$\omega$-FWI: Robust full-waveform inversion with Fourier-based metric

Full-waveform inversion is a cutting-edge methodology for recovering high-resolution subsurface models. However, one of the main conventional full-waveform optimization problems challenges is cycle-skipping, usually leading us to an inaccurate local minimum model. A highly investigated track to alleviate this challenge involves designing a more global measure of misfit between the observed and modelled data beyond the sample-to-sample comparison. However, most of these approaches admit relatively smooth inversion results. Here, we introduce a novel misfit function based on the Fourier-based metric. This metric has been successfully applied in molecular physics for solving the Boltzmann equation, and we adapt it to full-waveform inversion. This misfit function exploits the power spectrum information between the modelled and observed data to provide low-wavenumber velocity model updates early, and more high resolution updates as we approach the solution. Thus, it also can be reformulated as a weighted $\ell_{2}$-norm in a quadratic case, which can be seen as a simple extension for conventional full-waveform inversion. Thus, despite its robustness to cycle skipping, it is capable of delivering high-resolution models synonymous to conventional FWI. Considering its frequency domain utilization, we refer to this inversion method as $\omega$-FWI. Through the synthetic Marmousi model example, this method successfully recovers an accurate velocity model, starting from a linearly increasing model even for the case of noisy observed data and the lack of low frequencies below 3 Hz and 5Hz, in which the conventional $\ell_{2}$-norm full-waveform inversion suffers from cycle skipping.