  In this paper, we study the inverse boundary value problem for the wave
equation with a view towards an explicit reconstruction procedure. We consider
both the anisotropic problem where the unknown is a general Riemannian metric
smoothly varying in a domain, and the isotropic problem where the metric is
conformal to the Euclidean metric. Our objective in both cases is to construct
the metric, using either the Neumann-to-Dirichlet (N-to-D) map or
Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we
construct the metric in the boundary normal (or semi-geodesic) coordinates via
reconstruction of the wave field in the interior of the domain. In the
isotropic case we can go further and construct the wave speed in the Euclidean
coordinates via reconstruction of the coordinate transformation from the
boundary normal coordinates to the Euclidean coordinates. Both cases utilize a
variant of the Boundary Control method, and work by probing the interior using
special boundary sources. We provide a computational experiment to demonstrate
our procedure in the isotropic case with N-to-D data.
