Building a good initial model for full-waveform inversion using frequency shift filter

Accurate initial model or available low-frequency data is an important factor in the success of full waveform inversion (FWI). The low-frequency helps determine the kinematical relevant components, low-wavenumber of the velocity model, which are in turn needed to avoid FWI trap in local minima or cycle-skipping. However, in the field, acquiring data that <5 Hz is a challenging and expensive task. We attempt to find the common point of low- and high-frequency signal, then utilize the high-frequency data to obtain the low-wavenumber velocity model. It is well known that the instantaneous amplitude envelope of a wavelet is invariant under frequency shift. This means that resolution is constant for a given frequency bandwidth, and independent of the actual values of the frequencies. Based on this property, we develop a frequency shift filter (FSF) to build the relationship between low- and high-frequency information with a constant frequency bandwidth. After that, we can use the high-frequency information to get a plausible recovery of the low-wavenumber velocity model. Numerical results using synthetic data from the Marmousi and layer model demonstrate that our proposed envelope misfit function based on the frequency shift filter can build an initial model with more accurate long-wavelength components, when low-frequency signals are absent in recorded data.