You Can't Time Shift in the Time Domain.

You might wish to do migration in $(x,\,z,\,t)$-space with lateral velocity variation. Then the thin-lens stage would be implemented by time shifting instead of by multiplying by $\exp \{ i \omega\,[v(x,\,z)^{-1} - \bar v (z)^{-1} ] \Delta z \}$.Time shifting is a delightfully easy operation when what is needed is to shift data by an integral number of sample units. Repetitive time shifting by a fractional number of digital units, however, is a nightmare. Multipoint interpolation operators are required. Even then, pulses tend to disperse. So the lens term is probably best left in the frequency domain.