Born theory of wave-equation dip moveout

Wave-equation dip moveout (DMO) addresses the DMO amplitude problem of finding an algorithm which faithfully preserves angular reflectivity while processing data to zero offset. Only three fundamentally different theoretical approaches to the DMO amplitude problem have been proposed: (1) mathematical decomposition of a prestack migration operator; (2) intuitively accounting for specific amplitude factors; and (3) cascading operators for prestack migration (or inversion) and zerooffset forward modeling. Pursuing the cascaded operator method, wave-equation DMO for shot profiles has been developed. In this approach, a prestack common-shot inversion operator is combined with a zero-offset modeling operator. Both integral operators are theoretically based on theBorn asymptotic solution to the point-source, scalar wave equation. This total process, termed Born DMO, simultaneously accomplishes geometric spreading corrections, NMO, and DMO in an amplitudepreserving manner. The theory is for constant velocity and density, but variable velocity can be approximately incorporated. Common-shot Born DMO can be analytically verified by using Kirchhoff scattering data for a horizontal plane. In this analytic test, Born DMO yields the correct zero-offset reflector with amplitude proportional to the angular reflection coefficient. Numerical tests of common-shot Born DMO on synthetic data suggest that angular reflectivity is successfully preserved. In those situations where amplitude preservation is important, Born DMO is an alternative to conventional NM0 + DMO processing. INTRODUCTION carefully designed with amplitude preservation in mind, then amplitude interpretation becomes suspect. suggested for common offset (Deregowski and Rocca, 198 1; Many forms of kinematic dip moveout (DMO) have been Hale, 1984; Berg, 1985; Notfors and Godfrey, 1987; Bale and Jakubowicz, 1987; Fore1 and Gardner, 1987; Liner and Bleistein, 1988), and Biondi and Ronen (1987) have extended this work to common-shot DMO. For all their differences, these forms of DMO share a common origin in that they are ultimately based on the dip-corrected NM0 equation. Kinematic DMO, by definition, addresses only the issue of traveltimes. Here, I am concerned with the question of DMO amplitude. thing to amplitude. Some writers explicitly address this issue, while others, more interested in the imaging aspects, Every DMO process, kinematic or otherwise, does somedo not. There has been much recent work on DMO amplitude (Jorden, 1987; Beasley and Mobley, 1988; Black and Egan, 1988; Gardner and Fore], 1988; Liner, 1989). Although there may seem to be as many approaches as authors, all of this work actually follows from three fundamentally different approaches to the DMO amplitude problem. The first approach is operator splitting. Beginning with Yilmaz and Claerbout (1979), there have been several careful derivations of DMO from prestack full migration (PSFM). The idea is to manipulate the PSFM operator so that something between NM0 and poststack migration is isolated, and identify this as DMO. Many PSFM operators are known, but the one universally used in this approach to DMO amplitude should be of concern because, in practice, data amplitude is interpreted after DMO. This interpretation can be prestack amplitude-versus-offset (AVO) analysis or even poststack, postmigration as in bright spot work. If the entire processing stream, including DMO, is not Presented in part at the 59th Annual International Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor August 29, 1989; revised manuscript received August 9, 1990. *Formerly Golden Geophysical, 1748 Cole Blvd., Ste. 265, Golden, CO 80401; presently Department of Geosciences, University of Tulsa, 600 South College, Tulsa, OK 74104-3189.