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In this appendix, we derive , given by equation 6, in the
-domain. Using such an equation can avoid the process of mapping from
depth to time and back.
The vertical two-way traveltime, , is written as
| |
(29) |
where z corresponds to depth.
Similarly,
| |
(30) |
where corresponds to the new coordinate system.
Using the chain rule,
| |
(31) |
where extracted from equation (30) is given by
| |
(32) |
the partial derivative in is
| |
(33) |
Therefore, the
transformation from (, ) to (x, z) is governed
by the following Jacobian matrix in 2-D:
| |
(34) |
The inverse of Jc is
| |
(35) |
which should equal the Jacobian matrix for the transformation from (x, z) to (, ),
given by
| |
(36) |
As a result,
which is a convenient equation,
since we want to keep all fields, including velocity, in coordinates.
B
Next: The amplitude transport equation
Up: VTI processing in inhomogeneous
Previous: REFERENCES
Stanford Exploration Project
10/9/1997