Cascading DMO and inverse DMO allowed us to evaluate the summation path of AMO and the corresponding weighting function. However, this procedure is not sufficient for evaluating the third major component of the integral operator, that is, its aperture (range of integration). To solve this problem, Fomel and Biondi 1995c applied an alternative approach by defining AMO as the cascade of 3D common-offset common-azimuth migration and 3D modeling at a different azimuth and offset. I birefly discuss their derivations in this appendix as part of a complete definition of integral AMO.

The impulse response of the common-offset common-azimuth migration is a symmetric ellipsoid with the center in the input midpoint and axis of symmetry along the input-offset direction. Such an ellipsoid is described by the general formula

(90) |

(91) |

The impulse response of the AMO operator corresponds kinematically
to the reflections from the ellipsoid defined by
equation ellips.eq to a different azimuth and different
offset.
To constrain the AMO aperture, Fomel and Biondi 1995c
based their derivations on answering the
following question: *For a given elliptic
reflector defined by the input midpoint, offset, and time coordinates,
what points on the surface can form a source-receiver pair valid for a
reflection?* If a point in the output midpoint-offset space cannot be
related to a reflection pattern, it should be excluded from the AMO
aperture.

Fermat's principle provides a general method of solving the kinematic reflection problem Goldin (1986). The formal expression for the two-point reflection traveltime is given by

(92) |

(93) |

To find the solution of fermat.eq, it is convenient to decompose the reflection-point projection into three components: , where is parallel to the input offset vector , and is perpendicular to . The plane, drawn through the reflection point and the central line of ellipsoid ellips.eq, must contain the zero-offset (normally reflected) ray because of the cylindrical symmetry of the reflector. The fact that the zero-offset ray is normal to the reflector gives us the following connection between the zero-offset midpoint and the component of the reflection point :

(94) |

(95) |

(96) |

To find the third component of the reflection point projection , we substitute expression x02xi.eq into reflecttt.eq. Choosing a convenient parameterization , , where , and , we can rewrite the two-point traveltime function from reflecttt.eq in the form

(97) |

(98) |

Because the reflection point is contained inside the ellipsoid, its projection obeys the evident inequality

(99) |

1/18/2001