Frequency-domain full waveform inversion with an angle-domain wavenumber filter

To mitigate the local minima problem in full waveform inversion, the iteration process can be carried out in a multi-scale way, with the inversion starting from large-scale perturbations and gradually changing to small-scale structures. However, this technique is often limited due to the lack of low-frequency information in the observed data. In a scattering process, the scale length of the velocity perturbation is not only related to the frequency, but also associated with the scattering angles. We introduce a scattering angle filter in frequency-domain full waveform inversion. Choosing proper scattering angles in the iteration process can control the wavenumber components entering into the velocity updating, and accomplish multi-scale waveform inversion. Using the slant-stacking method, we decompose the source- and receiver-side waves into local plane waves, from which the scattering angles between the incident and scattering directions are calculated and used as the constraint to determine the wavenumber components to be retrieved. Small scattering angles are related to large-scale model perturbations. Therefore, by filtering out large scattering angles, we can invert the large-scale background perturbations first. Then, by gradually increasing the band pass of scattering angles, we can retrieve fine-scale structures. Numerical examples demonstrate that when the initial model has large velocity errors and the low-frequency information is unavailable in the data, the multi-scale strategy based on the angle-domain wavenumber filter can largely improve the convergence in the initial iteration stage and make the searching towards the global minimum.