To find the maximum of equation (8), we take its derivative with respect
to *p*_{h} and *p*_{x}, and set these derivatives to zero. The stationary point along the *p*_{h}-*p*_{x} plane
is obtained by solving the two new nonlinear equations in terms of *p*_{x} and *p*_{h}.
Since the source rayparameter, *p*_{s}, and the receiver rayparameter, *p*_{g}, are
linearly related to *p*_{x} and *p*_{h}, as follows

*p*_{s} = *p*_{x} - *p*_{h},

*p*_{g} = *p*_{x} + *p*_{h},

The stationary point solutions (Appendix A) are then given by

(9) |

For isotropic media, and equation (9) reduces to

(10) |

(11) |

(12) |

For traveltime calculation, equation (9) for *p*_{s} and *p*_{g} is
inserted into

(13) |

Equation (13) is the offset-midpoint (Cheop's pyramid) equation for VTI media. The derivation included the stationary phase (high frequency) approximation, as well as approximations corresponding to small . For =0, equation (13) reduces to the exact form (high-frequency limit) for isotropic media. However, for large the equation, as we will see later, is extremely accurate.

Figure 2

Figure 2 shows the traveltime calculated using equation (13) as a function of offset and midpoint for three values. The shape of the traveltime function resembles Cheop's pyramid, and as a result was given the name. Unlike the isotropic medium pyramid, the VTI ones include nonhyperbolic moveout along the offset and midpoint axis. Clearly, the higher horizontal velocity in the VTI media resulted in faster traveltime with increasing offset and midpoint than the isotropic case.

The stationary phase method also provides an amplitude factor given by the
second derivative of the phase function [equation (8)] with respect to
*p*_{s} and *p*_{g}. Specifically, the amplitude is proportional to the reciprocal of the square root
of the second derivative of the phase evaluated at the stationary point (Appendix C).

7/5/1998