This work is devoted to automatic determination of 2D background slowness from full waveform data. We have designed, in earlier works an original approach, the Migration-Based Travel Time (MBTT) reformulation of the waveform acoustic inversion problem and we give here its present status of development. After a brief recall of the principle of this approach, we address three important issues : first, we prove the legitimacy of the MBTT change of reflectivity unknowns, then we illustrate the ability of our MBTT criterion to be minimized with respect to background slowness by local techniques in the single shot case, and finally we show that, in the case of multiple shots, the firt gradient calculation is very promising for future minimization. We also show a validation of our gradient and we describe a Limited Memory Quasi-Newton minimization method.