Strong reverberation is a challenge for narrowband source localization, as most of the existing methods are based on times-of-arrival measurements, that is affected by boundaries. Amongst the methods that explicitly take into account the reverberation, wavefield separation projector processing (WSPP) splits the acoustic wave field into the direct path of the sources and the reverberation. However, WSPP requires a very large number of microphones, making this method impractical. This article studies three ways of alleviating this constraint, extending WSPP by adding different prior information on the wavefield. The first method is based on using the knowledge of the critical distance of the room to decrease the selectivity of the field separation. The second method adds constraints called "virtual measurements" when the room geometry is partially known. Finally, the last method requires a simple calibration step to estimate the Green's functions between each pair of microphones; this also extends the model to weakly inhomogeneous propagation media. It is shown numerically and experimentally that these methods allow a precise source localization, with a reduced number of microphones.
