Next: Commonazimuth downward continuation
Up: Biondi: Offset plane waves
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The full downward continuation of 3D prestack data
can be expressed in the frequencywavenumber domain
by the following phaseshift operator
 
(1) 
where is the temporal frequency,
is the midpointwavenumber vector,
is the offsetwavenumber vector,
and and are respectively the velocity at the source
and receivers locations.
The vertical wavenumber k_{z} is given
by the Double Square Root (DSR),
 

 (2) 
This operator is a function of the crossline component
of the offset wavenumber k_{yh},
while commonazimuth data are independent of k_{yh}
because they are different from zero only at y_{h}=0.
Therefore, the exact full downward continuation
is performed by applying a 5D operator on
a data set that is only 4D.
While accurate, this procedure is tremendously
wasteful of computational efforts,
because only a small subset of the 5D wavefield contributes
to the final image.
The final image is formed
by extracting the zerooffset cube
from the downwardcontinued wavefield.
This data extraction is equivalent to the summation of the wavefield
along both offsetwavenumber axes.
Most of the wavefield components
destructively interfere in the imaging step.
In fact, only a 4D slice of the 5D wavefield contributes
to the image when no multipathing occurs,
such as in constant velocity or in a vertically layered media.
Even when multipathing occurs,
most of the wavefield components
destructively interfere in the imaging step.
It is therefore natural to limit
the computational cost by reducing the dimensionality of
the downward continuation operator from 5D to 4D.
Both commonazimuth migration and offset plane wave migration
achieve this goal, though in different ways.
Next: Commonazimuth downward continuation
Up: Biondi: Offset plane waves
Previous: Introduction
Stanford Exploration Project
10/25/1999