Most of the approaches designed to avoid cycle skipping in full-waveform inversion (FWI) involve calculating a sequence of inversions in a multiscale fashion. We have adopted an alternative strategy, which is inverting a sequence of different misfit functions in the time domain. This is an implicit multiscale approach in the sense that the used misfit functions are sensitive to different wavelengths, but all of the inversion steps use the same modeling algorithm and the same model grid. In the first and third inversion steps, the transmitted (early arrivals) and reflected (late arrivals) components of the wavefield envelopes are respectively fitted. The second step promotes a smooth transition between the first and third steps, by using the envelope of the complete waveform. Because fitting just the envelope of the reflected waves has a minor effect on the misfit function of the whole data set, the phases of the reflected waves are mostly fitted in the fourth step, which is based on the waveform misfit function preserving only the late arrivals. The third and fourth steps are of crucial importance to fit the reflected events. We test the sequential inversion approach with the Marmousi model using data sets with different frequencies, obtaining better estimates of the velocity field than those obtained with the classic FWI. The solutions obtained with classic FWI and sequential inversion approach degrade with a progressively higher peak frequency data set, but the classic FWI solution degrades more rapidly.

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