** Next:** Remapping in
** Up:** Introduction
** Previous:** Introduction

The constant velocity **prestack migration**
in offset-midpoint coordinates can be formulated as:
| |
(1) |

where is the 3-D Fourier transform
of the field *p*(*t*,*y*,*h*,*z*=0) recorded at the surface, using
Claerbout's (1985) sign convention:
The constant is defined in the
dispersion relation as
| |
(2) |

The constant velocity **zero-offset migration**
in offset-midpoint coordinates can be formulated as:

| |
(3) |

where is the 2-D Fourier transform
of the field *p*(*t*,*y*,*z*=0).
The constant is defined in the
dispersion relation as
| |
(4) |

The goal of the ensuing derivation is to convert
equation (1) into a form similar to equation (3).
Using a change of variables and integrating over the
variable *k*_{h}, equation (1) will be transformed
into the form:

where represents the zero-offset data field.
Following Hale's (1983) derivation,
a new variable is introduced in order to
isolate the zero-offset migration operator.
The variable is expressed as
| |
(5) |

where is considered variable and *k*_{y}, *k*_{h} are
constant.
Substituting in equation (2),
the downward continuation operator
is transformed into
| |
(6) |

which now has the same form as the operator in
equation (4).
In order to isolate the zero-offset
migration operator, the variable
is substituted for in equation (1)
and the order
of integration is changed between and *k*_{h}.
The integration
boundaries have to be observed carefully
as they are modified after each change of variables
and integration order. However, for the sake of simplicity, I will
ignore in the following demonstration the integration
limits, which are discussed in Appendix A.
For simplicity I define the variables

By substituting the variable
with the expression in in equation (1)
and changing the integration order between and *k*_{h}
the prestack migration equation becomes
| |
(7) |

The new field represents a remapping
(interpolation) from to
of the field .The meaning of this remapping will be discussed
in the next section.
The field defined as
| |
(8) |

with the Jacobian
| |
(9) |

represents the zero-offset field.
The last equation in (7) represents zero-offset downward
continuation and imaging as introduced by Gazdag (1978)
or Stolt (1978). Equation (8) represents a
way of obtaining the zero-offset section from constant-offset
sections.

So far the operations needed to obtain the zero-offset
stacked section from the constant-offset field are:

- 1.
- Fourier transform the constant-offset field
.
- 2.
- Remap (interpolate) the axis into .
- 3.
- Multiply by the Jacobian.
- 4.
- Integrate over
*k*_{h}.
- 5.
- Inverse Fourier transform
.

** Next:** Remapping in
** Up:** Introduction
** Previous:** Introduction
Stanford Exploration Project

11/17/1997