** Next:** Separability of 3-d migration
** Up:** SPLITTING AND FULL SEPARATION
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The operator for migration of zero-offset
reflection seismic data in three
dimensions is expandable to second
order by Taylor series expansion
to the so-called 15 approximation

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(5) |

The most common case is when *v* is slowly variable
or independent of *x* and *y*.
Then the conditions of full separation do apply.
This is good news because it means that we
can use ordinary 2-D wave-extrapolation
programs for 3-D, doing the in-line
data and the out-of-line data in either order.
The bad news comes when we try for more accuracy.
Keeping more terms in the Taylor series
expansion soon brings in the cross
term .Such a term allows neither full separation nor splitting.
Fortunately, present-day marine data-acquisition techniques
are sufficiently crude in the out-of-line
direction that there is little
justification for out-of-line
processing beyond the 15 equation.
Francis Muir had the good idea of representing
the square root as
| |
(6) |

There may be justification for better approximations with land data.
Fourier transformation of at least one
of the two space axes will solve the computational problem.
This should be a good approach
when the medium velocity does not vary
laterally so rapidly as to invalidate application
of Fourier transformation.

** Next:** Separability of 3-d migration
** Up:** SPLITTING AND FULL SEPARATION
** Previous:** Application to lateral velocity
Stanford Exploration Project

10/31/1997