#### 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 $\left({\partial }_{t}^{2}-{\Delta }_{g}\right)u\left(t,x\right)=f\left(t,x\right)$, in ${ℝ}^{n}$, $u{|}_{{ℝ}_{-}×{ℝ}^{n}}=0$, where the metric $g={\left({g}_{jk}\left(x\right)\right)}_{j,k=1}^{n}$ is known outside an open and bounded set $M\subset {ℝ}^{n}$ with smooth boundary $\partial M$. We define a source as a sum of point sources, $f\left(t,x\right)={\sum }_{j=1}^{\infty }{a}_{j}{\delta }_{{x}_{j}}\left(x\right)\delta \left(t\right)$, where the points ${x}_{j},\phantom{\rule{1em}{0ex}}j\in {ℤ}_{+}$, form a dense set on $\partial M$. We show that when the weights ${a}_{j}$ are chosen appropriately, $u{|}_{ℝ×\partial M}$ determines the scattering relation on $\partial M$, 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 $u\left(t,x\right)$ contains the singularities produced by all point sources, but when ${a}_{j}={\lambda }^{-{\lambda }^{j}}$ for some $\lambda >1$, we can trace back the point source that produced a given singularity in the data. This gives us the distance in $\left({ℝ}^{n},g\right)$ between a source point ${x}_{j}$ and an arbitrary point $y\in \partial M$. In particular, if $\left(\overline{M},g\right)$ is a simple Riemannian manifold and $g$ is conformally Euclidian in $\overline{M}$, these distances are known to determine the metric $g$ in $M$. In the case when $\left(\overline{M},g\right)$ is nonsimple, we present a more detailed analysis of the wave fronts yielding the scattering relation on $\partial M$.

##### 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