Does it work?

The really interesting issue is whether this method works for data that is inadequately sampled in x, i.e. when the spectrum of the underlying wavefield is nonvanishing above the spatial Nyquist frequency of the sampled field. I think it might work if the spectrum has energy below spatial Nyquist as well as above. The method that chooses p might be insensitive to signal above spatial Nyquist because temporal frequencies below Nyquist are all agreed on the same p whereas temporal frequencies above the spatial Nyquist fit different values of p. After the wavefields are interpolated, perhaps by repetitive interlacing, the question of aliasing no longer arises. So, if you can bootstrap yourself up to the right solution, I think you will stay there.