Full waveform inversion based on the non-parametric estimate of the probability distribution of the residuals

In an attempt to overcome the difficulties of the full waveform inversion (FWI), several alternative objective functions have been proposed over the last few years. Many of them are based on the assumption that the residuals (differences between modelled and observed seismic data) follow specific probability distributions when, in fact, the true probability distribution is unknown. This leads FWI to converge to an incorrect probability distribution if the assumed probability distribution is different from the real one and, consequently it may lead the FWI to achieve biased models of the subsurface. In this work, we propose an objective function which does not force the residuals to follow a specific probability distribution. Instead, we propose to use the non-parametric kernel density estimation technique (KDE) (which imposes the least possible assumptions about the residuals) to explore the probability distribution that may be more suitable. As evidenced by the results obtained in a synthetic model and in a typical P-wave velocity model of the Brazilian pre-salt fields, the proposed FWI reveals a greater potential to overcome more adverse situations (such as cycle-skipping) and also a lower sensitivity to noise in the observed data than conventional L-2- and L-1-norm objective functions and thus making it possible to obtain more accurate models of the subsurface. This greater potential is also illustrated by the smoother and less sinuous shape of the proposed objective function with fewer local minima compared with the conventional objective functions.