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

Abstract

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...

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

“…1.1 This theorem follows from the recovery procedure that we have presented in sections 3 through 5. Indeed, from results in [8] applying step 1, which is described in section 3, for any x ∈ U we can recover the metric g in ( x, r) coordinates corresponding to Ψ t0 for any t 0 ∈ I. Then, in dimension 3 or higher, the argument of section 4 completes the proof, while in dimension 2 we must use the scalar curvature equation as described in section 5.…”

“…Recalling that F t0 n ( x, r) =γ t0 x (r), we also write r p j (t 0 ; x, r) = R p jnn (t 0 ; x, r), and for fixed t 0 8) for the directional curvature operator which we reconstruct in the first step of our procedure. Note that as with s, for any j r n j (r) = r j n (r) = 0, and so when we write r without indices we will actually be referring to the corresponding (n − 1) × (n − 1) matrix.…”

“…Following the geometric analysis of [8] we now describe the first step of the procedure which is itself broken up into a number of substeps below.…”

“…The data for our recovery are the matrix elements s k j ( x, t 0 ; 0, t), and their first three derivatives with respect to t, for 0 < t < t 0 and t not conjugate to 0 along γ t0 x . In [8], we obtain the following result: It is possible to uniquely determine the Riemannian metric g in a neighborhood of γ x0,η0 ([0, t 0 )) in Riemannian normal coordinates having origin at the point y t0 (this can be done for general metrics, not just ones which are conformally Euclidean). Here, we cast this result into an algorithm, and construct a conversion from the mentioned coordinates to Cartesian coordinates, which is the main contribution of this paper.…”

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

“…In [6] one considers the Spherical surface data consisting of the set U and the collection of all pairs (Σ, r) where Σ ⊂ U is a smooth (n−1) dimensional submanifold that can be written in the form…”