By taking equation (14) and changing the integration variable from to . In the same time the new variable becomes function of only two variables in ,in the form given by equation (11).

(16) |

Special attention should be given to the new field . As the variable is replaced with , each value of has to be associated with the appropriate value of the field

The new field represents a remapping (or interpolation) of the original field .Each value in the new field with coordinates corresponds to the value in the field with coordinates . A more detailed analysis of this mapping is done in a companion paper in this report: Popovici (1993).From equation (10) the Jacobian is

(17) |

In the next step we change the integration order

(18) |

(19) |

Equation (18) represents zero-offset downward continuation and imaging as introduced by Gazdag (1978) or Stolt (1978). Equation (19) represents a way of obtaining zero-offset section from constant-offset sections.

The operations needed to obtain the zero-offset stacked section from the constant-offset field described in equation (19) 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 .

11/17/1997