An analysis of 2D and 3D inverse scattering multiple attenuation
Authors: Ikelle Luc T. 
Published: October 2008

Abstract
Three dimensional formulations of free-surface multiple attenuation for multi-offset seismic data are well known. They are not yet used in practice because they require very dense source-receiver coverages which are still out of reach with existing seismic acquisition systems. We here present an alternative solution based on a "model and then subtract" multiple attenuation algorithm: inverse scattering multiple attenuation. We use the current 2D inverse scattering multiple attenuation algorithm to model free-surface multiples and then adjust for 3D wave propagation effects at the subtraction stage.
The 2D inverse scattering multiple attenuation algorithm predicts all free-surface multiples just like its 3D counterpart, but with some amplitude errors and small time shifts. The subtraction of predicted free-surface multiples from the data, which is generally carried out as an optimization for an inverse source signature, can be used to adjust for these 3D modeling errors as it is used today to compensate for other modelling errors such as the roughness of the sea surface. To accommodate for this new source of modelling errors, we assume that the source signature (or the inverse source signature) can vary as a function of directivity as well as of temporal frequency.
Numerical tests with synthetic and real data, including out of plane scattering, confirm that the 2D inverse scattering multiple attenuation Mgorithm with a source signature expressed as a function of directivity as well as of temporal frequency, can adequately attenuate a significant amount of 3D free-surface multiples.

INTRODUCTION
Recent developments on multiple attenuation, like the inverse scattering approach, have finally overcome the fundamental problem of attenuating free-surface multiples while preserving primary energy. So far, the application of these methods has been limited to 2D applications (Carvalho et al. (1991); Verschuur et al. (1992); Dragoset (1993); Ikelle et al. (1997a, 1997b); Weglein et al. (1997)). However, for these methods to become fully reliable for oil and gas exploration and production, we must develop their 3D cost-effective versions or alternatively we must construct accurate 2D to 3D corrections which will enable us to use 2D demultiple algorithms for 3D multi-offset data.
Solutions to 3D free-surface multiple attenuation problems are well known in theory, but they are not yet used in practice because they require very dense source-receiver coverage (in both horizontal directions) which is beyond the capability of present acquisition systems. Even if such coverages were to become possible, the application of these solutions would still require very large amounts of processing and data manipulation superior to anything we are doing today. One alternative solution is to construct 2D to 3D corrections which fully compensate for 3D propagation effects so that 2D demultiple algorithms can be used for 3D multi-offset data.
The 2D to 3D corrections to compensate for the 2D application are generally done through the ?t correction or Abel integral equations (Dampney (1971); Amundsen and Reitan (1994); Wapenaar et al. (1992)). These corrections are sufficient for reflection events over moderate dips. However, energy decays of diffraction events totally violate these rules.