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Model-space weighting functions

For an over-determined system of equations, the inverse problem can be summarized as follows - given a linear forward modeling operator , and some recorded data ,estimate a model such that .The model that minimizes the expected error in predicted data is given by:
 (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)
When we find a solution, we can then recover the model estimate, .If we choose the operator such that , then even simply applying the adjoint ()will yield a good model estimate; furthermore, gradient-based solvers should converge to a solution of the new system rapidly in only a few iterations. The problem then becomes: what is a good choice of ?

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 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
 (4)
will have the same units as .Furthermore, will be the ideal weighting function if the reference model equals the true model and we have the correct modeling/migration operator.

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.

Next: Three choices of reference Up: Rickett: Normalized migration Previous: Introduction
Stanford Exploration Project
4/29/2001