Inverse problems play a central role in science and engineering, such as ocean acoustic, medical imaging, nondestructive testing, remote sensing and radar.^[1,2] Especially, the inverse scattering problems of ocean acoustics have received much attention in recent years.^[3–13] A finite deep ocean is popularly modeled by a waveguide bounded by two parallel planes. Time-harmonic acoustic waves in the waveguide are usually modeled by the Helmholtz equation inside it with suitable boundary conditions.^[14] In this paper, we assume that the ocean has a pressure released surface and a rigid bottom, corresponding to the Dirichlet condition and Neumann condition, respectively. We concern with the inverse scattering problem of time-harmonic acoustic waves from a bounded obstacle in a planar waveguide at a fixed frequency. Let be the waveguide of thickness h > 0. The lower and upper boundaries of the waveguide are denoted by

Then the acoustic wave u traveling inside the waveguide satisfies the following equations:

Given incident fields, what we want to do is to get the information of the scattered field in the waveguide and to solve the inverse problem of reconstructing the shape and location of the domain D, which is occupied by the obstacle. This inverse problem is much more challenging than that in free-space. Due to the planar geometry, only a finite number of modes can propagate at a long distance, while the other modes are evanescent fields and undetectable away from the obstacles. This increases the ill posedness of the inverse problem. For this reason, the inverse scattering problem in a waveguide has received increasing attention.^[5,6,9,10] We refer to Ref. [15] for the generalized dual space method, Ref. [11] for reverse time migration method, and Refs. [4], [6], and [13] for the linear sampling method. The authors of Ref. [16] used the factorization method to recover the scattering surface of a periodic structure.

The purpose of this paper is to extend the factorization method^[17–19] to reconstruct the obstacle D in the waveguide scattering. In this paper, we will utilize the factorization method to reconstruct the shape of an unknown obstacle embedded in a planar acoustic waveguide using the measurement of the scattered field near the obstacle. Following the idea of Ref. [17], we construct an auxiliary outgoing-to-incoming operator to derive the factorization of data operator. The main principle of factorization method is that a point z in the waveguide belongs to the domain D if and only if the field generated by a special point source at z is in the range of a certain linear operator defined by the data, called the near-field operator. Finally, the range of this operator can be characterized by spectral properties.

The remaining part of this paper is organized as follows. In Section 2, we present the direct scattering problem in the 3D waveguide and some useful results. In Section 3, we derive the Fourier coefficients of the near-field operator. In Section 4, we give a justification of the factorization method for the case of a Dirichlet boundary condition in a waveguide. We define the outgoing-to-incoming operator in the first part and then give a description of the domain D. In Sections 5 and 6, we extend the factorization method with near-field data to the case of the impedance and Neumann boundary conditions and the inverse medium scattering case.

In this section, we will present some notation and basic results to be used throughout the paper. Recall that the waveguide is denoted by . Noting that the x₃-axis is orthogonal to the waveguide, we combine the first two coordinates to write

The Green’s function^[14] for the waveguide with respect to the boundary condition (3) is given by

Using the method of images,^[14] one can derive an equivalent representation of the Green’s function in ,

From Ref. [20], we know that the Green’s formulas are still holds in the waveguide , that is, for any function u satisfying Eqs. (2), (3), (4), and (10), it holds that

We remark that under some certain geometric assumptions on a sound-soft obstacle, the uniqueness of the solution to the acoustic waveguide scattering problem was proved in Ref. [21]. For the sound-hard obstacle, it has been proved the existence of non-trivial waves for the homogeneous problem, called trapped modes.^[22] The authors in Ref. [3] proved out that these scattering problems are uniquely solvable except possibly for an infinite series of exceptional values of the wavenumber with no finite accumulation point.

In this paper, we consider the point source wave as incident field u_i(x,y) = G(x, y) which is generated at the source position y ∈ C_R, and the corresponding scattered field is denoted by u^s(x, y) which depends on the point source y. Then the measurement data we need in this paper are taken on the cylinder C_R. Therefore, the set u^s(x, y): x, y ∈ C_R} are the data for the inverse problem, or equivalently, the so-called near-field operator N: L²(C_R) → L²(C_R) is given by

We have already known that the boundary condition (3) and the radiation condition (10) lead to a series representation of any radiating solution ,^[4]

Let J_n be the Bessel functions of order n. Set

Set

In this section, we will utilize the factorization method to reconstruct a sound-soft obstacle embedded in a planar acoustic waveguide using the measurement of the scattered field on C_R. The key ingredients in this part is the construction of an auxiliary unitary mapping T₁ and an appropriate factorization of the operator T₁.

For the sound-soft obstacle, we consider the general Dirichlet problem

Given f ∈ H^1/2(∂ D), we find a solution v to the following problem:

4.1. Factorization of near-field operator

First, we define the incidence operator

for the Dirichlet boundary problem in the waveguide as follows:

Here H_Dir is a Herglotz-like operator, which maps a superposition of the incident waves with the weight g_m,n into its trace on ∂D. The adjoint operator is

Recall the expression of the Bessel functions of order n:^[1]

We observe that . By the definition of k_m, we already know that k_m is real, if m ≤ M(k); otherwise, k_m is purely imaginary. Therefore,

Thus, we could avoid the conjugate of Bessel function J_n inside the integral of Eq. (37).

Let v be the unique outgoing solution to the Problem 1 with the boundary value f ∈ H^1/2(∂ D). Suppose that on C_R, v has the following expansion

Then the solution operator G_Dir: H^1/2(∂ D) → ℓ² is defined by the Fourier coefficients of v|C_R, i.e.,

From the definition of , H_Dir, and G_Dir, the following relation holds:

In the rest of this section, the single-layer operator and single-layer potential are essential tools. They are give by

for ψ ∈ H^{− 1/2}(∂ D), respectively. From the expansion of Green’s function (30), we obtain the expansion of single-layer potential

Together with the definition of G_Dir and the jump relation for single-layer potentials in the waveguide,^[20] we have the following relation

In order to find an appropriate factorization of , we observe Eqs. (37) and (41) that

with the operator T₀: ℓ² → ℓ² defined by

for g = {g_m,n: n ∈ ℤ⁺, m ∈ ℤ}. Here (T₀ g)_{m, n} represent the component of sequence T₀ g ∈ ℓ², m ∈ ℤ⁺, n ∈ ℤ. Moreover, it is seen from Eq. (48) that , that is, T₀ is unitary operator on ℓ².

From Eq. (42) and the second relation in Eq. (47), we observe that

Consequently, we obtain a factorization of the near-field operator N as follows.

4.2. Characteristic function of D

We first give the properties of the solution operator G_Dir defined by Eq. (41) and the modified solution operator .

Accordingly, the operator is compact, one-to-one with a dense range in L²(C_R)

From Ref. [3], we have the well-posedness of the three-dimensional waveguide scattering Problem 1 in . Together with the compact embedding property of H^1/2(C_R) into L²(C_R), we conclude that G_Dir is a compact operator.

Then we will prove that the operator G_Dir has a dense range in ℓ². For some P ∈ ℤ⁺, define the sequence g^(P) as

for g = {g_m,n: m ∈ ℤ*, n ∈ ℤ} ∈ ℓ². Thus, for any ε > 0 there exists a integer P_ε > 0 such that ||g^(P_ε) − g ||_ℓ² ≤ ε. Now we define the function v by

Obviously, v is a solution to the Helmholtz equation Δv + k² v = 0 in , and satisfies the outgoing radiation condition (10). From the definition of the solution operator G_Dir, we know that G_Dir (v|_∂D) = g^(P_ε), hence

Thus, we obtain the denseness of the range of G_Dir in ℓ².

Finally, we notice the fact that T₀ is an unitary operator in ℓ² and is an isomorphism. Then, the properties of follows.

Proof First, we assume that z ∈ D. Clearly, is analytic, hence it is the unique radiating solution to the Problem 1 with the boundary data f : = G(·, z) |_∂D. From the definition of G_Dir, , and T₁, it follows that

We notice that z ∈ D ⊂ Ω_R, then holds for x ∈ C_R. Thus, from the expansion (31) for the Green’s function G(x, z) with |x| > |z|, the complex conjugate of Green’s function can rewrite as

where M = M(k), such that k_m is real if m ≤ M while k_m is purely imaginary if m > M. The last equality holds with the fact . Observing Eq. (55), we find that

Combing Eqs. (54) and (55) yields

On the other hand, let and assume that . Since T₀ is unitary on ℓ², we have

Then,

The last identity follows from Eq. (56). Therefore, holds. Let v be the solution of Problem 1 with the boundary f such that

Due to the uniqueness of solutions to the Problem 1, we get v = G(·, z) in . Therefore, v = G(·, z) = holds by analytic continuation. If , this contradicts the fact that v is analytic in but G(·, z) is singular at z. It is impossible that z ∈ ∂ D since but . Hence, we conclude that z ∈ D, which complete the proof.

We now give the properties of the middle operator S.

Now, we have shown all prerequisites which are necessary to apply Theorem 2.15 in Ref. [18].

We make the following assumptions: i)

G is compact with dense range.

ii)

Re T has the form ReT = T₀ + T₁ with some coercive operator T₀ and some compact operator T₁ : X^* → X.

iii)

Im T is nonnegative on X, i.e., ⟨ φ, (ImT) φ ⟩ > 0.

iv)

T is injective.

Then the operator F♯ = |Re[e^itT] | + |Im F| is positive, self-adjoint, and compact, and the range of G: X → Y and coincide. Here |ReF| is defined in the standard way via the spectral representation for compact self-adjoint operators, and the parts ReF and ImF are defined by

From this result on range identities in factorizations applied to the factorization F = G TG^*, we obtain the final result of this section, which give an explicit reconstruction of D.

This theorem presents a sufficient and necessary computational criterion for precisely characterizing the region occupied by the sound-soft obstacles. In other words, the characteristic function χ_D of D is given by χ_D(z) = sign(W(z)). In conclusion, we establish the factorization method for reconstructing a sound-soft obstacle in the waveguide with the auxiliary outgoing-to-incoming operator T₁. When the boundary condition is sound-hard or impedance, the similar result can be deduced with the same means.

In this section we talk about the obstacle with respect to impedance boundary conditions. Then the scattered field in this case satisfies the following equations:

Given f ∈ H^–1/2(∂ D), find a solution to the problem:

Noting that for fixed , when we choose

We use the similar means in the case of Dirichlet boundary conditions, we first introduce the incidence operator ℓ² → H^{− 1/2}(∂ D) by

Define the layer-potential operators K, K′, and J, respectively, by

Since the Green’s function can be written as the superposition of the fundamental solution Φ(x, y) in free space and an analytic function, the layer-potential operators has the same properties as the ones with kernel Φ. It follows from Refs. [20] and [23] that the operators K: H^1/2(∂ D) → H^1/2(∂ D), K′: H^−1/2(∂ D) → H^−1/2(∂ D) and J: H^1/2(∂ D) → H^−1/2(∂ D) are all bounded.

(I) The operator is compact, one-to-one with a dense range in L²(S_R).

(II) The operator T_imp is an isomorphism from the space H^{1/2(∂ D)} into H^−1/2(∂ D).

(III) Let J_i be define by Eq. (64) with k = i. Then J_i is self-adjoint and coercive as an operator from H^1/2(∂ D) into H^−1/2(∂ D).

(IV) Im ⟨T_imp φ, φ ⟩ 0 for all φ ∈ H^−1/2(∂ D) with φ ≠ 0.

(V) The difference T_imp − J_i is compact from H^1/2(∂ D) into H^−1/2(∂ D).

Since the difference between the Green’s function in waveguide and the fundamental solution in free space is an analytic function, the theorem above is a trivial extension of Theorem 4.2 in Ref. [17].

Similarly to Theorem 3 for the sound-soft obstacle case, we can prove the following result.

We remark that when we take ρ(x) = 0, the factorization method with near-field data in the waveguide extends straightly to the Neumann boundary condition, provided for any m ∈ ℤ⁺, k ≠ (2 m − 1)π/2h.

In this section, we consider that D is a penetrable with the refraction index n(x). Assume that Re (n) ≥ 0, Im(n) ≥ 0, n = 1 in background and n ≠ 1 in D. Then the direct scattering in a planar waveguide with a penetrable medium is modeled simply by the equation

Given f ∈ L²(D), find the solution v to the following problems:

As in the case of sound-soft obstacle, we also define the solution operator G_pen : L²(D) → ℓ² and the incidence operator H_pen : ℓ² → L²(D) for the Problem 3 as follows:

The middle operator T_pen of the factorization (81) satisfies all the properties of Theorem 5 from Refs. [2], [20], and [24]. Using the similar argument as in Section 4, we have the following result.

In this paper, we establish the factorization method for the impenetrable and penetrable obstacles, with the idea of defining the outgoing-to-incoming operator T₁ to find an appropriate factorization of T₁ N. From the final results, Theorems 6, 9, and 10, we know that the characteristic functions of D only depend on the near-field operator, and T₁ is determined by the waveguide and wavenumber k. Finally, we observe that the inversion scheme is independent of the physical properties of the underlying obstacles. This is an advantage to practical application.