Direct and inverse acoustic scattering by a mixed-type scatterer (original) (raw)
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In this paper, we consider a problem of inverse acoustic scattering by an impenetrable obstacle embedded in a layered medium. We will show that the factorization method can be applied to recover the embedded obstacle; that is, the equationFg = φ z is solvable if and only if the sampling point z is in the interior of the unknown obstacle. Here,F is a self-adjoint operator related to the far field operator and φ z is the far field pattern of the Green function with respect to the problem of scattering by the background medium for point z. The validity of the factorization method is proven with the help of a mixed reciprocity principle and an application of the scattering operator. Due to the established mixed reciprocity principle, knowledge of the Green function for the background medium is no longer required, which makes the method attractive from the computational point of view. The paper is only concerned with sound-soft obstacles, but the analysis can be easily extended for sound-hard obstacles, or obstacles with separated sound-soft and soundhard parts. Finally, we provide an explicit example for a radially symmetric case and present some numerical examples.
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It is well-known that sampling type methods for solving inverse scattering problems fail if the wave number is an eigenvalue of a corresponding interior eigenvalue problem. By adding the far field patterns corresponding to an artificial ball lying within the obstacle and imposing an impedance boundary condition on the boundary of this ball we propose a modification of the factorization method which provides the characterization of the unknown obstacle for all wave numbers. Some numerical experiments are presented to demonstrate the feasibility and effectiveness of our method.
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We are concerned with the acoustic scattering by an extended obstacle surrounded by point-like obstacles. The extended obstacle is supposed to be rigid while the point-like obstacles are modeled by point perturbations of the exterior Laplacian. In the first part, we consider the forward problem. Following two equivalent approaches (the Foldy formal method and the Krein resolvent method), we show that the scattered field is a sum of two contributions: one is due to the diffusion by the extended obstacle and the other arises from the linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one. In the second part, we deal with the inverse problem. It consists in reconstructing both the extended and point-like scatterers from the corresponding far-field pattern. To solve this problem, we describe and justify the factorization method of Kirsch. Using this method, we provide several numerical results an...
Direct and Inverse Acoustic Scattering by a Collection of Extended and Point-Like Scatterers
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This paper concerns the acoustic scattering by an extended obstacle surrounded by point-like obstacles. The extended obstacle is supposed to be rigid, while the point-like obstacles are modeled by point perturbations of the exterior Laplacian. In the first part, we consider the forward problem. Following two equivalent approaches (the Foldy formal method and the Krein resolvent method), we show that the scattered field is a sum of two contributions: one is due to the diffusion by the extended obstacle, and the other arises from the linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one. In the second part, we deal with the inverse problem. It consists in reconstructing both the extended and point-like scatterers from the corresponding far-field pattern. To solve this problem, we describe and justify the factorization method of Kirsch. Using this method, we provide several numerical results and discuss the multiple scattering effect concerning both the interactions between the pointlike obstacles and between these obstacles and the extended one.
2014
This paper concerns the acoustic scattering by an extended obstacle surrounded by point-like obstacles. The extended obstacle is supposed to be rigid, while the point-like obstacles are modeled by point perturbations of the exterior Laplacian. In the first part, we consider the forward problem. Following two equivalent approaches (the Foldy formal method and the Krein resolvent method), we show that the scattered field is a sum of two contributions: one is due to the diffusion by the extended obstacle, and the other arises from the linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one. In the second part, we deal with the inverse problem. It consists in reconstructing both the extended and point-like scatterers from the corresponding far-field pattern. To solve this problem, we describe and justify the factorization method of Kirsch. Using this method, we provide several numerical results an...
Elastic scattering by an inhomogeneous medium with unknown buried obstacles
In this paper the direct and inverse scattering problem of time harmonic elastic waves by an inhomogeneous medium with buried objects inside is studied. Initially, the well-posedness of the direct scattering problem by the variational method in a suitable Sobolev space setting is presented and proved. Uniqueness, existence and continuity dependence of solution of the problem from the boundary data of the buried obstacles are established. Further, the corresponding inverse problem will be studied and in particular the factorization method for shape reconstruction and location of the support of the inhomogeneous medium will be exploited. In addition, an inversion algorithm for shape recovering of the medium is presented and proved as well. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one in elastic media are given.
Inverse acoustic scattering by a layered obstacle
1998
A uniqueness theorem is proved for the inverse acoustic scattering problem for a piecewise-homogeneous obstacle under the assumption that the scattering amplitude is known for all directions of the incident and the scattered field at a fixed frequency.
The inverse problem of an impenetrable sound-hard body in acoustic scattering
Journal of Physics: Conference Series, 2008
We study the inverse problem of recovering the scatterer shape from the far-field pattern(FFP) of the scattered wave in presence of noise. This problem is ill-posed and is usually addressed via regularization. Instead, a direct approach to denoise the FFP using wavelet technique is proposed by us. We are interested in methods that deal with the scatterer of the general shape which may be described by a finite number of parameters. To study the effectiveness of the technique we concentrate on simple bodies such as ellipses, where the analytic solution to the forward scattering problem is known. The shape parameters are found based on a least-square error estimator. Two cases with the FFP corrupted by Gaussian noise and/or computational error from a finite element method are considered. We also consider the case where only partial data is known in the far field.
On the Solution of Inverse Obstacle Acoustic Scattering Problems with a Limited Aperture
Mathematical and Numerical Aspects of Wave Propagation WAVES 2003, 2003
We present a computational methodology for retrieving the shape of a rigid obstacle from the knowledge of some acoustic far-field patterns. This methodology is based on the well-known regularized Newton algorithm, but distinguishes itself from similar optimization procedures by using (a) the far field pattern in a limited aperture, (b) a sensitivity-based and frequency-aware multi-stage solution strategy, (c) a computationally efficient usage of the exact sensitivities of the far-field pattern to the specified shape parameters, and (d) a numerically scalable domain decomposition method for the fast solution in a frequency band of direct acoustic scattering problems.