Local skewness attribute as a seismic phase detector

Sergey Fomel1 and Mirko van der Baan2

Abstract

We propose a novel seismic attribute, local skewness, as an indicator of localized phase of seismic signals.The proposed attribute appears to have a higher dynamical range and a better stability than the previously usedlocal kurtosis. Synthetic and real data examples demonstrate the effectiveness of local skewness in detectingand correcting time-varying, locally observed phase of seismic signals.

IntroductionWavelet phase is an important characteristic of

seismic signals. Physical causal systems exhibit mini-mum-phase behavior (Robinson and Treitel, 2000).For purposes of seismic interpretation, it is convenientto deal with zero-phase wavelets with minimum ormaximum amplitudes centered at the horizons of inter-est because it leads to the highest resolution as wellas more accurate estimates of reflection times andspacings (Schoenberger, 1974). Zero-phase correctionis therefore a routine procedure applied to seismicimages before they are passed to the interpreter(Brown, 1999).

It is important to make a distinction between phasesof propagating and locally observed wavelets (van derBaan et al., 2010a, 2010b). The former is the physicalwavelet that propagates through the earth, thereby sam-pling the local geology. It is subject to geometricspreading, attenuation, and concomitant dispersion.The latter is the wavelet as observed at a certain pointin space and time. Its immediate shape results from itsinteraction (convolution) with the reflectivity of theearth and the current shape of the propagating wavelet.For instance, a thin layer with a positive change in seis-mic impedance has opposite polarities of seismic reflec-tivities at the top and the bottom interface, which makeit act like a derivative filter and generate a wavelet withthe locally observed phase subjected to a 90° rotation(Zeng and Backus, 2005). In the absence of well log in-formation, it is usually difficult to separate unambigu-ously the locally observed phase from the phase ofthe propagating wavelet. Nevertheless, measuring thelocal phase can provide a useful attribute for analyzingseismic data (van der Baan and Fomel, 2009; Fomel andvan der Baan, 2010; van der Baan et al., 2010a; Xuet al., 2012).

Levy and Oldenburg (1987) propose a method ofphase detection based on maximization of the varimaxnorm or kurtosis as an objective measure of zero-phase-ness. By rotating the phase and measuring the kurtosisof seismic signals, one can detect the phase rotationsnecessary for zero-phase correction (van der Baan,2008). van der Baan and Fomel (2009) apply local kur-tosis, a smoothly nonstationary measure (Fomel et al.,2007), and demonstrate its advantages in measuringphase variations as compared with kurtosis measure-ments in sliding windows. Local kurtosis is an exampleof a local attribute (Fomel, 2007a) defined by usingregularized least-squares inversion.

In this paper, we revisit the problem of phase estima-tion and propose a novel attribute, local skewness, as aphase detector. Analogous to local kurtosis, local skew-ness is defined using local similarity measurements viaregularized least squares. This attribute is maximizedwhen the locally observed phase is close to zero. Advan-tages of the new attribute are a higher dynamical rangeand a better stability, which make it suitable for pickingphase-corrections. Using synthetic and field-data exam-ples, we demonstrate properties and applications of theproposed attribute.

Localized phase estimationOur goal is to estimate the time-variant, localized phase

from seismic data. What objective measure can indicatethat a certain signal has a zero phase? One classic mea-sure is the varimax norm or kurtosis (Wiggins, 1978; Levyand Oldenburg, 1987; White, 1988). Varimax is defined as

ϕ½s� ¼ NP

Nn¼1 s

4n�P

Nn¼1 s

2n

�2 ; (1)

1The University of Texas at Austin, Bureau of Economic Geology, John A. and Katherine G. Jackson School of Geosciences, Austin, Texas, USA.E-mail: sergey.fomel@beg.utexas.edu.

2University of Alberta, Department of Physics, Edmonton, Alberta, Canada. E-mail: Mirko.VanderBaan@ualberta.ca.Manuscript received by the Editor 10 June 2013; published online 7 February 2014. This paper appears in Interpretation, Vol. 2, No. 1 (February

2014); p. SA49–SA56, 12 FIGS.http://dx.doi.org/10.1190/INT-2013-0080.1. © 2014 Society of Exploration Geophysicists and American Association of Petroleum Geologists. All rights reserved.

t

Special section: Seismic attributes

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where s ¼ fs1; s2; : : : ; sNg represents a vector of seismicamplitudes inside a window of size N . Varimax is relatedto kurtosis of zero-mean signals.

The statistical rationale behind the Wiggins algo-rithm and its variants is that convolution of any filterwith a time series that is white with respect to all stat-istical orders makes the outcome more Gaussian. Theoptimum deconvolution filter is therefore one that en-sures the deconvolution output is maximally nonGaus-sian (Donoho, 1981). The constant-phase assumptionmade by Levy and Oldenburg (1987) and White(1988) reduces the number of free parameters to one,thereby stabilizing performance compared with theWiggins method. Wavelets derived in seismic-to-wellties often have a near-constant phase, thus justifyingthis assumption.

Noticing that the correlation coefficient of two se-quences an and bn is defined as

γ½a;b� ¼P

Nn¼1 anbnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

Nn¼1 a

2nP

Nn¼1 b

2n

q (2)

and the correlation of an with a constant is

γ½a;1� ¼P

Nn¼1 anffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

NP

Nn¼1 a

2n

q . (3)

Fomel et al. (2007) interpret the kurtosis measure inequation 1 as the inverse of the squared correlation co-efficient between s2n and a constant, ϕ½s� ¼ 1∕γ2½s2;1�.Well-focused or zero-phase signals exhibit low corre-lation with a constant and correspondingly higherkurtosis (Figure 1). This provides an alternative inter-pretation to the goal of making the deconvolutionoutcome maximally non-Gaussian for desired phase es-timation. Note that equation 2 is usually applied to zero-

mean sequences a and b. This is neglected in the der-ivation of expression 3.

In this paper, we suggest a different measure, skew-ness, for measuring the apparent phase of seismic sig-nals. Skewness of a sequence sn is defined as (Bulmer,1979)

κ½s� ¼1N

PNn¼1 s

3n�

1N

PNn¼1 s

2n

�3∕2 : (4)

In statistics, skewness is used for measuring asymmetryof probability distributions. Simple algebraic manipula-tions show that skewness squared can be represented as

κ2½s� ¼�P

Nn¼1 s

2n · sn

�2

PNn¼1 s

4nP

Nn¼1 s

2n

PNn¼1 s

4nP

Nn¼1 1

2�PNn¼1 s

2n

�2 ¼ γ2½s2; s�

γ2½s2;1�

¼ ϕ½s�γ2½s2; s�: (5)

In other words, squared skewness is equal to the kurtosismeasure modulated by the squared correlation coeffi-cient between the signal and its square. Zero-phase sig-nals tend to exhibit higher correlation with the squareand correspondingly higher skewness (Figure 2). Follow-ing experiments with synthetic and field data, we find itadvantageous to use sometimes the inverse skewness

1

κ2½s� ¼γ2½s2;1�γ2½s2; s� : (6)

Unlike kurtosis, which measures non-Gaussianity,skewness is related to asymmetry. Whereas convolu-tion of two non-Gaussian sequences makes the out-come more Gaussian, convolution of two asymmetricseries becomes more symmetric. Both phenomena

–0.4 –0.2 0Time (s) Time (s)

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20.

40.

60.

81

00.

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81

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a) b)

Figure 1. (a) Squared 0°-phase Ricker wavelet compared with a constant. (b) Squared 90°-phase Ricker wavelet compared with aconstant. The 90°-phase signal has a higher correlation with a constant and correspondingly a lower kurtosis.

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are a consequence of the central limit theorem. A zero-phase wavelet is more compact than a nonzero phaseone (Schoenberger, 1974), and therefore also moreasymmetric. Skewness-based criteria can thus detectthe appropriate wavelet phase by applying a series of

constant phase rotations to the data and then evaluatingthe angle that produces the most skewed distribution.

The two measures do not necessarily agree with oneanother, which is illustrated in Figures 3 and 4. For anisolated positive spike convolved with a compact zero-

–0.4 –0.2 0

a) b)

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1–0

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itude

Figure 2. (a) Here, 0°-phase Ricker wavelet compared with its square. (b) 90°-phase Ricker wavelet compared with its square. The0°-phase has a stronger correlation with its square and correspondingly a higher skewness.

–150 –100

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Degree (°) Degree (°)50 100 150 –150 –100 –50 0 50 100 150 –150 –100 –50 0 50 100 150

Figure 3. (a) Ricker wavelet rotated through different phases. (b) Skewness (solid line) and kurtosis (dashed line) as functions ofthe phase rotation angle. (c) Inverse skewness (solid line) and inverse kurtosis (dashed line) as functions of the phase rotationangle. The two measures agree in picking the signal at 0° and 180°. Note the higher dynamical range of skewness.

–150 –100 –50 0Degree (°)

50 100 150 –150 –100 –50 0Degree (°)

50 100 150 –150 –100 –50 0Degree (°)

50 100 1500.4

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plitu

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b) c)a)

Figure 4. (a) Ricker wavelet convolved with a double impulse and rotated through different phases. (b) Skewness (solid line) andkurtosis (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° in picking the optimal phase. The skewness attributepicks a better focused signal.

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phase wavelet, the two measures agree in the picking ofthe zero-phase result as having a high kurtosis and ahigh skewness (Figure 3). For a slightly more complexcase of a double positive spike convolved with the samewavelet (Figure 4), the two measures disagree: Kurtosispicks a signal rotated by 90° whereas skewness picksthe original signal. Note that, in both examples, skew-ness exhibits a significantly higher dynamical range,which makes it more suitable for picking optimal phaserotations.

Defining skewness as a local attributeThe method of local attributes (Fomel, 2007a) is a

technique for extending stationary or instantaneousattributes to smoothly varying or nonstationary attrib-utes by employing a regularized least-squares formu-lation. In particular, the scalar correlation coefficientγ in equation 2 is replaced with a vector, c, definedas a componentwise product of vectors c1 and c2,where

c1 ¼ ½λ2Iþ SðATA − λ2IÞ�−1SATb; (7)

c2 ¼ ½λ2Iþ SðBTB − λ2IÞ�−1SBTa: (8)

In equations 7 and 8, a and b are vectors composed ofan and bn, respectively; A and B are diagonal matricescomposed of the same elements; and S is a smoothingoperator. We use triangle smoothing (Claerbout, 1992)controlled by specifying the smoothing radius, whichcan be different in vertical and horizontal directions.

Regularized inversion appearing in equations 7 and 8is justified in the method of shaping regularization(Fomel, 2007b). The corresponding local similarityattribute has been used previously to align multi-component and time-lapse images (Fomel, 2007a; Fo-mel and Jin, 2009; Kazemeini et al., 2010; Zhanget al., 2013), to detect focusing of diffractions (Fomelet 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 localkurtosis (van der Baan and Fomel, 2009). In this paper,we apply it to zero-phasing seismic data using localskewness.

We illustrate the proposed zero-phase correctionprocedure in Figure 5. The input synthetic trace con-tains a set of Ricker wavelets with a gradually variablephase (Figure 5a). We start with phase rotations withdifferent angles, each time computing the local skew-ness. The result of this step is displayed in Figure 5band shows a clear low-skewness trend. After pickingthe trend, adding 90° to it, and performing the corre-sponding nonstationary trace rotation, we end up withthe phase-corrected trace, shown in Figure 5c. All theoriginal phase rotations are clearly detected and re-moved. The radius of the regularization smoothing inthis example was 100 samples or 0.4 s.

Application exampleThe input data set for our field-data example is the

Boonsville data set from the Fort Worth Basin in North-Central Texas, USA (Hardage et al., 1996a, 1996b). Theformations of interest are the Ellenburger Carbonates,the Barnett Shale, and the Bend Conglomerates (seeFigure 6 for a stratigraphic column). The EllenburgerCarbonates are of Ordovician age. Their karstificationdue to post-Ellenburger carbonate dissolution and sub-sequent cavern collapses has created sags in the over-lying formations, affecting sedimentation patterns andstructures in the overlying Barnett Shales and BendConglomerates. The collapse features look like verticalchimneys with roughly circular cross sections, extend-ing up to 600–760 m above the Ellenburger Carbonates(Hardage et al., 1996a), sometimes even reaching intothe Strawn Group above the Bend Conglomerates.

The Barnett Shales are of Mississippian age. They arethe target of much current exploitation in Texas asthese are tight-shale reservoirs (Pollastro et al.,2007). Zones with karst-induced cavern collapses forma drilling and completion hazard for the mainly horizon-tal drilling programs in these tight-shale reservoirsand must be mapped. They may affect local fracture

0

Input

0.5 1 1.5 2 2.5Time (s) Time (s)Inverse local skewness

Phase rotation (°)

Phase-corrected

Tim

e (s

)

3 3.5 4

–50 0

a) b) c)

50 100 150 200 250

0 0.5 1.51 2 2.5 3 3.5 443

21

0

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 nonstationary rotation to zero phase using picked phase.

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densities and thus permeabilities and reservoir drainagepositively, but can also lead to fluid barriers due to res-ervoir compartmentalization.

The shallower clastic Bend Conglomerates are ofMiddle Pennsylvanian (Atokan) age. The formationhas a thickness of 300–360 m in this area with depthsfrom 1370 to 1830 m. It was targeted throughout the1980s and 1990s as it contains several gas and oil-bear-ing reservoirs in a stacked fashion. Hardage et al.(1996a, 1996b) describe how the karstification hasgreatly impacted the system tracts and sedimentationpatterns in the Bend formation which were character-ized by low accommodation space. Resulting reservoir

compartmentalization is a significant challenge for thisformation and has also affected the reflector character.Reflection near the base of this formation display reflec-tor weakening and sometimes even polarity reversals inareas depressed due to local sagging. Acquisition andprocessing of this data set are described by Hardageet al. (1996a). A stacked section is shown in Figure 7.A zero-phase correction has been applied to the databut has left regions with variable localized phase.

Our processing sequence is similar to the one used inFigure 5. First, we apply phase rotations with differentangles and compute local skewness for each rotation.The regularization lengths in this examples were 500samples or 0.5 s in time and 50 traces in space. The re-sult is displayed in Figure 8. Next, we apply automaticpicking with the algorithm described by Fomel (2009)to extract the nonstationary phase rotation that maxi-mizes the local skewness. Finally, the phase-correctionis applied to the data, with the result displayed in

Figure 6. Stratigraphic column of the Fort Worth Basinwhere the Boonsville data set is located, after Pollastro et al.(2007). Karstification in the Ellenburger Carbonates hascaused local sags in the overlying Barnett and Bend Conglom-erate formations, creating reservoir compartmentalization.

50

21.

5T

ime

(s)

10.

50

100 150Trace

200

Figure 7. Input data: a section of the Boonsville data set.

0Phase rotation (°)

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)2

10

100

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200

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e90

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Figure 8. Inverse local skewness as a function of the phaserotation angle, with application to the section from Figure 7.Red colors correspond to high inverse similarity.

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Figure 9. A zoomed-in comparison shows the effects ofnonstationary phase correction: rotating major seismicevents to 0° and improving their continuity. These ef-fects can be useful for improving structural interpreta-tion, and for improved matching of seismic data andwell logs.

We applied the local phase detection to the 3D vol-ume in a window centered on the target horizon (Fig-ure 10). The estimated local phase variation along thetarget horizon is shown in Figure 11. Comparing the am-plitude before and after phase correction (Figure 12),we observe a noticeable improvement in event continu-ity. Once the processing and interpretation are done on

the zero-phase-corrected volume, it is easy to restorethe original phase by applying the inverse phaserotation.

DiscussionThe data example, as well as the previous case stud-

ies by van der Baan et al. (2010b), underline how analy-sis of the local phase can be used as a complementaryattribute to spectral decomposition to highlight varia-tions in wavelet character. There are two complimen-tary applications, namely analysis of the propagatingand locally observed wavelets. The targeted wavelet

type is determined by the chosen regu-larization length: The propagating wave-let is estimated by using long temporalregularization lengths, and the locallyobserved one from shorter lengths.The underlying assumption is that, forlong regularization lengths, variationsin the local geology are averaged out,revealing only the propagating wavelet.In this paper, we used relatively shortregularization lengths as the aim wasto highlight changes in the local reflec-tion character.

Well-log analyses have demonstratedthat the earth’s reflectivity series is non-Gaussian (Walden and Hosken, 1986)and white during first order (Walden andHosken, 1985). In addition, impedancestend to increase with depth, hence posi-tive reflection coefficients are slightlymore likely than negative ones, produc-ing an asymmetric reflectivity distribu-tion. Statistically, the skewness-basedcriterion assumes that the earth’s reflec-tivity series are white and asymmetric.This is in contrast to kurtosis used pre-

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)0.

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a) b)

6 8 10 12Trace

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14 16 18 20 22 24 42 6 8 10 12Trace

14 16 18 20 22 24

Figure 9. Zoomed-in comparison of the data before phase correction (a) andafter phase correction (b). Nonstationary phase correction helps in identifyingsignificant horizons and increasing their resolution in time.

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Figure 10. Boonsville data set windowed around the targethorizon.

80

120 –5

0A

ngle

(°)

050

Line

140

160

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200

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140 160 180 200

Figure 11. Local phase variation along the target horizon es-timated from the data shown in Figure 10.

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viously by van der Baan and Fomel (2009), which as-sumes a non-Gaussian reflectivity series. The non-Gaus-sianity and asymmetry assumptions seem warranted,but may fail if the local reflectivity series becomes, re-spectively, purely Gaussian or symmetric. The localskewness attribute has the advantage over kurtosis be-cause of its higher dynamic range, which facilitatespicking. Variance is the second statistical order, skew-ness is the third one, and kurtosis is related to the fourthorder. Estimation variances increases with the order ofa moment (Mendel, 1991). In other words, less samplesare needed to estimate skewness with the same accu-racy as kurtosis. We hypothesize that this contributesto the higher dynamic range of the skewness attribute.

ConclusionsWe have presented a novel approach to nonstation-

ary identification of apparent (locally observed) phase.Our approach is based on a new attribute, local skew-ness. In synthetic and field-data examples, local skew-ness exhibits a tendency to pick focused signals and ahigher dynamical range than the previously used localkurtosis. Its computation involves a local similarity be-tween the input signal and its square. Practical applica-tions of using local skewness for zero-phase correctionof seismic signals should be combined with well-loganalysis to better separate the locally observed phasefrom the propagating-wavelet phase.

AcknowledgmentsWe thank Milo Backus, Guochang Liu, Mike Perz,

and Hongliu Zeng for stimulating discussions.

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Figure 12. Amplitude along the target horizon before and after phase rotation.

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Sergey Fomel received a Ph.D.(2001) in geophysics from StanfordUniversity and worked previously atthe Institute of Geophysics in Russia,Schlumberger Geco-Prakla, and theLawrence Berkeley National Labora-tory. He is a professor at the Univer-sity of Texas at Austin, JacksonSchool of Geosciences, with a joint

appointment between the Bureau of Economic Geologyand the Department of Geological Sciences. He has re-ceived a number of professional awards, including the J.Clarence Karcher Award from SEG in 2001 and the ConradSchlumberger Award from EAGE in 2011. He is currentlydeveloping Madagascar, an open-source software packagefor geophysical data analysis.

Mirko van der Baan holds an M.Sc.(1996) from the University of Utrecht,the Netherlands, a Ph.D. (1999) withhonors from the Joseph Fourier Uni-versity, Grenoble, France, and ahabilitation (HDR, 2008) from the Uni-versity Denis Diderot, Paris, France.He was a Reader in Exploration Seis-mology at the University of Leeds, UK,

before he joined the Department of Physics at the Univer-sity of Alberta, Canada, as a professor. His research inter-ests include attenuation and velocity anisotropy, signalprocessing, and microseismicity. He is an Associate Editorfor GEOPHYSICS, serves on the SEG Research Committee,and a member of EAGE, CSEG, and SEG.

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