Vol. 5, No. 5, 2012

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An inverse problem for the wave equation with one measurement and the pseudorandom source

Tapio Helin, Matti Lassas and Lauri Oksanen

Vol. 5 (2012), No. 5, 887–912
Abstract

We consider the wave equation (t2 Δg)u(t,x) = f(t,x), in n, u|×n = 0, where the metric g = (gjk(x))j,k=1n is known outside an open and bounded set M n with smooth boundary M. We define a source as a sum of point sources, f(t,x) = j=1ajδxj(x)δ(t), where the points xj,j +, form a dense set on M. We show that when the weights aj are chosen appropriately, u|×M determines the scattering relation on M, that is, it determines for all geodesics which pass through M the travel times together with the entering and exit points and directions. The wave u(t,x) contains the singularities produced by all point sources, but when aj = λλj for some λ > 1, we can trace back the point source that produced a given singularity in the data. This gives us the distance in (n,g) between a source point xj and an arbitrary point y M. In particular, if ( M¯,g) is a simple Riemannian manifold and g is conformally Euclidian in M¯, these distances are known to determine the metric g in M. In the case when ( M¯,g) is nonsimple, we present a more detailed analysis of the wave fronts yielding the scattering relation on M.

Keywords
inverse problems, wave equation, single measurement, pseudorandom, Gaussian beams, scattering relation
Mathematical Subject Classification 2010
Primary: 35R30, 58J32
Secondary: 35A18
Milestones
Received: 10 November 2010
Accepted: 26 May 2011
Published: 29 December 2012
Authors
Tapio Helin
Johann Radon Institute for Computational and Applied Mathematics (RICAM)
Austrian Academy of Sciences
Altenbergerstrasse 69
A-4040 Linz
Austria
Matti Lassas
Department of Mathematics and Statistics
University of Helsinki
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FI-00014 Helsinki
Finland
Lauri Oksanen
Department of Mathematics and Statistics
University of Helsinki
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FI-00014 Helsinki
Finland