I present a general methodology for computing Angle-Domain Common Image Gathers (ADCIGs) in conjunction with anisotropic wavefield-continuation migration. The method is based on the transformation of the prestack image from the subsurface-offset domain to the angle domain by use of slant stacks. The processing sequence is the same as for the computation of ADCIGs for the isotropic case, though the interpretation of the relationship between the slopes measured in the prestack image and the aperture angles are more complex. I demonstrate that the slopes measured by performing slant stack along the subsurface-offset axis of the prestack image are a good approximation of the phase aperture angles, and that they are exactly equal to the phase aperture angles for flat reflectors in Vertical Transverse Isotropic (VTI) media. In the general case of dipping reflectors, the true aperture angles can be easily computed as a function of the reflector dip and anisotropic slowness at the reflector.
I derive the relationships between phase angles and slopes measured in the prestack image from both a ``plane-wave'' viewpoint and a ``ray'' viewpoint. The two derivations are consistent with each other, as demonstrated by the fact that in the special case of flat reflectors they lead to exactly the same expression. The ray-theoretical derivation is based on a novel generalization of kinematic migration to the computation of prestack images as a function of the subsurface offset. This theoretical development leads to the linking of the kinematics in ADCIGs with migration-velocity errors, and thus it enables the use of ADCIGs for velocity estimation.
I apply the proposed method to the computation of ADCIGs from the prestack image obtained by anisotropic migration of a 2-D line extracted from a Gulf of Mexico 3-D data set. The analysis of the error introduced by neglecting the difference between the true phase aperture angle and the angles computed through slant stack shows that these errors are negligible and can be safely ignored in realistic situations. On the contrary, group aperture angles can be quite different from phase aperture angles and thus ignoring the distinction between these two angles can be detrimental to practical applications of ADCIGs.
Angle-Domain Common Image Gathers (ADCIGs) have become a common tool for analyzing prestack images obtained by wavefield-continuation migration. They can be used for both updating migration velocity after wavefield-continuation migration Biondi and Sava (1999); Clapp and Biondi (2000), as well as the analysis of amplitudes as a function of aperture angle Wang et al. (2005).
All the present methods for computing ADCIGs in conjunction with wavefield migration 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. In a companion paper Biondi (2005) I derive the expressions for computing residual moveout in ADCIGs as a function of errors in the anisotropic parameters used for migration. This work is practically motivated by two current trends in the seismic exploration industry: 1) data are recorded with increasingly longer offsets, widening the range of propagation angles and thus making the inclusion of anisotropic effects crucial to the complete focusing of reflections, 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 (2004).
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, I will first address the question of which aperture angles we are 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 biases the estimates. This bias is caused by the difference in propagation velocity between the incident and the reflected waves, and thus for VTI media it is small unless the anisotropy is strong and the dips steep. The real-data example shown in this paper indicates that, for realistic values of anisotropy, the errors caused by the geological dips are small and can be neglected. This approximation greatly simplifies the computation of ADCIGs and thus makes their application more attractive. When the accuracy of this approximation is not sufficient, the true aperture angles can be easily computed from the measured slopes in the prestack image by iteratively solving a system of two non-linear equations, which usually converge to the solution in only few iterations.
The methodology developed in this paper is limited to the
the imaging of acoustic data and thus is limited to the
acoustic approximation of elastic anisotropic wave propagation.
Furthermore, the numerical examples are limited to VTI media
defined by their vertical velocity
and two of the three Thomsen parameters: .However, the basic concepts have a general validity
and the generalization to more general anisotropic media,
such as Tilted Transverse Isotropic (TTI) media
should be fairly straightforward,
though outside the scope of this paper.