Uniqueness for a seismic inverse source problem modeling a subsonic rupture

Hoop, Maarten V. de; Oksanen, Lauri; Tittelfitz, Justin

doi:10.1080/03605302.2016.1240183

Cited by 7 publications

(7 citation statements)

References 49 publications

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“…The result of [6,Theorem 4.2] is stated with source terms depending only on the time variable. To the best of our knowledge, except the result of [10], dealing with the recovery of discrete in time sources, and the result of the present paper, there is no result in the mathematical literature treating the recovery of a source term depending on both space and time variables appearing in hyperbolic equations.…”

Section: Known Resultsmentioning

confidence: 89%

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Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

Kian

2020

Inverse Problems &Amp; Imaging

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This paper is concerned with inverse source problems for the time-dependent Lamé system in an unbounded domain corresponding to the exterior of a bounded cavity or the full space R 3 . If the time and spatial variables of the source term can be separated with compact support, we prove that the vector valued spatial source term can be uniquely determined by boundary Dirichlet data in the exterior of a given cavity. Uniqueness and stability for recovering some class of time-dependent source terms are also obtained using partial boundary data.

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Section: Known Resultsmentioning

confidence: 89%

“…For hyperbolic equations, we refer to [10,41] where the recovery of some specific time-dependent source terms have been considered. For Lamé systems, [6,Theorem 4.2] seems to be the only result available in the mathematical literature where such a problem has been addressed for timedependent source terms.…”

Section: Known Resultsmentioning

confidence: 99%

Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

Kian

2020

Inverse Problems &Amp; Imaging

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“…This paper is concerned with an ISP for the wave equation, which has been extensively investigated for the deterministic case. The wellposedness and stability can be found in [7,9,11,16,29,30,44,45] and [20,29,30,[43][44][45], respectively. Some of the numerical results may be found in [8,12,19,34,39] and the references cited therein.…”

Section: Introductionmentioning

confidence: 95%

An inverse source problem for the stochastic wave equation

Feng¹,

Zhao²,

Li³

et al. 2021

Preprint

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This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. For the direct problem, it is shown to be well-posed with a unique mild solution. For the inverse problem, the uniqueness is proved for a certain class of functions and the instability is characterized. Numerical experiments are presented to illustrate the reconstructions by using a truncation-based regularization method.

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“…However, most of the above mentioned works dealt with recovery of time independent source terms. We refer to [9,36,2,18] where specific time-dependent source terms for hyperbolic equations were considered and to [29] for the recovery of some class of space-time-dependent source terms in the parabolic equation on a wave guide. In the time-harmonic case, inverse source problems with multi-frequency data have been extensively investigated.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

Kian

Zhao

2019

Acta Math. Appl. Sin. Engl. Ser.

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This paper is concerned with inverse acoustic source problems in an unbounded domain with dynamical boundary surface data of Dirichlet kind. The measurement data are taken at a surface far away from the source support. We prove uniqueness in recovering source terms of the form f (x)g(t) and f (x 1 , x 2 , t)h(x 3 ), where g(t) and h(x 3 ) are given and x = (x 1 , x 2 , x 3 ) is the spatial variable in three dimensions. Without these a priori information, we prove that the boundary data of a family of solutions can be used to recover general source terms depending on both time and spatial variables. For moving point sources radiating periodic signals, the data recorded at four receivers are prove sufficient to uniquely recover the orbit function. Simultaneous determination of embedded obstacles and source terms was verified in an inhomogeneous background medium using the observation data of infinite time period. Our approach depends heavily on the Laplace transform.problem for doubly hyperbolic equations arising from the nucleation rate reconstruction in the three-dimensional time cone model was analyzed in [32]. A Lipschitz stability result was proved for recovering the spatial component of the source term using interior data and an iterative thresholding algorithm (see also [26] with the final observation data) was tested. However, most of the above mentioned works dealt with recovery of time independent source terms. We refer to [9,36,2,18] where specific time-dependent source terms for hyperbolic equations were considered and to [29] for the recovery of some class of space-time-dependent source terms in the parabolic equation on a wave guide. In the time-harmonic case, inverse source problems with multi-frequency data have been extensively investigated. The increasing stability analysis in recovering spatial-dependent source terms has been carried out from both theoretical and numerical points of view (see e.g., [3,6,4,5,7,11,31,40]).In the time domain, it is very natural to transform the wave scattering problem governed by hyperbolic equations into elliptic inverse problems in the Fourier or Laplace domain with multi-frequency data; see e.g. [24] for determining sound-hard and impedance obstacles in a homogeneous background medium. In [7], the time-domain analysis helps for deriving an increasing stability to time-harmonic inverse source problems via Fourier transform. The same idea was used in [2,20,19] for recovering spatial-dependent sources as well as moving source profiles and orbits in elastodynamics and electromagnetism. The aim of this paper is to analyze the acoustic counterpart with new uniqueness results. Specially, this paper concerns the following four inverse problems with a single boundary surface data:1. Simultaneous determination of sound-soft obstacles and separable source terms in an inhomogeneous medium (Subsection 2.1).2. Simultaneous determination of sound-soft obstacles and general time-dependent source terms from a family of solutions (Subsection 2.2).3. Inverse moving point source pro...

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Uniqueness for a seismic inverse source problem modeling a subsonic rupture

Cited by 7 publications

References 49 publications

Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

An inverse source problem for the stochastic wave equation

Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

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