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Hale (1984) starts with the prestack migration equation in
common-midpoint and offset coordinates and tries to
isolate four major processing steps.
The four steps are
- 1.
- DMO,
- 2.
- NMO,
- 3.
- Stacking,
- 4.
- Zero-offset migration.

By considering steps 1, 2, and 3 as a single process (MZO), I isolate
only two steps:
- Migration to zero-offset.
- Zero-offset migration.

After extracting the zero-offset migration from the prestack migration
equation, we are left with the migration to zero-offset operator.
For a constant velocity,
downward continue the field *p*(*t*,*h*,*y*,*z*=0)
recorded at the surface, to a depth

| |
(7) |

for all values of .
The prestack migration imaging step consists of
summing for all
the values of and *k*_{h} after downward
continuation. To image for
*t*=0 and *h*=0:
| |
(8) |

where *k*_{z} is given by
| |
(9) |

Following Hale's (1984) derivation, introduce
a new variable in order to
isolate the zero-offset migration operator.
The substitution is based on the equation

| |
(10) |

Substituting in equation (9)
the downward continuation operator
is transformed into
| |
(11) |

The downward continuation operator now has the same form as the
one for zero-offset. In order to isolate the zero-offset
migration operator, the variable
is substituted for and then the order
of integration is changed between and *k*_{h}.
The integration boundaries have to be observed carefully.

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Stanford Exploration Project

11/17/1997