Uniqueness of Obstacles in Riemannian Manifolds from Travelling Times

Suppose that $K$ and $L$ are two disjoint unions of strictly convex obstacles with the same set of travelling times, contained in an $n$-dimensional Riemannian manifold $M$ (where $n\ge 2$). Under some natural curvature conditions on $M$, and provided that no geodesic intersects more than two components in $K$ or $L$, we show that $K=L$.