Direct 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 velocity variations, can be formulated as a nonstationary combination filter that avoids the change of variables. The result is a direct Fourier-domain process that, for each wavenumber, applies a filter matrix to a vector of input frequency samples to create a vector of output frequency samples. The filter matrix is analytically specified in the mixed domain of input frequency and migrated time. It is moved to the domain of input frequency and output frequency by a fast Fourier transform. For constant velocity, the v(z) f-k algorithm recreates the Stolt method but without the usual artifacts related to complex-valued frequency domain interpolation. Though considerably slower than the Stolt method, vertical velocity variations, through an rms velocity (straight ray) assumption, are handled with no additional cost. Greater accuracy at slight additional expense is obtained by extending the method to a W KBJ phase shift integral. This has the same accuracy as recursive phase shift and c be made to handle turning waves in the same way. Nonstationary filter theory allows the algorithm to be easily reformulated in other domains. The full Fourier domain method offers interesting conceptual parallels to Stolt’s algorithm. However, unless a m ore efficient method of calculating the Fourier filter matrix can be found, the mixed-domain method will be faster. The mixed-domain nonstationary filter moves the input data from the Fourier domain to the migrated time domain as it migrates. It is faster because the migration filter is known analytically in the mixed domain. INTRODUCTION There are two major forms of migration based on Fourier transforms and many derivatives of these. Stolt (1978) presented a migration method, called f-k migration, that is “exact” for constant velocity. Stolt’s method is called direct in that it constructs the migrated image directly from the unmigrated without intermediate products. Stolt also suggested an approximate extension to variable velocity but this has not proven popular because it is of low accuracy. Gazdag (1978) presented the other form, called phase-shift, and that is recursive rather than direct. The recursion is in the form of a loop over sampled epth. At each step the wavefield from the depth just above is downward extrapolated by phase shift. For constant velocity, phase shift is easily shown to be mathematically identical to f-k migration, though the latt r is much quicker to compute. For v(z), phase shift is superior to f-k migration and is known to converge to a famous approximate wave equation solution called the WKBJ solution (Aki and Richards, 1980).