  The article deals with a classical inverse problem: the computation of the
refractive index of a medium from ultrasound time-of-flight (TOF) measurements.
This problem is very popular in seismics but also for tomographic problems in
inhomogeneous media. For example ultrasound vector field tomography needs a
priori knowledge of the sound speed. According to Fermat's principle ultrasound
signals travel along geodesic curves of a Riemannian metric which is associated
with the refractive index. The inverse problem thus consists of determining the
index of refraction from integrals along geodesics curves associated with the
integrand leading to a nonlinear problem. In this article we describe a
numerical solver for this problem scheme based on an iterative minimization
method for an appropriate Tikhonov functional. The outcome of the method is a
stable approximation of the sought index of refraction as well as a
corresponding set of geodesic curves. We prove some analytical convergence
results for this method and demonstrate its performance by means of several
numerical experiments. Another novelty in this article is the explicit
representation of the backprojection operator for the ray transform in
Riemannian geometry and its numerical realization relying on a corresponding
phase function that is determined by the metric. This gives a natural extension
of the conventional backprojection from 2D computerized tomography to
inhomogeneous geometries.
