I present a general methodology for computing and analyzing Angle Domain Common Image Gathers (ADCIGs) in conjunction with anisotropic wavefield-continuation migration. I demonstrate that the aperture angles estimated by transforming prestack images using slant stacks along the subsurface-offset axis are a good approximation of the phase aperture angles, and that they are exactly equal to the phase aperture angles for flat events in VTI media.
I introduce a generalization of the concept of migration impulse response for the computation of prestack images function of the subsurface offset that enables a straightforward analytical analysis of the reflector movements caused by perturbations in anisotropic parameters. This analysis shows that the Residual Moveout (RMO) in migrated ADCIGs is function of both the phase aperture angle and the group aperture angle. The dependency of the RMO function on the group angles adds some complexity to the RMO analysis because the computation of group angles from phase angles, which are measured from the ADCIGs, depends on the local background anisotropic velocity at the reflector point.
Several numerical examples demonstrate the accuracy of the RMO function predicted by my kinematic analysis, and in contrast, that the approximation of the group angles by the phase angles may lead to substantial errors for events reflected at wide aperture angles.
Angle Domain Common Image Gathers (ADCIGs) are a useful tool for updating migration velocity after wavefield-continuation migration Biondi and Sava (1999); Clapp and Biondi (2000). When the migration velocity is not accurate, the inconsistency of the migrated events along the aperture-angle axis is proportional to the migration velocity errors and provides the quantitative information necessary to update the velocity function.
All the methods for computing ADCIGs currently available in the literature are limited to isotropic migration; this is true for both the methods applied during downward continuation before imaging Prucha et al. (1999), and the methods applied on the prestack migrated image as a post-processing operator Biondi and Tisserant (2004); Rickett and Sava (2002); Sava and Fomel (2003). Similarly, the quantitative analysis of the residual moveout measured in ADCIGs caused by migration-velocity errors is also limited to the isotropic case Biondi and Symes (2003); Biondi and Tisserant (2004).
In this paper I generalize the methodologies for computing and analyzing ADCIGs to prestack images obtained by wavefield-continuation anisotropic migration. This work is practically motivated by two current trends in the seismic exploration industry: 1) data are recorded with increasingly long offsets, improving the resolution and reliability of the estimation of anisotropic parameters from surface data, 2) anisotropic prestack depth migration is increasingly being used in areas, like near or under salt bodies, where the image quality, and consequently the velocity estimation process, could benefit from the use of wavefield-continuation migration Bear et al. (2003); Sarkar and Tsvankin (2004a). In this perspective, other papers in this report present complementary work that is aimed at developing methods for cost-efficient anisotropic 3-D prestack migration Sen and Biondi (2005), and overturned-events anisotropic 3-D prestack migration Shan and Biondi (2005a,b).
Sarkar and Tsvankin (2003, 2004b) analyze the effect of velocity errors on offset-domain CIGs produced by Kirchhoff migration. They demonstrate the effectiveness of their method by successfully applying it to a West Africa data set Sarkar and Tsvankin (2004a). In this paper, I provide the basic analytical tools necessary to perform anisotropic migration velocity analysis for data sets that benefit from imaging with wavefield-continuation migration instead of Kirchhoff migration.
The main conceptual differences between isotropic ADCIGs and anisotropic ADCIGs are related to the fact that in anisotropic wave-propagation the phase angles and velocities are different from the group angles and velocities Tsvankin (2001). Therefore, the first question that I will address is: which aperture angles are we measuring in the ADCIGs? I demonstrate that the transformation to angle domain maps the reflection into the phase-angle domain. Strictly speaking this mapping is exact only for events normal to the isotropic axis of symmetry (e.g. flat events for Vertical Transverse Isotropic (VTI) media), because the presence of dips skews the estimates in ways similar to when geological-dips bias the estimation of aperture angles while computing ADCIGs for converted events Rosales and Biondi (2005); Rosales and Rickett (2001). Fortunately, in the anisotropic case, the biases caused by geological dips are less likely to create problems in practical applications than in the converted waves case. The simple numerical examples shown in this paper seem to indicate that, for realistic values of anisotropy, the errors caused by the geological dips is small and can be neglected. This approximation greatly simplifies the computation of ADCIGs and thus makes their application more attractive.
The second question I address is: is the residual moveout
caused by velocity errors only function of the phase angles,
or does it also depend on the group angles?
In the second part of this paper I demonstrate
that the residual moveout is function
of both the angles and that neglecting its
dependency on the group angles
leads to substantial inaccuracy in the predicted RMO function.