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Recall equation (9)
for an ellipse centered at the origin.
| |
(19) |

where
| |
(20) |

| |
(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 *t*_{h}.
The equation for a circle of radius with center on the surface
at the source-receiver pair coordinate *x*=*b* is

| |
(22) |

where

| |
(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:

| |
(24) |

| |
(25) |

Eliminating *dz*/*dy* from equations (24) and (25) yields:

| |
(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).
| |
(27) |

Substituting the definitions (20), (21),
(23) of various parameter gives the
relation between zero-offset traveltime *t*_{0} and nonzero traveltime
*t*_{h}:

| |
(28) |

As with the Rocca operator, equation (28)
includes both dip moveout **DMO** and NMO.

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Stanford Exploration Project

12/26/2000