A unified method for interpolation andde-noising of seismic records

in the f-k domain

Mostafa Naghizadeh

University of Alberta(Currently at the University of Calgary)

SEG annual meetingDenver, Colorado19 October 2010

Outline

1 IntroductionMotivationsOverview

2 TheoryIdentifying dominant dips in f-k domainBuilding a mask functionLeast-squares fitting

3 ExamplesSynthetic dataReal data

4 Conclusions

1 IntroductionMotivationsOverview

2 TheoryIdentifying dominant dips in f-k domainBuilding a mask functionLeast-squares fitting

3 ExamplesSynthetic dataReal data

4 Conclusions

Goal

Utilizing information fromfull frequency band

for de-noising or interpolationof

any single frequencyin

the f-k domain.

1 IntroductionMotivationsOverview

2 TheoryIdentifying dominant dips in f-k domainBuilding a mask functionLeast-squares fitting

3 ExamplesSynthetic dataReal data

4 Conclusions

Interpolation methods

Signal processing based methodsf-x interpolation [Spitz, 1991]f-k interpolation [Gulunay, 2003]

Using low frequencies to interpolate high frequencies.Multi-step autoregressive [Naghizadeh and Sacchi, 2007]

Combining minimum weighted norm interpolation (MWNI)[Liu and Sacchi, 2004] and f-x interpolation.

Sparse Fourier inversion [Zwartjes and Sacchi, 2007]Pyramid transform [Guitton and Claerbout, 2010]Fourier-Radial Adaptive Thresholding [Curry, 2009]Curvelet interpolation [Hennenfent and Herrmann, 2008],[Naghizadeh and Sacchi, 2010]

Using coarser scales of curvelets to interpolate finer andaliased scales of curvelets.

De-noising 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 velocity filtering. . .

Principle of single frequency de-noising (I)

Principle of single frequency de-noising (II)

1 IntroductionMotivationsOverview

2 TheoryIdentifying dominant dips in f-k domainBuilding a mask functionLeast-squares fitting

3 ExamplesSynthetic dataReal data

4 Conclusions

f-k spectra of linear events (I)

f-k spectra of linear events (II)

f-k spectra of linear events (III)

Angular summation in the f-k domain

Search for dominant energy dipsd(t , x): Data in the t-x domainD(ω, k): Data in the f-k domain0 < ω < 0.5: Normalized frequencies−0.5 < k < 0.5: Normalized wavenumbersp: Slope of summation path in the f-k domain

M(p) =Nω∑

n=1

D(ωn, k = p.ωn − bp + 1

2c)

Schematic representation of angular summation in the f-k domain

Normalized Wavenumber

Norm

alized Frequency

0.25

0.5

0.0-0.5 0.0 0.5

p=0p=1

p=4

p=2p=-1

Thresholding for dominant energy dips

Identifying peak valuesLocating peak values above a threshold value andmarking them as dominant energy dips.

1 IntroductionMotivationsOverview

2 TheoryIdentifying dominant dips in f-k domainBuilding a mask functionLeast-squares fitting

3 ExamplesSynthetic dataReal data

4 Conclusions

A mask function for f-k domain

From dominant dips to mask function1 p1,p2, . . . ,pL are the identified dominant dips.2 Deploying rays of dominant dips in f-k domain.

Initiating H matrix with zeros,

H(ωn, k = pj .ωn − bpj + 1

2c) = 1, { n = 1,2, . . . ,Nω,

j = 1,2, . . . ,L.

3 Mask Widening to account for uncertainties.Convolving H with a 1D box car function, B(1,Lb),

W(ω, k) = H(ω, k) ∗ B,

Mask function

0.1

0.2

0.3

0.4

Nor

mal

ized

freq

uenc

y

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

1 IntroductionMotivationsOverview

2 TheoryIdentifying dominant dips in f-k domainBuilding a mask functionLeast-squares fitting

3 ExamplesSynthetic dataReal data

4 Conclusions

Regularized least-squares interpolation

A stable and unique solution can be found byminimizing the following cost function[Tikhonov and Goncharsky, 1987]

J = ||d− T FHWD||22 + µ2||D||22 .

d: Data in t-x domainD: Data in f-k domainT: Sampling functionF: Fourier transformW: Mask functionµ: Trade-off parameter

1 IntroductionMotivationsOverview

2 TheoryIdentifying dominant dips in f-k domainBuilding a mask functionLeast-squares fitting

3 ExamplesSynthetic dataReal data

4 Conclusions

Original data

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

Noisy data (SNR=1.0)

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

De-noised data using Canales f-x method

0.1

0.6

1.1

1.6

Tim

e (s

)Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

De-noised data using the proposed method

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

The mask function used for de-noising

0.1

0.2

0.3

0.4

Nor

mal

ized

freq

uenc

y

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

Irregularly sampled data

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

Interpolation of irregularly sampled data

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

Data with gap

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

Gap interpolation

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

Data which needs extrapolation

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

Extrapolated data

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

Removing alias from data (regular sampling)

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

Dealiased data

0.5

1.0

1.5

2.0

Tim

e (s

)

500 1000 1500 2000Distance (m)

t-x domain

0.1

0.2

0.3

0.4N

orm

aliz

ed fr

eque

ncy

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

1 IntroductionMotivationsOverview

2 TheoryIdentifying dominant dips in f-k domainBuilding a mask functionLeast-squares fitting

3 ExamplesSynthetic dataReal data

4 Conclusions

Original data from the Gulf of Mexico

1

2

3

Tim

e (s

)1000 4000 7000

Distance (m)

t-x domain

0.1

0.2

0.3

Nor

mal

ized

freq

uenc

y

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

Interpolated data from the Gulf of Mexico

1

2

3

Tim

e (s

)1000 4000 7000

Distance (m)

t-x domain

0.1

0.2

0.3

Nor

mal

ized

freq

uenc

y

-0.4 -0.2 0 0.2 0.4Normalized wavenumber

f-k domain

A time window of original data

2.1

2.3

2.5

2.7

2.9

Tim

e (s

)

1000 4000 7000Distance (m)

Original data

A time window of interpolated data

2.1

2.3

2.5

2.7

2.9

Tim

e (s

)

1000 4000 7000Distance (m)

Interpolated data

Ground-roll elimination by proposed method

1

2

3

Tim

e (s

)

500 1000 1500Distance (m)

Noisy data

1

2

3

500 1000 1500Distance (m)

De-noised (proposed method)

1

2

3

500 1000 1500Distance (m)

De-noised (f-k filtering)

Conclusions

For linear seismic events information from any bandof frequencies can be utilized to interpolate orde-noise any frequency.The assumption of linear events needs to befulfilled for the success of the proposed method.Therefore, proper spatial windowing is required foroptimal performance.The least-squares fitting of f-k and t-x domainsprevents the appearances of artifacts akin to f-xmethods.The thresholding criteria requires special careotherwise the algorithm can create artificial unrealisticevents.The Proposed method can be used for both randomand coherent noise elimination.

Acknowledgments

Sponsors of Signal Analysis and Imaging Group(SAIG) at the University of Alberta.Sponsors of CREWES at the University of Calgary.Dr. Mauricio D. SacchiDr. Sam T. Kaplan

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