Image warping waveform tomography

Imaging the change in physical parameters in the subsurface requires an estimate of the long wavelength component of the same parameters in order to reconstruct the kinematics of the waves propagating in the subsurface. The model is unknown and must be estimated from the same data. One can try to reconstruct the method by matching the recorded data with modeled waveforms extrapolated in a trial model of the medium. Alternatively, assuming a trial model, one can obtain a set of images of the reflectors from a number of seismic experiments and match the locations of the imaged interfaces. Apparent displacements between migrated images contain information about the velocity model and can be used for velocity analysis. A number of methods are available to estimate the displacement between images; in this paper we compare penalized local correlations and image warping. We show that the image warping vector field is a more reliable tool for estimating displacements between migrated images and leads to a more robust velocity analysis procedure. Using image warping, we define an optimization problem for migration velocity analysis. We derive the expressions for the objective function using the adjoint-method and we also propose an alternative, simpler solution for the computation of the model update based on the estimated perturbation of the imaged reflectivity and demigration. The second approach has straightforward implementation and reduced computational cost compared to the adjoint-state method calculations. We also discuss the weakness of migration velocity analysis in the migrated-shot domain in the case of highly refractive media, when the Born modeling operator is far from being unitary and thus its adjoint operator (the migration operator) poorly approximates the inverse.