Full waveform inversion (FWI) promises a generally automatic approach to obtain high resolution velocity models. It, however, suffers from the high non-linearity of the objective function due to cycle skipping whenever the initial velocity is far from the exact one with respect to the minimum frequency available in the data. In order to solve this problem, we propose an objective function that combines an extension of the normalized correlation in time (or space) lags with a data selection strategy. A weighting function that emphasizes the smaller lag correlation, but extends the comparison to the maximum user-defined lag, allows us to extend the base of attraction of the objective function to a wider range of velocities. A selective function allows us to mitigate any data the might negatively contribute to the objective function, like cross talk with the lag. The result is an efficient FWI implementation (similar cost to standard FWI) with pseudo global convergence capability. An application to simple examples as well as the Marmousi model demonstrate these features.