Rather than trying to solve the full inverse problem given by equation (), in this section I will look for a diagonal operator such that
can be applied to the migrated (adjoint) image with equation (); however, in their review of L2 migration, Ronen and Liner (2000) observe that normalized migration is only a good substitute for full (iterative) L2 migration in areas of high signal-to-noise. In areas of low signal-to-noise, can be used as a model-space preconditioner to the full L2 problem, as described in the previous section.
Claerbout and Nichols (1994) noticed that if we model and remigrate a reference image, the ratio between the reference image and the modeled/remigrated image will be a weighting function with the correct physical units. For example, the weighting function, , given by
Equation () forms the basis for the first part of this chapter. However, when following this approach, there are two important practical considerations to take into account: firstly, the choice of reference image, and secondly, the problem of dealing with zeros in the denominator.
Similar normalization schemes have been proposed for Kirchhoff migration operators [e.g. Biondi (1997); Chemingui (1999); Duquet et al. (2000); Slawson et al. (1995)]. In fact, both Nemeth et al. (1999) and Duquet et al. (2000) report success with using diagonal model-space weighting functions as preconditioners for Kirchhoff L2 migrations. Appendix explains how Kirchhoff normalization schemes fit into the framework of equation ().