Near-field imaging of an unbounded elastic rough surface with a direct imaging method
This paper is concerned with the inverse scattering problem of time-harmonic elastic waves by an unbounded rigid rough surface. A direct imaging method is developed to reconstruct the unbounded rough surface from the elastic scattered near-field Cauchy data generated by point sources. A Helmholtz-Kirchhoff-type identity is derived and then used to provide a theoretical analysis of the direct imaging algorithm. Numerical experiments are presented to show that the direct imaging algorithm is fast, accurate and robust with respect to noise in the data.
Key words. inverse elastic scattering, unbounded rough surface, Dirichlet boundary condition, direct imaging method, near-field Cauchy data
AMS subject classifications. 78A46, 35P25
Elastic scattering problems have received much attention from both the engineering and mathematical communities due to their significant applications in diverse scientific areas such as seismology, remote sensing, geophysics and nondestructive testing.
This paper focuses on the inverse problem of scattering of time-harmonic elastic waves by an unbounded rough surface. The domain above the rough surface is filled with a homogeneous and isotropic elastic medium, and the medium below the surface is assumed to be elastically rigid. Our purpose is to image the unbounded rough surface from the scattered elastic near-field Cauchy data measured on a horizontal straight line segment above the rough surface and generated by point sources. For simplicity, in this paper, we are restricted to the two-dimensional case. See Fig. LABEL:fig0 for the problem geometry. However, our imaging algorithm can be generalized to the three-dimensional case with appropriate modifications.
For inverse acoustic scattering problems by unbounded rough surfaces, many numerical reconstruction methods have been developed to recover the unbounded surfaces, such as the sampling method in [7, 37, 38, 39] for reconstructing periodic structures, the time-domain probe method in , the time-domain point source method in  and the nonlinear integral equation method in  for recovering sound-soft rough surfaces, the transformed field expansion method in [9, 10] for reconstructing a small and smooth deformation of a flat surface with Dirichlet, impedance or transmission conditions, the Kirsch-Kress method in  for a penetrable locally rough surface, the Newton iterative method in  for a sound-soft locally rough surface, and the direct imaging method in  for imaging impenetrable or penetrable unbounded rough surfaces from the scattered near-field Cauchy data.
Compared with the acoustic case, elastic scattering problems are more complicated due to the coexistence of the compressional and shear waves that propagate at different speeds. This makes the elastic scattering problems more difficult to deal with theoretically and numerically. For inverse scattering by bounded elastic bodies, several numerical inversion algorithms have been proposed, such as the sampling method in [6, 8, 16, 35, 40] and the transformed field expansion method in [28, 29, 30]. See also the monograph  for a good survey. However, as far as we know, not many results are available for inverse elastic scattering by unbounded rough surfaces. An optimization method was proposed in  to reconstruct an elastically rigid periodic surface, a factorization method was developed in  for imaging an elastically rigid periodic surface and in  for reconstructing bi-periodic interfaces between acoustic and elastic media, and the transformed field expansion method was given in  for imaging an elastically rigid bi-periodic surface.
The purpose of this paper is to develop a direct imaging method for inverse elastic scattering problems by an unbounded rigid rough surface (on which the elastic displacement vanishes). The main idea of a direct imaging method is to present a certain imaging function to determine whether or not a sampling point in the sampling area is on the obstacle. Direct imaging methods usually require very little priori information of the obstacle, and the imaging function usually involves only the measurement data but not the solution of the scattering problem and thus can be computed easily and fast. This is especially important for the elastic case since the computation of the elastic scattering solution is very time-consuming. Due to the above features, direct imaging methods have recently been studied by many authors for inverse scattering by bounded obstacles or inhomogeneity, such as the orthogonality sampling method , the direct sampling method [23, 24, 25, 34], the reverse time migration method [13, 14] and the references quoted there. Recently, a direct imaging method was proposed in  to reconstruct unbounded rough surfaces from the acoustic scattered near-field Cauchy data generated by acoustic point sources. This paper is a nontrivial extension of our recent work in  from the acoustic case to the elastic case since the elastic case is much more complicated than the acoustic case due to the coexistence of the compressional and shear waves that propagate at different speeds.
To understand why our direct imaging method works, a theoretical analysis is presented, based on an integral identity concerning the fundamental solution of the Navier equation (see Lemma LABEL:HK) which is established in this paper. This integral identity is similar to the Helmholtz-Kirchhoff identity for bounded obstacles in . In addition, a reciprocity relation (see Lemma LABEL:RR) is proved for the elastic scattered field corresponding to the unbounded rough surface. Based on these results, the required imaging function is then proposed, which, at each sampling point, involves only the inner products of the measured data and the fundamental solution of the Navier equation in a homogeneous background medium. Thus, the computation cost of the imaging function is very cheap. Numerical results are consistent with the theoretical analysis and show that our imaging method can provide an accurate and reliable reconstruction of the unbounded rough surface even in the presence of a fairly large amount of noise in the measured data.
The remaining part of the paper is organized as follows. In Section LABEL:sec2, we formulate the elastic scattering problem and present some inequalities that will be used in this paper. Moreover, the well-posedness of the forward scattering problem is presented, based on the integral equation method (see [3, 4, 5]). Section LABEL:sec3 provides a theoretical analysis of the continuous imaging function. Based on this, a novel direct imaging method is proposed for the inverse elastic scattering problem. Numerical examples are carried out in Section LABEL:sec4 to illustrate the effectiveness of the imaging method. In the appendix, we present two lemmas which are needed for the proof of the reciprocity relation.
We conclude this section by introducing some notations used throughout the paper. For any open set , denote by the set of bounded and continuous, complex-valued functions in , a Banach space under the norm . Given a function , denote by the (distributional) derivative , . Define , equipped with the norm . The space of Hölder continuous functions is denoted by with the norm . Denote by the space of uniformly Hölder continuously differentiable functions or the space of differentiable functions for which belongs to with the norm . We will also make use of the standard Sobolev space for any open set and provided the boundary of is smooth enough. The notations and denote the functions which are the elements of and for any , respectively.
2 Problem formulation
In this section, we formulate the elastic scattering problem considered in this paper and present its well-posedness results. Useful notations and inequalities used in the paper will also be given.
2.1 The forward elastic scattering problem
The propagation of time-harmonic waves with circular frequency in an elastic solid with Lamé constants is governed by the Navier equation
Here, denotes the displacement field and
We consider a two-dimensional unbounded rough surface , where is the surface profile function satisfying that . Denote by the region above . Throughout the paper, the letters and will be frequently used to denote certain real numbers satisfying that and . For any , introduce the sets
Let be the free-space Green’s tensor for the two-dimensional Navier equation:
with , where the scalar function is the fundamental solution to the two-dimensional Helmholtz equation given by
and the compressional wave number and the shear wave number are given by
Motivated by the form of the corresponding Green’s function for acoustic wave propagation, we define the Green’s tensor for the first boundary value problem of elasticity in the half-space by
Here, and the matrix function is analytic. For more properties of we refer to [3, Chapter 2.4].
Given a curve with the unit normal vector , the generalised stress vector on is defined by
Here, are real numbers satisfying and .
If we apply the generalised stress operator to and , we obtain the matrix functions and :
Given the incident wave , the scattered field is required to fulfill the upwards propagating radiation condition (UPRC) (see ):
with some .
We reformulate the elastic scattering problem by the unbounded rough surface as the following boundary value problem:
Given a vector field , find a vector field that satisfies
1) the Navier equation (LABEL:naviereq) in ,
2) the Dirichlet boundary condition on ,
3) the vertical growth rate condition: for some ,
4) the UPRC (LABEL:uprc).
In Problem LABEL:pb1, the boundary data is determined by the incident wave , that is, .
2.2 Some useful notations
We give some basic notations and fundamental functions that will be needed in the subsequent discussions.
The fundamental solution defined in (LABEL:Gamma) can be decomposed into the compressional and shear components :
The asymptotic behavior of these functions have been given in [2, Proposition 1.14]:
where , is defined in (LABEL:Pi1) with and for any vector
From [3, (2.14) and (2.29)] there exists a constant such that for
Further, we have the following asymptotic property of the Hankel functions and their derivatives : For fixed ,
By (LABEL:bessel) and (LABEL:bessel-), we can deduce that
for with , where is a constant. Combining this with (LABEL:pu)-(LABEL:Pi2) gives
Further, we also need the following inequalities (see [3, Theorems 2.13 and 2.16 (a)]:
for , where denotes any derivative with respect to of and . For detailed properties of , , and can be found in [3, Section 2.3-2.4].
2.3 Well-posedness of the forward scattering problem
The well-posedness of the elastic scattering problem reformulated as Problem LABEL:pb1 has been studied in [3, 4, 5] by the integral equation method or in [18, 19] by a variational approach. We present the well-posedness results obtained in [3, 4, 5] by the integral equation method, which will be needed in this paper. To this end, we introduce the elastic layer potentials. For a vector-valued density , we define the elastic single-layer potential by
and the elastic double-layer potential by
From the properties of and (see [3, Chapter 2]) it follows that the above integrals exist as improper integrals. Further, it is easy to verify that the potentials and are solutions to the Navier equation in both and . We now seek a solution to Problem LABEL:pb1 in the form of a combined single- and double-layer potential
where and is a complex number with . From [5, pp. 10] we know that is a solution to the boundary value Problem LABEL:pb1 if is a solution to the integral equation
Introduce the three integral operators , and :
where and . From [3, Theorems 3.11 (c) and 3.12 (b)], we know that these three operators are bounded mappings from into . Set . Then the integral equation (LABEL:eq1) is equivalent to
The unique solvability of (LABEL:eq2) was shown in [3, 4, 5] in both the space of bounded and continuous functions and , , yielding existence of solutions to Problem LABEL:pb1. We just present the conclusions that will be used in the next sections; see [3, Chapter 5] or [4, 5] for details.
For the original scattering problem formulated as a boundary value problem, Problem LABEL:pb1, we have the following well-posedness result.
3 The imaging algorithm
For and let be a horizontal line segment above the rough surface . Then our inverse scattering problem is to determine from the scattered near-field Cauchy data corresponding to the elastic incident point sources . Here, .
We will present a novel direct imaging method to recover the unbounded rough surface . To this end, we first establish certain results for the forward scattering problem associated with incident point sources. The imaging function for the inverse problem will then be given at the end of this section. In the following proofs, the constant may be different at different places.
Assume that is the solution to Problem LABEL:pb1 with the boundary data . If with , then for any and .
Proof. From (LABEL:ux), the scattered field can be written in the form
where is the unique solution to the boundary integral equation (LABEL:eq1).
The inequalities (LABEL:asyGammadh) and (LABEL:asyGammadh2) imply that
This, combined with (LABEL:usx) and Young’s inequality, implies that
Further, by (LABEL:usx) we deduce that for
Using this equation and the inequalities (LABEL:asyGammadh1) and(LABEL:asyGammadhx) and arguing similarly as above, we obtain that . The lemma is thus proved.
For let be the scattered field associated with the rough surface and the incident point source located at with polarization Then
for any .
Proof. We only prove the case . The proof of the case is similar. The asymptotic property (LABEL:asyGamma) implies that Thus, by Lemma LABEL:lem2301 we have that From (LABEL:asyGamma) and (LABEL:asyPi), we obtain that
Thus, choosing appropriate and yields
The proof is thus complete.
By (LABEL:phi) and the Funk-Hecke Formula (see ):
Taking the imaginary part of (LABEL:Gamma) and using (LABEL:111) yield
where and is defined as in (LABEL:otimes).
For any we have
for , where is defined in (LABEL:Pi1) with the unit normal on pointing into .
Proof. Let be the upper half-circle above centered at with radius and define . Denote by the bounded region enclosed by and . For any vectors , and satisfy the Navier equation in , so, by the third generalised Betti formula ([3, Lemma 2.4]) we have
Using the asymptotic properties (LABEL:aaaa)-(LABEL:cccc), we obtain that
Then (LABEL:hhhkkk) follows from (LABEL:eq222). The proof is thus complete.
We need the following reciprocity relation result.
(Reciprocity relation) For , let and be the scattered fields in associated with the rough surface and the incident point sources and , respectively. Then
Proof. For define