Zhang (1988) observed that in Hale's DMO the reflection point in the nonzero-offset case does not coincide with the reflection point in the zero-offset case. He shows a new formula for DMO which takes into account not only a time correction but also a mid-point correction.
The correction in constant velocity media as defined in Part 1 is written
We can isolate the NMO transformation from equation (12) which is only a time-shift
and write just for the DMO operator
where the sign of dy0 determines the sign of the angle.
The next steps follow exactly Hale's reasoning by defining another field p0(t0,y0,h), with the addition that not only the time variable is changed but also the common-midpoint variable. This accounts for the fact that the DMO transformation defined by Zhang moves the nonzero-offset reflection point to a zero-offset reflection point. Stacking after this transformation produces true common depth point gathers. In the transformation defined by Hale, the reflection point for the nonzero-offset is different from the reflection point in the zero-offset case.
The Fourier transform of the new field is
the phase becomes
which should be noted is the same phase as in Hale's equation (19).
Equation (21) becomes
The ratio between the two Jacobians is