Next: CONCLUSIONS Up: Zhang: Time-to-depth conversion Previous: Constant offset imaging

# EFFICIENT COMPUTATIONS OF IMAGE RAYS

The mapping functions for the time-to-depth conversion are previously computed by tracing image rays. The method gives the mapping functions along the image rays. Mapping time-migrated image along the image rays subsequently gives the depth-migrated image that has to be interpolated onto a regular grid. For complicated velocity models, image ray may cross each other or not penetrate shadow zones; the interpolation is thus cumbersome and computationally expensive. In this section, I describe an efficient algorithm for computing the mapping functions of the image-ray corrections on a regular grid,

To compute the functions and in equation (1), we consider two partial differential equations. Because all image rays shoot vertically when leaving the surface, they are associated with a vertically incident plane wavefront. The value of function is the traveltime for the wavefront to propagate from the surface to a subsurface point (x,z), which satisfies the eikonal equation:

 (12)
with the initial condition .The value of function is the initial surface position of the image ray that eventually reaches a subsurface point (x,z). This function is thus constant along each image ray. Using the orthogonal relation between rays wavefronts in a isotropic medium, we can show that this function satisfies the partial differential equation as follow:

 (13)
with the initial condition .

Two partial differential equations (12) and (13) can be solved simultaneously by using an up-wind finite difference method (Zhang, 1991), and the results are samples of the two mapping functions on a regular grid. Thus, the image-ray corrections can be implemented with an algorithm summerized, as follow:

The functions and p(x,z;xs) in equations (6) and (11) can be computed in a similar way. Two partial differential equations

 (14)
and

 (15)
are solved with the point-source initial conditions. The implementation of image-ray corrections for prestack imaging is similar to poststack imaging.

Next: CONCLUSIONS Up: Zhang: Time-to-depth conversion Previous: Constant offset imaging
Stanford Exploration Project
11/18/1997