We address the problem of the estimation of the sound speed
inside the Earth from time migration. The velocities chosen in the process
of time migration, known as time migration velocities,
are used as the input.
We have derived theoretical
relationships between the
time migration velocities and the true seismic velocities in 2D and 3D.
We have developed PDE's which allow us to reconstruct
the seismic velocities from the time migration velocities.
These PDE's are elliptic, while the physical setting allows us to pose
only a Cauchy problem which is well-known to be ill-posed.
Nonetheless we have developed some inversion techniques, tested them
on a collection of synthetic examples in 2D and 3D
and applied to field data with severe lateral inhomogeniety.
The theoretical components of the work are based on the seismic
ray tracing theory. The numerical components include Dijkstra-like solvers,
level set methods, finite difference methods with stabilizing error terms
and techniques for data smoothing.