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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 *v*_{0} 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 *v*_{0}=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
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Stanford Exploration Project

7/5/1998