General theory and comparative anatomy of dip moveout

Abstract

It is possible to derive a general formula for constant‐velocity, two‐dimensional dip moveout (DMO). This serves to unify the many published forms of DMO. Known results for common‐offset and shot profile DMO are special cases of the general formula. The analysis is based on the dip‐corrected NMO equation, and thus is a kinematic DMO theory. Using a logarithmic stretch of the time axis, efficient fast Fourier transform (FFT) common‐offset DMO algorithms can be derived. In the published versions, log variables are introduced into the NMO equation. It can be shown that the resulting impulse response departs from the DMO ellipse, which means that some dips have been improperly handled. A new log‐stretch formulation (exact log DMO) which preserves the DMO ellipse can be derived by starting with the analytic impulse response, rather than the NMO equation. Tests on field data indicate that exact log DMO handles shallow, steeply dipping events in agreement with Hale’s DMO, while the other log‐stretch algorithms degrade these events. In situations where constant‐velocity DMO is viable, exact log DMO can be used to create a Hale‐quality image at FFT speed.

Seismic structure and indicators of magma budget along the Southern East Pacific Rise

“…The across-axis reflection profiles were stacked using a velocity model determined from constant velocity stacks made every 2.5 km. Exact-log dip moveout (DMO) corrections [Liner, 1990] were applied prior to stacking. Finally, the profiles were migrated using a finite difference time 45 ø algorithm [Brysk, 1983] Interpolated from Scheirer and Macdonald [1993].…”

Theory of differential offset continuation

“…An analogous result can be obtained with the different definition of amplitude preservation proposed by Black et al (1993). In the time-andspace domain, the operator asymptotically analogous to (99) is found by applying either the stationary phase technique (Liner, 1990;Black et al, 1993) or Goldin's method of discontinuities (Goldin, 1988(Goldin, , 1990, which is the time-and-space analog of Beylkin's asymptotic inverse theory (Stovas and Fomel, 1996). The time-and-space asymptotic DMO operator takes the form…”

“…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).…”

“…As shown by Liner (1990), all of them but exact log DMO distort the geometry of reflection effects at large offsets. The distortion is caused by the implied approximations of the true phase function ψ. Bolondi's OC operator (Bolondi et al, 1982) …”

OC-seislet: Seislet transform construction with differential offset continuation

“…where The phase function ψ defined in equation A-7 corresponds to the analogous term in the exact-log DMO and AMO (Liner, 1990;Zhou et al, 1996;Biondi and Vlad, 2002).…”