(1) |

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 ,

(2) |

Rather than trying to solve the full inverse problem given by equation (1), I look for a diagonal operator such that

(3) |

can be applied to the migrated (adjoint)
image with equation (3); 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 these cases, can be used as a model-space
preconditioner to the full *L*2 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

(4) |

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 *L*2 migrations.
However, normalization schemes that work for Kirchhoff migrations are
not computationally feasible for recursive migration algorithms based
on downward-continuation.

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

4/29/2001