Sommerfeld radiation condition (3).
Remark 2. By using similar method in [1], we can obtain the existence and uniqueness of solution to the direct problem (6).Here we use the point source as incident wave, while the plane wave was used as the incident in [1].
According to Green's representation formula, we have On the boundary Ω \ Γ, we have Then, by substituting ( 9) into (8), we have By changing the order of  and , we have Then we have the following result.
Theorem 4. The near-field operator  defined by ( 7) is injective and has dense range.
Proof.From   (, ) =   (, ), the  2 adjoint of  is given by Then we have ( * ℎ)() = ()(), where () = ℎ().Thus, operator  is injective if and only if  * is injective.Since ( * ) ⊥ = ( 2 (Λ)) in a Hilbert space, our proof will be finished by only showing that operator  is injective.Let  = 0; we need to show that  = 0. Define It is easy to verify that V satisfies the exterior problem This exterior Dirichlet problem has only zero solution (see [9]).Then the unique continuation principle now yields V = 0 in  2 \ Γ.Therefore, Mathematical Problems in Engineering Now define By the boundary conditions ( 6) and (20) together with the jump relationship of the single-layer potential on the boundary Λ, we conclude that  satisfies the following problem: The problem has only zero solution which implies that Hence,  is injective.

Uniqueness for the Inverse Problem
Based on the idea of [10,11], we firstly conclude that Γ is uniquely determined from   | Λ without knowing  a priori.Secondly, we show that the surface impedance  can be uniquely determined by   | Λ (see [12]).
In the domain Then   satisfies problem (19) replacing the corresponding domain  with ; that is, where we used the condition   1 (, ) =   2 (, ) on Λ for all point sources  ∈ Λ.

The Linear Sampling Method
The inverse scattering problem in this paper is trying to determine the shape of the crack and the surface impedance coefficient from the near-field measurements of the scattered waves, while the source point is placed on a closed curve.In this part, we provide the mathematical basis to reconstruct the crack Γ from the knowledge of   (, ) for ,  ∈ Λ by using the linear sampling method; that is, we want to determine Γ from a knowledge of   (, ) for ,  ∈ Λ, where Λ is a circle centered at the origin; that is, Λ = { ∈  2 , || =  Λ > 0}.Based on the ideas of [2,3], we introduce the nearfield equation where and  is smooth nonintersecting arc and   () ∈ H−1/2 () and   () ∈ H1/2 ().
We want to characterize the crack Γ by using the behavior of an approximate solution  of the near-field equation (30).Now consider the following problem: for  ∈  1/2 (Γ) and  ∈  −1/2 (Γ).From [1], we know that this problem has a unique solution , where  > 0 is a constant and does not depend on  and .
To understand the near-field equation better, we define an operator  :  1/2 (Γ) ×  −1/2 (Γ) →  2 (Λ) which maps the boundary data (, ) to the solution  on Λ.We have the following conclusions about this operator .Theorem 6. Operator  is injective and compact and has dense range in  2 (Λ).
We now show that operator  is compact.Choose a disk  = { ∈  2 , || ≤  <  Λ } such that Ω ⊂  ⊂ .Using Green's representation formula for , we can decompose operator  as  =  1  2 , where The regularity of the solution to problem (33) implies that operator  2 is bounded.So, operator  is compact since operator  1 is compact.
operator corresponding to the scattering problem (6), then one has the following results: (1) If  ⊂ Γ, then for every  > 0 there exists a solution    ∈ where where  =  1  0 .Next, we assume that  ̸ ⊂ Γ.In this case, by Theorem 7, Φ  () for  ∈ Λ is not in the range of .But from Theorem 6 we know that operator  has dense range in  2 (Λ).Hence, for every  > 0, we can construct a unique Tikhonov regularized solution  , ∈ where  is the regularization parameter (chosen by a regular regularization strategy, e.g., the Morozov discrepancy principle).Then we have ‖ Since lim  → 0 () = 0, we have that lim  → 0 ‖ , ‖  1/2 (Γ)× −1/2 (Γ) → ∞.From (52), we have that lim  → 0 ‖  , ‖  1/2 (Γ)× −1/2 (Γ) → ∞.By the definition of operator  given by (39), we have that lim  → 0 ‖  , ‖  2 (Λ) → ∞.Then we complete the proof of this theorem.Remark 10.In numerical analysis, we can choose some suitable smooth arcs as a set such as L and then consider near-field equation  () = Φ  () ,  ∈ L. (54) If  ⊂ Γ, we can find a bounded solution to the near-field equation (30) with discrepancy , whereas if  ̸ ⊂ Γ, then there exists solution of the near-field equation (with discrepancy  + ) with arbitrary large norm in the limit as  → 0. Then the arc can be characterized by the behavior of this solution.But how to determine the surface impedance is a problem we need to study further.
Remark 11.Applying reciprocity gap functional method to reconstruct a crack, we need to know the near-field Cauchy data  and /] of the total field (see [6]).Qin and Colton used a method that may be called a modified linear sampling method to recover a cavity by using the near-field data (see [2,7]).We combine these two methods to recover a crack (which has empty inner product) by using the measurement of near-field data .In the process of recovering the crack, the near-field equation that we introduced is different.