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In this appendix, I derive the formula for calculating the Jacobian function
*J*(*x*,*z*) and the partial differential equation (14).

From equation (12), we have

The expression in the denominator of the last line is a cross-product
of two gradient vectors:
is the take-off angle of the ray that
reaches point(*x*,*z*), and is constant along each ray. Thus,
the gradient directions of are always orthogonal to rays. On
the other hand, the gradient directions of traveltimes are always
tangential to rays. Consequently, and are
orthogonal, which yields
and
In 2-D, equation (A.3) can be rewritten as
which is equation (14). From equations (A.1), (A.2) and (A.4),
we can derive
The eikonal equation states that or
. Applying this relation to equation (A.6), we get
equation (12):

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** Up:** Zhang: F-D calculation
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Stanford Exploration Project

12/18/1997