To attempt to speed convergence, we can always change model-space variables from to through a linear operator , and solve the following new system for ,
Rather than trying to solve the full inverse problem given by equation (1), I look for a diagonal operator such that
can be applied to the migrated (adjoint) image with equation (3); 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 these cases, can be used as a model-space preconditioner to the full L2 problem, as described in the introduction.
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, , whose square is given by
Equation (4) with forms the basis for the first part of this paper. 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 [e.g. Chemingui (1999); Duquet et al. (2000); Slawson et al. (1995)] have been proposed for Kirchhoff migration operators. 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. However, normalization schemes that work for Kirchhoff migrations are not computationally feasible for recursive migration algorithms based on downward-continuation.