Velocity-independent time-domain seismic imaging using local event slopes

$\tau$-$p$ NMO

One can choose to perform normal moveout in the slant-stack ($\tau$-$p$) domain rather than the original $t$-$x$ domain. In the $\tau$-$p$ domain, hyperbolas turn into ellipses but the moveout correction problem remains (Stoffa et al., 1981). Differentiating the $\tau$-$p$ moveout equation

$$\tau(p) = \tau_0\,\sqrt{1 - p^2\,v_n^2(\tau_0)}\;$$ (5)

leads to

$$r = {\frac{d \tau}{d p}} = - {\frac{p\,v_n^2(\tau_0)\,\tau_0^2}{\tau}}\;,$$ (6)

which resolves in the oriented velocity-free moveout equation analogous to equation 3

$$\tau_0 = \sqrt{\tau^2 - \tau\,r(\tau,p)\,p}\;.$$ (7)

Combining equations 5 and 7 produces the corresponding velocity mapping equation

$$v_n^2 = \frac{r(\tau,p)}{p^2\,r(\tau,p) - p\,\tau}\;.$$ (8)