Normalization of Kirchhoff operators

Kirchhoff operators are more complex than interpolators: each output is a weighted-sum of an entire summation surface, rather than just a few nearby points. The process of calculating rows of ${\bf A}_{\rm Kirch}' \, {\bf A}_{\rm Kirch}$ by modeling and migrating a reference model full of ones will give the ideal weighting function if the true model is smooth on the scale of an impulse response of ${\bf A}_{\rm Kirch}' \, {\bf A}_{\rm Kirch}$ (the familiar bow-tie shape). This is true for both time and depth migrations.

The validity of operator fold is less well-defined. Dimensionally, it has the units of ${\bf A}_{\rm Kirch}^{-1}$ as opposed than ${\bf A}_{\rm Kirch}^{-2}$ which one would expect. Furthermore, if we generate synthetic data over a model-space full of ones, we no longer get simply a data-space full of ones. Although for a v(z) model, the data will be a constant function of midpoint, it may vary as a function of offset and time depending on the implementation of the modeling algorithm.

We can still ask, however, what reference model would generate a constant function in data-space? Restricting ourselves to v(z) problems (time-migration, DMO, AMO etc.), it is clear that the reference model would have to be constant along any midpoint axes, but may potentially vary along the time and offset (or angle) axes. However, rapid variations along these axes are unlikely.

Starting with such a reference model, we can go ahead and define the row-based operator fold as the migration response to a data-space full of ones: ${\bf A}_{\rm Kirch}' \, {\bf 1}$.The correct fold [from equation ()] will be the result of dividing this by the original model; but since the original model did not vary along the midpoint axis, and presumably varied slowly along the other axes, the operator fold should be close to the correct fold. In fact, because the original model was a constant value of midpoint, the relative fold of flat events at constant offset will remain unchanged. Hence calculating operator fold via a data-space full of ones is in some sense equivalent to ``flat-event calibration'' for time migration operators Chemingui (1999).

For Kirchhoff depth migration operators, it is not clear what reference model will produce a data-space full of ones. However, the operator fold calculated by migrating such a dataset will still captures the effects of an incomplete recording geometry; and so normalization by operator fold may still be useful (albeit approximate) technique.

Duquet et al. (2000) calculate illumination appropriate for Kirchhoff depth imaging by independently modeling and migrating point scatterers at every location in the model-space. This explicitly evaluates the diagonal elements of ${\bf A}_{\rm Kirch}' \, {\bf A}_{\rm Kirch}$.As for the case of linear interpolation, this approach is not equivalent to normalization by the operator fold, which includes a summation along the rows of ${\bf A}_{\rm Kirch}'$. Considering only the diagonal of ${\bf A}_{\rm Kirch}' \, {\bf A}_{\rm Kirch}$ will lead to similar problems that are observed in Figures  and . The diagonal may approach the ideal weighting function if the true model consists of point scatterers that are isolated on the scale of the bow-tie (${\bf A}_{\rm Kirch}' \, {\bf A}_{\rm Kirch}$)impulse responses; however, for any kind of reflectors, as opposed to diffractors, this situation is not realized.