Fourier migration for vertical velocity variations

The Stolt (f-k) migration algorithm is a direct (i.e. non-recursive) Fourier-domain technique based on a change of variables (or equivalently, a mapping) that converts the input spectrum to the output spectrum. The algorithm is simple and efficient but limited to constant velocity. A v(z)(f-k) migration method, capable of very high accuracy for vertical variations of velocity, can be formulated as a non-stationary combination filter that avoids the change of variables.
In this article, we compare the efficiency of Stolt (f-k) migration (eq. 1) with two non-stationary filters based on v(z)(f-k) migration methods.
(1)
The result of applying v(z)(f-k) migration is a direct Fourier-domain process that for each wavenumber applies a non-stationary migration filter to a vector of input frequency samples to create a vector of output frequency samples (eq. 2).
(2)

The filter matrix is analytically specified in the mixed domain of input frequency and migrated time. It can be moved to the full Fourier domain of input frequency and output frequency by a fast Fourier transform. When applied for seismic traces the v(z)(f-k) algorithm is slower than the Stolt method but without the usual artifacts related to complex-valued frequency domain interpolation.
We used two different schemes to consider the variations of velocity with depth. Vertical variations through an rms velocity (straight-ray) assumption are handled by v(z)(f-k) method with no additional cost. Greater accuracy at slight additional expense is obtained by extending the method to a WKBJ phase shift integral. We tested the efficiency of these methods on synthetic seismic records. Finally v(z)(f-k) method is applied to a real seismic section and the result are presented.