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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 *dy*_{0} determines the sign of the angle.
The next steps follow exactly Hale's reasoning
by defining another field *p*_{0}(*t*_{0},*y*_{0},*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 *t*_{0} and *y*_{0} in equation (21)
with the known variables *t*_{n} and *y*. Fortunately it is
not necessary to express explicitly *t*_{n}=*t*_{n}(*t*_{0},*y*_{0}) and
*y*=*y*(*t*_{0},*y*_{0}), 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