Uniqueness to inverse acoustic scattering from coated polygonal obstacles with a single incoming wave

Abstract

It is proved that a connected polygonal obstacle coated by thin layers together with its surface impedance function can be determined uniquely from the far field pattern of a single incident plane wave. As a by-product, we prove that the wave field cannot be real-analytic on each corner point lying on the convex hull of the scatterer. Our arguments are based on the Schwarz reflection principle for the Helmholtz equation satisfying the impedance boundary condition on a flat boundary.

Factorization method with one plane wave: from model-driven and data-driven perspectives

“…By the Schwartz reflection principle of the Helmholtz equation, u D can be extended onto R 2 (see also [14] for discussions on the conductivity equation). We remark that u D ≡ 0 when the angle of Σ is irrational and that the impedance case follows from the arguments in [19]. This implies that the scattered field u s D is an entire radiating solution.…”

“…In the impedance case, the impedance function is supposed to be a constant. As one can imagine, the proof of Lemma 3.4 is closely related to uniqueness in determining a convex polygonal obstacle with a single incoming wave (see e.g., [6],[8, Theorem 5.5] and [19]). In fact, the result of Lemma 3.4 implies that a convex polygonal obstacle of sound-soft, soundhard or impedance type can be uniquely determined by one far-field pattern.…”

“…The aim of this paper is to address a framework of the one-wave factorization method, which was earlier proposed in [13] for inverse elastic scattering from rigid polygonal rigid bodies and later discussed in [17] for inverse acoustic source problems in an inhomogeneous medium. Our arguments are motivated by the existing one-wave sampling methods mentioned above and the recently developed corner scattering theory for justifying the absence of non-scattering energies and non-radiating sources (see [2,3,11,12,18,19,30,32,38]). The corner scattering theory implies that the wave field cannot be analytically continued across a strongly or weakly singular point lying on the scattering interface.…”

Factorization method with one plane wave: from model-driven and data-driven perspectives

“…In the impedance case, the impedance function is supposed to be a constant. As one can imagine, the proof of lemma 3.4 is closely related to uniqueness in determining a convex polygonal obstacle with a single incoming wave (see e.g., [7], [9, theorem 5.5] and [20]). In fact, the result of lemma 3.4 implies that a convex polygonal obstacle of sound-soft, sound-hard or impedance type can be uniquely determined by one far-field pattern.…”

“…The aim of this paper is to address a framework of the one-wave factorization method, which was earlier discussed in [13] for inverse elastic scattering from rigid polygonal bodies and also in [18] for inverse acoustic source problems in an inhomogeneous medium but without numerical verifications. Our arguments are motivated by the existing one-wave sampling methods mentioned above and the recently developed corner scattering theory for justifying the absence of non-scattering energies and non-radiating sources (see [3,4,11,12,19,20,32,34,39]). The corner scattering theory is closely connected to the concept of 'scattering support' explored by [32].…”