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Migration is based on the one-way wave equation
| |
(1) |

which has the analytic solution
| |
(2) |

where
| |
(3) |

and *P* is the wavefield, *z* is the depth coordinate,
is the frequency, *v* is the velocity,
and *k*_{x} and *k*_{z} are the horizontal and vertical wavenumbers.
For small *k*_{x}, *k*_{z} can be expanded in a series such as
| |
(4) |

Retaining the first two terms in the expansion results in
the 15-degree equation
| |
(5) |

Because the error term is negative, the approximate is
always greater
than the physical *k*_{z}. Migration is the process of dephasing (Mo, 1992),
which means that, imaging condition (time *t*=0) is satisfied
when the entire exploding reflector upward propagation phase is exhausted.
For |*k*_{x}|>0, that is, for waves with propagation angles greater than zero,
less distance is traveled when the imaging operation is performed.
And the larger *k*_{x} is, the larger the error term,
which results in a smaller propagation distance for the migration response.
This explains why the migration impulse response swarms around the vertical
axis (Claerbout, 1985, Figure 4.2-2; Yilmaz, 1987, Figure 4-90).
Thus the parabolic approximation of the exact one-way wave equation causes
undermigration - the specification error.
This phenomenon was defined by Claerbout (1985) as anisotropy dispersion.

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Stanford Exploration Project

11/17/1997