Multiple attenuation using Abelian group theory

Lewei Mo

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ABSTRACT

This paper assumes a flat layered earth model and offset recording geometry. We apply velocity transform on the space-time shot gather into a domain, each of whose attributes forms an Abelian group. An attribute that is addable irrespective of ordering forms an Abelian (or commutative) group. Several attributes form Abelian groups. We implement two of them at this stage, the zero-offset two way travel time T, and Dix's parameter T.V_nmo². Addability of attributes dictates that in this domain sets of multiples - both primary and multiples - lie at even intervals along parallel straight lines. The next most important property of an Abelian group is closure; in other words, linear combination of different Abelian group attributes also forms an Abelian group. We form linear combination of the T and T.V_nmo² to steer the waterbottom multiples and peglegs straight down the T axis. The periodic character of reverberation events in this domain helps design predictive multiple attenuation operators. We apply predictive deconvolution on this domain to attenuate the waterbottom multiples and peglegs. Then we conjugate velocity transform the deconned result back to the space-time domain. We test the method on a Solid elastic model synthetic dataset and on real data. The results show the efficacy of the method to attenuate waterbottom multiples and peglegs simultaneously. Theoretically, recursive application of the procedure will attenuate all deep earth intrabed multiples. However, problems remain for this method in velocity transform, and we point out them in this paper.