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# Problem formulation

The post-stack velocity continuation process is governed by a partial differential equation in the domain, composed by the seismic image coordinates (midpoint x and vertical time t) and the additional velocity coordinate v. Neglecting some amplitude-correcting terms Fomel (1996), the equation takes the form Claerbout (1986b)
 (1)
Equation (1) is linear and belongs to the hyperbolic type. It describes a wave-type process with the velocity v acting as a time-like'' variable. Each constant-v slice of the function P(x,t,v) corresponds to an image with the corresponding constant velocity. The necessary boundary and initial conditions are
 (2)
where v0 is the starting velocity, T=0 for continuation to a smaller velocity and T is the largest time on the image (completely attenuated reflection energy) for continuation to a larger velocity. The first case corresponds to modeling''; the latter case, to seismic migration.

Mathematically, equations (1) and (2) define a Goursat-type problem Courant (1962). Its analytical solution can be constructed by a variation of the Riemann method in the form of an integral operator Fomel (1994, 1996):
 (3)
where , m=1 in the 2-D case, and m=2 in the 3-D case. In the case of continuation from zero velocity v0=0, 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 (1985), regarded as the most efficient 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 spectral approaches are discussed in the next two sections.

Next: Fourier approach Up: Fomel: Spectral velocity continuation Previous: Introduction
Stanford Exploration Project
7/5/1998