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

TL;DR

This paper demonstrates how a single measurement of wave propagation with a pseudorandom source can determine the scattering relation and metric inside a Riemannian manifold, even in non-simple cases.

Contribution

It introduces a method to recover the scattering relation and metric from wave data generated by a pseudorandom point source with specific weights.

Findings

Wave data determines the scattering relation on the boundary.

Appropriate weights allow tracing back the source of singularities.

Distances between sources and boundary points can be recovered.

Abstract

We consider the wave equation $(\p_t^2-\Delta_g)u(t,x)=f(t,x)$, in $\R^n$, $u|_{\R_-\times \R^n}=0$, where the metric $g=(g_{jk}(x))_{j,k=1}^n$ is known outside an open and bounded set $M\subset \R^n$ with smooth boundary $\p M$. We define a deterministic source $f(t,x)$ called the pseudorandom noise as a sum of point sources, $f(t,x)=\sum_{j=1}^\infty a_j\delta_{x_j}(x)\delta(t)$, where the points $x_j,\ j\in\Z_+$, form a dense set on $\p M$. We show that when the weights $a_j$ are chosen appropriately, $u|_{\R\times \p M}$ determines the scattering relation on $\p M$, 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 $a_j=\lambda^{-\lambda^{j}}$ 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 $(\R^n, g)$ between a source point $x_j$ and an arbitrary point $y \in \p M$. 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 non-simple we present a more detailed analysis of the wave fronts yielding the scattering relation on $\p M$.