If we want to model seismic data correctly, propagation operator W should be a time extrapolation since wave fields extrapolate in time physically. However, the time extrapolation is not efficient in computation. Instead, depth extrapolation is widely used, which is efficient in computation. If we use depth extrapolation as a propagation operator, however, we cannot model the correct amplitude of the wave field because the energy of the wave field in a lossless medium is preserved in time, not in depth.
This does not mean that the amplitude of wave field obtained by depth extrapolation is totally incorrect because most of the energy recorded in reflection seismic travels near the vertical direction and the wave field becomes a plane wave in the far distance where the energy of the wave field is preserved both in time and depth. Therefore, if the source wave field is a plane wave, the amplitude of the wave field after depth extrapolation behaves similarly to that of the wave field obtained by time extrapolation.