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The goal of this appendix is to prove the equivalence between the
result of the zero-offset velocity continuation from zero velocity and
the conventional post-stack migration. After solving the velocity
continuation problem in the frequency domain, I transform the
solution back to the time-and-space domain and compare it with the
famous Kirchhoff migration operator.
Zero-offset migration based on velocity continuation is the solution
of the boundary problem for equation (55) with the
boundary condition

| |
(82) |

where *P*_{0}(*t*_{0},*x*_{0}) is the zero-offset seismic section, and
*P*(*t*,*x*,*v*) is the continued wavefield. In order to find the solution
of the boundary problem composed of (55) and (82), it is
convenient to apply the function transformation , the time coordinate transformation , and,
finally, the double Fourier transform over the squared time coordinate
and the spatial coordinate *x*:
| |
(83) |

With the change of domain, equation (55) transforms
to the ordinary differential equation
| |
(84) |

and the boundary condition (82) transforms to the initial
value condition
| |
(85) |

where
| |
(86) |

and . The unique solution of the initial value
(Cauchy) problem (84) - (85) is easily found to be
| |
(87) |

We can see that, in the transformed domain, velocity continuation is a
unitary phase-shift operator. An immediate consequence of this
remarkable fact is the cascaded migration decomposition of post-stack
migration Larner and Beasley (1987):

| |
(88) |

Analogously, three-dimensional post-stack migration is decomposed
into the two-pass procedure Jakubowicz and Levin (1983):
| |
(89) |

The inverse double Fourier transform of both sides of equality
(87) yields the integral (convolution) operator

| |
(90) |

with the kernel *K* defined by
| |
(91) |

where *m* is the number of dimensions in *x* and *k* (*m* equals 1
or 2). The inner integral on the wavenumber axis *k* in formula
(91) is a known table integral Gradshtein and Ryzhik (1994). Evaluating this
integral simplifies equation (91) to the form
| |
(92) |

The term is the spectrum of the anti-causal
derivative operator of the order *m*/2. Noting
the equivalence
| |
(93) |

which is exact in the 3-D case (*m*=2) and asymptotically correct in
the 2-D case (*m*=1), and applying the convolution theorem, we can
transform operator (90) to the form
| |
(94) |

where , and . Operator (94) coincides with the Kirchhoff operator
of the conventional post-stack time migration Schneider (1978).
C

** Next:** FINITE-DIFFERENCING POST-STACK VELOCITY CONTINUATION
** Up:** Fomel: Velocity continuation
** Previous:** DERIVING THE KINEMATIC EQUATIONS
Stanford Exploration Project

11/12/1997