UNIQUENESS IN INVERSE ELASTIC SCATTERING FROM UNBOUNDED RIGID SURFACES OF RECTANGULAR TYPE

Consider the two-dimensional inverse elastic scattering problem of recovering a piecewise linear rigid rough or periodic surface of rectangular type for which the neighboring line segments are always perpendicular. We prove the global uniqueness with at most two incident elastic plane waves by using near-field data. If the Lamé constants satisfy a certain condition, then the data of a single plane wave is sufficient to imply the uniqueness. Our proof is based on a transcendental equation for the Navier equation, which is derived from the expansion of analytic solutions to the Helmholtz equation. The uniqueness results apply also to an inverse scattering problem for non-convex bounded rigid bodies of rectangular type.


1.
Introduction.This paper is concerned with the inverse scattering of timeharmonic elastic waves from rigid unbounded periodic and rough surfaces of rectangular type (see Sections 2.1 and 3 for a precise description), which has a wide field of applications, particularly in geophysics, seismology and nondestructive testing.For instance, identifying fractures in sedimentary rocks has significant impact on the production of underground gas and liquids by employing controlled explosions.The sedimentary rock under consideration can be regarded as a homogeneous transversely isotropic elastic medium with periodic vertical fractures which can be extended to infinity in one of the horizontal directions.Using an elastic plane wave as an incoming source, we thus obtain a two-dimensional inverse problem of recovering a rectangular interface from the knowledge of near-field data measured above the periodic structure (diffraction grating); see [17].The associated direct scattering problem is formulated as a Dirichlet boundary value problem for the time-harmonic Navier equation in the unbounded domain above the surface, which can be considered as a simple model problem in linear elasticity.
We refer to [2] for the first uniqueness result in inverse elastic scattering from rigid periodic surfaces.It was proved that a smooth (C 2 ) surface can be uniquely determined from incident pressure waves for one incident angle and an interval of wave numbers.Furthermore, a finite set of wave numbers is enough if a priori information about the height of the grating curve is known.This extends the periodic version of Schiffer's theorem by Hettlich and Kirsch (see [11]) to the case of inverse elastic diffraction problems.The application of the Kirsch-Kress optimization scheme with one or several incident elastic plane waves can be found in [8], where the reconstruction of rectangular rigid surfaces was also treated.The factorization method established in [13] gives rise to uniqueness results by utilizing only the compressional or shear components of the scattered field corresponding to all quasi-periodic incident plane waves with a common phase-shift.
Other studies on the uniqueness have been carried out within the class of piecewise linear periodic and rough surfaces using a single plane or point source wave.Global uniqueness results for the Helmholtz equation were first shown in [10] within the rectangular periodic structures under the Dirichlet or Neumann condition.Relying on the reflection principles for the Helmholtz, Navier and Maxwell equations, one can find out and classify several extremely rare sets of unidentifiable polygonal or polyhedral periodic structures by one incident plane wave.Thus, the global uniqueness with one incoming wave holds within the piecewise linear periodic structures excluding all unidentifiable sets; see [6,1,7].In particular, sending a single incident point source wave always leads to the uniqueness of the inverse problem within polygonal periodic or rough surfaces; see [12] for the Helmholtz equation.However, such an argument applies so far only to the third or fourth kind boundary value problems of the Navier equation, and it still remains a challenging problem to prove the uniqueness under the more practical Dirichlet or Neumann-type boundary conditions, due to the lack of corresponding reflection principles.
In this paper, we restrict our discussions to the unbounded rigid periodic and rough surfaces of rectangular type in R 2 .Instead of using reflection principles, our approach to the uniqueness in the inverse scattering problem is based on the expansion of analytic solutions to the Navier equation with zero Dirichlet data on two perpendicular lines.The main ingredient in the uniqueness proof is the study of a transcendental equation for the Navier equation, which has been already used in [3,14,18] to analyze corner singularities of the Lamé equation (i.e., Navier equation without the zeroth order term) in a sector.We show the uniqueness with a single incident plane wave in the case of no integer roots to the resulting transcendental equation.If an integer root exists, then we further verify that the dimension of the solution space to the Navier equation is at most one, giving rise to a uniqueness result with at most two incident angles for both periodic and non-periodic scattering surfaces.We conjecture that non-rectangular piecewise linear surfaces can be uniquely determined by sending a finite number of incident plane waves, provided some a priori information on the angles of the interface is available.Moreover, our uniqueness results are extended to non-convex bounded rigid bodies of rectangular type by using far-field measurements of at most two incident directions.
The rest of the paper is organized as follows.In Section 2, we state and prove the uniqueness results for diffraction gratings.The transcendental equation with a general angle is studied in Section 2.2, and the equation in the case of the right angle is utilized for justifying our uniqueness with at most two incident directions in Section 2.1.Finally in Section 3, the proof of the uniqueness in periodic structures is carried over to the case of rough surfaces.
2.1.Mathematical formulation and main result.Consider the elastic scattering problem from a rigid diffraction grating Λ in R 2 .It is supposed that Λ is of rectangular type, i.e., the neighboring line segments are always perpendicular.
More precisely, we assume that for some b > 0 the scattering surface Λ belongs to the following admissible class: The angle between any two neighboring line segments is π/2. .
We emphasize that Λ is allowed to be a non-graph profile, and the line segments of Λ are not necessarily parallel or perpendicular to the coordinate axes; see Figure 1 (right).We formulate the direct scattering problem following the lines in [15] for the Helmholtz equation and [5] for the Navier equation.Denote by Ω Λ the unbounded periodic region above Λ and assume, for simplicity, that Ω Λ is occupied by a linear isotropic and homogeneous elastic material with mass density one.Suppose an incident pressure wave (with the incident angle θ ∈ (−π/2, π/2)) given by (1) is incident on Λ from the region above.Here, k p := ω/ √ 2µ + λ is the compressional wave number, λ and µ denote the Lamé constants satisfying µ > 0 and λ + µ > 0, ω > 0 is the angular frequency of the harmonic motion, and the symbol (•) T stands for the transpose of a vector in R 2 .The shear wave number is defined as k s := ω/ √ µ.
Recall that a function v is called quasi-periodic with phase-shift α (or α-quasiperiodic) in Ω Λ , if exp(−iαx 1 ) v(x 1 , x 2 ) is 2π-periodic with respect to x 1 , or equivalently, Obviously, the incident pressure wave u in p is α-quasi-periodic with α = k p sin θ in Ω Λ .If the scattered field u sc is supposed to be quasi-periodic with the same phaseshift as that of u in , then the direct scattering problem, due to the incident pressure wave (1), aims to find the quasi-periodic scattered field and that satisfies the Rayleigh expansion ( [5]) Inverse Problems and Imaging Volume 9, No. 1 (2015), 127-141 for all x 2 ≥ Λ + := max (x1,x2)∈Λ x 2 .Here, the constants A p,n , A s,n ∈ C are called the Rayleigh coefficients, α n := α + n and Since β n and γ n are real for at most a finite number of indices n ∈ Z, only a finite number of plane waves in (5) propagate into the far field, with the remaining evanescent waves (or surface waves) decaying exponentially as x 2 → +∞.The above expansion (5) converges uniformly with all derivatives in the half-plane {x ∈ R 2 : The uniqueness and the existence of quasi-periodic solutions to (3)-( 5) were verified in [5] by the variational argument for grating profiles given by step functions (see Figure 1 (Left)) or Lipschitz functions.If the scattering surface is given by a general Lipschitz curve, existence can always be proved at arbitrary incident frequencies, although there is no uniqueness in general.The solvability results for pressure wave incidence extend directly to the incident shear wave with θ := (sin θ, − cos θ) T and θ⊥ := (cos θ, sin θ) T , for which the phase-shift of the scattered field is α = k s sin θ.This differs from the case of pressure wave incidence given in (1).The incident wave in our paper is also allowed to be a general elastic plane wave of the form for which the unique solution belongs to the sum of a k p sin θ and a k s sin θ-quasiperiodic Sobolev space, since the scattered field depends linearly on the incident field.
In this paper we are interested in the inverse problem of recovering an unknown periodic scattering surface Λ ∈ A from the knowledge of the scattered near-field measured on Γ b := {(x 1 , x 2 ) : x 2 = b, 0 < x 1 < 2π}, where b > Λ + is given as in the definition of the admissible class A. We state the uniqueness results with at most two incident angles as follows: Theorem 2.1.Let the incident elastic wave be given by ( 8).
(i): If the Lamé constants satisfy then Λ can be uniquely determined by for some odd number n 0 ∈ N, (10) then Λ can be uniquely determined by u sc (x; We shall carry out the proof of Theorem 2.1 in Section 2.3, relying on some lemmas to be established in Section 2.2. 2.2.Key lemmas.For x = (x 1 , x 2 ), let (r, ϕ) be the polar coordinates of x in R 2 .For notational convenience, we set N 0 := N ∪ {0}.We first derive the power series expansion of analytic solutions to the Helmholtz equation around the origin.Lemma 2.2.Assume (∆ + k 2 )u = 0 in a neighborhood of the origin.Then we can expand u = u(r, ϕ) into a convergent power series around the origin, where u ± n,m ∈ C satisfy the recurrence relations Remark 1.The expansion (11) is nothing else than the reformulation of the corresponding expansion in terms of Bessel functions (see e.g., [4,Chapter 3.4]).Note that (11) reduces to the power series for harmonic functions if k = 0.
Proof of Lemma 2.2.We begin with the Taylor expansion of u around the origin Performing the change of variables for some B n,m ∈ C.Moreover, u can be reformulated in the form (11) with some u ± n,m ∈ C. Applying the Laplace operator to u, we have Since u is a solution of the Helmholtz equation, the coefficients u ± n,m have to satisfy the recurrence relations (12).
In the following we study a transcendental equation for the Navier equation with the Dirichlet boundary condition.This equation has been used to compute corner singularities of solutions to the Lamé equation; see e.g., [3,14,18].
has no integer roots z = n ∈ N. Then it holds that u ≡ 0 in R 2 .
Proof.Since the Navier equation is rotationally invariant and u is analytic in a neighborhood of the corner point, without loss of generality we may assume the positive x 1 -axis coincides with the half-line {(r, ϕ) : ϕ = (ϕ 2 + ϕ 1 )/2} and ψ = 2ϕ 0 for some ϕ 0 ∈ (0, π/2).This implies that ϕ 1 = ϕ 0 and ϕ 2 = −ϕ 0 .For x = r(cos ϕ, sin ϕ), set x = x/r = (cos ϕ, sin ϕ), and x⊥ = (− sin ϕ, cos ϕ).We decompose u into its compressional and shear parts by where the two curl operators in R 2 are defined by and the two scalar functions v and w satisfy the Helmholtz equations It is easy to check that This, together with (14), enables us to define the functions with the vanishing data since u = 0 on ϕ = ±ϕ 0 .Observing that v and w are solutions to the homogeneous Helmholtz equation in R 2 , by Lemma 2.2 we may expand them into the series in a small neighborhood of the origin, where v ± n,m , w ± n,m ∈ C satisfy the recurrence relations for all n, m ∈ N 0 .By unique continuation, it is now sufficient to prove v ± n,m = w ± n,m = 0 for all n, m ∈ N 0 , if the transcendental equation ( 13) has no integer roots.
Inserting (18) into the definitions of F and G in (16) yields with and for all N ∈ N 0 .We proceed by equating coefficients of r N in (20).If N = 0, then we have the indexes n = 1, m = 0. Hence, it follows from ( 22) and (21) that then n = 2 and m = 0.By arguing as the previous case we find Making use of the recurrence relations (19), the equalities v ± 1,0 = ∓w ∓ 1,0 and the definitions of f ± 1,1 and g ± 1,1 (see ( 21)), we represent f ± 1,1 and g ∓ 1,1 in terms of v ± 1,0 as (see also (28) with j = 0 ) Combining ( 23) and (24), and using the fact that g − 3,0 = −f + 3,0 , g + 3,0 = f − 3,0 , we may transform the equations in (23 Simple calculations yield that the determinant of A ± 0 takes the form Thus, Det(A ± 0 ) = 0 if and only if This can be guaranteed by assuming that the number z = 2 is not an integer root of (13).Therefore, we obtain v ± 1,0 = f ± 3,0 = 0. Consequently, it holds that w ± 1,0 = g ± 3,0 = 0, and thus In summary, we have proved that for j = 1, Now, assuming that (25) is valid for some fixed j ∈ N, we show that (25) also holds with j replaced by j + 1.
Remark 2. Lemma 2.4 implies that the dimension of the solution space to the Navier equation in R 2 with vanishing data on two perpendicular straight lines is at most one.
c. Since f + j+3,0 = 0, we also get the linear dependence of w − j+3,0 and that of w − j+3,m , m ∈ N 0 on c. Repeating the above procedure, we finally conclude that Since ṽ+ n,m and w− n,m satisfy the recurrence relation ( 19), v 0 and w 0 are solutions to the Helmholtz equations in (15).Hence u 0 satisfies the Navier equation and u = cu 0 .The proof of Lemma 2.4 is complete.
2.3.Proof of Theorem 2.1.Relying on the properties of the Navier equation shown in Lemma 2.4, we prove the uniqueness results in Theorem 2.1 for diffraction gratings by contradiction.Let the incident elastic plane wave be given as in (8) with the incident angle θ.Assume there are two distinct scattering surfaces Λ 1 , Λ 2 ∈ A generating the same near-field data on Γ b : By the well-posedness of the direct scattering problem for a flat profile, we get the coincidence of u 1 and u 2 in x 2 > b, and the unique continuation of solutions to the Navier equation leads to where Ω denotes the unbounded connected component of Ω Λ1 ∩ Ω Λ2 .We consider two cases.
Since the convex hull of the corner points coincides with a strip and both profiles are bounded in the x 2 -direction, the line segments lying on them must be parallel to the coordinate axes in Case 1.Therefore, the horizontal line segments of Λ j (j = 1, 2) lie on two straight lines Γ b1 and Γ b2 for some −b < b 2 < b 1 < b, whereas the vertical segments are identical (see Figure 2 ).Without loss of generality, we suppose Γ b1 to be the x 1 -axis, i.e., b 1 = 0. Recalling the Dirichlet boundary conditions on Λ 1 and Λ 2 , we get u = 0 on Γ 0 .This suggests that u is the total field corresponding to the rigid scattering surface x 2 = 0 due to the incident plane wave (8).By linear supposition, it is not difficult to get the explicit expression of u in x 2 ≥ 0 as follows: u = (c p /k p ) U p + (c s /k s )U s , where c p and c s are the coefficients attached to the incident plane pressure and shear waves, respectively, and with Since u consists of finitely many terms only, it extends analytically to the whole space R 2 .Hence, u must also vanish on at least one vertical straight line, for instance {x 1 = 0}, which can be extended to infinity in the x 2 -direction.This implies that c p = c s = 0, which is a contradiction.Hence, Λ 1 = Λ 2 .
Case 2. The corners of Λ 1 and Λ 2 do not coincide.First we consider Case (a): there exists a corner point O j of Λ j in Ω Λj+1 for j = 1 or j = 2, where Ω Λ3 = Ω Λ1 .Without loss of generality, we suppose that Case (a) occurs with j = 1; see Figure 3 (left).It follows from (36) and the Dirichlet boundary condition of u 1 on Λ 1 that u 2 vanishes on the two perpendicular line segments of Λ 1 meeting at O 1 in Ω Λ2 .Moreover, u 2 satisfies the Navier equation in a small neighborhood D 1 ⊂ Ω Λ2 of O 1 .Applying Lemma 2.4 to u 2 yields: (i): u 2 (x; θ) ≡ 0 under the condition (9).This contradiction implies Λ 1 = Λ 2 , and thus uniqueness with a single incident plane wave holds.(ii): u 2 (x; θ) = c u 0 (x), x ∈ D 1 , under the condition (10).By arguing in the same manner we get u and by unique continuation also in x 2 > b.This contradicts the linear independence of u 2 (x; θ) and u 2 (x; θ ) in x 2 > b 2 which can be readily justified using the Rayleigh expansions of u sc 2 (x; θ) and u sc 2 (x; θ ).Now we conclude that Λ 1 = Λ 2 if the near-field data coincide for two distinct incident angles.
If Case (a) is excluded, we may suppose the existence of a corner point O 1 of Λ 1 lying on a certain line segment l ⊂ Λ 2 ; see Figure 3 (right).In this case, l must be perpendicular to a line segment of Λ 1 passing through O 1 , and l coincides partly with another line segment of Λ 1 .Since l is an analytic boundary part of Ω Λ2 and u 2 = 0 on l, u 2 is analytic in Ω Λ2 up to l (see [16, Theorem A]) and thus u 2 has analytic Cauchy data on l.Applying the Cauchy-Kowalewski theorem, we Let the incident plane wave be given as in (8), and define S h := Ω Λ \{x 2 ≥ h} .It was shown in [9] that the forward two-dimensional scattering problem admits a unique total field u = u in + u sc in the following weighted Sobolev space V h, := (1 + x 2 1 ) − /2 H 1 0 (S h ) 2 for all h ≥ b, −1 < < −1/2, (39) provided the scattering surface Λ is given by the graph of a bounded and uniformly Lipschitz continuous function.Note that the space H 1 0 (S h ) denotes the functions

Figure 2 .
Figure 2. Examples of rectangular diffraction gratings sharing the same corners.