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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 kx and kz are the horizontal and vertical wavenumbers.
For small kx, kz 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 kz. 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 |kx|>0, that is, for waves with propagation angles greater than zero,
less distance is traveled when the imaging operation is performed.
And the larger kx 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.
Next: OVERMIGRATION CAUSED BY THE
Up: Mo: Numerical analysis
Previous: Introduction
Stanford Exploration Project
11/17/1997