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For insight into the appearance of reflector images in the dip-angle domain, let us consider the case of a hyperbolic reflector (Fomel and Kazinnik, 2013). A special property of hyperbolic reflectors is that they can transform to plane dipping reflectors or point diffractors with an appropriate choice of parameters.
Reflector depth is given by the function
![\begin{displaymath}
z(x)=\sqrt{z_0^2+(x-x_0)^2 \tan^2\beta}\;,
\end{displaymath}](img35.png) |
(11) |
and zero-offset reflection traveltime is given by
![\begin{displaymath}
t(y)=\frac{2}{v} \sqrt{z_0^2+(y-x_0)^2 \sin^2\beta}\;.
\end{displaymath}](img36.png) |
(12) |
When the reflector is imaged by time migration in the dip-angle domain
(Sava and Fomel, 2003) using velocity
, point
in
the data domain migrates to
in the image domain
according to
where
is the migration dip angle. Eliminating
from
equations A-3 and A-4, we arrive at the equation
![\begin{displaymath}
t_m(x_m) = \frac{2 \cos{\alpha}}{v} \frac{\gamma (x_m-x_...
...n^2{\beta} + \sqrt{(x_m-x_0)^2 \sin^2{\beta}+z_0^2 D}}{D}\;,
\end{displaymath}](img45.png) |
(15) |
where
and
. Equation A-5
describes the shape of the image of the hyperbolic
reflector (A-1) in the dip-angle domain.
When the dip of the migrated event, imaged at a correct velocity (
),
![\begin{displaymath}
\tan{\alpha_m} = \displaystyle \frac{v}{2} t_m'(x_m) =
\...
...m-x_0}{\sqrt{(x_m-x_0)^2 \sin^2{\beta}+z_0^2 D}}\right]\;,
\end{displaymath}](img49.png) |
(16) |
is equal to the dip of the image (
), it also becomes
equal to the true dip of the reflector (
), where
![\begin{displaymath}
\tan{\alpha_0} = z'(x_m) = \frac{(x_m-x_0) \tan^2{\beta}}{\sqrt{z_0^2+(x-x_0)^2 \tan^2\beta}}\;.
\end{displaymath}](img52.png) |
(17) |
We can specify these conditions for two special cases described next.
Subsections
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Next: Point diffractor
Up: Klokov and Fomel: Optimal
Previous: Acknowledgments
2014-03-25