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

Stripping equations

The first alternative to the Dix inversion is what we call stripping equations (Casasanta and Fomel, 2010). Starting from the integral equation 7 for $\tau$ -$p$  reflection moveout and employing the chain rule (equation C-4), we first deduce an expression for slope $R_{\tau}$ (equation C-5) and curvature $Q_{\tau}$ (equation C-6) using $\tau$ -derivatives, that now depend on the interval parameters. Then, solving for $\hat{V}_{N}$ and $\hat{V}_{H}$ , we obtain the following expressions:

$$\hat{V}_{N}^{2}(\tau, p) = -\frac{1}{p}\frac{{16R_{{\tau }}^{3}}}{{\hat{N}\hat{D}}},$$ (24)

$$\hat{V}_{H}^{2}(\tau, p) = \frac{1}{{p^{2}}}\dfrac{{\hat{N}-4R}_{{\tau }}}{{\hat{N}}}% ,$$ (25)

and

$$\hat{\eta}(\tau, p)=\frac{1}{p}\frac{{\hat{N}(4{R}_{{\tau }}-\hat{D})}}{{32\tau {R}_{{<tex2html_comment_mark>239 \tau }}^{3}}},$$ (26)

which provide an estimate for the interval parameters. In the above equations, $\hat{N}={pQ}_{{\tau }}{+ 3R}_{{\tau }}{-3p{R}_{{\tau }}^{2}}$ and $\hat{D}=p{Q}_{{\tau }}+3{R}_{{\tau }}+p{{R}_{{\tau }}^{2}},$ which corresponds to the interval values of the numerator $N$ and denominator $D$ of the square root in equation 15. These relations are very similar to those previously derived for the effective parameters (equations 16-18). However, they require the $\tau$ derivative of the slope and curvature fields (Table 1). This result agrees with the discussion above about layer stripping in $\tau$ -$p$ . In this domain, layer stripping reduces to computing traveltime differences (equation 4) at each horizontal slowness $p$ . Therefore, differentiating the effective slope $R$ and curvature $Q$ fields in $\tau$ provides the necessary information to access the interval parameters directly. This is the power of the $\tau$ -$p$  domain as opposed to $t$ -$X$  , where the only practical path to interval parameters is through Dix inversion that requires the knowledge of effective parameters. The zero-slope time $\tau _0$ is needed to map the interval parameter estimates to the correct vertical time (Table 1).

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