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Offset plane wave migration reduces
the computational complexity of downward continuation
of commonazimuth data
one step further than commonazimuth migration does.
It approximates the application of the full 5D operator
expressed in equation (2)
with the application of several 3D downward continuation
operators.
These 3D operators are applied to commonazimuth data
after planewave decomposition along the offset axis.
The first step of the method is thus the decomposition
of the commonazimuth data into offset plane waves.
Each plane wave is then independently
downward continued.
Full downward continuation of offset plane waves could be
performed by applying
following operator
 
(6) 
where the vertical wavenumber k_{z} is
now function of the offset plane wave parameters
and
;that is,
 

 (7) 
Strictly speaking,
only in vertically layered media
can each plane wave be downward continued independently.
The plane waves should be allowed to mix at each depth step
when lateral velocity variations occur.
Therefore, the computationally attractive
feature of imaging each plane wave independently
also causes limitations in accuracy.
These limitations are difficult to study analytically,
and thus in a following section
I will analyze their effects
by comparing migration results below a complex
overburden (eg. a salt body).
In practice,
because commonazimuth data has
no crossline offset axis,
the plane wave decomposition is performed only as a function
of the inline offset ray parameter p_{xh},
and the crossline offset ray parameter p_{yh} is
assumed to be zero.
This assumption introduces another approximation
in the migration operator,
that can be studied analytically.
When p_{yh} is set to zero,
equation (7)
becomes:
 

 (8) 
This equation is equivalent to the offset plane wave equation
presented by Mosher et al. 1997.
^{}
It is easy to verify that if we assume
,the dispersion relation of equation (8)
can be expressed as the cascade of
a zerooffset downward continuation along the crossline
direction:
 
(9) 
and prestack downward continuation
along the inline direction:
 
(10) 
The interpretation of this decomposition
is similar to the one discussed above for commonazimuth migration.
A constant velocity offset plane wave migration
that uses the dispersion relation of equation (8)
is equivalent to a constantvelocity crossline zerooffset migration,
followed by a constantvelocity inline prestack migration.
The order between these migrations
is thus reversed with respect to the correct order.
We analyze the implications of this
order reversal in the following section.
Next: Comparing constantvelocity migration operators
Up: Efficient waveequation migrations of
Previous: Commonazimuth downward continuation
Stanford Exploration Project
10/25/1999