Gazdag's (1978) phase-shift method for depth extrapolation of a seismic wave-field in the frequency-wavenumber domain is given 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
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
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