Theory of Multisource Crosstalk Reduction by Phase-Encoded Statics

Theory of Multisource CrosstalkTheory of Multisource Crosstalk

Reduction by Phase-Encoded StaticsReduction by Phase-Encoded Statics

G. Schuster, X. Wang, Y. Huang, C. BoonyasiriwatG. Schuster, X. Wang, Y. Huang, C. BoonyasiriwatKing Abdullah University Science & TechnologyKing Abdullah University Science & Technology

OutlineOutline1.1. Seismic Experiment:Seismic Experiment:

L m = d

L m = d1 1

L m = d2 2...N N

2. Standard vs Phase Encoded Least Squares Soln.2. Standard vs Phase Encoded Least Squares Soln.

LL11

LL22

dd 11

dd 22m =m = vs N L + N LN L + N L11 2211 22[ ]mm = [N = [N dd + N + N dd ] ]11 2211 22

3. Theory Noise Reduction3. Theory Noise Reduction 4. Summmary and Road Ahead4. Summmary and Road Ahead

Gulf of Mexico Seismic SurveyGulf of Mexico Seismic Survey

mm

L m = d

L m = d1 1

L m = d2 2...N N

Time (s)

6 X (km)

4

0

d

Goal:Goal: Solve overdetermined Solve overdeterminedSystem of equations for mSystem of equations for m

Predicted data Observed data

Problem:Problem: Expensive, one Expensive, one migration/shot gathermigration/shot gather

Solution:Solution: Supergather Supergather migrationmigration

Brief History Multisource Brief History Multisource Phase Encoded ImagingPhase Encoded Imaging

Romero, Ghiglia, Ober, & Morton, Geophysics, (2000)

Krebs, Anderson, Hinkley, Neelamani, Lee, Baumstein, Lacasse, SEG, (2009)

Virieux and Operto, EAGE, (2009)

Dai and Schuster, SEG, (2009)

Migration

Waveform Inversion and Least Squares Migration

Biondi et al., SEG, (2009)


L m = d

L m = d1 1

L m = d2 2...N N


LL11

LL22

dd 11


3. Theory Noise Reduction3. Theory Noise Reduction 4. Summmary and Road Ahead4. Summmary and Road Ahead

(k)(k)

Conventional Least Squares Conventional Least Squares Solution: Solution: L=L= & & d = d =

GivenGiven: : LLm=dm=d

FindFind: m : m s.t.s.t. min|| min||LLm-d||m-d||22

SolutionSolution: m = [: m = [L LL L] ] LL d d TT TT-1-1

m = m – m = m – LL ( (LLm - d) m - d) TT(k+1)(k+1) (k)(k) (k)(k)(k)(k)

or if or if LL is too big is too big

ProblemProblem::

LL11

LL22

dd 11

dd 22

= m – = m – LL ( (L L m - d ) m - d ) (k)(k)

+ L+ L ( (L L m - d ) m - d ) 11 11 22 22 2211

TTTT[ ]

In general, hugedimension matrix

Note: subscripts agree



FindFind: m : m s.t.s.t. min|| min||LLm-d||m-d||22



ProblemProblem::

LL11

LL22

dd 11

dd 22

= m – = m – LL ( (L L m - d ) m - d ) (k)(k)

+ L+ L ( (L L m - d ) m - d ) 11 11 22 22 2211

TTTT[ ]


Problem: Expensive, FD solve/CSGSolution: Blend+encode Data

(k)(k)

Blending+Phase EncodingBlending+Phase Encoding

22 dd = = N d + N d + N dN d + N d + N d221111 3333

PhasePhasePhasePhaseBlendingBlending

Encoding MatrixEncodedsupergather

LL = = NN L + L + NN L L + + NN L L33 3322 2211 11m [ ]m

dd 11LL mm==11

Encoded supergather modeler

dd 33LL mm==33dd 22LL mm==22

BlendingBlending11

e i in in domain domain

(k)(k)(k)(k)(k) (k) (k) (k)

++

Blended Phase-Encoded Least Squares Solution Blended Phase-Encoded Least Squares Solution

LL = = && dd = = N d + N dN d + N d

GivenGiven: : LLm=m=ddFindFind: m s.t. min||: m s.t. min||LLm-m-dd||||22

SolutionSolution: m = [: m = [LL LL] ] LL dd TT TT-1-1

TT(k+1)(k+1) (k)(k) (k)(k)(k)(k)


11N L + N LN L + N L22 11

= m – = m – LL ( (L L m - d ) m - d ) (k)(k)

+ L+ L ( (L L m - d ) m - d ) 11 11 22 22 2211

TTTT[ ]

11 22 11 22 22

+ Crosstalk+ L+ L N N ( N N (L L m - d ) m - d ) 22

TT

11 L L N N (N N (L L m - d ) m - d ) 11

TT

11 2222 11 11 22 22

** **

(k)(k)

In general, SMALLdimension matrix

(k)(k) (k)(k)

Iterations are proxyFor ensemble averaging

(k+1)(k+1) (k)(k) (k)(k)

(k)(k) (k)(k)

(k)(k)(k)(k)(k)(k)(k)(k) (k)(k) (k)(k)

(k)(k) (k)(k)

m = m – m = m – LL ( (LLm - m - dd) )


L m = d

L m = d1 1

L m = d2 2...N N


LL11

LL22

dd 11


3. Theory Noise Reduction3. Theory Noise Reduction

4. Summary 4. Summary

Ensemble Average of Crosstalk TermEnsemble Average of Crosstalk Term

With Random Time ShiftsWith Random Time Shifts

++Crosstalk: LL N N ( N N (L L m - d ) m - d ) 22

TT

1111 L L N N (N N (L L m - d ) m - d ) 11

TT

11 2222 11 11 22 22

** **

N NN N1122

** = e e -i 2i 1 e i

1 2< > < > >= <

e i1 2= e

1 2

-2

dd1 2

~ e 2 2

e -i 2

Gaussian PDF

Noise

Crosstalk term decreaseswith increasing and

<Noise>:

Crosstalk Prediction FormulaLL ( (L L m - d ) m - d ) 22

TT

1111 + L+ L ( (L L m - d ) m - d ) 11

TT

2222 e-2 2

O( )~X =

Pt. Scatt. Stand. Mig. Pt. Scatt.. Mig. of Supergathers.

Pt. Scatt.. Mig. of Supergathers.Pt. Scatt.. Mig. of Supergathers.

= .05 s = .1 s

= .01 s

X Offset

Ensemble Average of Crosstalk TermEnsemble Average of Crosstalk Term

With Random PolarityWith Random Polarity

++Crosstalk: LL N N ( N N (L L m - d ) m - d ) 22

TT

1111 L L N N (N N (L L m - d ) m - d ) 11

TT

11 2222 11 11 22 22

** **

N NN N1122

** 1 2<Noise>: < > = < > = 0

sgn(

Noise

sgn(sgn(

Conclusion: Conclusion: Random polarity better than Random polarity better than random time shiftsrandom time shifts

Further Analysis: Further Analysis: Variance of the crosstalk noise Variance of the crosstalk noise

says that random polarity & random time shifts says that random polarity & random time shifts can be almost twice better than polarity alone can be almost twice better than polarity alone

< >N NN N1122

**( )22

0 6.75X (km)

a) Standard migration (320 CSG) b) Time static σ = 0.1 s

f) SNR

c) Noise

0 6.75X (km) 0 6.75X (km)

0 6.75X (km) 0 6.75X (km)

e) Noise

0.01 0.1Time static σ (s)

polarity

Time static

Polarity and time static

Polarity+Time Statics+Location StaticsPolarity+Time StaticsPolarity

< < +/-+/- < < +/-+/-, , < < xx Time Statics

a) Polarity b) Noise d) Source polarity & static

Key Theory+Num. Results for 320 CSG SupergatherKey Theory+Num. Results for 320 CSG Supergather(Xin Wang, Yunsong Huang)(Xin Wang, Yunsong Huang)

Key Results Theory of Multisource Imaging Key Results Theory of Multisource Imaging of Encoded Supergathers of Encoded Supergathers (Xin Wang)(Xin Wang)

Sig/Noise = GI < GINSig/Noise = GI < GIN

# geophones/supergather

# subsupergatherss

0 6.7X (km)

a) Image of 1 stack

Iteration Number

c) Image of 50 stacks

0 6.75

X (km)

0 6.7X (km)

b) Image of 5 stacks

1 115

d) SNR vs Iterations

Observed

Prediction

# iterations

Bulk shift

Standard Migration SNR

GS# geophones/CSG# geophones/CSG

# CSGs# CSGs

SNR= ...

migrate

SNR=

d(t) =d(t) =Zero-mean white noise

[s(t) +n(t) ][s(t) +n(t) ] Neglect geometric spreading



Assume:

migrate+++

stack

S1

SGS G~~

iterate

GI

Iterative Multisrc. Mig. SNR

# iterations# iterations

SNR=

Cost ~ O(S)

Cost ~ O(I)


L m = d

L m = d1 1

L m = d2 2...N N


LL11

LL22

dd 11



4. Summary 4. Summary

SNR: VS3. 3. GS GI

2. 2.

11. .


< < +/- +/- < < +/-+/-, , < < xx Time Statics

LL11

LL22

dd 11

dd 22m =m = N L + N LN L + N L

11 2211 22[ ]mm = [N = [N dd + N + N dd ] ]

11 2211 22

Summary

vs

LL ( (L L m - d ) m - d ) 22

TT

1111 + L+ L ( (L L m - d ) m - d ) 11

TT

2222 e-2 2

O( )~< >

4. Passive Seismic Interferometry = Multisrc4. Passive Seismic Interferometry = Multisrc Imaging Imaging

IO 1 1/320

Cost ~

Resolution dx 1 1

SNR~

Stnd. Mig Multsrc. LSMStnd. Mig Multsrc. LSM

Less 1

1 <1/10

Cost vs QualityCost vs Quality

Summary

1

LL11

LL22

dd 11

dd 22m =m = N L + N LN L + N L

11 2211 22[ ]mm = [N = [N dd + N + N dd ] ]

11 2211 22

SN

R0

1 Number of Iterations 300

7The SNR of MLSM image grows as the square root of the number of iterations.

SNR = GI

0 6.75X (km)

a) Standard migration (320 CSG) b) Time static σ = 0.1 s

f) SNR

c) Noise

0 6.75X (km) 0 6.75X (km)

0 6.75X (km) 0 6.75X (km)

e) Noise

0.01 0.1Time static σ (s)

polarity

Time static

Polarity and time static


< +/- < +/-, < +/- < +/-, < < xx Time Statics

a) Polarity b) Noise d) Source polarity & static

Key Theory+Num. Results for 320 CSG SupergatherKey Theory+Num. Results for 320 CSG Supergather(Xin Wang, Yunsong Huang)(Xin Wang, Yunsong Huang)

Key Results Theory of Multisource Imaging Key Results Theory of Multisource Imaging of Encoded Supergathers of Encoded Supergathers (Xin Wang)(Xin Wang)



# subsupergatherss

0 6.7X (km)

a) Image of 1 stack

Iteration Number

c) Image of 50 stacks

0 6.75

X (km)

0 6.7X (km)

b) Image of 5 stacks

1 115

d) SNR vs Iterations

Observed

Prediction

# iterations

Key Results Theory of Multisource Imaging Key Results Theory of Multisource Imaging of Encoded Supergathers of Encoded Supergathers (Boonyasiriwat)(Boonyasiriwat)


3.5 km

Dynamic QMC TomogramDynamic QMC Tomogram (99 CSGs/supergather)(99 CSGs/supergather)

Dynamic Polarity TomogramDynamic Polarity Tomogram(1089 CSGs/supergather)(1089 CSGs/supergather)

1/1000 1/300


# iterations

# subsupergatherss

Multisource Migration:Multisource Migration: mmmigmig=L=LTTdd

Forward Model:Forward Model:

Multisource Phase Encoded ImagingMultisource Phase Encoded Imaging

d +d +dd =[ =[L +L +LL ]m ]m11 222211

LL{dd{

=[=[L +L +LL ]( ](dd + + dd ) ) 11 222211

TT TT

= = L d +L d +L dL d + + 11 222211

TT TT

LL dd + +L L dd22 112211

Crosstalk noiseCrosstalk noiseStandard migrationStandard migration

TT TT

m = m +(k+1) (k)


Relative Merits of 4 Encoding StrategiesRelative Merits of 4 Encoding Strategies < < +/- +/- < < +/-+/-, , < < xx

dd 11LL mm==11 dd 33LL mm==33dd 22LL mm==22

Time Statics

supergather

Supergather #2

Supergather #3

Supergather #4

Supergather #1

< < +/- +/- < < +/-+/-, , < < xx



Phase Encoded Multisource Migration Phase Encoded Multisource Migration

d +d +dd =[ =[L +L +LL ]m ]m11 222211

LL{dd{

=[=[L +L +LL ]( ](dd + + dd ) ) 11 222211

TT TT

= = L d +L d +L dL d + + 11 222211

TT TT

LL dd + +L L dd22 112211


TT TTmmmigmig

= = L d +L d +L dL d + + 11 222211

TT TT

LL dd + +LL dd22 112211

TT TTmmmigmig

mmmigmig

= = L d +L d +L dL d11 222211

mmmigmig

+ +



Phase Encoded Multisrce Phase Encoded Multisrce Least Squares Least Squares Migration Migration

d +d +dd =[ =[L +L +LL ]m ]m11 222211

LL{dd{

=[=[L +L +LL ]( ](dd + + dd ) ) 11 222211

TT TTmmmigmig

= = L d +L d +L dL d + + 11 222211

TT TT

LL dd + +L L dd22 112211


TT TT

m = m +(k+1) (k)


L m = d

L m = d1 1

L m = d2 2...N N


LL11

LL22

dd 11



4. Numerical 4. Numerical

TestsTests

RTM & FWI Problem & Possible Soln.RTM & FWI Problem & Possible Soln.

• ProblemProblem:: RTM & FWI computationally costly RTM & FWI computationally costly

• Solution:Solution: Multisource LSM & FWI Multisource LSM & FWI

Preconditioning speeds up by factor 2-3Preconditioning speeds up by factor 2-3

LSM reduces crosstalkLSM reduces crosstalk



Multisource Least Squares Migration Multisource Least Squares Migration

d +d =[d +d =[L +L ]mL +L ]m11 222211

LL{dd{Standard migration

Crosstalk term

Phase encodingPhase encoding

Kirchhoff kernelKirchhoff kernel

Multisource Least Squares Migration Multisource Least Squares Migration Crosstalk term



FindFind: m s.t. min||: m s.t. min||LLm-d||m-d|| 22




ProblemProblem::

LL11

LL22

dd 11

dd 22

= m – = m – LL ( (L L m - d ) m - d ) (k)(k)

+ L+ L ( (L L m - d ) m - d ) 11 11 22 22 2211

TTTT[ ]


Note: subscripts agree

Key Results Theory of Multisource Imaging Key Results Theory of Multisource Imaging of Encoded Supergathers of Encoded Supergathers (Boonyasiriwat)(Boonyasiriwat)


3.5 km

Dynamic QMC TomogramDynamic QMC Tomogram (99 CSGs/supergather)(99 CSGs/supergather)

Dynamic Polarity TomogramDynamic Polarity Tomogram(1089 CSGs/supergather)(1089 CSGs/supergather)

1/1000 1/300


# iterations

# subsupergatherss

Theory of Multisource Crosstalk Reduction by Phase-Encoded Statics

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