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Recall equation (9)
for an ellipse centered at the origin.
| ![\begin{displaymath}
0 \eq {y^{2}\over{A^{2}}} + {z^{2}\over{B^{2}}} -1 .\end{displaymath}](img57.gif) |
(19) |
where
| ![\begin{displaymath}
A \eq v_{\rm half}\, t_h ,\end{displaymath}](img58.gif) |
(20) |
| ![\begin{displaymath}
B^2 \eq A^2 - h^2 .\end{displaymath}](img59.gif) |
(21) |
The ray goes from the shot at one focus of the ellipse
to anywhere on the ellipse,
and then to the receiver in traveltime th.
The equation for a circle of radius
with center on the surface
at the source-receiver pair coordinate x=b is
| ![\begin{displaymath}
R^2 \eq (y - b)^2 + z^{2} ,\end{displaymath}](img61.gif) |
(22) |
where
| ![\begin{displaymath}
R \eq t_0 \, v_{\rm half}.\end{displaymath}](img62.gif) |
(23) |
To get the circle and ellipse tangent to each other,
their slopes must match.
Implicit differentiation of equation (19) and (22)
with respect to y yields:
| ![\begin{displaymath}
0 \eq {y \over{A^2}} + {z \over{B^2}}
\ {dz \over dy}\end{displaymath}](img63.gif) |
(24) |
| ![\begin{displaymath}
0 \eq (y-b) + z
\ {dz \over dy}\end{displaymath}](img64.gif) |
(25) |
Eliminating dz/dy from equations (24) and (25) yields:
| ![\begin{displaymath}
y \eq {b\over 1 - {B^{2}\over{A^{2}}}} .\end{displaymath}](img65.gif) |
(26) |
At the point of tangency the circle and the ellipse should coincide.
Thus we need to combine equations to eliminate x and z.
We eliminate z from equation (19) and (22)
to get an equation only dependent on the y variable.
This y variable can be eliminated by inserting equation (26).
| ![\begin{displaymath}
R^2 \eq B^2 \left( {A^2 - B^2 - b^2 \over{A^2 - B^2}} \right).\end{displaymath}](img66.gif) |
(27) |
Substituting the definitions (20), (21),
(23) of various parameter gives the
relation between zero-offset traveltime t0 and nonzero traveltime
th:
| ![\begin{displaymath}
t_0^2\eq
\left(t_h^2-{h^{2}\over v_{\rm half}^2}\right)
\left(1-{b^2\over h^2}\right).\end{displaymath}](img67.gif) |
(28) |
As with the Rocca operator, equation (28)
includes both dip moveout DMO and NMO.
Next: DMO IN THE PROCESSING
Up: GARDNER'S SMEAR OPERATOR
Previous: GARDNER'S SMEAR OPERATOR
Stanford Exploration Project
12/26/2000