We show in this paper how the application of convex duality leads to various reformulations of the classical non optimizable least squares approach to waveform prestack inversion. These reformulations search all for i) a background velocity model and ii) a reflectivity model defined in the time domain, and linked to the usual depth reflectivity model by prestack depth migration. We show that the data themselve are a good approximation of this optimal time reflectivity unknown when a quantitative migration is used.\\ \indent Hence duality provides a synthetic view of various approaches to prestack inversion, building up a bridge between concepts as different a priori as waveform inversion by minimization of data misfit and migration velocity analysis.\\ \indent From a pratical point of view, it provides objective functions for the determination of the velocity background (data misfit for the Migration Based Travel Time reformulation, norm of the stack of the prestack migrated sections for the Multiple Migration reformulation), whose evaluation and optimization by local gradient technique is now feasible, thus eliminating the need for travel time picking or analysis of coherency panels.