![]() |
![]() |
![]() |
![]() | Local skewness attribute as a seismic phase detector | ![]() |
![]() |
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
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 and
is defined as
![]() |
---|
kur
Figure 1. (a) Squared ![]() ![]() ![]() |
![]() ![]() ![]() |
![]() |
---|
sqr
Figure 2. (a) ![]() ![]() ![]() |
![]() ![]() ![]() |
In this paper, we suggest a different measure, skewness, for
measuring the apparent phase of seismic signals. Skewness of a
sequence is defined
as (Bulmer, 1979)
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
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 ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() |
---|
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 ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | Local skewness attribute as a seismic phase detector | ![]() |
![]() |