In order to prove the validity of equation (), it is
convenient to transform it to the coordinates of the initial shot
gathers: *s*=*y*-*h*, *r*=*y*+*h*, and . The transformed equation takes the form

(145) |

Let *S* and *R* be the source and the receiver locations, and *O* be a
reflection point for that pair. Note that the incident ray *SO* and
the reflected ray *OR* form a triangle with the basis on the offset
*SR* (*l*=|*SR*|=|*r*-*s*|). Let be the angle of *SO* from the
vertical axis, and be the analogous angle of *RO* (Figure
). The law of sines gives us the following explicit
relationships between the sides and the angles of the triangle *SOR*:

(146) | ||

(147) |

(148) |

(149) | ||

(150) |

(151) |

ocoray
Reflection rays in a constant
velocity medium (a scheme).
Figure 2 |

Thus we have proved that equation (), equivalent to (), is valid in constant velocity media independently of the reflector geometry and the offset. This means that high-frequency asymptotic components of the waves, described by the OC equation, are located on the true reflection traveltime curves.

The theory of characteristics can provide other ways to prove the kinematic validity of equation (), as described by Fomel (1994) and Goldin (1994).

12/28/2000