  We consider the inverse problem for the wave equation on a compact Riemannian
manifold or on a bounded domain of $\R^n$, and generalize the concept of {\em
domain of influence}. We present an efficient minimization algorithm to compute
the volume of a domain of influence using boundary measurements and
time-reversed boundary measurements. Moreover, we show that if the manifold is
simple, then the volumes of the domains of influence determine the manifold.
For a continuous real valued function $\tau$ on the boundary of the manifold,
the domain of influence is the set of those points on the manifold from which
the travel time to some boundary point $y$ is less than $\tau(y)$.
