There are many potential applications for the AMO operator. In this paper we illustrate the use of AMO to reduce the computational cost of 3-D prestack depth imaging by partial stacking. We show that the application of AMO significantly improves, with respect to conventional methods, the result of partial stacking a 3-D marine data set over a range of offsets and azimuths. By reducing the amount of data to be migrated, partial stacking reduces the cost of 3-D prestack imaging, because the cost of migration is approximately proportional to the amount of data to be migrated. However, for partial stacking to enhance reflections and suppress noise, reflections need to be coherent across the traces to be stacked. Normal moveout (NMO) increases the coherency of reflections over offsets by a first-order correction of their traveltime. Therefore, NMO is often applied to traces before partial stacking Hanson and Witney (1995). However, a simple trace-to-trace transformation such as NMO is insufficient when the reflections have conflicting dips or diffractions occur. By correctly moving the dipping energy across midpoints, insures the preservation of all the dips in the data during partial stacking.
For reducing the computational cost of prestack migration, an alternative to partial stacking is to migrate only a subset of the available traces. To minimize the effects of data aliasing caused by the data subsampling, the input traces can be selected according to a quasi-random selection criterion Zhou and Schuster (1995). This method can be attractive in high signal-to-noise areas when all the data offsets are stacked during migration. However, when the signal-to-noise ratio is low, and/or when a prestack analysis of migration results is desired, either for velocity estimation or for AVO purposes, partial stacking is more robust with respect to noise, either coherent or uncoherent, because it uses all the available traces to improve the signal-to-noise ratio. A combination of the two methods, that is, the synthesis by AMO and partial stacking of a quasi-randomly sampled data set, has the potential of reducing the cost of imaging even further, but its testing is beyond the scope of this paper.
AMO can be considered a generalization of dip moveout (DMO) Deregowski and Rocca (1981); Hale (1984), in the sense that it transforms prestack data into equivalent data with an arbitrary offset and azimuth; in contrast, DMO is only capable of transforming non zero-offset data to zero-offset data. AMO is derived by analytically evaluating the operator that is equivalent to the the cascade of a 3-D prestack imaging operator with the corresponding 3-D prestack modeling. Any 3-D prestack imaging operator can be used for defining AMO. However, our analytical closed-form expression for AMO was derived only by use of constant-velocity DMO and constant velocity prestack migration and modeling. Goldin 1994 and Hubral et al. 1996 have recently presented a very general theory for cascading 3-D imaging operators; however the implementation of their theory would require an expensive numerical evaluation of the cascaded operators. The derivation of AMO from constant-velocity operators has the further advantage of making the kinematics of the operator velocity-independent. Nonwithstanding the constant-velocity assumption underlying its derivation, AMO can be effectively applied to data recorded on a complex velocity model, as the data example in this paper demonstrates. The first order effects of velocity variations are removed by NMO, which is applied before AMO, as it is also usually assumed when applying DMO. But AMO can successfully transform data to nearby offsets and azimuth, when velocity variations are too strong for DMO to transform data correctly all the way to zero-offset. Because AMO is correct to the first order, its results are accurate if the amounts of azimuth rotation and offset continuation are sufficiently small.
In addition to the data reduction application presented in this paper, the AMO operator has a wide spectrum of potential applications in the processing of 3-D seismic data. A promising application is the transformation of narrow-azimuth marine surveys to effective common-azimuth data. Common-azimuth data can be efficiently depth-imaged by new 3-D prestack migration methods Biondi and Palacharla (1996); Canning and Gardner (1996b). AMO can also improve the amplitude accuracy of prestack imaging wide-azimuth data recorded with irregular geometry by applying it before full prestack imaging to regularize the data geometry (). For some applications, such as the synthesis of 2-D lines from 3-D data, AMO is related to the 3-D data regularization method proposed by Canning and Gardner 1996a, which is based on the successive application of DMO and inverse DMO. However, AMO can be applied to a wider set of problems and data sets because the geometry of the output data is not constrained to be common-azimuth. Therefore, in addition to transforming marine surveys to effective common-azimuth data, AMO can be applied to more general data-regularization problems, as well as to data reduction and interpolation. Furthermore, the application of AMO as a single-step procedure has two benefits: substantial computational savings because of the small size of the AMO operator when azimuth rotation and offset continuation are small, and simplified handling of large data sets because only one pass through the data is required.
The next section introduces the AMO operator and analyzes the characteristics of the AMO impulse response. The second section introduces a transformation of the midpoint coordinates that is crucial to an efficient and accurate implementation of AMO as an integral operator. Finally, the third section of the paper presents the results obtained when AMO was applied to the partial stacking of a 3-D marine survey. The appendices present the derivations of the main analytical results presented in the paper; that is, the expressions for the kinematics, the amplitudes, and the aperture extent of the AMO impulse response.