An accurate formulation of log‐stretch dip moveout in the frequency‐wavenumber domain

Abstract

Dip moveout (DMO) processing is a partial prestack migration procedure that has been widely used in seismic data processing. The DMO process has been described in Deregowski (1986), Hale (1991) and Liner (1990). Many different DMO algorithms have been developed over the past decade. These algorithms have been designed to improve either the accuracy or the computational speed of the DMO process. Hale (1984) developed a method for performing DMO via Fourier transforms that is accurate for all reflector dips (assuming constant velocity). Hale’s method is computationally expensive because his DMO operator is temporally nonstationary, but its accuracy and simplicity have made it an industry standard. It has become a benchmark by which results from other DMO algorithms are judged. Of all the methods used to make the frequency‐domain DMO computationally efficient, the technique of logarithmic time stretching, first suggested in Bolondi et al. (1982), is widely used. After logarithmic stretching of the time axis, the DMO operator becomes temporally stationary which enables replacement of the slow temporal Fourier integration with a fast Fourier transform combined with a simple phase shift. Bale and Jakubowicz (1987) presented a log‐stretch DMO operator (hereafter referred to as Bale’s DMO) in the frequency‐wavenumber (F-K) domain without approximations, while Notfors and Godfrey (1987) suggested an approximate version of log‐stretch DMO operator (hereafter referred to as Notfors’s DMO). Surprisingly, Bale’s full log‐stretch DMO operator produces a less satisfactory impulse response than Notfors’s approximate log‐stretch DMO scheme (see Liner, 1990). Liner (1990) attributed this characteristic to the fact that Bale’s DMO derivation implicitly assumes that the Fourier transform frequency in the log‐stretch domain is not time‐dependant. He presented an exact log‐stretch DMO operator (hereafter referred to as Liner’s DMO) which was derived by transforming the time log‐stretched Hale’s (t, x) DMO impulse response into the Fourier domain. Its derivation is relatively complicated, but Liner has shown that his DMO does generate good DMO impulse responses.