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| Velocity continuation by spectral methods | |
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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 and vertical time ) and the additional
velocity coordinate . Neglecting some amplitude-correcting terms
(Fomel, 1996), the equation takes the form
(Claerbout, 1986b)
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(1) |
Equation (1) is linear and belongs to the hyperbolic type. It
describes a wave-type process with the velocity acting as a
``time-like'' variable. Each constant- slice of the function
corresponds to an image with the corresponding constant
velocity. The necessary boundary and initial conditions are
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(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''; 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, 1996,1994):
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(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
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(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.
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| Velocity continuation by spectral methods | |
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Next: Fourier approach
Up: Fomel: Spectral velocity continuation
Previous: Introduction
2013-03-03