multicomponent f-x seismic random noise attenuation via...

Multicomponent f-x seismic random noise attenuation viavector autoregressive operators

Mostafa Naghizadeh and Mauricio Sacchi

ABSTRACT

We propose an extension of the traditional frequency-space (f-x ) random noise at-tenuation method to 3-component seismic records. For this purpose, we develop a3-component vector autoregressive (VAR) model in the f-x domain that is applied tothe multicomponent spatial samples of each individual temporal frequency. VAR modelparameters are estimated using the least-squares minimization of forward and backwardprediction errors. VAR modeling effectively identifies the potential coherencies betweenvarious components of multicomponent signal. We use the squared coherence spectrumof VAR models as an indicator to determine these coherencies. Synthetic and real dataexamples are provided to show the effectiveness of the proposed method.

INTRODUCTION

In exploration seismology we sometimes deal with multicomponent seismic records. Forinstance, 3-component geophones simultaneously record two horizontal components andone vertical component of the incident wave-field. The common approach in dealing withmulticomponent data is to process each component separately. However, vector autore-gressive (VAR) modeling presents a promising application for processing multicomponentdata. VAR modeling can provide not only a robust analysis of each individual componentbut also valuable information about the coherency between each component (Pagano, 1978;Hrafnkelsson and Newton, 2000). In this article we will investigate the application of VARmodeling to random noise attenuation.

A large class of de-noising methods utilize the Fourier transform. The strategy behindFourier-based de-noising methods is to retain a few dominant harmonics by preserving afinite number of frequency or wavenumber components in the Fourier domain (Naghizadehand Sacchi, 2010). Frequency-space (f-x ) domain methods comprise a large group of seis-mic data interpolation and de-noising methods. For instance, prediction filters are used byCanales (1984) and Spitz (1991) in the f-x domain for de-noising and data interpolation, re-spectively. Other methods such as projection filters (Soubaras, 1994), noncausal predictionfilters (Gulunay, 2000), Singular Value Decomposition (Trickett, 2003), Cadzow de-noising(Cadzow and Ogino, 1981; Trickett and Burroughs, 2009), and Singular Spectrum Analysis(Oropeza and Sacchi, 2009) have also been used for random noise attenuation in the f-xdomain. All of the f-x de-noising methods are based on the assumption that the spatialsignals at each single frequency are composed of a superposition of a limited number ofcomplex harmonics (Sacchi and Kuehl, 2000).

In this article we introduce a VAR modeling method for multicomponent signal en-hancement. The details of computing VAR models and their spectral interpretations are

Vector Autoregressive

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presented. The proposed de-noising method is applied to seismic records in the f-x do-main. Synthetic and real seismic examples are provided to examine the performance of theproposed VAR de-noising method. We find that VAR filtering is a more powerful noiseattenuator than standard AR filtering when components are correlated.

THEORY

Vector Autoregressive (VAR) operators

For a multicomponent signal of length N , we define the M -order forward VAR model as(Leonard and Kennett, 1999)

gk =

M∑j=1

Ajgk−j , k = M + 1, . . . , N. (1)

For a 3-component signal the vector autoregressive model is represented by 3× 3 matricesof the form

Aj =

aj11 aj12 aj13aj21 aj22 aj23aj31 aj32 aj33

, (2)

and gk = (gx, gy, gz)Tk is a 3-component vector at spatial sample k. For instance, expanding

equation 1 for order M = 2 we have gxgygz

k

=

a111 a112 a113a121 a122 a123a131 a132 a133

gxgygz

k−1

+

a211 a212 a213a221 a222 a223a231 a232 a233

gxgygz

k−2

. (3)

We can also define backward VAR modeling via the following expression

g∗k =

M∑j=1

Ajg∗k+j , k = 1, . . . , N −M, (4)

where ∗ represents the complex conjugate. The elements of A can be estimated using theleast-squares method by simultaneously minimizing the forward and backward predictionerrors in equations 1 and 4 (Marple, 1987).

Spectral analysis of VAR operators

The spectral density matrix of a VAR model is defined as (Hrafnkelsson and Newton, 2000)

F(η) = G−1(η)G−H(η), −0.5 ≤ η ≤ 0.5, (5)

where

G(η) = I−M∑l=1

Ale−i2πlη, (6)


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G−H is the inverse of the hermitian of G, i =√−1, and I is the identity matrix. The

variable η indicates normalized temporal frequency when the original signal evolves in time,whereas it indicates wavenumber for space dependent signals. A set of spectral attributescan be extracted from the spectral density matrix of the VAR models, the main two beingthe squared coherence and phase coherence spectra of the data. The squared coherencespectrum of the VAR model is

Wij(η) =<2{Fij(η)}+ =2{Fji(η)}

Fii(η)Fjj(η). (7)

Similarly, the phase coherence spectrum is

Φij(η) = arctan

(={Fji(η)}<{Fij(η)}

). (8)

where Fji(η) represents the (i, j)th element of F(η). The symbols < and = represent thereal and imaginary parts of a complex function, respectively.

To gain understanding of VAR modeling we start with a simple 1D multivariate example.We created three signals of 256 samples each and plotted them in Figures 1a-c. Figures 1d-fshow the Fourier spectra of the data in Figures 1a-c, respectively. Each signal contains twoharmonics and each signal has one harmonic in common with each of the other two signals.We estimated the VAR model parameters of the 3-component signal using Equations 1 and4 and then computed the diagonal spectrum, squared coherence spectrum, and phase coher-ence spectrum of the VAR model. Figures 2a, 2b, and 2c show the spectrum of the diagonalelements of VAR model (F11, F22, F33) for signals in Figures 1a-c, respectively. These plotsare equivalent to the spectrum of the ordinary autoregressive (AR) modeling applied indi-vidually to each signal. Figures 3a, 3b, and 3c show the squared coherence spectra betweeneach pair of signals (W12,W13,W23) shown in Figures 1a-1b, 1a-1c, and 1b-1c, respectively.Notice that the squared coherence spectrum has peaks at the common frequencies of eachpair of 3-component data. We have also shown the phase coherence spectra (Φ12,Φ13,Φ23,)in Figures 4a, 4b, and 4c, respectively. Notice that the ±π discontinuities in the plots arethe result of the well-known phase wrapping phenomenon.

The spectral estimator derived from VAR models (equation 7) provides a measure toforesee if signal enhancement can be achieved via VAR modeling. The squared coherenceamplitude shows how one component of data can benefit from other components. If thesquared coherence spectra between two components is close to one, VAR modeling canincrease the signal quality. In contrast, if the squared coherence spectra is near zero, VARmodeling will lead to results quite similar to those obtained via classical autoregressivemodeling of individual components.

Random noise elimination using VAR operators

To enhance the signal-to-noise ratio of a multicomponent signal, we first estimate the VARoperator from the noisy data and represent it by Aj . Next, the forward estimate of de-noiseddata can be obtained using

gfk =M∑j=1

Ajgk−j , k = M + 1, . . . , N. (9)


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between the signals in Figures 1b and 1c.


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Similarly the backward estimate of de-noised data is given by

gbk =M∑j=1

A∗jgk+j , k = 1, . . . , N −M, (10)

where A∗j represents the complex conjugate of the VAR operator. The final estimate of the

de-noised data is given by averaging forward and backward estimators

gtk =gfk + gbk

2. (11)

In seismic data, the noise attenuation is performed in the f-x domain for the spatialsamples of each individual frequency. Clearly, the proposed method extends Canales’ f-xrandom noise attenuation technique (Canales, 1984) to the multivariate case. Notice thatCanales’ noise attenuation method, also called f-x deconvolution, uses the scalar versionof equations 1 and 4 for the estimation of a scalar prediction filter. It also assumes thescalar version of equations 9-11 for noise attenuation. In addition, the complex conjugateconstraint between the forward and backward prediction filter in Canales’s methods imposesa constraint on the lateral gain variations that may exist in the data. Such f-x filters areunable to predict lateral changes in amplitudes. One may have to design non-casual f-xprediction filters for such cases (Gulunay, 2000).

EXAMPLES

1D synthetic example

To examine the performance of VAR de-noising we started with the 3-component signal.We added random noise to the original data in Figures 1a-c to produce the noisy data inFigures 5a-c. The signal to noise ratio (SNR) for all 3 components is equal to 0.75. Figures6a-c show the de-noised data using the VAR operator. The very high amplitude randomnoise has been successfully eliminated from data in Figures 5a-c.

Synthetic seismic example

We test the VAR de-noising method for a synthetic multicomponent seismic record com-posed of the three linear events shown in Figures 7a-c. Notice that the linear events in eachseismic component have different amplitudes and polarities but they have the same dipstructure. Figures 7d-f show the f-k spectra of the data in Figures 7a-c, respectively. Eachseismic section in Figure 7 is contaminated with random noise of SNR = 1.0 to obtain thenoisy seismic sections in Figure 8. Figures 8d-f show the f-k spectra of the noisy seismicsections in Figures 8a-c, respectively. Figures 9a-c show the results of VAR de-noising ofthe seismic sections in Figures 8a-c. We used 3-component VAR operators of order M = 4to de-noise the data. Figures 9d-f show the f-k spectra of the VAR de-noised seismic sec-tions in Figures 9a-c, respectively. Figures 10a-c show the results of Canales’ f-x de-noisingmethod applied individually to the seismic sections in Figures 8a-c. Figures 10d-f show thef-k spectra of the seismic sections in Figures 10a-c, respectively. A detailed comparison


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Figure 5: a)-c) are the noisy data created from data in Figures 1a-c by adding noise with SNR =0.75.


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Figure 6: a)-c) are the de-noised data from Figures 5a-c using VAR modeling, respectively.


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of Figures 9 and 10 reveals that VAR de-noising method performs better for de-noisingmulticomponent data since it simultaneously uses information from all of the components.

Application of the VAR de-noising method to seismic sections with curved events re-quires spatial windowing in order to satisfy the linear seismic events assumption (Naghizadehand Sacchi, 2009). The presence of linear events in the t-x domain implies having a fewdominant harmonics at a given frequency in the f-x domain. Therefore, VAR modelingcan be used to identify dominant harmonics present in the multicomponent signal. Figures11a-c show a simulated 3-component seismic record with one vertical and two horizontalcomponents. The data are contaminated by random noise with SNR = 1.0. For illustra-tion purposes we plotted only every 4 traces of the original data in each section. Figures11d-f depict the f-k spectra of the data in Figures 11a-c, respectively. Figures 12a-c showthe de-noised data using VAR modeling. Figures 12d-f show the f-k spectra of the data inFigures 12a-c, respectively.


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Figure 8: a)-c) are the noisy data created form the data in Figures 7a-c with SNR = 1.0. d)-f) arethe f-k spectra of a-c, respectively.


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Figure 9: a)-c) are the de-noised data using VAR modeling for the data in Figures 8a-c. d)-f) arethe f-k spectra of a-c, respectively.


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Figure 10: a)-c) are the de-noised data using Canales’ f-x denoising method for the data in Figures8a-c. d)-f) are the f-k spectra of a-c, respectively.


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Figure 11: a)-c) are the three components of simulated multicomponent seismic data with SNR =1.0. d)-f) are the f-k spectra of a-c, respectively.


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Figure 12: a)-c) are the de-noised data using VAR modeling for the data in Figures 11a-c. d)-f)are the f-k spectra of a-c, respectively.


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Real data example

In order to examine the performance of VAR de-noising on real seismic data we use a shotrecord from an ocean bottom cable (OBC) survey. The data are composed of two horizontaland one vertical component records. Figures 13a-c show the vertical and the two horizontalcomponents of the OBC shot record, respectively. The vertical component of data (Figure13a) contains a noticeable amount of random noise. Figures 13d-f show the result of VARde-noising method applied to the data in Figures 13a-c, respectively. The order of VARoperator used to de-noise the data was M = 3. The results show that random noise waseliminated from the vertical component (Figure 13d).

To further analyze the performance of VAR de-noising, we de-noised the vertical com-ponent of the data (Figure 13a) using Canales’ f-x de-noising method. We used the samespatial window size and operator order for both Canales’ and VAR denoising. Figures14a-14c show the results of VAR de-noising and Canales’ f-x de-noising for the verticalcomponent and their difference, respectively. While both methods has been successful ineliminating the random noise, the VAR de-noising method has preserved the signal better.

To examine the effectiveness of VAR de-noising in the presence of noise in all three com-ponents we added random noise to the original field data to obtain the noise-contaminateddata in Figure 15a-c. Figures 15d-f show the results of VAR de-noising applied to the datain Figures15a-c, respectively. Here one can clearly see that the random noise has beeneffectively removed from all three components of the data.

DISCUSSION

In this article we used VAR modeling for analysis of multicomponent signals. Special em-phasis is placed on the noise elimination of multicomponent noisy data. The VAR modelingpreforms better in comparison to ordinary AR modeling, the basis of f-x random noise re-duction, when there are common harmonics present in multicomponent data. If the squaredcoherency term between all data components is zero over all harmonic range, then the VARde-noising does not provide extra benefits. In addition, our experiments show that for lowSNR, higher orders of VAR models improve the quality of de-noising process.

We use the conjugate gradient (CG) algorithm to estimate the VAR operators. Thisway we can control the trade-off between misfit and model norm by adjusting the numberof iterations for the CG algorithm. This is important because for noisy data over-fittingthe data may result in ineffective VAR parameter estimation for de-noising purposes. Ourexperiments show that an optimal estimate of VAR de-noising operators can be achievedwith a small number of CG iterations (≈ 10).

VAR modeling can be extended effectively to the multidimensional case. Possible ap-plications are de-noising and interpolation of multicomponent seismic records with multiplespatial dimensions. Another interesting application for VAR modeling is the band-widthextension of seismic records. VAR operators can be used to extract information from wide-band components in order to extend the band-width of narrow-band components.


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Figure 13: a) Vertical component of OBC data. b) and c) are the horizontal components of OBCsurvey. d)-f) show the data in a-c after applying the VAR de-noising method, respectively. g)-i)show the diffence sections between a-d, b-e, and c-f, respectively.


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Figure 14: a) The vertical component of OBC data de-noised using VAR modeling. b) The verticalcomponent of OBC data de-noised using Canales’ f-x method. c) The difference between a and b.

CONCLUSIONS

We introduced and investigated the performance of VAR modeling for seismic data de-noising purposes by extending f-x random noise attenuation to the multivariate case. Weused the least-squares method to estimate the optimal VAR operators and also described thespectral interpretation of the VAR model. The performance of VAR modeling was analyzedfor de-noising of 1D multicomponent noisy data. We showed that VAR de-noising algorithmimproves the noise elimination if common harmonics are present in different componentsof the signal. The synthetic and real seismic data examples showed the effectiveness of theproposed VAR de-noising algorithm.

ACKNOWLEDGMENTS

We acknowledge financial support by the sponsors of the Signal Analysis and Imaging Groupat the University of Alberta. We also thank Dr. Keith Louden and Mr. Omid Aghaei fromDalhousie University for kindly providing OBC gathers utilized for our real data tests.

REFERENCES

Cadzow, J. A. and K. Ogino, 1981, Two-dimensional spectral estimation: IEEE Transac-tions on Acoustics, Speech, and Signal processing, 29, 396–401.

Canales, L. L., 1984, Random noise reduction: 54th Annual International Meeting, SEG,Expanded Abstracts, Session:S10.1.

Gulunay, N., 2000, Noncausal spatial prediction filtering for random noise reduction on 3-dpoststack data: Geophysics, 65, 1641–1653.

Hrafnkelsson, B. and H. J. Newton, 2000, Asymptotic simultaneous confidence bands forvector autoregressive spectra: Biometrika, 87, 173–182.


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Figure 15: a)-c) are the vertical and two horizontal components, respectively, of an OBC surveyafter adding random noise with SNR = 2. d)-f) show a-c after applying VAR de-noising.


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Leonard, M. and B. L. N. Kennett, 1999, Multi-component autoregressive techniques forthe analysis of seismograms: Physics of the Earth and Planetary Interiors, 113, 247–263.

Marple, S. L., 1987, Digital spectral analysis with applications: Prentice-Hall Inc.Naghizadeh, M. and M. D. Sacchi, 2009, f-x adaptive seismic-trace interpolation: Geo-

physics, 74, V9–V16.——– 2010, On sampling functions and Fourier reconstruction methods: Geophysics, 75,

WB137–WB151.Oropeza, V. E. and M. D. Sacchi, 2009, Multifrequency singular spectrum analysis: SEG,

Expanded Abstracts, 29, 3193– 3197.Pagano, M., 1978, On periodic and multiple autoregressions: The Annals of Statistics, 6,

1310–1317.Sacchi, M. and H. Kuehl, 2000, FX ARMA filters: 70th Annual International Meeting,

SEG, Expanded Abstracts, 2092–2095.Soubaras, R., 1994, Signal-preserving random noise attenuation by the f-x projection: 64th

Annual International Meeting, SEG, Expanded Abstracts, 1576–1579.Spitz, S., 1991, Seismic trace interpolation in the F-X domain: Geophysics, 56, 785–794.Trickett, S. R., 2003, F-xy eigenimage noise suppression: Geophysics, 68, 751–759.Trickett, S. R. and L. Burroughs, 2009, Prestack rank-reducing noise suppression: theory:

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