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## Zero-offset phase-shift migration

Gazdag's (1978) phase-shift method for depth extrapolation of a seismic wave-field in the frequency-wavenumber domain is given by
 (3)
(Hale, 1992), where W is the wave-field, is the angular frequency, kx is the horizontal component of the wave number, z is the depth, and is the angle defined, for isotropic media, by

where px is the horizontal component of slowness. A(px,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 Cartesian-coordinate ray tracing (non-dynamic). 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 frequency-wavenumber domain (the Helmholtz's equation).

For migration of zero-offset 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 kx 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 zero-offset phase-shift migration method in isotropic media (Hale, 1992).

Output data after time migration, however, are usually presented as a function of two-way 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 phase-shift 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 P-wave velocity (=VP0). Setting (The velocity-normalized vertical component of slowness), equation (6) becomes
 (8)
and equation (5) becomes
 (9)
which concisely summarizes the zero-offset phase-shift migration method in VTI media.

Next: Prestack phase-shift migration Up: Time migration Previous: Time migration
Stanford Exploration Project
11/11/1997