TY - JOUR

T1 - Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

AU - de Hoop, Maarten

AU - Holman, Sean

AU - Iversen, Einar

AU - Lassas, Matti

AU - Ursin, Bjørn

N1 - This research was supported by National Science Foundation grant CMG DMS- 1025318, the members of the Geo-Mathematical Imaging Group at Purdue University, the Finnish Centre of Excellence in Inverse Problems Research, Academy of Finland project COE 250215, the Research Council of Norway, and the VISTA project. The research was initialized at the Program on Inverse Problems and Applications at MSRI, Berkeley, during the Fall of 2010.

PY - 2014/10/30

Y1 - 2014/10/30

N2 - We 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 subset M and the pairs (t; Sigma) where Sigma subset 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, 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 subset 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 subset 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.

AB - We 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 subset M and the pairs (t; Sigma) where Sigma subset 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, 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 subset 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 subset 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.

U2 - 10.1016/j.matpur.2014.09.003

DO - 10.1016/j.matpur.2014.09.003

M3 - Article

SN - 0021-7824

JO - Journal de Mathematiques Pures et Appliquees

JF - Journal de Mathematiques Pures et Appliquees

ER -