Reconstruction of a Conformally Euclidean Metric from Local Boundary Diffraction Travel Times

Hoop, Maarten V. de; Holman, Sean; Iversen, Einar; Lassas, Matti; Ursin, Bjørn

doi:10.1137/130931291

Cited by 15 publications

(18 citation statements)

References 16 publications

(26 reference statements)

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“…In the case when the wave speed is isotropic, it is possible to transform the reconstruction from the travel time coordinates to the Euclidean coordinates. This transformation is considered in [6]. We also note that as an intermediate step in the proof of Theorem 2, we show that the metric can be recovered from S U locally in Fermi coordinates along geodesics passing through U .…”

Section: Theorem 2 Let M and M Be Two Smooth (Compact Or Complete) Rmentioning

confidence: 73%

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Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

Hoop

Holman

Iversen

et al. 2015

Journal de Mathématiques Pures et Appliquées

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MSC: 35R30 86A15 53C21Keywords: Geometric inverse problems Riemannian manifold Shape operatorWe analyze the inverse problem, if a manifold and a Riemannian metric on it can be reconstructed from the sphere data. The sphere data consist of an open set U ⊂M and the pairs (t, Σ) where Σ ⊂ U is a smooth subset of a generalized metric sphere of radius t. This problem is an idealization of a seismic inverse problem, originally formulated by Dix [8], of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of waves. In this problem, one considers a domain M with a varying and possibly anisotropic wave speed which we model as a Riemannian metric g. For our data, we assume that M contains a dense set of point diffractors and that in a subset U ⊂M , we can measure the wave fronts of the waves generated by these. The inverse problem we study is to recover the metric g in local coordinates anywhere on a set M ⊂M up to an isometry (i.e. we recover the isometry type of M ). To do this we show that the shape operators related to wave fronts produced by the point diffractors within M satisfy a certain system of differential equations which may be solved along geodesics of the metric. In this way, assuming that we know g as well as the shape operator of the wave fronts in the region U , we may recover g in certain coordinate systems (e.g. Riemannian normal coordinates centered at point diffractors). This generalizes the method of Dix to metrics which may depend on all spatial variables and be anisotropic. In particular, the novelty of this solution lies in the fact that it can be used to reconstruct the metric also in the presence of the caustics. (M.V. de Hoop), sean.holman@manchester.ac.uk (S.F. Holman), Einar.Iversen@norsar.com (E. Iversen), matti.lassas@helsinki.fi (M. Lassas), bjorn.ursin@ntnu.no (B. Ursin). http://dx.doi.org/10.1016/j.matpur.2014.09.003 0021-7824/© 2014 Elsevier Masson SAS. All rights reserved. M.V. de Hoop et al. / J. Math. Pures Appl. 103 (2015) 830-848 831 r é s u m éOn analyse un problème inverse, si une variété riemannienne peut être reconstruite à partir des données sphère. Les données sphère sont constituées d'un ensemble ouvert U ⊂M et les paires (t, Σ), où Σ ⊂ U set un sous-ensemble lisse d'une sphère métrique généralisée. Ce problème est une idéalisation d'un problème sismique inverse, à l'origine formulé par Dix [8], consistant à reconstruire la vitesse d'onde dans un domaine à partir des mesures aux frontières associées à la dispersion simple des ondes sismiques. On considère un domaine M avec une vitesse d'onde variable et éventuellement anisotrope modélisée par une métrique riemannienne g. On suppose que M contient une densité élevée de points diffractants et que dans un sous-ensemble U ⊂M , correspondant à un domaine contenant les instruments de mesure, on peut mesurer les fronts d'onde de la diffusion simple des ondes diffractées depuis les points diffractants. Le problème inverse étudié consiste à reconstruire la métrique g en coordon...

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Section: Theorem 2 Let M and M Be Two Smooth (Compact Or Complete) Rmentioning

confidence: 73%

“…Note that at the point γ x,η (r) we have ∂ r =γ x,η (r). Let r k j (r) = r k j (x, η, r) be the coefficients defined in (6), that is,…”

Section: Jacobi and Riccati Equationsmentioning

confidence: 99%

Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

Hoop

Holman

Iversen

et al. 2015

Journal de Mathématiques Pures et Appliquées

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“…Which completes the demonstration of (14). Notice that we only require Λf | R , since, for t ∈ [0, T ], Λf (t) vanishes outside of R by finite speed of propagation.…”

Section: 2mentioning

confidence: 62%

Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction

Hoop¹,

Kepley²,

Oksanen³

2018

SIAM J. Appl. Math.

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In this paper, we study the inverse boundary value problem for the wave equation with a view towards an explicit reconstruction procedure. We consider both the anisotropic problem where the unknown is a general Riemannian metric smoothly varying in a domain, and the isotropic problem where the metric is conformal to the Euclidean metric. Our objective in both cases is to construct the metric, using either the Neumann-to-Dirichlet (N-to-D) map or Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we construct the metric in the boundary normal (or semi-geodesic) coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. Both cases utilize a variant of the Boundary Control method, and work by probing the interior using special boundary sources. We provide a computational experiment to demonstrate our procedure in the isotropic case with N-to-D data.

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“…In [5] a special case of problem [6] is considered. The authors study a setup where M ⊂ R n , n ≥ 2 and the metric tensor g| M = v −2 e, for some smooth and strictly positive function v. Let x ∈ R n \ M and y := γ x,ξ (t 0 ) ∈ M , for some ξ ∈ S x N and t 0 > 0, such that y is not a conjugate point to x along γ p,ξ .…”

Section: 23mentioning

confidence: 99%

“…The authors study a setup where M ⊂ R n , n ≥ 2 and the metric tensor g| M = v −2 e, for some smooth and strictly positive function v. Let x ∈ R n \ M and y := γ x,ξ (t 0 ) ∈ M , for some ξ ∈ S x N and t 0 > 0, such that y is not a conjugate point to x along γ p,ξ . The main theorem of [5] is that, if the coefficient functions of the shape operators of generalized spheres Σ y,r,W are known in a neighborhood V of γ x,ξ ([0, t 0 )) and the wave speed v is known in (N \ M ) ∩ V , then the wave speed v can be determined in some neighborhood V ⊂ V of γ x,ξ ([0, t 0 ]).…”

Section: 23mentioning

confidence: 99%

Reconstruction of a compact manifold from the scattering data of internal sources

Lassas¹,

Saksala

Zhou

2018

Inverse Problems &Amp; Imaging

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Given a smooth non-trapping compact manifold with strictly convex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. These data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, the scattering data measured on the boundary determine the Riemannian manifold up to isometry.

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Reconstruction of a Conformally Euclidean Metric from Local Boundary Diffraction Travel Times

Cited by 15 publications

References 16 publications

Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction

Reconstruction of a compact manifold from the scattering data of internal sources

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