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-
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Applying the chain rule, we rewrite the Dix inversion formula
(Dix, 1955) as follows:
![$\displaystyle \hat{\mu}=\frac{d}{{d\tau }}\left[ {\tau _{0}(\tau )\mu (\tau )}\...
...] <tex2html_comment_mark>193 \left[ \frac{d{\tau }_{0}}{{d\tau }}\right] ^{-1}.$](img98.png) |
(22) |
where
is a vertically-variable interval parameter,
is the corresponding effective parameter, and
is the
zero-slope time mapping function. Substituting the LHS of
equations 9 and 10 as
and equation 15 as
, we deduce expressions for the interval NMO velocity
, horizontal velocity
, and the
anellipticity parameter
. The derivations and the final formulas
are detailed in appendix B. In order to retrieve interval parameters
by slope-based Dix inversion, one needs as inputs the slope
and
the curvature
fields as well as their derivatives along the time axis
(see Table 1). This confirms that, even in
-
,
an application of Dix's formula requires the knowledge of the
effective quantities which, in this context, are mathematically
represented by the slope
and curvature
. The
mapping
field is also needed to map the estimated VTI parameters to the
correct imaging time. The Dix inversion route does not seem very
practical in the
-
domain because the equations (derived in
appendix B) appear cumbersome.
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![](icons/left.png) | Velocity-independent
-
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in a horizontally-layered VTI medium | ![](icons/right.png) |
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Next: Claerbout's straightedge method
Up: Estimation of interval parameters
Previous: Estimation of interval parameters
2011-06-25