Low frequency data are crucial for the success of full waveform inversion (FWI) because it defines the low wavenumber component of the velocity model and determines the kinemat- ics of the seismic waves at higher frequencies. When low frequency data are missing, FWI with inaccurate starting models suffers severely from cycle skipping and yields erroneous inversion results. However, acquiring low frequency data from the field is a challenging and expensive task. In this paper, we explore the possibilities of synthesizing low frequencies from high frequency data. Low frequency extrapolation from the band-limited recordings is feasible by first separating the seismic record into individual events and extrapolating their frequency content to a lower frequency band based on the nondispersive assumption. The extrapolated low frequency data are reasonable approximations to the true low frequency data, with higher fidelity in phase than in amplitude. To demonstrate the reliability of the extrapolated low frequencies for FWI, we initialize FWI at low frequencies using the extrapolated phase information. The resulting low wavenumber models are then used to initialize full waveform inversion at higher frequencies. We demonstrate that FWI with extrapolated low frequency data produces reasonable estimations of the low wavenumber velocity model, which further help FWI escape the local minima.

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