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 ${\bf A}$, and some recorded data ${\bf d}$,estimate a model ${\bf m}$ such that ${\bf A} \, {\bf m} \approx {\bf d}$.The model that minimizes the expected error in predicted data is given by:  

$${\bf m}_{L2} = ({\bf A}'\, {\bf A})^{-1} \; {\bf A}' \, {\bf d}.$$ (1)

To attempt to speed convergence, we can always change model-space variables from ${\bf m}$ to ${\bf x}$ through a linear operator ${\bf P}$, and solve the following new system for ${\bf x}$,

$${\bf d}={\bf A \, P \, x} = {\bf B \, x}.$$ (2)

When we find a solution, we can then recover the model estimate, ${\bf m}_{L2}={\bf P} \, {\bf x}$.If we choose the operator ${\bf P}$ such that ${\bf B}' {\bf B} \approx {\bf I}$, then even simply applying the adjoint (${\bf B}'$)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 ${\bf P}$?

Rather than trying to solve the full inverse problem given by equation (1), I look for a diagonal operator ${\bf W}_{\rm m}$ such that  

$${\bf W}_{\rm m}^{2} \; {\bf A}' \, {\bf d} \approx {\bf m}_{L2}.$$ (3)

${\bf W}_{\rm m}$ 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, ${\bf W}_{\rm m}$ 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, ${\bf W}_{\rm m}$, whose square is given by  

$${\bf W}_{\rm m}^{2} = \frac{ {\rm\bf diag} ({\bf m}_{\rm ref... ...m}_{\rm ref}) } \approx \left( {\bf A}'\, {\bf A}\right)^{-1},$$ (4)

will have the same units as ${\bf A}^{-1}$.Furthermore, ${\bf W}_{\rm m}^{2}$ 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.