Data regularization is cast as a least-squares inversion problem. The model space is a five-dimensional (t,cmpx, cmpy, hx, hy) hypercube. The regularization minimizes the difference between various (t, cmpx, cmpy) cubes by applying a filter that acts in (hx,hy) plane. Azimuth Move-out is used transform the cubes to the same ( hx,hy) before applying the filter. The methodology is made efficient by a Fourier-domain implementation, and preconditioning the problem. The methodology, along with two approximations is demonstrated on 3-D dataset from the North Sea. The inversion result proves superior at a reasonable cost.
The irregularity of seismic data, particularly 3-D data, in both the model domain (in terms of subsurface position and reflection angle) and the data domain (in terms of midpoint, offset, and time) cause imaging problems. The most effectively family of multiple removal methods, SRME Verschuur et al. (1992) rely on data regularity. The standard marine acquisition technology results in empty bins in the cross-line direction.
Migration methods also desire a greater level of regularity than is often present in seismic surveys. There are two general approaches to deal with this problem. One approach is to treat the imaging problem as inverse problem. In this case is the adjoint of the migration operator. Duquet and Marfurt (1999); Prucha et al. (2000); Ronen and Liner (2000) cast the problem as such and then try to solve it with an iterative solver. These approaches have shown promise but are in many cases prohibitively expensive, and rely on an accurate subsurface velocity model.
Another approach is to try to regularize the data. AMO provides an effective regularization tool Biondi et al. (1998). AMO is generally applied as an adjoint to create a more regularized volume. These regularized volumes still often contain in `acquisition footprint' or more subtle amplitude effects. Chemingui (1999) used a logstretch transform to make the AMO operator stationary in time. He then cast the regularization problem as a frequency by frequency inversion problem using a Kirchoff-style AMO operator. He showed that acquisition footprint could be significantly reduced. The downside of this approach is relatively high cost of Kirchoff implementation and the difficulty with a frequency-by-frequency approach to a global inversion problem.
Biondi and Vlad (2001) built on the work of Fomel (2001). They set up an inverse problem relating the irregular input data to a regular model space. They regularized the problem by enforcing consistency between the various (t,cmpx,cmpy) cubes. The consistency took two forms. In the first a simple difference between two adjacent in-line offset cubes was minimized. In the second the difference was taken after transforming the cubes to the same offset AMO. For efficiency the model was preconditioned with the inverse of the regularization operator Fomel et al. (1997). Instead of solving the least squares inverse problem, the Hessian is approximated by a diagonal operator computed from a reference model Claerbout and Nichols (1994); Clapp (2003a); Rickett (2001).
In this paper I examine and extend the work in Biondi and Vlad (2001) and Clapp (2003b). I compare the result of using the AMO operator as an adjoint, using a diagonal operate to approximate the Hessian, and doing a full inverse. I show that applying the inverse proves to be significantly better. In the paper I begin the paper by going the general methodology, I then discuss how to implement it effectively on a Beowulf cluster.