Velocity-independent $\tau$ -$p$ moveout in a horizontally-layered VTI medium

Dix Inversion

Applying the chain rule, we rewrite the Dix inversion formula (Dix, 1955) as follows:

$$\hat{\mu}=\frac{d}{{d\tau }}\left[ {\tau _{0}(\tau )\mu (\tau )}\... ...] <tex2html_comment_mark>193 \left[ \frac{d{\tau }_{0}}{{d\tau }}\right] ^{-1}.$$ (22)

where $\hat{\mu}$ is a vertically-variable interval parameter, ${\mu}$ is the corresponding effective parameter, and $\tau_{0}(\tau)$ is the zero-slope time mapping function. Substituting the LHS of equations 9 and 10 as $\mu$ and equation 15 as $\tau _0$ , we deduce expressions for the interval NMO velocity $\hat{V}_{N}$ , horizontal velocity $\hat{V}_{H}$ , and the anellipticity parameter $\hat{\eta}$ . 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 $R$ and the curvature $Q$ fields as well as their derivatives along the time axis $\tau$ (see Table 1). This confirms that, even in $\tau$ -$p$  , an application of Dix's formula requires the knowledge of the effective quantities which, in this context, are mathematically represented by the slope $R$ and curvature $Q$ . The $\tau _0$ 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 $\tau$ -$p$  domain because the equations (derived in appendix B) appear cumbersome.

Velocity-independent $\tau$ -$p$ moveout in a horizontally-layered VTI medium