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Theory of Multisource Crosstalk Reduction by Phase-Encoded Statics. G. Schuster, X. Wang, Y. Huang, C. Boonyasiriwat King Abdullah University Science & Technology. Outline. Seismic Experiment:. L m = d. 1. 1. L m = d. 2. 2. L m = d. N. N. - PowerPoint PPT Presentation
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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)
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
(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
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)
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
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)
or if or if LL is too big is too big
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) )
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. 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
Standard Migration SNR
Standard Migration SNR
Assume:
migrate+++
stack
S1
SGS G~~
iterate
GI
Iterative Multisrc. Mig. SNR
# iterations# iterations
SNR=
Cost ~ O(S)
Cost ~ O(I)
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. Summary 4. Summary
SNR: VS3. 3. GS GI
2. 2.
11. .
Polarity+Time Statics+Location StaticsPolarity+Time StaticsPolarity
< < +/- +/- < < +/-+/-, , < < 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
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
Key Results Theory of Multisource Imaging Key Results Theory of Multisource Imaging of Encoded Supergathers of Encoded Supergathers (Boonyasiriwat)(Boonyasiriwat)
Sig/Noise = GI < GINSig/Noise = GI < GIN
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
# geophones/supergather
# 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)
Polarity+Time Statics+Location StaticsPolarity+Time StaticsPolarity
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
Multisource Migration:Multisource Migration: mmmigmig=L=LTTdd
Forward Model:Forward Model:
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
Crosstalk noiseCrosstalk noiseStandard migrationStandard migration
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
+ +
Multisource Migration:Multisource Migration: mmmigmig=L=LTTdd
Forward Model:Forward Model:
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
Crosstalk noiseCrosstalk noiseStandard migrationStandard migration
TT TT
m = m +(k+1) (k)
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. 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 Migration:Multisource Migration: mmmigmig=L=LTTdd
Forward Model:Forward Model:
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
Conventional Least Squares Conventional Least Squares Solution: Solution: L=L= & & d = d =
GivenGiven: : LLm=dm=d
FindFind: m s.t. min||: m s.t. 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
Key Results Theory of Multisource Imaging Key Results Theory of Multisource Imaging of Encoded Supergathers of Encoded Supergathers (Boonyasiriwat)(Boonyasiriwat)
Sig/Noise = GI < GINSig/Noise = GI < GIN
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
# geophones/supergather
# iterations
# subsupergatherss