Rather than trying to solve the full inverse problem given by equation (), in this section I will look for a diagonal operator such that

(94) |

can be applied to the migrated (adjoint)
image with equation (); however,
in their review of *L*2 migration, Ronen and Liner (2000) observe that
normalized migration is only a good substitute for full (iterative)
*L*2 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 *L*2 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

(95) |

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 *L*2 migrations.
Appendix explains how Kirchhoff normalization
schemes fit into the framework of equation ().

- Three choices of reference image
- Stabilizing the denominator
- Numerical comparison
- Computational cost

5/27/2001