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

Claerbout's straightedge method

Claerbout (1978) suggested that interval velocity in an isotropic ( $\hat{V}_N=\hat{V}_H$ ) layered medium could be estimated with a pen and a straightedge by measuring the offset difference $\Delta x$ between equal slope $p$ points on two reflection events (Figure 1a). The computation of $\Delta x$ is straightforward after a CMP gather has been transformed into the $\tau$ -$p$  domain. In fact, the $\tau$ -$p$  transform naturally aligns seismic events with equal slope along the same trace (Figure 1b). Moreover, local slopes $R(\tau,p)=d\tau/dp$ are related to the emerging offset $x=-R$ (van der Baan, 2004) , therefore Claerbout's inversion formula can be expressed as

$$\hat{V}_N^{2}(\tau,p)=\dfrac{R_{\tau}}{p^2R_{\tau}-p},$$ (23)

where $R_{\tau}=\partial R(\tau,p)/\partial \tau$ and $\hat{V}_N$ is the NMO interval velocity that we map back to zero-slope time $\tau _0$ using the isotropic velocity independent $\tau$ -$p$  NMO, as suggested by Fomel (2007b). The details of the derivation are in appendix C.

The two methods discussed next can be thought of as two alternative extensions for VTI media of the original Claerbout's straightedge method.

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