A source of low frequencies - Smoothing the wavefield

Waveform inversion is usually plagued with the high nonlinearity of the wavefield in complex velocity models. The Born series exposes the fact that smoother perturbations in the velocity model result in smoother more linear changes in the wavefield. As a result, we apply a lateral smoothing operator to the Born scattering formulation, and under the assumption that background velocity does not change laterally within the smoothing window, the lateral smoothing of the scattered wavefield can be linearly dependent on an equivalent smoothing to the velocity perturbation. As a result, using full waveform inversion (FWI), we smooth the data along the shot direction for the same offset and expect the resulting velocity gradients to be as smooth. This suggests an FWI implementation that starts with a vertically varying velocity model (i.e. from a well), and based on this linear relation, we smooth the data over a large window (maybe over the whole survey), and then slowly reduce such smoothing as we converge to conventional FWI without any smoothing. An application of this approach on the Marmousi model for a data frequency of 3 Hz demonstrates its ability to produce clean smooth results.