rays
Reflection rays in a constant-velocity
medium: a scheme.
Figure 1 |

Let us consider a simple reflection experiment in an effectively
constant-velocity medium, as depicted in Figure 1. The
pair of incident and reflected rays and the line between the source
*s* and the receiver *r* form a triangle in space. From the
trigonometry of that triangle we can derive simple relationships among
all the variables of the experiment
Fomel (1995, 1996a, 1997).

Introducing the dip angle and the reflection angle ,the total reflection traveltime *t* can be expressed from the law of
sines as

(1) |

Additionally, by following simple trigonometry, we can connect the
half-offset *h* with the depth of the reflection point *z*, as
follows:

(2) |

Finally, the horizontal distance between the midpoint *x* and the
reflection point is

(3) |

Equations (1-3) completely define the kinematics of angle-gather migration. Regrouping the terms, we can rewrite the three equations in a more symmetric form:

(4) | ||

(5) | ||

(6) |

(7) | ||

(8) | ||

(9) |

The lines of constant reflection angle and variable dip angle for a given position of a reflection (diffraction) point have the meaning of summation curves for angle-gather Kirchhoff migration. The whole range of such curves for all possible values of covers the diffraction traveltime surface - ``Cheops' pyramid'' Claerbout (1985) in the space of seismic reflection data. As pointed out by Fowler (1997), this condition is sufficient for proving the kinematic validity of the angle-gather approach. For comparison, Figure 2 shows the diffraction traveltime pyramid from a diffractor at 0.5 km depth. The pyramid is composed of common-offset summation curves of the conventional time migration. Figure 3 shows the same pyramid composed of constant- curves of the angle-gather migration.

coffset
Traveltime pyramid, composed of
common-offset summation curves.
Figure 2 |

cangle
Traveltime pyramid, composed of
common-reflection-angle summation curves.
Figure 3 |

The most straightforward Kirchhoff algorithm of angle-gather migration can be formulated as follows:

- For each reflection angle and each dip angle ,

(10) |

4/20/1999