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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 *J*_{c} 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

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Stanford Exploration Project

10/9/1997