In this chapter I introduce an alternative method to SRME and data space Radon demultiple. The method attenuates the multiples in the image space rather than in the data space. Prestack wave-equation depth migration accurately handles the complex wave propagation of primaries (, )3DI, to the extent that the presence of the multiples allows an accurate estimation of the migration velocities. The residual moveout of primaries in angle-domain common-image gathers (ADCIGs), therefore, is likely to be flat. It is not immediately obvious, however, what the residual moveout of the over-migrated multiples is in ADCIGs. In order to maximize the separation of primaries and multiples in the Radon domain, the kernel of the Radon transform should approximate the functional dependency of the residual moveout of the multiples as a function of the aperture angle as much as possible. () and () used the tangent-squared approximation of () assuming that the residual moveout of the multiples is the same as that of primaries migrated with faster velocity. The tangent-squared approximation, however, is a straight ray approximation that is appropriate for events, such as primaries, whose migration velocity is likely to be close to the actual propagation velocity. Multiples, on the other hand, given their large difference in velocity with respect to that of the primaries, are likely to be severely over-migrated and the straight ray approximation is not appropriate for them.
Primaries are migrated to zero subsurface offsets in SODCIGs and with flat moveout in ADCIGs (, )3DI. I show that 2D specular water-bottom multiples, even from dipping water-bottom, are focused by wave-equation migration similar to primaries. Hence, if migrated with constant water velocity, they too are mapped to zero subsurface-offset in SODCIGs and with flat moveout in ADCIGs. When migrated with the velocity of the primaries, specular water-bottom multiples are over-migrated and thus do not map to zero subsurface offsets. For off-end geometry, they are mapped to subsurface offsets with the opposite sign to that of their surface offsets. I derive the moveout curve of these multiples in SODCIGs and ADCIGs. I then take the special case of the residual moveout of a specular multiple from flat water-bottom in ADCIGs and use it to design a Radon transform that accounts for ray-bending of the multiple raypath at the multiple-generating interface. This Radon transform improves the separation of primaries and multiples in the Radon domain compared with a Radon transform based on the tangent-squared approximation.
Water-bottom diffracted multiples do not migrate as primary reflections. That is, they do not focus to zero subsurface offset even if migrated with constant water velocity (, )Alvarez05. These multiples migrate to both positive and negative subsurface offsets in SODCIGs depending on the relative position of the diffractor with respect to the receiver (for receiver-side diffracted multiples). In ADCIGs, these multiples have their apex at non-zero aperture angle, similar to their behavior in data space (CMP gathers) (, )Alvarez05. I propose to attenuate these multiples with an apex-shifted Radon transform similar to that used by () but replacing the tangent-squared Radon kernel with the new equation that I derive in this chapter for the residual moveout of the multiples in ADCIGS. Apex-shifted Radon Apex-shifted transforms were introduced for data interpolation by () and for attenuation of diffracted multiples in data space by ().
In this chapter I limit the application of the method to 2D data. In Chapter I show the extension of the method to 3D data using the methodology developed by () to compute 3D ADCIGs. Unlike the 2D ADCIGs used here, the 3D ADCIGs are a function not only of the aperture angle but also the reflection azimuth angle. More important, 3D specular multiples have their apex shifted shifted away from zero aperture angle if the water-bottom or the multiple-generating interface have crossline dip. These multiple, therefore, behave as diffracted multiples and can be attenuated with a modified version of the apex-shifted Radon transform presented in this chapter.