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## MZO from prestack migration

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 kh, 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 ky, kh 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 kh. 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 kh 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 kh.
5.
Inverse Fourier transform .

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Stanford Exploration Project
11/17/1997