The Phase-shift operator in a prestack Fourier-domain is described
by the double-square-root equation, which is a function of the angular frequency ,the midpoint rayparameter *p*_{x}, the offset rayparameter *p*_{h}, and the velocity, *v*.
Constant-velocity prestack migration, with output provided in two-way vertical time (),
in offset-midpoint coordinates Yilmaz (1979), is given by:

(3) |

(4) |

For VTI media, the phase factor is given by a more complicated equation Alkhalifah (1997b),

(5) |

Kirchhoff migration is typically applied by smearing an input trace, after the proper traveltime shifts, over the output section in a summation process. To obtain the response of inserting a single trace into the prestack phase-shift migration [equation (3)], we multiply the input data by a Direc-delta function in midpoint and offset axes as follows

(6) |

(7) |

(8) |

The number of integrals in Equation (7) can be reduced by recognizing areas in the integrand that contribute
the most to the integrals in *k*_{h} and *k*_{x}.
Since the integrand is an oscillatory function its biggest contributions
take place when the oscillations are stationary, when the phase function is either
minimum or maximum. This approach is referred to as the stationary phase method (Appendix C). The stationary
points (*p*_{x} and *p*_{h}) correspond to
the minimum or maximum of equation (8). In fact, the phase has
a dome-like shape as a function of *p*_{x} and *p*_{h} (see Figure 1). Thus, to calculate the stationary
points, we must set the derivative of equation (8) with respect to *p*_{h} and
*p*_{x} to zero, and solve the two equations for these two parameters. An easier approach is discussed next and
in Appendix A.

Figure 1

7/5/1998