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2012 An inverse problem for the wave equation with one measurement and the pseudorandom source
Tapio Helin, Matti Lassas, Lauri Oksanen
Anal. PDE 5(5): 887-912 (2012). DOI: 10.2140/apde.2012.5.887


We consider the wave equation (t2Δg)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 Mn 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 yM. 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.


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Tapio Helin. Matti Lassas. Lauri Oksanen. "An inverse problem for the wave equation with one measurement and the pseudorandom source." Anal. PDE 5 (5) 887 - 912, 2012.


Received: 10 November 2010; Accepted: 26 May 2011; Published: 2012
First available in Project Euclid: 20 December 2017

zbMATH: 1329.35350
MathSciNet: MR3022845
Digital Object Identifier: 10.2140/apde.2012.5.887

Primary: 35R30, 58J32
Secondary: 35A18

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.5 • No. 5 • 2012
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