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

Helin, Tapio; Lassas, Matti; Oksanen, Lauri

doi:10.2140/apde.2012.5.887

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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

DOI: 10.2140/apde.2012.5.887

Abstract

We consider the wave equation $(\partial_{t}^{2} - Δ_{g}) u (t, x) = f (t, x)$ , in $ℝ^{n}$ , $u |_{ℝ_{-} \times ℝ^{n}} = 0$ , where the metric $g = {(g_{j k} (x))}_{j, k = 1}^{n}$ 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, $f (t, x) = \sum_{j = 1}^{\infty} a_{j} δ_{x_{j}} (x) δ (t)$ , where the points $x_{j}, j \in ℤ_{+}$ , form a dense set on $\partial M$ . We show that when the weights $a_{j}$ are chosen appropriately, $u |_{ℝ \times \partial M}$ 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 $u (t, x)$ contains the singularities produced by all point sources, but when $a_{j} = λ^{- λ^{j}}$ 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 $(ℝ^{n}, g)$ between a source point $x_{j}$ and an arbitrary point $y \in \partial M$ . In particular, if $(\bar{M}, g)$ is a simple Riemannian manifold and $g$ is conformally Euclidian in $\bar{M}$ , these distances are known to determine the metric $g$ in $M$ . In the case when $(\bar{M}, g)$ 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

Vol. 5, No. 5, 2012