  We analyze the inverse problem, originally formulated by Dix in geophysics,
of reconstructing the wave speed inside a domain from boundary measurements
associated with the single scattering of seismic waves. We consider a domain
$\tilde M$ with a varying and possibly anisotropic wave speed which we model as
a Riemannian metric $g$. For our data, we assume that $\tilde M$ contains a
dense set of point scatterers and that in a subset $U\subset \tilde M$,
modeling a region containing measurement devices, we can measure the wave
fronts of the single scattered waves diffracted from the point scatterers. The
inverse problem we study is to recover the metric $g$ in local coordinates
anywhere on a set $M \subset \tilde M$ up to an isometry (i.e. we recover the
isometry type of $M$). To do this we show that the shape operators related to
wave fronts produced by the point scatterers within $\tilde M$ satisfy a
certain system of differential equations which may be solved along geodesics of
the metric. In this way, assuming we know $g$ as well as the shape operator of
the wave fronts in the region $U$, we may recover $g$ in certain coordinate
systems (e.g. Riemannian normal coordinates centered at point scatterers). This
generalizes the well-known geophysical method of Dix to metrics which may
depend on all spatial variables and be anisotropic. In particular, the novelty
of this solution lies in the fact that it can be used to reconstruct the metric
also in the presence of the caustics.
