Local skewness attribute as a seismic phase detector

Localized phase estimation

Our goal is to estimate the time-variant, localized phase from seismic data. What objective measure can indicate that a certain signal has a zero phase? One classic measure is the varimax norm or kurtosis (Wiggins, 1978; Levy and Oldenburg, 1987; White, 1988). Varimax is defined as

$$\phi[\mathbf{s}] = \frac{\displaystyle N \sum_{n=1}^N s_n^4}{\displaystyle \left(\sum_{n=1}^{N} s_n^2\right)^2}\;,$$ (1)

where $\mathbf{s}=\{s_1,s_2,\ldots,s_N\}$ represents a vector of seismic amplitudes inside a window of size $N$. Varimax is simply related to kurtosis of zero-mean signals.

The statistical rationale behind the Wiggins algorithm and its variants is that convolution of any filter with a time series that is white with respect to all statistical orders makes the outcome more Gaussian. The optimum deconvolution filter is therefore one that ensures the deconvolution output is maximally non-Gaussian (Donoho, 1981). The constant-phase assumption made by Levy and Oldenburg (1987) and White (1988) reduces the number of free parameters to one, thereby stabilizing performance compared with the Wiggins method. Wavelets derived in seismic-to-well ties often have a near-constant phase, thus justifying this assumption.

Noticing that the correlation coefficient of two sequences $a_n$ and $b_n$ is defined as

$$\gamma[\mathbf{a},\mathbf{b}] = {\frac{\displaystyle \sum_{... ...displaystyle \sqrt{\sum_{n=1}^N a_n^2 \sum_{n=1}^N b_n^2}}}$$ (2)

and the correlation of $a_n$ with a constant is

$$\gamma[\mathbf{a},\mathbf{1}] = {\frac{\displaystyle \sum_{n=1}^N a_n}{\displaystyle \sqrt{N \sum_{n=1}^N a_n^2}}}\;,$$ (3)

Fomel et al. (2007) interpreted the kurtosis measure in equation 1 as the inverse of the squared correlation coefficient between $s_n^2$ and a constant, $\phi[\mathbf{s}] = 1/\gamma^2[\mathbf{s}^2,\mathbf{1}]$. Well-focused or zero-phase signals exhibit low correlation with a constant and correspondingly higher kurtosis (Figure 1). This provides an alternative interpretation to the goal of making the deconvolution outcome maximally non-Gaussian for desired phase estimation. Note that equation 2 is usually applied to zero-mean sequences ${\bf a}$ and ${\bf b}$. This is neglected in the derivation of expression 3.

kur
Figure 1. (a) Squared $0^{\circ }$-phase Ricker wavelet compared with a constant. (b) Squared $90^{\circ }$-phase Ricker wavelet compared with a constant. The $90^{\circ }$-phase signal has a higher correlation with a constant and correspondingly a lower kurtosis.

sqr
Figure 2. (a) $0^{\circ }$-phase Ricker wavelet compared with its square. (b) $90^{\circ }$-phase Ricker wavelet compared with its square. The $0^{\circ }$-phase has a stronger correlation with its square and correspondingly a higher skewness.

In this paper, we suggest a different measure, skewness, for measuring the apparent phase of seismic signals. Skewness of a sequence $s_n$ is defined as (Bulmer, 1979)

$$\kappa[\mathbf{s}] = \frac{\displaystyle \frac{1}{N} \sum\l... ...left(\frac{1}{N} \sum\limits_{n=1}^{N} s_n^2\right)^{3/2}}\;.$$ (4)

In statistics, skewness is used for measuring asymmetry of probability distributions. Simple algebraic manipulations show that skewness squared can be represented as

$$\kappa^2[\mathbf{s}] = \frac{\displaystyle \left(\sum\limits... ...{1}]} = \phi[\mathbf{s}] \gamma^2[\mathbf{s}^2,\mathbf{s}]\;.$$ (5)

In other words, squared skewness is equal to the kurtosis measure modulated by the squared correlation coefficient between the signal and its square. Zero-phase signals tend to exhibit higher correlation with the square and correspondingly higher skewness (Figure 2). Following experiments with synthetic and field data, we find it advantageous to use sometimes the inverse skewness

$$\displaystyle \frac{1}{\kappa^2[\mathbf{s}]} = \frac{\gamma^2[\mathbf{s}^2,\mathbf{1}]}{\gamma^2[\mathbf{s}^2,\mathbf{s}]}\;.$$ (6)

Unlike kurtosis which measures non-Gaussianity, skewness is related to asymmetry. Whereas convolution of two non-Gaussian sequences makes the outcome more Gaussian, convolution of two asymmetric series becomes more symmetric. Both phenomena are a consequence of the central limit theorem. A zero-phase wavelet is more compact than a nonzero phase one (Schoenberger, 1974), and therefore also more asymmetric. Skewness-based criteria can thus detect the appropriate wavelet phase by applying a series of constant phase rotations to the data and then evaluating the angle that produces the most skewed distribution.

The two measures do not necessarily agree with one another, which is illustrated in Figures 3 and 4. For an isolated positive spike convolved with a compact zero-phase wavelet, the two measures agree in the picking of the zero-phase result as having both a high kurtosis and a high skewness (Figure 3). For a slightly more complex case of a double positive spike convolved with the same wavelet (Figure 4), the two measures disagree: kurtosis picks a signal rotated by $90^{\circ }$ whereas skewness picks the original signal. Note that, in both examples, skewness exhibits a significantly higher dynamical range, which makes it more suitable for picking optimal phase rotations.

ricker-all,ricker-sq-corr,ricker-sq-corr-inv
Figure 3. (a) Ricker wavelet rotated through different phases. (b) Skewness (solid line) and kurtosis (dashed line) as functions of the phase rotation angle. (c) Inverse skewness (solid line) and inverse kurtosis (dashed line) as functions of the phase rotation angle. The two measures agree in picking the signal at $0^{\circ }$ and $180^{\circ }$. Note the higher dynamical range of skewness.

ricker2-all,ricker2-sq-corr,ricker2-sq-corr-inv
Figure 4. (a) Ricker wavelet convolved with a double impulse and rotated through different phases. (b) Skewness (solid line) and kurtosis (dashed line) as functions of the phase rotation angle. (c) Inverse skewness (solid line) and inverse kurtosis (dashed line) as functions of the phase rotation angle. The two measures disagree by $90^{\circ }$ in picking the optimal phase. The skewness attribute picks a better focused signal.

Local skewness attribute as a seismic phase detector