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

Abstract

We consider the wave equation $\left({\partial }_t^2-{\Delta }_g\right)u\left(t,x\right)=f\left(t,x\right)$, in ${ℝ}^n$, $u{|}_{{ℝ}_{-}\times {ℝ}^n}=0$, 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 $x_j,j\in {\mathbb{Z}}_{+}$, 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 $u\left(t,x\right)$ 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 $\partial M$.