Direct sampling methods for inverse elastic scattering problems
Abstract
We consider the inverse elastic scattering of incident plane compressional and shear waves from the knowledge of the far field patterns. Specifically, three direct sampling methods for location and shape reconstruction are proposed using the different component of the far field patterns. Only inner products are involved in the computation, thus the novel sampling methods are very simple and fast to be implemented. With the help of the factorization of the far field operator, we give a lower bound of the proposed indicator functionals for sampling points inside the scatterers. While for the sampling points outside the scatterers, we show that the indicator functionals decay like the Bessel functions as the sampling point goes away from the boundary of the scatterers. We also show that the proposed indicator functionals continuously dependent on the far field patterns, which further implies that the novel sampling methods are extremely stable with respect to data error. For the case when the observation directions are restricted into the limited aperture, we firstly introduce some data retrieval techniques to obtain those data that can not be measured directly and then use the proposed direct sampling methods for location and shape reconstructions. Finally, some numerical simulations in two dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed sampling methods, even for multiple multiscale cases and limitedaperture problems.
Key words. Elastic scattering; sampling method; far field pattern; limitedaperture problem
AMS subject classifications. 35P25, 45Q05, 78A46, 74B05
1 Introduction
Scattering of elastic waves plays an important role in such different areas as seismic imaging, nondestructive testing, geophysical exploration, and medical diagnosis. In the last twenty years, noniterative sampling methods for location and shape reconstruction in inverse elastic scattering problems have attracted a lot of interest. Typical examples include the Linear Sampling Method by Arens [2], the Factorization Method by Alves and Kress [1], Arens [2], Charalambopoulos & Kirsch et al. [6], and recently by Hu, Kirsch & Sini [13], and the method based on topological derivative by Guizina & Chikivhev [11]. The basic idea is to design an indicator which is big inside the underlying scatterer and relatively small outside. Recently, in [7], Chen & Huang introduced the Reverse Time Migration where the indicator functional is defined as the imaginary part of the crosscorrelation of the weighted elastic Green function and the weighted backpropagated elastic wave field. Different to the previous sampling methods, the indicator functional decays as the sampling points go away from the boundary. We also refer to Li, Wang et al. [18] and Bao & Hu et al. [3] for iterative methods by using multiple frequency data. For the readers interested in a more comprehensive treatment of the inverse elastic scattering problems, we suggest consulting Bonnet and Constantinescu [5] on this subject.
The purpose of this paper is to introduce some novel indicator functionals for inverse elastic scattering problems with the far field patterns. These indicator functionals have a lower bound for sampling points inside the scatterers and decay like the Bessel functions as the sampling point goes away from the boundary of the scatterers. Besides, some data retrieval techniques will be introduced for limited aperture problems. We will focus on the case of twodimensional scattering and present both the theoretical and numerical results for this case, illustrating the fast, effective and robust reconstructions, even for the limited aperture problems.
We begin with the notations used throughout this paper. All vectors will be denoted in bold script. For a vector , we introduce the two unit vectors and obtained by rotating anticlockwise by . For simplicity, we write for the usual partial derivative . Then, in addition to the usual differential operators and , we define two auxiliary differential operators and by and , respectively. It is easy to deduce the differential indentities .
Let be an open and bounded Lipschitz domain such that the exterior of is connected. Here and throughout the paper we denote the closure of the set . A confusion with the complex conjugate of is not expected. The propagation of timeharmonic elastic wave equation in an isotropic homogeneous media outside is governed by the reduced Navier equation
\hb@xt@.01(1.1) 
where denotes the total displacement field and is the circular frequency, and are Lam constants satisfying . It is well known that any solution of the Navier equation (LABEL:elastic) has a decomposition in the form
where
are known as the compressional (longitudinal) and shear (transversal) parts of , respectively. Here, and denote the compressional wave number and the shear wave number, respectively. It is clear that
For a rigid body , the total displacement field satisfies the first (Dirichlet) boundary condition
\hb@xt@.01(1.2) 
For a cavity, we impose the second (Neumann) boundary condition
\hb@xt@.01(1.3) 
where denotes the surface traction operator defined by
in terms of the exterior unit normal vector on .
Let denote the unit circle in . In elastic scattering, an important case is the scattering of a plane wave with incident direction which takes the form
where is a plane compressional wave and is a plane shear wave, respectively. The scatterer gives rise to a scattered field . The scattered field is a solution to (LABEL:elastic) and has a decomposition with and . The scattered field satisfies the Kupradze’s radiation conditions
\hb@xt@.01(1.4) 
uniformly in all directions as . For the unique solvability of the scattering problems (LABEL:elastic)(LABEL:KupradzeRC) in the space we refer to Kupradze [17] and Li, Wang et al. [18].
It is well known that every radiating solution to the Navier equation has an asymptotic behaviour of the form
\hb@xt@.01(1.6)  
uniformly in all direction . The functions and are known as the compressional and shear far field pattern of , and are analytic functions on . We want to remark here that, to simplify the subsequent representations, the coefficients in (LABEL:usasymptotic) are slightly different to those given in [2, 21]. We also have the asymptotic behaviour [2]
\hb@xt@.01(1.8)  
where denotes the surface traction operator on a circle centered on the origin of radius .
Throughout this paper, we will denote the pair of far field patterns
of the corresponding scattered field by , indicating the dependence on the observation direction , the incident direction and the pair of coefficients . We want to remark that the dependence on the coefficients is linear. For convenience, we use the following notations
Let be a subset of with nonempty interior. Then, we are interested in the following three inverse elastic scattering problems.

IPFF: Determine the location and shape of the scatterer from the knowledge of the full far field pattern for all observation directions , all incident directions and for and .

IPPP: Determine the location and shape of the scatterer from the knowledge of the compressional far field pattern for all observation directions , and all incident directions .

IPSS: Determine the location and shape of the scatterer from the knowledge of the shear far field pattern for all observation directions , and all incident directions .
Clearly, the measurement data in IPPP and IPSS is much less than those in IPFF. For uniqueness of IPFF, we refer to Hähner and Hsiao [12], while the corresponding result of IPPP and IPSS can be found in the recent works by Gintides & Sini [10] and Hu, Kirsch & Sini [13]. Due to analyticity, for uniqueness it suffices to know the far field pattern on the subset . However, as one would expect, the quality of the reconstructions decreases drastically for this so called limitedaperture problem. In particular, the traditional sampling type methods studied in Alves and Kress [1], Arens [2], Charalambopoulos & Kirsch et al. [6], Hu, Kirsch & Sini [13], and Sevroglou [21] fail to work. In this paper, we will introduce some data retrieval techniques to compute those data that can not be measured directly. Combining this and using the proposed sampling methods, the limitedaperture problems are desired to be solved partially.
This paper is organized as follows. In the next section, the theoretical analysis of the proposed reconstruction scheme will be established. A lower bound of the indicators for the sampling points inside the scatterers is obtained with the help of an infcriterion characterization. The decay behavior of the indicators will then be studied for sampling points away from the boundary of the scatterers. A stability statement will also be established to reflect the important feature of the reconstruction scheme. For the limitedaperture problems, a data retrieval scheme will be introduced to numerically obtain the far field patterns that can not be measured directly. One then may combine the data retrieval technique and the previous sampling methods for inverse elastic scattering problems. Some numerical simulations in two dimensions will be presented in the last section to indicate the efficiency and robustness of the proposed methods.
2 Novel direct sampling methods and their mathematical basis
For any and any polarization define the two functions and by
\hb@xt@.01(2.1) 
respectively. Then for IPFF, IPPP and IPSS we introduce the indicator functionals
\hb@xt@.01(2.2)  
\hb@xt@.01(2.3)  
\hb@xt@.01(2.4) 
respectively.
2.1 Lower bound estimate of , and for
We will use the shorthand . For any , we adopt the notation with
The Hilbert space will be equipped with the inner product
\hb@xt@.01(2.5) 
It is clear that there holds the decomposition
where and . For later use, we introduce the orthogonal projection operators and , i.e., for all , and . Their adjoint operators and are just the inclusions from and , respectively, to .
Consider the elastic Herglotz wave function
\hb@xt@.01(2.6) 
with a vector Herglotz kernel . We now introduce the far field operator defined by
\hb@xt@.01(2.7)  
\hb@xt@.01(2.8) 
i.e., is the far field pattern for the scattering of elastic Herglotz wave function with kernel . The far field operator plays an essential role in the investigations of the sampling type methods for inverse problems, we refer to [2] for a survey on the state of the art of its properties and applications.
For any , recall the two test functions and , we define by
\hb@xt@.01(2.9) 
By interchanging orders of integration, we may rewrite our indicator given by (LABEL:Indicatorff) in a very simple form
\hb@xt@.01(2.10) 
where is the inner product of the space given in (LABEL:innerproduct). Similarly, by defining and , we found that the indicators and given by (LABEL:Indicatorpp) and (LABEL:Indicatorss) can be written as a very simple form,
\hb@xt@.01(2.11) 
with and , respectively.
Recall the Green’s tensor
of the Navier equation in in terms of the identity matrix and the Hankel function of the first kind of order zero. For any and any polarization , an elastic point source with source point and polarization is given by . From the asymptotics for the Hankel function it follows that the far field pattern of the point source is given by
\hb@xt@.01(2.12)  
\hb@xt@.01(2.13) 
Now we introduce the elastic singlelayer operator , given by
\hb@xt@.01(2.14) 
Then the following property of the singlelayer operator is important for our subsequent analysis.
Lemma 2.1
Assume that is not a Dirichlet eigenvalue of in . Then there exist with
where denotes the duality pairing in .
Proof. This property follows immediately from Lemma 1.17 of [15] in combination with Lemma 4.2 of [2].
Define the datatopattern operator by
\hb@xt@.01(2.15) 
where is the far field pattern of the solution to the Dirichlet boundary value problem with boundary data . By a standard argument, we have that the datatopattern operator is compact, injective with dense range in . To characterize the scatterer by the corresponding datetopattern operator, we have the following lemma.
Lemma 2.2
For any and , define the function by (LABEL:phiz). Then the following holds.

if and only if .

if and only if .

if and only if .
Proof. The first result has been proved by Arens in Theorem 4.7 of [2]. The other two can be done analogously in combination with the definitions of the projection operators.
We introduce the Hergoltz wave operator by setting
is the trace on of the elastic Herglotz wave function (LABEL:eHerglotz) with vector Herglotz kernel . Its adjoint is given by
We note that is exactly the far field pattern of the elastic singlelayer potential , hence
where and denote the adjoints of and , respectively. On the other hand, we observe that is the far field pattern of the solution of the exterior Dirichlet problem with boundary data , which implies . Combining the previous operator equality, we deduce the factorization of the far field operator
\hb@xt@.01(2.16) 
Consequently, we have the factorizations of the operators and ,
\hb@xt@.01(2.17) 
For all and polarization , define a subspace by
where is the test function given in (LABEL:phiz) and is the inner product of defined by (LABEL:innerproduct). Now we are in a position to state the main result of this subsection.
Theorem 2.3
Consider the inverse elastic scattering by a rigid body . Assume that is not a Dirichlet eigenvalue of in . Let be the polarization. Then , if and only if,
\hb@xt@.01(2.18) 
Furthermore, we have the lower bound estimate
\hb@xt@.01(2.19) 
for some constant which is independent of . Similar, we have
\hb@xt@.01(2.20) 
for some constant which is independent of .
Proof. First, (LABEL:infcreterior) follows directly by applying Theorem 1.16 in [15] to the factorization (LABEL:ffactorization), Lemma LABEL:Gproperty and noting the fact that the operator is coercive by Lemma LABEL:Scoercive. Furthermore, note that , using Theorem 1.16 of [15] again we deduce that
for some constant which is independent of . Second, we observe that
\hb@xt@.01(2.21) 
which implies . From this we derive that
for some constant which is independent of . Thus, the lower bound estimate (LABEL:estimateff) for the indicator in has been proved. The other two lower bound estimates in (LABEL:estimateppss) can be shown analogously using the factorizations (LABEL:ppssfactorization).
We conclude this subsection by a remark that the analogous result of Theorem LABEL:theorem1 holds for the Neumann boundary condition. Our analysis rely on the factorization of the far field operator. The datatopattern operator is now defined to map into the far field pattern of the exterior Neumann boundary value problem with boundary data . We introduce the hypersingular integral operator by
for . Then one can derive the factorization of the far field operator for Neumann problem in the form
Based on this, the analogous results of Lemmas LABEL:Scoercive, LABEL:Gproperty and Theorem LABEL:theorem1 can be derived.
Finally, we want to remark that the assumption on interior eigenvalues in Lemma LABEL:Scoercive and Theorem LABEL:theorem1 has only its theory interest. It is wellknown that the classical sampling type methods for solving inverse scattering problems fail if the wave number is an eigenvalue of a corresponding interior eigenvalue problem. We refer to a modification proposed by Kirsch & Liu[16] to avoid the eigenvalues. However, our sampling methods are independent of the interior eigenvalues from the numerical point of view.
2.2 Indicator behavior for the sampling points away from the boundary
We have known from the previous subsection that the values of the indicator functionals can not be small arbitrarily for sampling points inside. In this subsection, we study the behavior of the indicator functionals for sampling points away from the boundary.
To simplify the subsequent representations, we introduce
Straightforward calculations show that . For the scattering problem (LABEL:elastic)(LABEL:KupradzeRC), the farfield pattern has the following representation [2]
\hb@xt@.01(2.23)  
\hb@xt@.01(2.25)  
By straightforward calculations, it can be seen that, for all ,
and
Inserting this into (LABEL:upinfty)(LABEL:usinfty), we find that
\hb@xt@.01(2.27)  
\hb@xt@.01(2.29)  
Now we introduce the following auxiliary functions
Then the indicators , and involve the terms
\hb@xt@.01(2.30)  
\hb@xt@.01(2.31) 
Noting that and , by the well known RiemannLebesgue Lemma, we obtain that all the expressions in (LABEL:G1)(LABEL:G2) go to as , so we have