We introduce a computationally efficient and robust method to regularize acquisition geometries of 3-D prestack seismic data before prestack migration. The proposed method is based on a formulation of the geometry regularization problem as a regularized least-squares problem. The model space of this least-squares problem is composed of uniformly sampled common offset-azimuth cubes. The regularization term fills the acquisition gaps by minimizing inconsistencies between cubes with similar offset and azimuth. To preserve the resolution of dipping events in the final image, the regularization term includes a transformation by Azimuth Moveout (AMO) of the common offset-azimuth cubes. The method is computationally efficient because we applied the AMO operator in the Fourier-domain, and we precondition the least-squares problem. Therefore, no iterative solution is needed and excellent results are obtained by applying the adjoint operator followed by a diagonal weighting in the model domain.
We tested the method on a 3-D land data set from South America. Subtle reflectivity features are better preserved after migration when the proposed method is employed as compared to more standard geometry regularization methods. Furthermore, a dipping event at the reservoir depth (more than 3 km) is better imaged using the AMO regularization as compared to a regularization operator that simply smoothes the data over offsets.
Irregular acquisition geometries are a serious impediment to the accurate imaging of the subsurface. When the data are irregularly sampled, images are often affected by amplitude artifacts and phase distortions, even if the imaging algorithm employed is designed to preserve amplitudes. Addressing this problem becomes more crucial when the goal is to use information contained in the image amplitudes. In these cases, the application of a simple imaging sequence that relies on standard `adjoint' imaging operators is likely to produce misleading results. Amplitude-preserving imaging of irregular geometries is thus one area of seismic processing that can greatly benefit from the application of inverse theory, and extensive research have been carried out in this direction.
There are two distinct approaches that can be used to apply inverse theory to the problem. The first one attempts to regularize the data geometry before migration Duijndam et al. (2000), while the second one attempts to correct for the irregular geometries during migration Albertin et al. (1999); Audebert (2000); Bloor et al. (1999); Duquet et al. (1998); Nemeth et al. (1999); Rousseau et al. (2000). The main strength of the latter approach is also its main weakness; it is model based; in particular, it depends on an accurate knowledge of the interval velocity model. If the model is well known, the methods based on the inversion of imaging operators have the potential of being accurate, because they exploit the intrinsic correlation between seismic traces recorded at different locations. However, when the uncertainties on the model are large, these methods can be also unreliable. Furthermore, a full prestack migration is an expensive process. Its substitution with an inversion process, even if iterative or approximate, might be beyond the practical reach.
In this paper we propose a method that has the advantages of both approaches. We regularize the data geometry before migration, but to fill the acquisition gaps we use a partial migration operator that exploits the intrinsic correlation between prestack seismic traces. The imaging operator is Azimuth Moveout (AMO) Biondi et al. (1998), that depends on a priory knowledge of RMS velocity. RMS velocity can be estimated from the data much more robustly than interval velocity.
Ronen 1987 was the first to use a partial migration operator to improve the estimate of a regularized data set. His method uses dip moveout (DMO) to regularize stacked cubes. Chemingui and Biondi have previously inverted AMO to create regularly sampled common offset-azimuth cubes. The main advantages of the method proposed in this paper over the previous methods are: a) it is based on a Fourier-domain implementation of AMO Vlad and Biondi (2001), as opposed to a Kirchhoff implementation, and thus it is computationally efficient and its implementation is straightforward, b) it uses AMO in the regularization equation (model styling) formulation of a regularized least-squares inverse problem, instead that in the modeling equation. In this formulation, the regularization term can be effectively preconditioned, with a substantial gain in computational efficiency, c) it approximates the solution of the preconditioned least-squares problem by applying normalization weights to the model vector after the application of the adjoint operator. Therefore, it avoids the costs and pitfalls of iterative solutions.
Our formulation of the geometry regularization problem as a regularized least-squares problem is similar to the formulation that Fomel presented in his Ph.D. thesis 2001. He uses a finite difference implementation of offset continuation where we use a Fourier implementation of AMO. These two operators are kinematically equivalent, and their computational efficiency is similar. However, the methods are different with respect to items b) and c) listed above. Our method should be more efficient because it explicitly preconditions the regularization term by inverting it. The inversion is fast because exploits the fact that the regularization matrix can be factored into the product of a block lower-diagonal matrix with a block upper-diagonal matrix, which are easily invertible by recursion. The preconditioning substantially improves the conditioning of the problem; therefore, a simple diagonal normalization of the model vector yields a good and fast solution to the problem.