Model-space weighting functions

Although the matrix ${\bf A}' {\bf A}$ is not strictly diagonal, a diagonal model-space weighting function may still reduce illumination-related amplitude artifacts caused by wave-propagation in the overburden or incomplete recording geometry.

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

$${\bf W}_{\rm m}^{2} \; {\bf A}' \, {\bf d} \approx {\bf m}_{L2}.$$ (94)

${\bf W}_{\rm m}$ can be applied to the migrated (adjoint) image with equation (); 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 areas of low signal-to-noise, ${\bf W}_{\rm m}$ can be used as a model-space preconditioner to the full L2 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, ${\bf W}_{\rm m}$, given by  

$${\bf W}_{\rm m}^{2} = \frac{ {\rm\bf diag} ({\bf m}_{\rm ref... ... \; {\bf m}_{\rm ref}) } \approx \frac{1}{{\bf A}'\, {\bf A}},$$ (95)

will have the 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 operator.

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 L2 migrations. Appendix  explains how Kirchhoff normalization schemes fit into the framework of equation ().