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A
This appendix presents a derivation of the expression for the
weighting Jacobian in the case of variable velocity.
The dispersion relation
| |
(43) |
can be approximated using a Taylor series expansion around
the constant reference slowness (s0):
| |
(44) |
We then take the derivative of
kz with respect to the frequency
and if we note that
we obtain
| |
(45) |
We continue by evaluating the derivatives with respect to
on the right hand side. With little algebra, we obtain
therefore
| |
(46) |
The prestack weighting Jacobian is:
| |
(47) |
which, in constant slowness, takes the simple form
| |
(48) |
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Stanford Exploration Project
4/30/2001