Time migration velocity analysis by velocity continuation |

The post-stack velocity continuation process is governed by a partial
differential equation in the domain, composed by the seismic image
coordinates (midpoint and vertical time ) and the additional
velocity coordinate . Neglecting some amplitude-correcting terms
(Fomel, 2003), the equation takes the form
(Claerbout, 1986)

where is the starting velocity, for continuation to a smaller velocity and is the largest time on the image (completely attenuated reflection energy) for continuation to a larger velocity. The first case corresponds to ``modeling'' (demigration); the latter case, to seismic migration.

Mathematically, equations (1) and (2) define a
Goursat-type problem (Courant and Hilbert, 1989). Its analytical solution can be
constructed by a variation of the Riemann method in the form of an
integral operator (Fomel, 2003,1994):

which is equivalent (up to scaling coefficients) to Stolt migration (Stolt, 1978), regarded as the most efficient constant-velocity migration method.

If our task is to create many constant-velocity slices, there are
other ways to construct the solution of problem (1-2).
Two alternative approaches are discussed in the next two
subsections.

Time migration velocity analysis by velocity continuation |

2013-03-03