Full waveform inversion starts being used as a standard stage of the seismic imaging workflow, at the exploration scale, for the reconstruction of high resolution wave velocity models. However, its successful application still relies on the estimation of an accurate enough initial velocity model, as well as on the design of a suitable hierarchical workflow, allowing to feed the inversion process progressively with data. These two requirements are mandatory to avoid the cycle skipping or phase ambiguity problem when comparing observed and synthetic data. This difficulty is due to the definition of the full waveform inversion problem as the least-squares minimization of the data misfit. The resulting misfit function has local minima which corresponds to the interpretation of the seismic data up to one or several phase-shifts. In this article, we review an alternative formulation of full waveform inversion based on the optimal transport distance we have proposed in recent studies. We propose to use a particular instance of the optimal transport problem, which is adapted to the interpretation of real seismic data, and for which we design an efficient low-complexity numerical strategy. Numerical results in 2D and 3D configurations (BP 2004, Chevron 2014 benchmark model, SEG/EAGE overthrust model) show that this reformulation should yield a more convex misfit function, less prone to cycle skipping. In this study, we present a simple illustration on the Marmousi model which illustrates how this new distance strongly relaxes the requirement on the initial model design. Starting from a rather simplistic approximation of the initial model, the method is able to reconstruct a meaningful estimation of the Marmousi model, while the standard least-squares formulation is trapped into a local, meaningless minimum.