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Previous: Example 2: point diffractor
The third example (the right side of Figure 4) is the
curious case of a focusing elliptic reflector. Let
be the center
of the ellipse and
be half the distance between the foci of the
ellipse. If both foci are on the surface, the zero-offset
traveltime curve is defined by the so-called ``DMO smile''
(Deregowski and Rocca, 1981):
![\begin{displaymath}
t_0\left(y_0\right)={t_n \over h}\,\sqrt{h^2-\left(y-y_0\right)^2}\;,
\end{displaymath}](img115.png) |
(40) |
where
, and
is the small semi-axis of the ellipse.
The time-ray equations are
![\begin{displaymath}
y_1\left(t_1\right)=y+{h^2\over {y-y_0}}\,{{t_1^2-t_n^2} \ov...
... \left(y-y_0\right)^2}\,{{t_1^2-t_n^2} \over t_n^2}
\right)\;.
\end{displaymath}](img117.png) |
(41) |
When
coincides with
, and
coincides with
, the
source and the receiver are in the foci of the elliptic reflector, and
the traveltime curve degenerates to a point
. This remarkable
fact is the actual basis of the geometric theory of dip moveout
(Deregowski and Rocca, 1981).
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Next: Proof of amplitude equivalence
Up: Offset continuation geometry: time
Previous: Example 2: point diffractor
2014-03-26