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

Fowler's equations

The second alternative to get VTI interval parameters comes from the integral formulation of the $\tau$ -$p$  moveout signature in equation 7. The derivation is detailed in Appendix C. We first compute $\tau_{0,\tau}=\partial\tau_{0}/\partial\tau$ and then, applying the chain rule, $R_{\tau}$ . Solving for $\hat{V}_{N}$ and $\hat{V}_{H}$ , we arrive at the following relations:

$$\hat{V}_{N}^{2}(\tau, p) = -\frac{[ \tau_{0,\tau}^{2}-1]^{2}}{% p^{3}\tau_{0,\tau}^{2} R_{\tau} },$$ (27)

$$\hat{V}_{H}^{2}(\tau, p) = \frac{\tau_{0,\tau}^{2}[1-R_{\tau}]+1}{p^{3}\tau_{0,\tau}^{2} R_{\tau} },$$ (28)

which are equivalent to those proposed previously by Fowler et al. (2008). According to equations 27 and 28, the gradients of offset $x$ and the zero-slope time $\tau _0$ measured at common slope locations $p$ on two consecutive seismic event return the VTI interval parameters for the layer bounded by these two events (Figure 1a). Fowler et al. (2008) first pick traveltime curves in $t$ -$X$  domain, and then differentiate those curves in offset to compute slopes $p$ . Finally, for any given $p$ value on each seismic event, they determine the corresponding $\Delta x$ and $\Delta \tau_0$ values (Figure 1a). The main practical limitation in this inversion scheme is the difficulty of picking seismic events accurately.

The processing becomes easier if it is accomplished in $\tau$ -$p$  with automatic slope estimation. First, $\tau$ -$p$   transform unveils the position of equal slope events. Second, $\tau_{0,\tau}$ and $R_{\tau}$ are measured automatically (without event picking) on the $\tau$ -$p$  transformed CMP gather. The quantity $\tau_{0,\tau}$ can be estimated as the $\tau$ finite difference of $\tau _{0}$ values computed according to velocity-independent moveout equation 15. The zero-slope time $\tau _0$ function is still needed to map the interval parameter estimated using equations 27 and 28 to the correct vertical time (Table 1).

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