Velocity Transform

Velocity transform is another form of hyperbolic stacking with the summation path  

$$\widehat{\theta}(h;t_0,s) = \sqrt{t_0^2 + s^2\,h^2}\;,$$ (81)

where h corresponds to the offset, s is the stacking slowness, and t₀ is the estimated zero-offset traveltime. Hyperbolic stacking is routinely applied for scanning velocity analysis in common-midpoint stacking. Velocity transform inversion has proved to be a powerful tool for data interpolation and amplitude-preserving multiple supression Ji (1994a); Lumley et al. (1994); Thorson (1984).

Solving equation (81) for t₀, we find that the asymptotic inverse and adjoint operators have the elliptic summation path

$$\theta(s;t,h) = \sqrt{t^2 - s^2\,h^2}\;.$$ (82)

The weighting functions of the asymptotic pseudo-unitary velocity transform are found using formulas (34) and (35) to have the form

$$\begin{eqnarray} w^{(+)} & = & {1\over{\left(2\,\pi\right)^{1/2}}} \, \left\ver... ...\sqrt{\pi}}} \, {{\sqrt{s\,h}\,\sqrt{t_0/t}} \over {\sqrt{t}}}\;.\end{eqnarray}$$ (83) (84)

The factor $\sqrt{s\,h}$ for pseudo-unitary velocity transform weighting has been discovered empirically by Claerbout .