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Up: Hyperbolic and nonhyperbolic CRS
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In the case of 3-D multi-azimuth acquisition, both
and
become
two-dimensional vectors. A natural way to extend
approximation (8) is to replace it with
![$\displaystyle \widehat{\theta}_{CRS}(\mathbf{d},\mathbf{h};t_0) = \sqrt{F(\mathbf{d}) + \mathbf{h}^T \mathbf{B}_2 \mathbf{h}}\;,$](img45.png) |
(14) |
where
,
is a
two-dimensional vector, and
and
are
two-by-two symmetric matrices (Tygel and Santos, 2007). A similar approach
works for extending approximation (12) to
![$\displaystyle \widehat{\theta}(\mathbf{d},\mathbf{h};t_0) = \sqrt{\frac{F(\math...
...,\mathbf{h} + \sqrt{F(\mathbf{d}-\mathbf{h}) F(\mathbf{d}+\mathbf{h})}}{2}}\;,$](img50.png) |
(15) |
where
.
In the 3-D case, we have not found a simple connection between
approximation (15) and the analytical reflection
traveltime for a 3-D hyperbolic reflector.
2013-03-02