Angle-Domain Common Image Gathers (ADCIGs) are an essential tool for Migration Velocity Analysis (MVA). We present a method for computing ADCIGs in 3-D from the results of wavefield-continuation migration. The proposed methodology can be applied before or after the imaging step in a migration procedure. When computed before imaging, 3-D ADCIGs are functions of the offset ray parameters ;we derive the geometric relationship that links the offset ray parameters to the aperture angle and the reflection azimuth .When computed after imaging, 3-D ADCIGs are directly produced as functions of and .
The mapping of the offset ray parameters into the angles depends on both the local dips and the local interval velocity; therefore, the transformation of ADCIGs computed before imaging into ADCIGs that are function of the actual angles is difficult in complex structure. In contrast, the computation of ADCIGs after imaging is efficient and accurate even in presence of complex structure and a heterogeneous velocity function. On the other hand, the estimation of the offset ray parameters is less sensitive to velocity errors than the estimation of the angles .When ADCIGs that are functions of the offset ray parameters are adequate for the application of interest (e.g. ray-based tomography), the computation of ADCIGs before imaging might be preferable.
Errors in the migration velocity cause the image point in the angle domain to shift along the normal to the apparent geological dip. By assuming stationary rays (i.e. small velocity errors), we derive a quantitative relationship between this normal shift and the traveltime perturbation caused by velocity errors. This relationship can be directly used in a MVA procedure to invert depth errors measured from ADCIGs into migration velocity updates. In this paper, we use it to derive an approximate 3-D Residual Moveout (RMO) function for measuring inconsistencies between the migrated images at different and .We tested the accuracy of our kinematic analysis on a 3-D synthetic data set with steeply dipping reflectors and a vertically varying propagation velocity. The tests confirm the accuracy of our analysis and illustrate the limitations of the straight-rays approximation underlying our derivation of the 3-D RMO function.
Wavefield-continuation migration methods have the potential of producing high-quality images even when complex overburden severely distorts the wavefield. However, as for all migration methods, the quality of the final image is strongly dependent on the accuracy of the velocity model. In complex area the velocity model is usually estimated in an iterative process called Migration Velocity Analysis (MVA). At each iteration of an MVA process the velocity is updated based on the information extracted from the current migrated image, and, in particular, from the Common Image Gathers (CIGs). The computation of accurate CIG is thus crucial for any MVA method. Most of the current MVA methods for wavefield-continuation migration employ Angle-Domain CIGs (ADCIGs) Biondi and Sava (1999); Clapp and Biondi (2000); Liu et al. (2001); Mosher et al. (2001).
The computation of ADCIGs with wavefield-continuation migration is based on a plane-waves decomposition of the wavefield either before imaging Mosher et al. (1997); Prucha et al. (1999); Xie and Wu (2002); de Bruin et al. (1990), or after imaging Biondi and Shan (2002); Rickett and Sava (2002); Sava and Fomel (2003). The methods previously presented in the literature are limited to the computation of 2-D ADCIGs that are functions of the aperture angle only. These methods have been applied to 3-D marine data by assuming zero-azimuth reflections; this assumption is approximately correct for marine data that have been acquired with a narrow-azimuth acquisition geometry.
In this paper we present the computation and the geometric interpretation of full 3-D ADCIGs that decompose the image not only according to the aperture angle , but also to the reflection azimuth .We extend to 3-D both methods for computing ADCIGs: before and after imaging. Our analysis shows that the 2-D equation that relates the in-line offset ray parameter pxh to the aperture angle is also valid in 3-D for zero-azimuth reflections, when used in combination with common-azimuth migration Biondi and Palacharla (1996). This result supports the previous use of 2-D ADCIGs computed before imaging for MVA of 3-D marine data Biondi and Vaillant (2000); Clapp (2001); Liu et al. (2001); Mosher et al. (2001). In contrast, the 2-D transformation to angle domain performed after imaging Sava and Fomel (2003) is not correct in 3-D, not even in the case of zero-azimuth reflections; its use in presence of cross-line dips leads to the overestimation of the reflection-aperture angle . In either case (before or after imaging), when the azimuth of the reflections is not oriented along the in-line direction, we must use the full 3-D methodology to obtain accurate ADCIGs.
The geometrical understanding we developed when generalizing 2-D ADCIGs to 3-D ADCIGs enables us to generalize the analysis of the kinematics of 2-D ADCIGs in the presence of migration-velocity errors that was presented by Biondi and Symes (2003). However their purely ray-theoretical analysis of 2-D ADCIGs cannot be directly extended because in 3-D the source and receiver rays are not guaranteed to be coplanar. Fortunately, a plane-wave interpretation of 3-D ADCIGs overcomes these difficulties because the two plane waves corresponding to the source and receiver rays define a plane of coplanarity that passes through the angle-domain image point. Once this plane is defined, the 2-D kinematic analysis carries over to 3-D and we can define a quantitative relationship between the reflectors' movement along the apparent geological dip and the traveltime perturbations caused by velocity errors. This relationship can be directly used in a tomographic inversion of 3-D ADCIGs; we use it to define a 3-D RMO function for measuring kinematic errors from 3-D ADCIGs.
ADCIGs have been introduced also for integral migration methods (e.g. Kirchhoff and Generalized Radon Transform); Xu et al. (2001) defined and applied them in 2-D, Brandsberg-Dahl et al. (2003) defined them in 3-D and applied them in 2.5-D, and Brandsberg-Dahl et al. (1999) applied 2-D ADCIGs to MVA. ADCIGs computed by Kirchhoff-like migration share many properties with ADCIGs computed by wavefield-continuation migration. However, in complex media, the two types of ADCIG have subtle kinematic differences, as clearly demonstrated by Stolk and Symes (2004). A thorough comparison of these two types of ADCIG would be of theoretical and practical value, but we consider it beyond the scope of this paper.