Seismic full-waveform inversion (FWI), as proposed by Lailly and Tarantola in the 1980s (Lailly, 1983, Tarantola, 1987), consists of minimizing the misfit between observed and computed data. This data fitting approach is attractive because, theoretically, it allows us to directly invert for the Earth parameters, without using concepts like angle reflectivity and scale separation between background velocity and reflectivity. However, the applicability of waveform inversion is limited by at least two factors: the numerical cost of the solutions of the elastodynamic equations needed to obtain the computed data and the ill-posedness of the inverse problem because of the presence of local minima caused by cycle skipping between observed and computed data. The large increase in computing power became an enabler for this technique, even if it is still expensive to numerically solve the visco-elastic wave equations and most of the current waveform inversion implementations make severe approximations, notably the acoustic assumption in exploration geophysics. Because of the computing cost of solving the wave equations, the minimization of the misfit function has to be carried out with a local (gradient) optimization technique. The current implementations to invert 3D seismic data are then sensitive to the initial values of the Earth parameters. This sensitivity depends on the type of data (Gauthier et al., 1986, and Virieux and Operto, 2009, for a review), which means that improvements in acquisition greatly impact the quality of the results obtained with FWI.

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