Next: MZO by Fourier transform Up: DMO BY FOURIER TRANSFORM Previous: Hale's DMO

## Zhang's improved DMO

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.

ZhangDMO
Figure 13
Geometry for a dipping reflector in a constant velocity medium. Notice that the reflection point for the nonzero-offset ray R is the same as the one for the zero-offset ray JR. The dipping angle is .

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
 (20)
To clarify the sign convention for the spatial coordinate in equation (20) note that the y axis is increasing to the left, and the angles are positive if they dip toward the right and negative if they dip toward the left. In Figure the angle is negative. This can be also derived from equation (16)

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
 (21)
We need to replace the variables t0 and y0 in equation (21) with the known variables tn and y. Fortunately it is not necessary to express explicitly tn=tn(t0,y0) and y=y(t0,y0), though we assume the respective dependencies. From equation (20) we can express the differentials of the new variables
 (22)
and introduce them in equation (21). After using equation (16) to express

the phase becomes

which should be noted is the same phase as in Hale's equation (19).

Equation (21) becomes
 (23)
Comparing equation (23) with Hale's equation (19) we notice that surprisingly the phase term is the same and the only difference is in the amplitude term. The two Jacobians are:

The ratio between the two Jacobians is
 (24)

Next: MZO by Fourier transform Up: DMO BY FOURIER TRANSFORM Previous: Hale's DMO
Stanford Exploration Project
11/17/1997