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The stretch factor in time

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