Spectral factorization of linear modeling operators

Most industrial-strength geophysics involves filtering the recorded data with the adjoint of the physical process that created it. If the true earth model is ${\bf m}$ and ${\bf A}$ is the physical forward modeling operator, then the image we compute is ${\bf A}' \,{\bf A} \,{\bf m}$. In the final part of this thesis, I form approximations to the ${\bf A}' \,{\bf A}$ operators associated with prestack depth migration that are diagonal in physical space. Since the approximations are diagonal, I can easily compute two factors simply by taking the square root of their diagonal elements. These factors are also diagonal, and hence easily invertible. These diagonal factors can then be applied directly to the migrated image to produce an image whose amplitude more closely resembles those of the true earth model. Alternatively, the factors can be applied in concert with the original operator, to produce a new dimensionless composite operator, which is more easily invertible with iterative linear solvers.

In Chapter , I discuss how to calculate the shot-illumination cheaply during shot-profile migration. I then show that for sparse-shot geometries with dense receiver coverage this weighting function can completely compensate for illumination problems on flat events.

Lastly, in Chapter , I compare alternative methods of computing appropriate diagonal model-space and data-space weighting functions appropriate for generic linear operators, and discuss how model-space and data-space weights can be calculated and applied simultaneously.