The time‐domain Lippmann–Schwinger equation and convolution quadrature

Lechleiter, Armin; Monk, Peter

doi:10.1002/num.21921

Cited by 11 publications

(13 citation statements)

References 19 publications

(76 reference statements)

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“…Using the definitions given in (2), the total pressure field can be decomposed into an unperturbed wave p 0 and a scattered wave p s such that p=p 0 +p s is the unique solution to (1). It follows that if the unperturbed pressure field p 0 satisfies…”

Section: Formulation Of the Direct Acoustic Scattering Problemmentioning

confidence: 99%

“…Of fundamental importance to scattering theory is the Lippmann-Scwhinger equation (e.g. [1]), which explains not only primary (or single) scattered waves, but all multiply scattered waves as well. The Lippmann-Schwinger equation provides an exact representation of the scattered field in terms of a weighted superposition of the impulse response of the background medium over the region containing the scatterer.…”

Section: Introductionmentioning

confidence: 99%

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An acoustic Lippmann-Schwinger inversion method: applications and comparison with the linear sampling method

Prunty

Snieder

2020

J. Phys. Commun.

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We introduce an imaging method based on solving the Lippmann-Schwinger equation of acoustic scattering theory. We compare and contrast the proposed Lippmann-Schwinger inversion with the well-established linear sampling method using numerical examples. We demonstrate that the two imaging methods are physically grounded in different but related wave propagation problems: Lippmann-Schwinger inversion seeks to reconstruct the space and time dependence of a scatterer based on the observed scattered field in a performed physical experiment, whereas the linear sampling method seeks to focus wave fields in a simulated virtual experiment by estimating the space and time dependence of an inverse source function that cancels the effects of the scatterer at a specified focusing point. In both cases, the medium in which the waves propagate is the same; however, neither method requires prior knowledge or assumptions on the physical properties of the unknown scatterer-only knowledge of the background medium is needed. We demonstrate that the linear sampling method is preferable to Lippmann-Schwinger inversion for target-oriented imaging applications, as Lippmann-Schwinger inversion gives nonphysical results when the chosen imaging domain does not contain the scatterer.

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Section: Formulation Of the Direct Acoustic Scattering Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An acoustic Lippmann-Schwinger inversion method: applications and comparison with the linear sampling method

Prunty

Snieder

2020

J. Phys. Commun.

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“…We begin by discussing the well-posedness of (1). This is well known, and to discuss it precisely we follow [11,20,30], introducing some space-time Sobolev spaces described through the Fourier-Laplace transform. This will allow us to introduce and state the well-posedness of a time domain weak scattering approximation and its frequency domain counterpart.…”

Section: Forward Model and The Born Approximationmentioning

confidence: 99%

“…In order to make these equations precise, we recall the appropriate space-time Sobolev spaces, following [20,30]. To this end, we first introduce the Fourier-Laplace transform.…”

Section: Forward Model and The Born Approximationmentioning

confidence: 99%

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Direct imaging of small scatterers using reduced time dependent data

Cakoni

Rezac

2017

Journal of Computational Physics

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We introduce qualitative methods for locating small objects using time dependent acoustic near field waves. These methods have reduced data collection requirements compared to typical qualitative imaging techniques. In particular, we only collect scattered field data in a small region surrounding the location from which an incident field was transmitted. The new methods are partially theoretically justified and numerical simulations demonstrate their efficacy. We show that these reduced data techniques give comparable results to methods which require full multistatic data and that these time dependent methods require less scattered field data than their time harmonic analogs.

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Theory of the linear sampling method for time-dependent fields

Prunty

Snieder

2019

Inverse Problems

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The linear sampling method seeks to localize the unknown source of an observed, time-dependent field. The unknown source could be, for example, a scatterer embedded within a medium, or an impulsive excitation such as an earthquake or explosion. The source of the observed field is localized by means of solving the so-called near-field equation and mapping the obtained solutions through an indicator functional over a test region assumed to contain the source. In its current formulation, however, the linear sampling method suffers from an ambiguous time parameter that strongly influences its ability to localize the unknown source. Our paper consists of two fundamental results central to the theoretical understanding of the linear sampling method and its numerical implementation. First, we prove the so-called blowup behavior of solutions to the near-field equation for a general source function that is separable in space and time. Second, we show that the linear sampling method can be formulated such that the ambiguous time parameter is irrelevant. We demonstrate that a dependence of the linear sampling method on the time parameter arises from an incorrect implementation of a convolution-type operator found in the near-field equation. When the operator is implemented correctly, the dependence on the time parameter vanishes. We provide detailed algorithms for efficient and proper implementations of the convolutional operator in both the time and frequency domains. The crucial result of the improved implementations is that they allow the linear sampling method to be completely automated, as one does not need to know the space-time dependence of the unknown source. We demonstrate the effectiveness of the improved time-and frequency-domain implementations using several numerical examples applied to imaging scatterers.

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The time‐domain Lippmann–Schwinger equation and convolution quadrature

Cited by 11 publications

References 19 publications

An acoustic Lippmann-Schwinger inversion method: applications and comparison with the linear sampling method

An acoustic Lippmann-Schwinger inversion method: applications and comparison with the linear sampling method

Direct imaging of small scatterers using reduced time dependent data

Theory of the linear sampling method for time-dependent fields

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