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| A fast algorithm for 3D azimuthally anisotropic velocity scan | |
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The right-hand side of equation 5 is a quotient of two (discrete) generalized Radon transforms (Beylkin, 1984). They can be expressed in a unified way as (to simplify the notation, we write
,
in this and next subsections)
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(7) |
where
is
or some composite function of
.
To construct the fast algorithm, we first rewrite equation 7 in the frequency domain as
|
(8) |
where
is frequency and
is the Fourier transform of
in time. We next perform a linear transformation to map all discrete points in
and
domains to points in the unit cube
; i.e., a point
minmaxminmaxminmax
is mapped to
via
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maxminmin |
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maxminmin |
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maxminmin |
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a point
minmaxminmaxminmax
is mapped to
via
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maxminmin |
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maxminmin |
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maxminmin |
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If we define a phase function
as
|
(9) |
then equation 8 can be recast as
|
(10) |
(throughout the paper,
and
are used to denote either sets of discrete points or the cubic
domains containing them).
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|
|
| A fast algorithm for 3D azimuthally anisotropic velocity scan | |
|
Next: Fast 3D butterfly algorithm
Up: Theory
Previous: Theory
2015-03-27