As the search for oil concentrates on ever more complicated areas of the subsurface, we find ourselves needing to balance the benefits of obtaining a better image with the cost of obtaining that better image. To obtain an ideal image, we would have to use an imaging operator that is the inverse of the physical operator propagating our seismic signal into the ground. However, imaging operators such as migration are adjoints rather than inverses Claerbout (1995), so in complex areas the resulting image may not be as good as it could be. Unfortunately, finding an operator that is an inverse in complex areas is almost impossible, so we generally approximate the inverse through a process like least-squares inversion Duquet and Marfurt (1999); Nemeth et al. (1999); Ronen and Liner (2000). When using such an iterative technique, the result of iterating to convergence can be thought of as the ``ideal'' model.
Iterative inversion schemes often have trouble with problems that are unstable or where the mapping operator has a null space Claerbout (1991). These issues can be overcome by regularizing the problem Fomel (1997); Harlan (1986); Tikhonov and Arsenin (1977). However, our regularization operators, which are usually roughening operators, tend to be small. Their influence at any single iteration is limited in range. When our mapping operator has large areas that do not correspond to any data locations this can be especially troublesome. A solution to this problem is to perform a change of variables, turning it into a preconditioned problem Fomel et al. (1997). Using the helix transform Claerbout (1998), we can apply the inverse of our small regularization operator, which will be a smoothing operator, whose influence extends a large distance. The advantage of this approach is that we quickly define our solution at all model points. The disadvantage of this approach is that our preconditioning operator dominates early iterations, creating a model that is often too low in frequency. What we ideally would like is a process where the solution is defined everywhere without the reduction in frequency content.
In Prucha and Biondi (2002), we presented a scheme that met these requirements by using the result of the preconditioned inversion as an initial model for a regularized inversion. From that example, and for the purposes of this paper, I will define an ``improved'' model as one that has a solution defined at every point and has a frequency content comparable to that of the ``ideal'' model. The combined inversion using preconditioning and regularization allows me to obtain an improved model with fewer iterations than would be needed using preconditioning or regularization alone. In this paper, I will take a closer look at the process of combined inversion.
In order to efficiently examine combined inversion with preconditioning and regularization (CIPR), this paper solves an interpolation problem which is much simpler than the imaging problem in Prucha and Biondi (2002). I will begin by explaining the constructed problem and the operator that is used for interpolation. Then I will present and discuss the results. Finally, I will explain my future plans for this combined inversion scheme.