Smooth objective functionals for seismic velocity inversion ∗ Christiaan

In seismic inverse scattering, the data are divided into subsets, each of which is used to reconstruct the medium discontinuities by linearized inversion. The reconstructions depend on an a priori unknown, smoothly varying background medium (velocity model). The semblance principle, which states that the images must agree, is the basis for the reconstruction of the background medium. Several estimators for the background medium have been proposed, based on optimization of different objective functionals. Use of local, gradient-based optimization methods requires that the functionals to be optimized are smooth. Such a smoothness requirement essentially implies that the objective functional is of differential semblance type. The proof involves a characterization of pseudodifferential operators as having L continuous repeated commutators with order 1 pseudodifferential operators.