It is obvious that this equation in 3-D cannot be solved analytically for any kind of velocity model. Since the Eikonal equation relates the gradient of the traveltime to the velocity structure, several methods have been formulated to study the classical Eikonal equation (3) for a velocity structure sampled at discrete points in a 3-D space (3-D grid) and using finite-difference algorithms: M. Reshef and D. Kosloff 1986 integrate a finite-difference approximation of the Eikonal equation with a Runge-Kutta method; J. Vidale 1988 also uses a finite-difference algorithm but solves directly for traveltimes using a planar or circular wavefront extrapolation; and J. Van Trier and W. Symes 1990 derive a conservation law from the Eikonal equation depicting the first arrival time field.
Each differential term in Eikonal equation (14) can be approximated with finite difference to give the traveltime at a particular point of a grid with respect to the previously calculated traveltimes. To apply a finite-difference scheme to this equation, we need a 3-D grid of velocities, and since I use GOCAD to handle 3-D surfaces and structures, I present in this report a method to compute a 3-D grid of velocities from a GOCAD model Berlioux (1993a).
The aim of this project is to find how we can obtain a de-migration process from this Eikonal equation for time-migration. Nevertheless, to find an ``inverse'' process to the time migration method presented here, it is necessary to understand clearly how it is performed. I will therefore try to operate the time migration on several synthetic models of zero-offset data.