    Time migration velocity analysis by velocity continuation  Next: Finite-difference approach Up: Fomel: Velocity continuation Previous: Introduction

# Numerical velocity continuation in the post-stack domain

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) (1)

Equation (1) is linear and belongs to the hyperbolic type. It describes a wave-type process with the velocity acting as a propagation variable. Each constant- slice of the function corresponds to an image with the corresponding constant velocity. The necessary boundary and initial conditions are (2)

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): (3)

where , in the 2-D case, and in the 3-D case. In the case of continuation from zero velocity , operator (3) is equivalent (up to the amplitude weighting) to conventional Kirchoff time migration (Schneider, 1978). Similarly, in the frequency-wavenumber domain, velocity continuation takes the form (4)

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.

Subsections    Time migration velocity analysis by velocity continuation  Next: Finite-difference approach Up: Fomel: Velocity continuation Previous: Introduction

2013-03-03