Geophysical Prospecting, 2013, 61 (Suppl. 1), 1–9 doi: 10.1111/j.1365-2478.2012.01090.x

Coherent and random noise attenuation via multichannel singularspectrum analysis in the randomized domain

Stephen K. Chiu∗ConocoPhillips, Houston, Texas, United States

Received November 2011, revision accepted March 2012

ABSTRACTThe attenuation of coherent and random noise still poses technical challenges inseismic data processing, especially in onshore environments. Multichannel SingularSpectrum Analysis (MSSA) is an existing and effective technique for random-noisereduction. By incorporating a randomizing operator into MSSA, this modificationcreates a new and powerful filtering method that can attenuate both coherent andrandom noise simultaneously. The key of the randomizing operator exploits the factthat primary events after NMO are relatively horizontal. The randomizing operatorrandomly rearranges the order of input data and reorganizes coherent noise intoincoherent noise but has a minimal effect on nearly horizontal primary reflections.The randomizing process enables MSSA to suppress both coherent and random noisesimultaneously. This new filter, MSSARD (MSSA in the randomized domain) alsoresembles a combination of eigenimage and Cadzow filters. I start with a syntheticdata set to illustrate the basic concept and apply MSSARD filtering on a 3D cross-spread data set that was severely contaminated with ground roll and scattered noise.MSSARD filtering gives superior results when compared with a conventional 3D f-kfilter. For a random-noise example, the application of MSSARD filtering on time-migrated offset-vector-tile (OVT) gathers also produces images with higher signal-to-noise ratios than a conventional f-xy deconvolution filter.

Key words: Randomizing operator, Eigenimage, Deconvolution

INTRODUCTION

Coherent noise often coexists with random noise in seismicfield data. Noise attenuation plays a key role in data pro-cessing to enhance the signal content of the data. The char-acteristics of coherent noise are considerably different thanrandom noise. A general approach requires designing filteringtechniques based upon the characteristics of noise and appliesnoise suppression to target each noise type separately. This isa standard denoising procedure in the industry. However, itis desirable to have a filtering method that can attenuate bothcoherent and random noise simultaneously. The main objec-

∗E-mail: Stephen.K.Chiu@conocophillips.com

tive of this paper is to introduce a new rank-based-reductiondenoising algorithm to perform both filtering operations con-currently.

Numerous filtering methods have been developed to sup-press various types of coherent noise; however, coherent-noise attenuation is still an active research area. For exam-ple, the suppression of surface-wave noise including groundroll and scattered noise is a difficult problem. Ground rolland near-surface scattered energy are the most troublesomeforms of coherent noise. Ground roll is characterized by low-velocity events with high amplitude and dispersive low fre-quencies. Near-surface scattered energy often exhibits compli-cated diffraction patterns associated with the spatial positionsof point scatters. Strong coherent noise often overwhelmsand obscures primary reflections. Classical ‘global filtering’

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2 S.K. Chiu

methods such as f-k and tau-p have shown limited success inattenuating ground roll or scattered energy because of spatialaliasing and irregular spacing in the data. In addition, globalfiltering tends to create amplitude artefacts by smearing theamplitudes spatially. Recent methods on ground-roll attenu-ation employ transform algorithms to localize the coherentnoise in the transform domains to achieve a better separationbetween the signals and noise. The transform methods includewavelet transform (Deighan and Watts 1997), curvelet trans-form (Yarham and Herrmann 2008) and S-transform (Askariand Siahkoohi 2008). However, the transform methods mayproduce suboptimal filtering results, if the data are spatiallyaliased. Eigenimage filtering (Chiu and Howell 2008; Caryand Zhang 2009) using singular-value decomposition pro-vides another alternative to suppress the surface-wave noise.It can handle irregular sampling and aliased noise with mini-mal spatial smearing of amplitudes. The eigenimage filter alsorelates to a class of denoising techniques that use rank reduc-tion of a Hankel matrix to approximate the signals (Trickettand Burroughs 2009).

Random noise attenuation is a critical step to improvesignal-to-noise ratios for prestack and post-stack data.Median-based filtering methods are widely used in the in-dustry to suppress random or spike-like noise. Some of themedian-filtering methods include a 2D multistage medianfilter (Liu et al. 2006), lineament-preserving filtering (Al-Dossary and Marfurt 2007) and a 1D time-varying medianfilter (Liu, Liu and Wang 2009). Canales (1984) proposeda signal-enhancement algorithm in the f-x domain (f-x de-convolution), which became a popular algorithm for randomnoise attenuation. If the noise level is high, this algorithmcan distort the signals considerably (Harris and White 1997).Trickett and Burroughs (2009) applied a Cadzow filter toovercome the low signal-to-noise ratio of prestack data andthey summarized the key properties of Cadzow filtering. Fourof the desirable properties are: ability to handle low signal-to-noise data, preservation of AVO and AVAZ effects, no cre-ation of artefacts and flexibility to allow filtering to be strongor mild. Sacchi (2009) and Oropeza (2010) adopted an al-ternative method based on Multichannel Singular SpectrumAnalysis (MSSA) for random-noise suppression but it turnsout that Cadzow filtering is equivalent to MSSA. Both filtersbelong to the same class of rank-reduction based denoisingtechniques. Another advantage of this class of filtering is theability to easily extend from a 2D filter to a multi-dimensionalfilter. However, the Cadzow filter or MSSA is for random-noise attenuation only. It cannot distinguish between coherent

noise and signals. As a result, it treats the coherent noise assignals.

This paper presents a novel method by incorporating a ran-domizing operator into the f-xy MSSA. The randomizing op-erator is the key in creating a new and powerful denoisingtechnique that attenuates both coherent and random noisesimultaneously. This technique resembles a combination ofeigenimage and Cadzow filters and is referred to as Multichan-nel Singular Spectrum Analysis in the randomized domain(f-xy MSSARD). To shorten the term ‘f-xy MSSARD’,MSSARD is used for the entire paper instead of f-xyMSSARD. I will first describe the basic theory and use a 3Dsynthetic example to illustrate the concept. The application ofthe MSSARD on a 3D cross-spread data set that was severelycontaminated with ground roll and scattered noise clearlydemonstrates its effectiveness in suppressing strong aliasednoise when compared with a 3D f-k filter. For a random-noiseexample, I apply f-xy deconvolution and MSSARD filters onmigrated offset-vector-tile (OVT) gathers and show the com-parison of both filter performances.

THEORY

Multichannel Singular Spectrum Analysis

I begin with a discussion of the basic Multichannel Singu-lar Spectrum Analysis (MSSA) theory and then present theMSSARD (MSSA in the randomized domain) algorithm. Tosimplify the MSSA discussion, let us assume a square datacube but a rectangular cube will work the same way. For eachsquare cube, the data are transformed into the frequency do-main to form an n-by-n grid of complex values. The spatialcoordinates of the in-line and cross-line of a 3D post-stackseismic volume represent an n-by-n grid example. d(i,j) corre-sponds to a complex data value at a location in this frequencyslice. The data grid at a given frequency can be represented bya matrix D.

D =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

d(1, 1) d(1, 2) d(1, 3) .... d(1, n)

d(2, 1) d(2, 2) d(2, 3) .... d(2, n)

d(3, 1) d(3, 2) ... .... d(3, n): : :: : :d(n, 1) d(n, 2) d(n, 3) .... d(n, n)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (1)

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Coherent and random noise attenuation in the randomized domain 3

MSSA first constructs a Hankel matrix, Hi, for each row ofmatrix D.

Hi =

⎡⎢⎢⎢⎢⎢⎢⎣

d(i, 1) d(i, 2) .... d(i, n − k + 1)d(i, 2) d(i, 3) .... d(i, n − k + 2): : :: : :d(i, k) d(i, k + 1) .... d(i, n)

⎤⎥⎥⎥⎥⎥⎥⎦

, (2)

where k is chosen to be n/2 to make the Hankel matrix tobe approximately square (Trickett and Burroughs 2009). TheHankel matrix does not need to be a square; it can be rectan-gular. Since each row of matrix D creates a Hankel matrix,it generates a series of Hankel matrices ranging from H1 toHn. By arranging each Hi into a block Hankel matrix, theblock Hankel matrix, M, forms a signal matrix to be used inderiving a 2D spatial filter. M is defined as

M=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

H1 H2 .... Hn−k+1

H2 H3 .... Hn−k+2

: : :: : :Hk Hk+1 .... Hn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (3)

The noise attenuation is achieved through a low-rank re-construction of the matrix M. Singular value decomposition(Golub and Van Loan 1996) provides a simple way to com-pute this low-rank matrix by selecting a range of eigenimages(Freire and Ulrych 1998). The coherent signals tend to fallinto the first few eigenimages, while the remaining eigenim-ages correspond to random noise.

After singular value decomposition, M becomes

M = USVH, (4)

where U is a matrix of eigenvectors that span the data space,V is a matrix of eigenvectors that span the model space, S isa diagonal matrix whose diagonal elements organized in de-scending order are called singular values and H is a conjugatetranspose operator.

By keeping the first k largest singular values, the k-rankreduction matrix of M is

Mk = UkSkVHk . (5)

The choice of k depends on the noise characteristics andfiltering objective. The determination of the k value is oftenby trial and error. In general, the stronger the filter, the smallerthe k value.

The filtered data are recovered by averaging along the anti-diagonal elements of the Mk matrix (Oropeza and Sacchi

2010). Letting ζ A be an averaging operator, the filtered data,S, after MSSA filtering are as follows:

S = ζAMk.

Multichannel Singular Spectrum Analysis in the randomizeddomain

MSSARD assumes a 3D data volume to have NMO applied.The 3D data set is then divided into overlapping smaller 3Dcubes. Let ζ M be an operator that performs the MSSA filter-ing, ζ R be a randomizing operator and ζ R

−1 be an inverserandomizing operator. The use of a randomizing operator ex-ploits the fact that primary events after NMO are relativelyhorizontal. The randomizing operator randomly rearrangesthe order of the input data and reorganizes the coherent noiseinto incoherent noise but has minimal effect on the nearly hor-izontal primary reflections. This randomizing process enablesMSSA to suppress both coherent and random noise simultane-ously. MSSARD works surprisingly well even when primaryreflections have residual moveout, because it uses a small datacube for the analysis. Within a small data window, primaryreflections are considered to be locally horizontal.

However, if the primary reflections have considerable cur-vatures, it is crucial to determine an optimal-analysis window(data cube to be used for MSSARD). Typically, poor NMOvelocities can cause the primary reflections of the prestackdata to deviate significantly from the horizontal; the geologystructures also contribute to the curvatures of seismic eventson the post-stack data. In the case of primary events havingconsiderable curvatures, the analysis window should be smallenough to approximate the primary events to be locally hor-izontal in order to preserve the data integrity. If the analysiswindow is too small, there is little noise attenuation but it pre-serves the primary events. If the analysis window is too large, itsuppresses the noise but it potentially attenuates primary sig-nals. The analysis-window size is a trade-off between signalpreservations and noise attenuation; it is critical to determinean optimal-analysis window through testing to minimize thepotential loss of the primary signals.

MSSARD filtering at a given frequency is

Df = ζ−1R ζMζRD, (6)

where D is the input data matrix and Df is the data matrixafter MSSARD filtering at a given frequency. The randomizingoperator can be either applied in the time or frequency domain

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Figure 1

Figure 2

but it is computationally more convenient to apply in thefrequency domain.

The MSSARD algorithm consists in the following steps:� Transform 3D data into the frequency domain.

� Apply a randomizing operator to the frequency data.� Apply MSSA to the random data.� Inverse the randomizing operator.� Transform the frequency data back to the time domain.

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Coherent and random noise attenuation in the randomized domain 5

Figure 3

Synthetic example

The typical first step to evaluate a new algorithm is to use asynthetic data set to validate the concept and to understandhow it works. The 3D synthetic model consists in two primaryevents contaminated with linear and random noise. Figure1(a) shows an in-line section of a 25 trace by 25 trace stackvolume with two primary reflections. The first primary reflec-tion simulates residual moveout after NMO and the secondis horizontal. Two linear noise trains and random noise areadded to the data to generate a 3D noise contaminated dataset (Fig. 1b,c). MSSARD recovers both primary reflectionscorrectly (Fig. 1d) and preserves the amplitudes of primaryreflections remarkably well (Fig. 1e). To further illustrate itsperformance, Fig. 2(a) shows a panel of four in-line sectionsof this noise contaminated 3D synthetic data. The MSSARDfilter does an excellent job to attenuate both coherent and

random noise, even when the linear-noise trains vary spa-tially. The primary reflections extracted from the noisy back-ground match the ideal primary reflections remarkably well(Fig. 2b,c). This demonstrates the effectiveness of this new 3Dfilter.

Another way in understanding how the randomizing oper-ator works is to examine this synthetic example in the fre-quency domain. As an illustrated example, I chose a 31 Hzfrequency-data slice. The amplitude map of the two primaryreflections shows little variation (Fig. 3a) because the ampli-tudes only vary between 12.9–13.7 in units of amplitude perhertz but the use of the overall plotting scale ranging from1–40 in units of amplitude per hertz considerably reduces theamplitude variations visually. However, the amplitude mapcontaminated with random noise and two linear noise trainsshows a series of linear events (Fig. 3b). MSSA filtering sup-presses the random noise but it also enhances the linear-noise

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Figure 4

trends (Fig. 3c). The distinctive linear events become incoher-ent (Fig. 3d) after randomly reordering the amplitude map ofFig. 3(b). MSSA filtering on Fig. 3(d) attenuates both randomand coherent noise (Fig. 3e). A small residual noise imprint(Fig. 3f) is still visible after applying the inverse randomizingoperator on Fig. 3(e). This residual noise in the frequency do-main also reflects the residual noise in the time domain as inFig. 2(c).

Cross-spread example

A cross spread is a simulated single-fold 3D gather formedby a source line and a receiver line. The ground roll in thisdomain mainly lies on the surface of a circular cone (Steinand Langston 2007). A 3D filter can drastically suppress theground roll in a cross spread. The cross-spread data in this ex-ample were severely contaminated with ground roll, scatterednoise and random noise. Figure 4(a) shows a typical in-linesection after NMO correction. This in-line example is at thecentre of the cross spread where the noise contamination isthe worst. The target horizon is between 2300–2400 ms. Thedominant ground roll and scattered noise severely mask the

primary reflections at the target horizon. The application ofthe 3D f-k filter attenuates a significant portion of the noisebut residual ground roll and scattered noise are still notice-able after 3D f-k filtering (Fig. 4b). This is because the f-kfilter cannot properly handle aliased noise. Conflicting dip-ping noise trains are still visible after f-k filtering and theamplitudes of primary reflections are not well preserved (Fig.4e). The MSSARD filter, on the other hand, is able to han-dle the aliased-coherent noise and has the flexibility to useeither a mild or a strong filter on the data. The mild filter,in this case, uses the first 4 largest singular values to recon-struct the low-rank matrix but the strong filter only uses thefirst 2 largest singular values. The overlapping 3D data cubeuses a 125 sample of a time window and 11 by 11 of thetwo spatial dimensions. Figure 4(c) and Fig. 4(f) show the fil-tering result using a mild MSSARD filter. The mild filter inthis case uses the first 4 largest eigenimages to approximatethe data cube. For example, if there are only two primaryevents present in the data cube, the first two largest eigen-images model these two primary events; the two remainingeigenimages model the noise. The filtering result composes oftwo primary events and a portion of noise. When the number

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Coherent and random noise attenuation in the randomized domain 7

Figure 5

of eigenimages used in the filtering exceeds the number of pri-mary events present in the data, some residual ground roll andscattered noise are still visible after filtering but the differencebetween the input and filtered data shows minimal distortionof the primary reflections. The ground roll and scattered noiseare not observable after the strong MSSARD filter and thedifference plot also shows minimal distortion of the primaryreflections (Fig 4d and Fig. 4g). After strong MSSARD filter-ing, some residual random noise still remains. One approachto attenuate the residual random noise is to apply another passof random-noise attenuation but a preferable approach is toallow stacking or migration to suppress the residual randomnoise. Comparing the time slices of all filtering results at thetarget horizon is another way to assess the performance ofeach filter. The circular patterns typically relate to ground rollon a cross spread (Fig. 5a). Circular ground-roll patterns arenot noticeable (Fig. 5d) after the strong MSSARD filter. Onthe contrary, residual ground roll and scattered noise after a

3D f-k filter or a mild MSSARD filter are still apparent on thetime slice (Fig. 5b,c). Although we can apply a hasher 3D f-kfilter to suppress the noise more aggressively, the aggressivefilter also causes more spatial smearing of the primary signals,especially when the data are spatially aliased. The aggressiveMSSARD filter, in contrast, helps to suppress the coherentnoise and avoids the spatial-amplitude smearing associatedwith the f-k filter. The flexibility to use either a mild or strongMSSARD filter is a key advantage of this algorithm.

Example of migrated offset vector tiles

Offset-vector-tile is a natural extension of cross spreads. AnOVT is a collection of traces that have a limited range ofoffsets and azimuths from a subset of cross spreads (Ver-meer 2007). ConocoPhillips has acquired a high-density wide-azimuth land 3D seismic survey. The survey was designedto acquire data with dense spatial sampling rich in azimuth

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8 S.K. Chiu

Figure 6

Figure 7

and offset for azimuthal analysis. After prestack time mi-gration, the high-amplitude random noise severely degradedthe primary events especially between 0–1.8 s (Fig. 6a). F-xy deconvolution is a standard processing step for random-

noise attenuation. The high-amplitude random noise, in thisdata set, degrades the effectiveness of f-xy deconvolution.Thus, the f-xy deconvolution filtering result is suboptimal,with noticeable residual random noise remaining after the

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Coherent and random noise attenuation in the randomized domain 9

filter (Fig. 6b). In contrast, the MSSARD filter derived fromthe first 2 largest singular values produces significantly betterimages that have higher signal-to-noise ratios (Fig. 6c). We ex-amine the filtered data in more detail by zooming in the targetzone between 2.0–2.6 s (Fig. 7a). F-xy deconvolution helpsto improve the signal content of the primary reflections butresidual random noise is apparent after filtering (Fig. 7b andFig. 7d). The MSSARD filter, on the other hand, greatly re-duces the incoherent noise and the primary reflections becomemore visible and continuous (Fig. 7c and Fig. 7e).

CONCLUSIONS

Multichannel Singular Spectrum Analysis in the randomizeddomain (MSSARD) provides a new and powerful filteringmethod that can attenuate both coherent and random noisesimultaneously. It handles low signal-to-noise data includingaliased data, does not create artefacts and allows filtering tobe mild or strong. It outperforms the classical 3D f-k filter on across-spread 3D field data set that was severely contaminatedwith coherent noise. For the random-noise example, f-xy de-convolution helps to improve signal content of the primaryreflections but the MSSARD filter produces significantly bet-ter images with higher signal-to-noise ratios. This MSSARDfilter provides a new and valuable tool in data processing tosuppress unwanted noise.

ACKNOWLEDG EME N T S

The author thanks ConocoPhillips for permission to publishthis paper. I would also like to thank Brad Bankhead of Cono-coPhillips Subsurface Technology; Christopher Barrett and Ja-son Stein of ConocoPhillips Lower 48 Business unit for theirsupport; Kruger Corn and Leo Ji for their assistance in soft-ware development; Michael Davidson, Bob Olson and JackHowell for their assistance and useful discussions in data pro-cessing. The author especially thanks Vicente Oropeza andMauricio Sacchi of Signal Analysis and Imaging Group forsharing insightful discussions on the MSSA algorithm; JeffMalloy and Larry Morley for their critical reviews of thispaper.

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