We recognized that the envelope fluctuation and decay of seismic records carries ultra low-frequency (ULF, i.e., the frequency below the lowest frequency in the source spectrum) signals that can be used to estimate the long-wavelength velocity structure. We then developed envelope inversion for the recovery of low-wavenumber components of media (smooth background), so that the initial model dependence of waveform inversion can be reduced. We derived the misfit function and the corresponding gradient operator for envelope inversion. To understand the long-wavelength recovery by the envelope inversion, we developed a nonlinear seismic signal model, the modulation signal model, as the basis for retrieving the ULF data and studied the nonlinear scale separation by the envelope operator. To separate the envelope data from the wavefield data (envelope extraction), a demodulation operator (envelope operator) was applied to the waveform data. Numerical tests using synthetic data for the Marmousi model proved the validity and feasibility of the proposed approach. The final results of combined EI+WI (envelope-inversion for smooth background plus waveform-inversion for high-resolution velocity structure) indicated that it can deliver much improved results compared with regular full-waveform inversion (FWI) alone. Furthermore, to test the independence of the envelope to the source frequency band, we used a low-cut source wavelet (cut from 5 Hz below) to generate the synthetic data. The envelope inversion and the combined EI+WI showed no appreciable difference from the full-band source results. The proposed envelope inversion is also an efficient method with very little extra work compared with conventional FWI.

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