Three choices of reference image

The ideal reference image would be the true subsurface model. However, since we do not know what that is, we have to substitute an alternative model. I experiment with three practical alternatives, which I will denote ${\bf m}_{\rm 1}$, ${\bf m}_{\rm 2}$, and ${\bf m}_{\rm 3}$.

Claerbout and Nichols (1994) attribute to Bill Symes the idea of using the adjoint (migrated) image as the reference model. The rationale for this is that migration provides a robust estimate of the true model. As the first alternative I take Symes' suggestion, so that ${\bf m}_{\rm 1}={\bf A}' {\bf d}$. A potential problem with this choice is that it may depend too much on the data: the weighting function may be poorly determined in areas with little or no signal, and it will be difficult to separate data problems from operator problems.

The second alternative is to try a reference image of purely random numbers: ${\bf m}_{\rm 2}={\bf r}$, where ${\bf r}$ is a random vector. This is has the advantage of not being influenced by the data, but has the disadvantage that different realizations of ${\bf r}$ may produce different weighting functions.

The third alternative (denoted ${\bf m}_{\rm 2}$) that I consider is using a monochromatic reference image consisting of purely flat events: literally flat-event calibration as proposed by Black and Schleicher (1989) and discussed further in Appendix .