Local skewness attribute as a seismic phase detector

Defining skewness as a local attribute

The method of local attributes (Fomel, 2007a) is a technique for extending stationary or instantaneous attributes to smoothly varying or nonstationary attributes by employing a regularized least-squares formulation. In particular, the scalar correlation coefficient $\gamma$ in equation 2 is replaced with a vector, $\mathbf{c}$, defined as a componentwise product of vectors $\mathbf{c}_1$ and $\mathbf{c}_2$, where

$$\mathbf{c}_1 = \left[\lambda^2 \mathbf{I} + \mathbf{S} \left(\mathbf{A}^T \... ...bda^2 \mathbf{I}\right)\right]^{-1} \mathbf{S} \mathbf{A}^T \mathbf{b}\;,$$ (7)

$$\mathbf{c}_2 = \left[\lambda^2 \mathbf{I} + \mathbf{S} \left(\mathbf{B}^T \... ...bda^2 \mathbf{I}\right)\right]^{-1} \mathbf{S} \mathbf{B}^T \mathbf{a}\;.$$ (8)

In equations 7-8, $\mathbf{a}$ and $\mathbf{b}$ are vectors composed of $a_n$ and $b_n$, respectively; $\mathbf{A}$ and $\mathbf{B}$ are diagonal matrices composed of the same elements; and $\mathbf{S}$ is a smoothing operator. We use triangle smoothing (Claerbout, 1992) controlled by specifying the smoothing radius, which can be different in vertical and horizontal directions.

Regularized inversion appearing in equations 7 and 8 is justified in the method of shaping regularization (Fomel, 2007b). The corresponding local similarity attribute has been used previously to align multicomponent and time-lapse images (Kazemeini et al., 2010; Zhang et al., 2013; Fomel, 2007a; Fomel and Jin, 2009), to detect focusing of diffractions (Fomel et al., 2007), to enhance stacking (Liu et al., 2009,2011a), to create time-frequency distributions (Liu et al., 2011b), and to perform zero-phase correction with local kurtosis (Van der Baan and Fomel, 2009). In this paper, we apply it to zero-phasing seismic data using local skewness.

trace,tskew,trace0
Figure 5. (a) Input synthetic trace with variable-phase events. (b) Inverse local skewness as a function of the phase rotation angle. Red colors correspond to high inverse similarity. (c) Synthetic trace after non-stationary rotation to zero phase using picked phase.

We illustrate the proposed zero-phase correction procedure in Figure 5. The input synthetic trace contains a set of Ricker wavelets with a gradually variable phase (Figure 5a). We start with a number of phase rotations with different angles, each time computing the local skewness. The result of this step is displayed in Figure 5b and shows a clear high-similarity trend. After picking the trend, adding $90^{\circ }$ to it, and performing the corresponding nonstationary trace rotation, we end up with the phase-corrected trace, shown in Figure 5c. All the original phase rotations are clearly detected and removed. The radius of the regularization smoothing in this example was 100 samples or 0.4 s.

Local skewness attribute as a seismic phase detector