Full-waveform inversion with an exponential filter in wavenumber domain

The gradient of full-waveform inversion (FWI) can be decomposed into a tomographic part, which is smooth (lowwavenumbers) and an image part, which is more sharp (highwavenumbers). Most of the FWI procedures need to update the smooth part first and gradually include the details to mitigate the cycle-skipping problem. The scattering angle filter for the gradient is one way to isolate the low-wavenumber part of the gradient at the early stage. However, it could be costly because of the extensions necessary for precise scattering angle control. Due to the relationship between the wavenumber and the scattering angle, the large scattering angles correspond to the low wavenumber component, which is usually what we want to update in our initial inversion steps. Thus, we use an exponential filter in the wavenumber domain and update the model in the wavenumber domain from low to high. The numerical result indicates that the inversion benefits from the wavenumber domain filter.