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Gazdag's (1978) phaseshift
method for depth extrapolation of a seismic wavefield in the
frequencywavenumber domain is given by
 
(3) 
(Hale, 1992), where W is the wavefield, is the angular frequency,
k_{x} is the horizontal component of the wave number, z is the depth,
and is the angle defined, for isotropic media, by
where p_{x} is the horizontal component of slowness.
A(p_{x},z) is an amplitude factor that corrects for the v(z) influence and is
often omitted, partially because it goes to infinity as z approaches
the turning point, that depth where . This erroneous infinite amplitude is similar to
that encountered when performing Kirchhoff migration with WKBJ
amplitudes determined by Cartesiancoordinate ray tracing (nondynamic). I will
also omit this amplitude factor in the rest of this paper.
Equation (3) is the WKBJ solution (e.g., Aki and Richards,
1980, page 416) of the differential equation,
which is the wave equation, expressed in the
frequencywavenumber domain (the Helmholtz's equation).
For migration of zerooffset seismic data f(t,x), we identify
as the Fourier transformed data recorded
at the earth's surface (z=0).
Inverse Fourier transformation of equation (3) from wavenumber
k_{x} to distance x gives
 
(4) 
and then evaluation of the inverse Fourier transformation from frequency
to time t at t=0 yields the subsurface image
 

 (5) 
Equation (5) concisely summarizes the zerooffset phaseshift migration
method in isotropic media (Hale, 1992).
Output data after time migration, however, are usually presented as a function
of twoway vertical traveltime, , rather than depth. Substituting
and into
equations (4) and (5) yields
 
(6) 
and
 
(7) 
where .
Kitchenside (1991) and Gonzalez et.al. (1991) showed the earliest implementations
of poststack phaseshift migration in anisotropic media.
In VTI media, velocity varies with phase angle, , and, therefore,
where V is the phase velocity, and
where is the vertical Pwave velocity (=V_{P0}). Setting
(The velocitynormalized vertical component
of slowness), equation (6) becomes
 
(8) 
and equation (5) becomes
 
(9) 
which concisely summarizes the zerooffset phaseshift migration
method in VTI media.
Next: Prestack phaseshift migration
Up: Time migration
Previous: Time migration
Stanford Exploration Project
11/11/1997