Mostafa Naghizadeh and Mauricio D. Sacchimostafan/Files/Presentations/...Vector AR ﬁlters: Extending f-x random noise attenuation to the multicomponent case Mostafa Naghizadeh and

Vector AR filters:Extending f-x random noise attenuation to

the multicomponent case

Mostafa Naghizadeh and Mauricio D. Sacchi

Signal Analysis and Imaging Group (SAIG)

SEG annual meeting, San Antonio22 September 2011

Multicomponent Seismic Data Reconstruction Using The Quaternion Fourier Transform and POCS

Aaron Stanton and Mauricio Sacchi,September 2011

Outline

1 IntroductionMotivationOverview

2 Vector autoregressive (VAR) modelingVAR model estimationSpectrum analysis of VAR model parametersVAR de-noising algorithm

3 Examples1D synthetic examplesSynthetic seismic dataReal seismic data

4 Conclusions




4 Conclusions

Goal

Usingcoherent information

betweenmulticomponent signals

fornoise attenuation.




4 Conclusions

Seismic random noise attenuation methods

Including but not limited tof-x prediction filter [Canales, 1984]f-x projection filter [Soubaras, 1994]Singular Value Decomposition [Trickett, 2003]Cadzow methods [Trickett and Burroughs, 2009]or Singular Spectrum Analysis [Oropeza and Sacchi, 2009]Empirical Mode Decomposition[Bekara and Van der Baan, 2009]f-k domain dominant dips [Naghizadeh, 2010]. . .

Main assumptionIn the f-x domain each frequency slice of seismic data iscomposed of few dominant complex harmonics.

Principle of single frequency de-noising (I)

Principle of single frequency de-noising (II)




4 Conclusions

Estimating VAR model for 3-component signal (I)

For a multicomponent signal of length N, we define the M-orderforward VAR model as [Leonard and Kennett, 1999]

gk =M∑

j=1

Ajgk−j , k = M + 1, . . . ,N.

For a 3-component signal the vector autoregressive model isrepresented by 3× 3 matrices of the form

Aj =

aj11 aj12 aj13aj21 aj22 aj23aj31 aj32 aj33

,

gk = (gx ,gy ,gz)Tk : 3-component vector at time sample k .

Estimating VAR model for 3-component signal (II)

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

.

The backward VAR modeling is defined as

g∗k =

M∑j=1

Ajg∗k+j , k = 1, . . . ,N −M,

The elements of A can be estimated using the least-squaresmethod by simultaneously minimizing the forward and backwardprediction errors. [Marple, 1987]




4 Conclusions

VAR model spectral density matrix

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,

where

G(η) = I−M∑

k=1

Ak e−i2πkη,

G−H : Inverse of the hermitian of G

i =√−1,

I: Identity matrix,

η : Normalized frequency/wavenumber

Spectral attributes of VAR model

The squared coherence spectrum of the VAR model:

Wij (η) =<2{Fij (η)}+ =2{Fji (η)}

Fii (η)Fjj (η).

The phase coherence spectrum of the VAR model:

Φij (η) = arctan(={Fji (η)}<{Fij (η)}

).

Fji (η): (i , j)th element of F(η),

< and =: Real and imaginary parts of a complex function.

Synthetic 3-component example

50 100-2-1012

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plit

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Fourier domain

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VAR model parameters

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ase

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Spectrum coherency


-2-1012Phase coherency

0 0.2 0.4-101234

0 0.2 0.42-3

0.5

1.0


-2-1012




4 Conclusions

VAR noise attenuation

The VAR operator, Aj , is estimated from the noisy data.

The forward estimate of de-noised data can be obtained using

gfk =

M∑j=1

Ajgk−j , k = M + 1, . . . ,N.

The backward estimate of de-noised data is given by

gbk =

M∑j=1

A∗j gk+j , k = 1, . . . ,N −M.

The final de-noised data is given by averaging forward andbackward estimators

gtk =

gfk + gb

k2

.




4 Conclusions

Synthetic 3-component signal

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Original data

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Synthetic 3-component signal (cont.)

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Noisy data

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Synthetic 3-component signal (cont.)

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Denoised data using VAR Modeling

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4 Conclusions

VAR de-noising for seismic data

Linear seismic events

1 Transform the original data to f-x domain.

2 Apply VAR de-noising to each frequency of multicomponentdata.

3 Transform back the results of step 2 to t-x domain.

Curved seismic events

1 Divided the seismic data into small spatial windows with properoverlap between them.

2 Apply VAR de-noising of linear events for each patch of data.

3 Put the de-noised patches together with proper averaging inoverlapped areas.

Synthetic linear seismic events

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Noisy linear seismic events

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VAR de-noising

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Canales de-noising

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f-k spectra of original data

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f-k spectra of noisy data

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Channel 3

f-k spectra of VAR de-noised data

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Channel 2

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Channel 3

f-k spectra of Canales de-noised data

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No

rmal

ized

fre

qu

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Channel 1

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qu

ency


Channel 2

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Channel 3

Hyperbolic Synthetic data

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Noisy hyperbolic Synthetic data

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VAR de-noised data

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f-k spectra of original data

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rmal

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qu

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Channel 1

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rmal

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Channel 2

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f-k spectra of noisy data

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f-k spectra of VAR de-noised data

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4 Conclusions

Original OBC data

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VAR de-noised OBC data

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Difference between original and de-noised data

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VAR vs. Canales de-noising for channel 1

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VAR de-noising

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Canales de-noising

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Difference

Noisy OBC data with SNR=2

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VAR de-noised data

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Noisy OBC data with SNR=2

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VAR de-noised data

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Conclusions

VAR noise attenuation is an extension f-x random noiseattenuation [Canales, 1984] to the multivariate case.

The least-squares method was used to estimate the optimalVAR operators and the spectral interpretation of the VARmodel was investigated.

VAR de-noising algorithm improves the noise elimination ifcommon harmonics are present in different components of thesignal.

VAR de-noising for seismic data is implemented in the f-xdomain individually for each given frequency.

The VAR modeling can be extended effectively to themultidimensional cases. It can be used for de-noising andinterpolation of multicomponent seismic records with multiplespatial dimensions.

Acknowledgments

Sponsors of SAIG at the University of Alberta.

Dr. Keith Louden and Mr. Omid Aghaei from DalhousieUniversity for kindly providing OBC gathers.

Bekara, M. and M. Van der Baan, 2009, Random and coherent noise attenuation by empirical modedecomposition: Geophysics, 74, V89–V98.

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

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

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

Marple, S. L., 1987, Digital spectral analysis with applications: Prentice-Hall Inc.

Naghizadeh, M., 2010, A unified method for interpolation and de-noising of seismic records in the f-k domain:SEG Technical Program Expanded Abstracts, 29, 3579–3583.

Oropeza, V. E. and M. D. Sacchi, 2009, Multifrequency singular spectrum analysis: SEG, ExpandedAbstracts, 29, 3193– 3197.

Soubaras, R., 1994, Signal-preserving random noise attenuation by the f-x projection: 64th AnnualInternational Meeting, SEG, Expanded Abstracts, 1576–1579.

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: SEG, ExpandedAbstracts, 29, 3332– 3336.

Documents

Mostafa Naghizadeh and Mauricio D. Sacchimostafan/Files/Presentations/...Vector AR ﬁlters: Extending f-x random noise attenuation to the multicomponent case Mostafa Naghizadeh and