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.

Accurate and efficient shot‐gather dip moveout processing in the log‐stretch domain1

“…They provided a corrected derivation of the F-K DMO which will produce more reliable DMO results, especially with respect to the preservation of correct amplitude. Based on the DMO algorithm of Black et al (1993), Zhou, Mason andGreenhalgh (1995) presented an accurate F-K DMO formulation for common-offset sections in the time log-stretch domain. They concluded that the subtle flaw in Hale's derivation is largely responsible for the inaccurate DMO impulse responses of Bale and Jacubowicz's DMO scheme (1937).…”

Theory of differential offset continuation

“…The phase function ψ defined in (112) coincides precisely with the analogous term in Liner's exact log DMO (Liner, 1990), which provides the correct geometric properties of DMO. Similar expressions for the log-stretch phase factor ψ were derived in different ways by Zhou et al (1996) and Canning and Gardner (1996). However, the amplitude term F ( ) differs from the previously published ones because of the difference in the amplitude preservation properties.…”

Converted-wave azimuth moveout

“…This new PS-DMO operator distributes the amplitudes correctly along strong dip events. This operator is just the extension of Zhou et al (1996) for PS data. Using the improved operator presented by Rosales (2002) the filter F( , k, h i ) takes the form: Figure 4 compares the PP-AMO impulses response obtained with the filter in both equation (10) (top) and equation (11) (bottom).…”