Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1...
Transcript of Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1...
![Page 1: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/1.jpg)
University of British ColumbiaSLIM
Felix Oghenekohwo
Randomized sampling “without repetition’’ in time-lapse seismic surveys
Released to public domain under Creative Commons license type BY (https://creativecommons.org/licenses/by/4.0).Copyright (c) 2014 SLIM group @ The University of British Columbia.
![Page 2: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/2.jpg)
University of British ColumbiaSLIM
Felix Oghenekohwo
Randomized sampling “without repetition’’ in time-lapse seismic surveys
Team : Rajiv Kumar, Haneet Wason, Ernie Esser, Felix Herrmann
![Page 3: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/3.jpg)
Randomized undersampling – examples from industry (ConocoPhilips)
Deliberate*&*natural*randomness*in*acquisition*(thanks*to*Chuck*Mosher)
Compressive Sensing and Seismic Acquisition Sparse Transforms Optimal Sampling Data Reconstruction Compressive Sensing = Acquisition Efficiency
Land / Node
Marine
TSuRBSb *Data Sparsity Sampling
Mosher Et Al, SEG 2012; Li Et Al, SEG 2013
2011 Trial 1: Non-Uniform Optimal Sampling (NUOS)
Designed Actually Acquired
Non-Uniform Acquisition Common Offset Reconstruction
Mosher, C. C., Keskula, E., Kaplan, S. T., Keys, R. G., Li, C., Ata, E. Z., ... & Sood, S. (2012, November). Compressive Seismic Imaging. In 2012 SEG Annual Meeting. Society of Exploration Geophysicists.
![Page 4: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/4.jpg)
Time-lapse seismic
• Current*acquisition*paradigm:*‣ repeat*expensive*dense.acquisitions*&*“independent”.processing*‣ compute*differences*between*baseline*&*monitor*survey(s)*‣ hampered*by*practical*challenges*to*ensure*repetition
Haneet Wason and Felix J. Herrmann, “Time-jittered ocean bottom seismic acquisition”!in SEG Technical Program Expanded Abstracts, 2013, p. 1-6
Hassan Mansour, Haneet Wason, Tim T.Y. Lin, and Felix J. Herrmann, “Randomized marine acquisition with compressive sampling matrices”, !Geophysical Prospecting, vol. 60, p. 648-662, 2012.
![Page 5: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/5.jpg)
Time-lapse seismic
• Current*acquisition*paradigm:*‣ repeat*expensive*dense.acquisitions*&*“independent”.processing*‣ compute*differences*between*baseline*&*monitor*survey(s)*‣ hampered*by*practical*challenges*to*ensure*repetition*
!
• New*compressive*sampling*paradigm:*‣ cheap+subsampled.acquisition,*e.g.*via*timeMjittered*marine*undersampling+
‣ may*offer*possibility*to*relax*insistence*on*repeatability.‣ exploits*insights*from*distributed.*compressive*sensing
Haneet Wason and Felix J. Herrmann, “Time-jittered ocean bottom seismic acquisition”!in SEG Technical Program Expanded Abstracts, 2013, p. 1-6
Hassan Mansour, Haneet Wason, Tim T.Y. Lin, and Felix J. Herrmann, “Randomized marine acquisition with compressive sampling matrices”, !Geophysical Prospecting, vol. 60, p. 648-662, 2012.
![Page 6: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/6.jpg)
Framework
Sparsity-‐promoting recovery
sampling matrix observed subsampled measurements
x̃ = argmin
x
kxk1 subject to Ax = b
Ax = b
![Page 7: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/7.jpg)
subsampled baseline data
subsampled monitor data
Framework in 4-D
should A1 = A2 ?
what if A1 ⇡ A2 ?
what if A1 6= A2 ?
A1x1 = b1
A2x2 = b2
Question : To repeat survey design or not
![Page 8: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/8.jpg)
Baseline
Idealized synthetic time-lapse data
Monitor 4-‐D signal
CMP position (km)
Tim
e (s
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0
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CMP position (km)
Tim
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1 1.5 2 2.5
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Tim
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![Page 9: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/9.jpg)
Structure - curvelet representation
![Page 10: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/10.jpg)
Observations
Time-lapse data has structure - significant correlations
4-D signal has structure - increased sparsity
Can we exploit the structure in the vintages and the difference simultaneously ?
![Page 11: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/11.jpg)
Distributed compressive sensing – joint recovery model (JRM)
common+component
differences
vintages
x2 = z0 + z2
Az }| {A1 A1 0A2 0 A2
�zz }| {2
4z0z1z2
3
5=
bz }| {b1
b2
�x1 = z0 + z1
baseline
monitor
Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B. Wakin , Richard G. Baraniuk. An Information-Theoretic Approach to Distributed Compressed Sensing (2005)!
![Page 12: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/12.jpg)
Distributed compressive sensing – joint recovery model (JRM)
!
!
!
!
!
!
• Key+idea:+‣ use*the*fact*that*different*vintages*share*common*information*‣ invert*for*common.components*&*differences*w.r.t.*the*common*components*with*sparse*recovery
common+component
differences
vintages
x2 = z0 + z2
Az }| {A1 A1 0A2 0 A2
�zz }| {2
4z0z1z2
3
5=
bz }| {b1
b2
�x1 = z0 + z1
baseline
monitor
Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B. Wakin , Richard G. Baraniuk. An Information-Theoretic Approach to Distributed Compressed Sensing (2005)!
![Page 13: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/13.jpg)
Sparsity-promoting recovery
Joint recovery model (JRM)
Independent reconstruction
x̃i = argmin
xi
kxik1 subject to Aixi = bi, for i = 1, 2
z̃ = argmin
zkzk1 subject to Az = b
![Page 14: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/14.jpg)
Interpretation of the model – w/ & w/o repetition
!
• In+an+ideal+world+‣ JRM*simplifies*to*recovering*the*difference*from*‣ expect*good*recovery*when*difference*is*sparse*‣ but*relies*on*“exact”*repeatability...
(A1 = A2)(b2 � b1) = A1(x2 � x1)
![Page 15: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/15.jpg)
Interpretation of the model – w/ & w/o repetition
!
• In+an+ideal+world+‣ JRM*simplifies*to*recovering*the*difference*from*‣ expect*good*recovery*when*difference*is*sparse*‣ but*relies*on*“exact”*repeatability...*
!
• In+the+real+world++‣ no*absolute*control*on*surveys*‣ calibration*errors*‣ noise...
(A1 = A2)(b2 � b1) = A1(x2 � x1)
(A1 6= A2)
![Page 16: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/16.jpg)
Stylized Examples
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Sparse baseline, monitor and time-lapse signals
z0
z1
z2
x1
x2
x1 � x2
common component
“difference”
“difference”
baseline
monitor
time-‐lapseSignal length N = 50
![Page 18: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/18.jpg)
Stylized experiments!
Conduct*many*CS*experiments*to*compare*‣ joint*vs*parallel.recovery*of*signals*and*the*difference*‣ recovery*with*completely*independent**‣ random*acquisition*with*different*numbers*of*samples**
A1, A2
![Page 19: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/19.jpg)
Stylized experiments!
Conduct*many*CS*experiments*to*compare*‣ joint*vs*parallel.recovery*of*signals*and*the*difference*‣ recovery*with*completely*independent**‣ random*acquisition*with*different*numbers*of*samples**
A1, A2
n n
N N
A1 A2n = 10
b1 = A1x1 b2 = A2x2
![Page 20: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/20.jpg)
Stylized experiments!
Conduct*many*CS*experiments*to*compare*‣ joint*vs*parallel.recovery*of*signals*and*the*difference*‣ recovery*with*completely*independent**‣ random*acquisition*with*different*numbers*of*samples**
A1, A2
n n
N N
A1 A2n = 10
b1 = A1x1 b2 = A2x2
Run 2000 different experiments
Compute Probability of recovery
![Page 21: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/21.jpg)
Results: independent versus joint recovery
Recovery of vintages Recovery of difference
![Page 22: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/22.jpg)
Observations
Joint recovery is better than independent
Improved recovery of the vintages and the difference
Requires fewer samples
![Page 23: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/23.jpg)
With exact repetition
A1 = A2
A1 A2n = 10
N N
n n
b1 = A1x1 b2 = A2x2
Run*1000*different*experimentsCompute*Probability*of*recovery
REPEAT EXPERIMENT AS BEFORE
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Results: independent versus joint recovery
Recovery of vintages Recovery of difference
![Page 25: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/25.jpg)
Recovery of vintages Recovery of difference
WITH Repetition
Recovery of differenceRecovery of vintages
![Page 26: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/26.jpg)
Recovery of difference
WITHOUT Repetition
Recovery of differenceRecovery of vintages
![Page 27: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/27.jpg)
Observations• Recovery*of*vintages*themselves*improves*without+repetition.• Recovery*of*difference*improves*with+repetition.because.‣ difference.is*sparse*compared*to*sparsity*of*vintages.‣ does*not*recover*the*vintages*themselves
![Page 28: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/28.jpg)
Observations• Recovery*of*vintages*themselves*improves*without+repetition.• Recovery*of*difference*improves*with+repetition.because.‣ difference.is*sparse*compared*to*sparsity*of*vintages.‣ does*not*recover*the*vintages*themselves.
!
• Do5the5acquisitions5really5have5to5overlap?
![Page 29: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/29.jpg)
Observations• Recovery*of*vintages*themselves*improves*without+repetition.• Recovery*of*difference*improves*with+repetition.because.‣ difference.is*sparse*compared*to*sparsity*of*vintages.‣ does*not*recover*the*vintages*themselves.
!
• Do5the5acquisitions5really5have5to5overlap?
n
A1 A2
40%Overlap
REPEAT EXPERIMENT AS BEFORE
![Page 30: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/30.jpg)
Results: recovery and overlap dependency
Recovery of vintages Recovery of difference
![Page 31: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/31.jpg)
Interpretation from the stylized example
• Joint.recovery.model.(JRM).is.always.superior.to.the.independent.or.parallel.method.!
• As.the.degree.of.overlap.between.the.sampling.increases,.the.recovery.of.the.signals.gets.worse..!
• TimeGlapse.signal.recovery.benefits.from.some.overlap
![Page 32: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/32.jpg)
Seismic example
Time-jittered source in marine
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x (m)
z (m
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baseline
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z (m
)
monitor
0 500 1000 1500 2000 2500 3000
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Velocity (m/s)
1500 2000 2500 3000 3500 4000 4500 5000
Method• Velocity*and*density*model*provided*by*BG,*taken*as*baseline*
• High*permeability*zone*identified*at*a*depth*of*~*1300m*
• Fluid*substitution*(gas/oil*replaced*with*brine)*simulated*to*derive*monitor*velocity*model*
•Wavefield*simulation*to*generate*synthetic*timeMlapse*data
Baseline Model Monitor Model
x (m)
z (m
)
baseline
1400 1600 1800 2000 2200
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monitor
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4D (difference)
1400 1600 1800 2000 2200
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Simulated original data – time-domain finite differences
Baseline Monitor 4-D signal
time samples: 512 receivers: 100 sources: 100 !sampling time: 4.0 ms receiver: 12.5 m source: 12.5 m
Source position (m)Ti
me
(s)
0 250 500 750 1000
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Source position (m)
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Source position (m)
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Source position (m)
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Conventional vs. time-jittered sources – undersampling ratio = 2, 2 source arrays
jittered acquisition 1 (for baseline)conventional
jittered acquisition 2 (for monitor)
shorter*acquisition*time
geometry*is*not*the*same
![Page 36: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/36.jpg)
Measurements – undersampled and blended
Receiver position (m)
Reco
rdin
g tim
e (s
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0 500 1000
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Receiver position (m)Re
cord
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baseline monitor
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Stacked sections
CMP (km)
Tim
e (s
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1 1.5 2 2.5 3
0.5
1
1.5
2
CMP (km)
Tim
e (s
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1 1.5 2 2.5 3
0.5
1
1.5
2
Original baseline Original 4-D signal
![Page 38: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/38.jpg)
Stacked sections
CMP (km)
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Original 4-D signal
CMP (km)
Tim
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Original 4-D signal
![Page 39: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/39.jpg)
Stacked sections – 50% overlap in acquisition matrices
Parallel (9.7 dB)
Joint (18.2 dB)
CMP (km)
Tim
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CMP (km)
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NRMS Plot
![Page 41: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/41.jpg)
Seismic example
An extension to model space
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Example : Stacking
b = M
HN
HS
HC
Hx}
m = C
Hx
M
H
midpoint-‐offset
normal move-‐out
stacking
sparsifying operator
adjoint
observedundersampledmeasurements
stacked section
S
Nb = Ax
C
![Page 43: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/43.jpg)
Baseline
Idealized synthetic time-lapse data
Monitor 4-‐D signal
CMP position (km)
Tim
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CMP position (km)
Tim
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CMP position (km)
Tim
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![Page 44: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/44.jpg)
Method
•Acquisition‣ Subsampled baseline and monitor data, with independent and randomly missing shots
•Processing‣ Independent processing of the observed data (Parallel)‣ Joint processing (JRM)
![Page 45: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/45.jpg)
Method
•Acquisition‣ Subsampled baseline and monitor data, with independent and randomly missing shots
•Processing‣ Independent processing of the observed data (Parallel)‣ Joint processing (JRM)
Compare Parallel versus Joint Repeat for a “partial” dependence in geometry
![Page 46: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/46.jpg)
Baseline recovery- 0% overlap in acquisition matrices
Parallel Jointresidual residual
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
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1.5
2
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
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1.5
2
CMP position (km)
Tim
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1 1.5 2 2.5
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(9.62 dB) (10.08 dB)
![Page 47: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/47.jpg)
Monitor recovery- 0% overlap in acquisition matrices
Parallel Jointresidual residual
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
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1 1.5 2 2.5
0
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(10.08 dB) (10.02 dB)
![Page 48: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/48.jpg)
4-D recovery- 0% overlap in acquisition matrices
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
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Parallel JointIdeal (True)
CMP position (km)
Tim
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1 1.5 2 2.5
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CMP position (km)
Tim
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(3.45 dB) (7.53 dB)
![Page 49: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/49.jpg)
Baseline recovery- 50% overlap in acquisition matrices
CMP position (km)
Tim
e (s
)
1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
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1 1.5 2 2.5
0
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Parallel Jointresidual residual(9.62 dB) (9.79 dB)
![Page 50: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/50.jpg)
Monitor recovery- 50% overlap in acquisition matrices
Parallel Jointresidual residual
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
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CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
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(9.69 dB) (9.80 dB)
![Page 51: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/51.jpg)
4-D recovery- 50% overlap in acquisition matrices
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
e (s
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1 1.5 2 2.5
0
0.5
1
1.5
2
CMP position (km)
Tim
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1 1.5 2 2.5
0
0.5
1
1.5
2
Parallel JointIdeal (True)(8.07 dB)(7.51 dB)
![Page 52: Randomized sampling “ without repetition ’’ in time- …...3 5= z }|b {b 1 b 2 x 1 = z 0 + z 1 baseline monitor Dror Baron , Marco F. Duarte , Shriram Sarvotham , Michael B.](https://reader036.fdocuments.net/reader036/viewer/2022062506/5fb77abd61d21c5f9e7f66b7/html5/thumbnails/52.jpg)
Conclusions
Randomized sampling techniques can be extended to time-‐lapse seismic surveys and processing.
Process time-‐lapse data jointly, not independently, in order to exploit the shared information.
We can work with subsampled data, and recover densely sampled vintages and time-‐lapse differences.
Provided we understand the physics of our model, we can safely work with subsampled data from randomized sampling ideas.
TAKE HOME
Think randomized sampling in seismic surveys!! It saves cost!!!
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Acknowledgements
This work was in part financially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant (22R81254) and the Collaborative Research and Development Grant DNOISE II (375142-‐08). This research was carried out as part of the SINBAD II project with support from the following organizations: BG Group, BGP, BP, CGG, Chevron, ConocoPhillips, ION, Petrobras, PGS, Statoil, Total SA, Sub Salt Solutions, WesternGeco, and Woodside.
Thank you for your attention!