Signal and noise separation in prestack seismic data using velocity-dependent seislet transform

VD-slope pattern for pegleg multiples

In a laterally homogeneous model, the NMO equation 6 flattens primary events on a CMP gather with offset $x$ and time $t$ to its zero-offset traveltime $t_0$. Brown and Guitton (2005) use an analogous NMO equation for pegleg multiples under locally 1D earth assumption. For example, first-order pegleg can be kinematically approximated by pseudo-primary with the same offset but with an additional zero-offset traveltime $\tau$. The NMO equation for an $m$th-order pegleg multiple is generalized to

$$t_m(x) = \sqrt{(t_0+m\tau)^2 + \frac{x^2}{v_m^2(t_0)}}\;,$$ (8)

where $t_m(x)$ is the corresponding multiple traveltime recorded at offset $x$ and the effective RMS velocity $v_m$ is defined according to Dix's equation as:

$$v_m(t_0) = \sqrt{\frac{t_0v^2(t_0)+m\tau\,v^2(\tau)}{t_0+m\tau}}\;.$$ (9)

In marine seismic data, $v(\tau)$ is constant water velocity, and it assumes that we are able to pick zero-offset traveltime $\tau$ of the water bottom. According to the definition of slopes for primaries (equation 7), slopes for pegleg multiples can be calculated analogously by:

$$\sigma_m(t,x) = {\frac{x}{t_m(x)\,v_m^2(t_0,x)}}\;.$$ (10)

Equation 10 provides the estimation of multiple slopes, which we use to define VD-seislet frame for representing pegleg multiples of different orders.

Signal and noise separation in prestack seismic data using velocity-dependent seislet transform