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Both Hale and Zhang separate the NMO correction from the DMO
correction. However for variable velocity media this can not
be done, and in order to generalize the NMO DMO operation
the migration to zero-offset (MZO) concept has to be used.
Migration to zero-offset is the process that transforms a
nonzero-offset (constant offset) section into a zero-offset
section. In constant velocity media MZO = NMO+DMO.
**MZOdip
**

Figure 1 Geometry for a dipping reflector in a constant velocity medium.
The zero-offset ray **JR** and the nonzero-offset ray bounce
at the same reflection point **R**.
The dipping angle is .

Taking into account the geometry in Figure
the MZO correction in constant velocity media is

| |
(1) |

where *h* is the half offset, *t*_{0} is twice the zero-offset travel
time from the reflection point to the surface, *v* is the velocity
of the medium and *t*_{h} is the constant-offset travel time.
The MZO correction for a given dip is

| |
(2) |

To clarify the sign convention for the spatial coordinate in equation
(2), we can notice that the
*y*-axis is oriented to the left, and
the angles are positive if the reflector dips toward the right
and negative if the reflector
dips toward the left. In Figure the angle
is negative. This can be also derived from the equation
where the sign of *dy*_{0} determines the sign of the angle .
The differentials of the new variables are

| |
(3) |

The 2-D Fourier transform of the zero-offset field is

| |
(4) |

Following Hale's (1984) technique,
replace the variables *t*_{0} and *y*_{0} in
equation (4)
with their expression in equation (2).
Fortunately, it is
not necessary to express explicitly *t*_{h}=*t*_{h}(*t*_{0},*y*_{0}) and
*y*=*y*(*t*_{0},*y*_{0}), though the respective dependencies are assumed.
After replacing the variables *y*_{0} and *t*_{0} in
equation (4), the migration to zero-offset transformation
becomes

| |
(5) |

Note that by replacing the quantity
in equation (5), we get the same phase as the one obtained
by Hale and Zhang, but the Jacobian differs from Zhang's Jacobian
by a factor
| |
(6) |

which corresponds to the Jacobian of the NMO transformation.

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** Up:** Introduction
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Stanford Exploration Project

11/17/1997