If we then rewrite the eikonal equation () in the
time-source-receiver coordinate system as (),
we can easily^{} verify that the explicit
expression for the phase of the Kirchhoff integral
kernel () satisfies the eikonal equation for any
scattering point *x*. Here, is related to as *t*
is related to *t*_{n} in equation ().

The general solution of the amplitude equation () has the form

(214) |

(215) |

Again, we exploit the assumption that
the signal *f* has the form of the delta function.
In this case, the amplitudes
before and after the NMO correction are connected according to the
known properties of the delta function, as follows:

(216) |

(217) |

(218) |

It is apparent that the OC differential equation () and the Kirchhoff representation have the same effect on reflection data because the amplitude and phase of the former match those of the latter. Thus, we see that the amplitude and phase of the Kirchhoff representation for arbitrary offset correspond to the point diffractor WKBJ solution of the offset-continuation differential equation. Hence, the Kirchhoff approximation is a solution of the OC differential equation when we hold the reflection coefficient constant. This means that the solution of the OC differential equation has all the features of amplitude preservation, as does the Kirchhoff representation, including geometrical spreading, curvature effects, and phase shift effects. Furthermore, in the Kirchhoff representation and the solution of the OC partial differential equation by WKBJ, we have not used the 2.5-D assumption. Therefore the preservation of amplitude is not restricted to cylindrical surfaces as it is in Bleistein's and Cohen's 1995 true-amplitude proof for DMO.

12/28/2000