Data-driven inversion/depth imaging derived from approximations to one-dimensional inverse acoustic scattering

Amundsen, Lasse; Reitan, Arne; Helgesen, Hans Kristian; Arntsen, Børge

doi:10.1088/0266-5611/21/6/002

Cited by 16 publications

(16 citation statements)

References 45 publications

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“…where H (1) n is the Hankel function of the first kind with order n. Let B : H 1/2 (∂B) → H −1/2 (∂B) be the Dirichlet-to-Neumann (DtN) operator defined as follows: for any ψ ∈…”

Section: Variational Problemmentioning

confidence: 99%

Inverse scattering problems with multi-frequencies

Bao¹,

Li²,

Lin³

et al. 2015

Inverse Problems

189

167

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This paper is concerned with computational approaches and mathematical analysis for solving inverse scattering problems in the frequency domain. The problems arise in a diverse set of scientific areas with significant industrial, medical, and military applications. In addition to nonlinearity, there are two common difficulties associated with the inverse problems: ill-posedness and limited resolution(diffraction limit). Due to the diffraction limit, for a given frequency, only a low spatial frequency part of the desired parameter can be observed from measurements in the far field. The main idea developed here is that if the reconstruction is restricted to only the observable part, then the inversion will become stable. The challenging task is how to design stable numerical methods for solving these inverse scattering problems inspired by the diffraction limit. Recently, novel recursive linearization based algorithms have been presented in an attempt to answer the above question. These methods require multi-frequency scattering data and proceeds via a continuation procedure with respect to the frequency from low to high. The objective of this paper is to give a brief review of these methods, their error estimates and the related mathematical analysis. More attention is paid on the inverse medium and inverse source problems. Numerical experiments are included to illustrate the effectiveness of these methods.

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“…where H (1) n is the Hankel function of the first kind with order n. Let B : H 1/2 (∂B) → H −1/2 (∂B) be the Dirichlet-to-Neumann (DtN) operator defined as follows: for any ψ ∈…”

Section: Variational Problemmentioning

confidence: 99%

Inverse scattering problems with multi-frequencies

Bao¹,

Li²,

Lin³

et al. 2015

Inverse Problems

189

167

View full text Add to dashboard Cite

show abstract

“…Inverse acoustic scattering has undergone a long study and the methods developed are extensive. The principal state of the art inverse methods can be divided into two categories: (1) the linearized approximation inversion (e.g., Cohen and Bleistein 1977, Bleistein 1984, Clayton and Stolt 1981, Amundsen et al 2005 , which usually uses the Born approximation or Rytov approximation of the Lippmann-Schwinger equation to develop a forward equation relating the measured data to the scattering potential. The limitation of the method is that it entails the small-contrast assumption which means the reference and actual medium should be close.…”

Section: Introductionmentioning

confidence: 99%

“…The real data contains both primaries and multiples. Many inverse scattering methods process only the primaries (Shaw 2005, Amundsen et al 2005, which require the recorded data to undergo a pre-processing step to attenuate all multiples. However, we stress here that our method does not require the data to go through a multiple attenuating procedure.…”

Section: Introductionmentioning

confidence: 99%

One dimensional acoustic direct nonlinear inversion using the Volterra inverse scattering series

et al. 2014

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Direct inversion of acoustic scattering problems is nonlinear. One way to treat the inverse scattering problem is based on the reversion of the Born–Neumann series solution of the Lippmann–Schwinger equation. An important issue for this approach is the radius of convergence of the Born–Neumann series for the forward problem. However, this issue can be tackled by employing a renormalization technique to transform the Lippmann–Schwinger equation from a Fredholm to a Volterra integral form. The Born series of a Volterra integral equation converges absolutely and uniformly in the entire complex plane. We present a further study of this new mathematical framework. A Volterra inverse scattering series (VISS) using both reflection and transmission data is derived and tested for several acoustic velocity models. For large velocity contrast, series summation techniques (e.g., Cesàro summation, Euler transform, etc) are employed to improve the rate of convergence of VISS. It is shown that the VISS method with summation techniques can provide a relatively good estimation of the velocity profile. The method is fully data-driven in the respect that no prior information of the model is required. Besides, no internal multiple removal is needed. This one dimensional VISS approach is useful for inverse scattering and serves as an important step for studying more complicated and realistic inversions.

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“…The Taylor series for at is given by (10) From (1), we know that (11) Inserting (9) and (11) into (10), (10) can be rewritten as (12) Dividing both sides of (12) by , we can obtain (7). Lemma 1 confirms the intuitive belief that is closely related with which has the meaning of reflection.…”

Section: B Asymptotic Expansionmentioning

confidence: 99%

The Schur Algorithm Applied to the One-Dimensional Continuous Inverse Scattering Problem

Choi

Chun

Kim

et al. 2013

IEEE Trans. Signal Process.

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The one-dimensional continuous inverse scattering problem can be solved by the Schur algorithm in the discrete-time domain using sampled scattering data. The sampling rate of the scattering data should be increased to reduce the discretization error, but the complexity of the Schur algorithm is proportional to the square of the sampling rate. To improve this tradeoff between the complexity and the accuracy, we propose a Schur algorithm with the Richardson extrapolation (SARE). The asymptotic expansion of the Schur algorithm, necessary for the Richardson extrapolation, is derived in powers of the discretization step, which shows that the accuracy order (with respect to the discretization step) of the Schur algorithm is 1. The accuracy order of the SARE with the -step Richardson extrapolation is increased to with comparable complexity to the Schur algorithm. Therefore, the discretization error of the Schur algorithm can be decreased in a computationally efficient manner by the SARE.

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Data-driven inversion/depth imaging derived from approximations to one-dimensional inverse acoustic scattering

Cited by 16 publications

References 45 publications

Inverse scattering problems with multi-frequencies

Inverse scattering problems with multi-frequencies

One dimensional acoustic direct nonlinear inversion using the Volterra inverse scattering series

The Schur Algorithm Applied to the One-Dimensional Continuous Inverse Scattering Problem

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