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

VD-slope pattern for primary reflections

The kinematic description of a seismic event is an essential step for several developments in seismic data processing. Local slope is one important kinematic pattern for seismic data in the time-space domain. PWD provides a constructive algorithm for estimating local slopes (Chen et al., 2013b; Schleicher et al., 2009; Claerbout, 1992; Fomel, 2002; Chen et al., 2013a) and can combine with a seislet framework to implement the PWD-seislet. Local slant stack (Ottolini, 1983a) is another standard tool for calculating slopes.

Under 1D earth assumption, one can consider the classic hyperbolic model of primary reflection moveouts at near offsets (Dix, 1955):

$$t(x) = \sqrt{t_0^2 + \frac{x^2}{v^2(t_0)}}\;,$$ (6)

where $t_0$ is the zero-offset traveltime, $t(x)$ is the corresponding primary traveltime recorded at offset $x$, and $v(t_0)$ is the stacking, or root mean square (RMS) velocity, which comes from a standard velocity scan. As follows from equation 6, the traveltime slopes $\sigma= dt/dx$ in CMP gathers are given by

$${\sigma(t,x)} = {\frac{x}{t(x)\,v^2(t_0,x)}}\;.$$ (7)

This calculation is reverse to the one used in NMO by velocity-independent imaging (Ottolini, 1983b; Fomel, 2007). To calculate local slopes of primaries, we need to know $v(t_0, x)$ at each time-space location ($t_0,x$). This can be accomplished by simultaneously scanning both $t_0$ and $v(t_0, x)$ according to the hyperbolic NMO equation at each $x$-coordinate position or by time-warping. In this paper, we use the time-warping algorithm to calculate $v(t_0, x)$. Time warping performs mapping between different coordinates: if one has sampled functions $f(x)$ and $y(x)$, the mapping operation finds sampled $f(y)$ (Casasanta and Fomel, 2011; Burnett and Fomel, 2009).

After the VD-slope pattern of primaries is calculated, we can design pattern-based prediction and update operators $\mathbf{R}_k$ by using plane-wave construction for the VD-seislet transform to represent only primary reflections. When VD-seislet transform is applied to a CMP gather, random noise spreads over different scales while the predictable reflection information gets compressed to large coefficients at small scales. A simple thresholding operation can easily remove small coefficients. Finally, applying the inverse VD-seislet transform reconstructs the signal while attenuating random noise.

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