Short Note
AMO regularization: Effective approximate inverses for amplitude preservation

Robert G. Clapp

bob@sep.stanford.edu

Amplitude preservation in imaging is becoming increasingly important. 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) can have deleterious effects on amplitude behavior. There have been several general approaches to correct for this irregularity. The imaging problem is fundamentally an inverse problem, relating some model $\bf m$ to some data $\bf d$ through a linear operator $\bf L$, which 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.

The problem is further complicated in that many migration algorithms assume the data is lying on regular mesh (downward continuation and finite difference schemes for example). Biondi and Vlad (2001) dealt with the problem of mapping the irregular data to a regular mesh for downward continuation migration. 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 (time,cmp_x,cmp_y) cubes. The consistency took two forms. In the first a simple difference between two adjacent inline offset cubes was minimized. In the second the difference was taken after transforming the cubes to the same offset through Azimuth Moveout (AMO) Biondi et al. (1998). 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 (2003); Rickett (2001).

In this paper I examine and extend the work in Biondi and Vlad (2001). I show that approximating the inverse matrix with a simple diagonal operator is not sufficient. The resulting regular dataset has artificial amplitude anomalies. I replace the simple derivative operators with a filter that smooths along not only ${\rm offset}_x$, but also ${\rm offset}_y$.I conclude by discussing how the problem can be effectively parallelized and the computational and storage challenges of various estimation schemes.