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The first anelliptic approximation has a certain lack of symmetry.
There are two vertical control parameters,
Vz and ,but only a single horizontal control parameter
Vx.
If our data
includes sources and receivers separated both horizontally and
vertically, it makes sense
to use an approximation that is symmetric between x and z.
We can do this by generalizing to the following template:
| |
(14) |
If equation () reduces to
the first anelliptic form given by equation ();
if equation () reduces
to the original elliptical
form given by equation ().
Following this newer template,
equation () (the ray equation) becomes
| |
(15) |
where
,,and as before M indicates slowness squared
(Mx = 1/ Vx2,
,,and Mz = 1/ Vz2).
Similarly,
equation () (the dispersion relation) becomes
| |
(16) |
where
,,and as before W indicates velocity squared
(Wx = Vx2,
,,and Wz = Vz2).
Note that the subscript indicates
moveout velocity measured for near-vertical propagation; i.e.,
is the square NMO velocity we use
every day in surface-to-surface data processing.
(If an x subscript seems confusing for a paraxial measurement about
the vertical, remember that in an elliptic world it is a horizontal velocity
that surface moveout measures.)
The subscript indicates
moveout velocity measured for near-horizontal propagation, such
as might be found in a cross-borehole experiment.
Next: ANELLIPTIC PARAMETERS FOR TI
Up: Dellinger, Muir, & Karrenbach:
Previous: Consistency
Stanford Exploration Project
11/17/1997