The determination of background velocity by Full Waveform Inversion (FWI) is known to be hampered by the local minima of the data misfit caused by the phase shifts associated with background perturbations. Attraction basins for the underlying optimization problems can be computed around any nominal velocity model and guarantee that the misfit functional has only one (global) minimum. The attraction basins are further associated with tolerable error levels representing the maximal allowed distance between the (observed) data and the simulations (i.e., the acceptable noise level). The estimates are defined a priori, and only require the computation of (possibly many) first- and second-order directional derivatives of the (model to synthetic) forward map. The geometry of the search direction and the frequency influence the size of the attraction basins, and complex frequency can be used to enlarge the basins. The size of the attraction basins for the perturbation of background velocities in the classical FWI (global model parametrization) and the data-space reflectivity reformulation (MBTT) are compared: the MBTT reformulation increases substantially the size of the attraction basins (by a factor of four to fifteen). Practically, this reformulation compensates for the lack of low-frequency data. Our analysis provides guidelines for a successful implementation of the MBTT reformulation.