Long-wavelength FWI updates beyond cycle skipping

Full Waveform Inversion (FWI) success depends on producing seamless updates of the short- and long wavelength features missing in the starting velocity model while avoiding cycle skipping. The use of cross-correlation gradients in FWI can lead to updates with the reflectivity imprint (high-wavenumbers) before the long wavelength updates have been constructed. In addition, the use of L2-norm to measure the data misfit is prone to cycle skipping. This may conduct to a local-minimum if the data lacks of low frequency information and/or the initial model is far from the true earth model. We offer a solution to these two FWI fundamental problems that combines a robust implementation of the velocity sensitivity kernel and the optimal transport norm to measure the data misfit. The new scheme can retrieve the long wavelength updates and reduce the cycle skipping problem. The velocity kernel eliminates the migration isochrones emphasizing the long wavelength updates produced by the diving waves and the rabbit ears provided by reflections. The optimal transport norm accentuates those long-wavelength updates while minimizing the cycle skipping. We demonstrate the advantages of our implementation on synthetic and field data examples.