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) |

(13) |

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*;*x*_{s}) in equations (6)
and (11) can be computed in a similar way. Two partial
differential equations

(14) |

(15) |

11/18/1997