- 1.
- Full prestack migration to a depth model. An input data point in a constant offset section, is migrated into an ellipse. In other words a single impulse on a trace can be generated by a reflection in any point on the reflecting elliptic surface.
- 2.
- Zero-offset modeling. Each point on the ellipse can be modeled into a hyperbola, which is obtained by considering the chosen point as a point diffractor.

As a result, an impulse in a constant offset section will be moved in a zero offset section accounting also for the effect of the dipping reflector. The element which fixes the position of the migrated point on the ellipse is the dipping angle . The coordinates of the points on the ellipse are functions of the dip angle .

The mapping of the input point (*t*_{h},*x*_{h})
from a constant offset section into
the point (*t _{0}*,

Given the geometry in Figure 1, the coordinates of the point P can be
expressed as a function of the semiaxes of
the ellipse (**a** and **b** ) and the dip angle .The semiaxis are **a** = *t*_{h}*V* and **b**
, where V is half the earth velocity.
The equation of the ellipse is:

(1) |

*z ^{2}*

The equation of the tangent to the ellipse is where and .

(2) |

Squaring equation 2 and multiplying both sides with we obtain:

As the intersection of the tangent to the ellipse should satisfy both equation (1) and (2), by equalizing both sides of

(3) |

(4) |

From the previous equation (4) we obtain the value for *x*_{m},
and introducing this value in equation 1, we obtain *z*_{m}.

(5) |

(6) |

Each point on the ellipse corresponding to a certain parameter (dipping angle), can be considered a point diffractor and therefore generate a hyperbola. For a zero-offset section the point will be situated on the modeling hyperbola with the following coordinates:

(7) |

(8) |

Inserting equations (5,6) in (7,8) we obtain:

(9) |

(10) |

In the end the DMO operator contoured for only half space will look like in Figure 3. We demonstrate in Appendix A that the operator follows the envelope of the hyperbolas.

1/13/1998