waveform inversion by stochastic optimizationwaveform inversion by stochastic optimization thursday,...
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SLIMUniversityofBritishColumbia
Consortium2010
TristanvanLeeuwen&SashaAravkin
Waveform inversion by Stochastic optimization
Thursday, December 9, 2010
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CostsperiterationofFWIgrowslinearlywiththenumberofshots.
Thecostscanbereducedby(randomly)combiningshots....
Motivation
[Krebsetal’09;Hermann’09;Dai’10;Li’10;Moghaddan’10;Symes’10]
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example
Motivation
allsources onesimultaneoussource
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‣Randomizedtraceestimation‣Stochasticoptimization‣Numericalresults‣Conclusions‣Openproblems&Roadahead
Overview
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Werepresentthedataasfrequencyslices
Source blending
sourcepos.
rec.pos. D w =
fulldata
encoding
blendeddata
[Beasly’98;Ikelle’07;Berkhout’08;]Thursday, December 9, 2010
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Evaluatingthemisfit
isatraceestimationproblem
Trace estimation
[Hutchinson’89;Avron’10,Haberetal’10]
||S ||2F = t race(S T S )
![c] =!
!
|| D(c)!Dobs
" #$ %S
||2F
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Picksuchthatandthen:
andfor
Trace estimationEw
!wwT
"= Iw Ew
!w} = 0
trace(A) = Ew
!trace(wTAw)
"
! 1N
N!1#
i=0
wTi Awi
P [error ! !] = 1" " N ! a!!2 ln(b/")
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Trace estimation
row index
colu
mn
inde
x
50 100 150 200 250 300
50
100
150
200
250
300
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Trace estimation
100 101 102 1030
0.2
0.4
0.6
0.8
1
K
P(E ! ")
HutchinsonGaussPhase−encoded
! = 0.2
100 101 102 1030
0.2
0.4
0.6
0.8
1
K
P(E ! ")
HutchinsonGaussPhase−encoded
! = 0.1
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Themisfitcanbewrittenas
whichcanbeapproximatedatthecostofsimulations
Trace estimation
N
![c] = Ew
!"
!
||(D(c)!Dobs)w||2F
#
!N [c] =1N
N!1!
i=0
!
!
||(D(c)!Dobs)wi||2F
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Evaluatethemisfitandgradientintheusualway
Trace estimation
H[c]u = Qw
D[c] = Pu
H![c]v = P !(D[c]!Dobsw)
"![c] =!
!
"2u!v
[Tarantola’84;Pra^’98;Plessix’06]Thursday, December 9, 2010
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misfit
Trace estimation!N [c0 + "s] , s = !"!N [c0]
N = 1 N = 5 N = 10
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Trace estimation N = 1
N = 5 N = 10
N
|!!"!
!N
|
full
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CanonicalSOproblem:
Wediscerntwodistinctapproaches:
1.Sample Average Approximation (SAA)2.Stochastic approximation (SA)
Stochastic optimization
minc
Ew{![c;w]}
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Replacetheexpectationbyanensembleaverage
Thenuseyoufavoriteoptimizationmethod
Stochastic optimization I
!N [c] =1N
N!1!
i=0
!
!
||(D(c)!Dobs)wi||2F
[Nemerovski’00,’09;Shapiro’03,’05;]Thursday, December 9, 2010
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basicsteepestdescent
Stochastic optimization I
while not converged dos! "#!N [ci] //search directionsolve min! !N [ci + "s] //linesearchci+1 ! ci + "s //update modeli! i + 1
end while
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Considerasa`noisy’measurementof.Inparticularonerequiresthatandlikewiseforthegradient.
Stochastic optimization II
[Robbins’51;Bertsekas’96,’00;Nesterov’96]
![c, w]![c]
Ew{![c, w]} = ![c]
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BasicSAalgorithm:
Stochastic optimization II
while not converged dodraw wi from a pre-scribed distributions! "#![ci, wi] //search directionsolve min! ![ci + "s, wi] //linesearchci+1 ! 1
n+1
!"ii!n ci + "s
#//averaging
i! i + 1end while
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61shots/receivers,7frequencies[5‐30]Hz,10HzRickerwavelet,additiveGaussiannoise
Numerical results
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Numerical results: full
nonoise SNR=20dB SNR=10dB
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100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10K=20full
Numerical results: SAA
nonoise
iterahon#
mod
elerror
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100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10K=20full
Numerical results: SAA
SNR=20dB
iterahon#
mod
elerror
Thursday, December 9, 2010
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100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10K=20full
Numerical results: SAA
SNR=10dB
iterahon#
mod
elerror
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N=20Numerical results: SAA
nonoise SNR=20dB SNR=10dB
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Numerical results: SAAfull
nonoise SNR=20dB SNR=10dB
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nonoise
Numerical results: SA
noaveraging
n=10
n=500
100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10full
100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10full
100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10full
Thursday, December 9, 2010
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Numerical results: SASNR=20dB
noaveraging n=10 n=500
100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10full
100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10full
100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10full
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Numerical results: SASNR=10dB
n=500n=10
noaveraging
100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10full
100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10full
100 101 102
10−1
iteration #
|! m
|
K=1K=5K=10full
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N=5,historysize=10
Numerical results: SA
nonoise SNR=20dB SNR=10dB
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Numerical results: SAfull
nonoise SNR=20dB SNR=10dB
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‣SAAneedslargerbatchsize,canbeusedwithsecondorderoptimizationmethods‣SAisabletomatchfullresultsformodestbatchsizes,evenincaseofnoise‣RenewalsandaveragingareimportantintheSAapproach
Conclusions
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‣UseofsecondorderinformationinSAapproach‣TradeoffbetweenSAAandSA‣Marineacquisition
Open problems & Road ahead
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Marine acquisition
+ =
observeddata modeleddatadobs1 + dobs
2 PH!1(q1 + q2)
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Acknowledgements
ThisworkwasinpartfinanciallysupportedbytheNaturalSciencesandEngineeringResearchCouncilofCanadaDiscoveryGrant(22R81254)andtheCollaborativeResearchandDevelopmentGrantDNOISEII(375142‐08).ThisresearchwascarriedoutaspartoftheSINBADIIprojectwithsupportfromthefollowingorganizations:BGGroup,BP,Chevron,ConocoPhillips,Petrobras,TotalSA,andWesternGeco.
• EldadHaberandMarkSchmidtforusefulldiscussions
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ReferencesAvron, H., and S. Toledo, 2010, Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix: submitted to ACM.Beasley, C. J., R. E. Chambers, and Z. Jiang, 1998, A new look at simultaneous sources: SEG Technical Program Expanded Abstracts, 17, 133–135.Berkhout, A. J. G., 2008, Changing the mindset in seismic data acquisition: The Leading Edge, 27, 924–938.Haber, E., M. Chung, and F. J. Herrmann, 2010, An effective method for parameter estimation with PDE constraints with multiple right hand sides: Technical Report TR-2010-4,UBC-Earth and Ocean Sciences Department.Hutchinson, M., 1989, A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines: Communications in Statistics - Simulation and Computation, 18, 1059–1076Ikelle, L., 2007, Coding and decoding: Seismic data modeling, acquisition and processing: SEG Technical Program Expanded Abstracts, 26, 66–70.Krebs, J. R., J. E. Anderson, D. Hinkley, R. Neelamani, S. Lee, A. Baumstein, and M.-D.Lacasse, 2009, Fast full-wavefield seismic inversion using encoded sources: Geophysics, 74, WCC177–WCC188.Li, X., and F. J. Herrmann, 2010, Fullwaveform inversion from compressively recovered model updates: SEG Expanded Abstracts, 29, 1029–1033.Moghaddam, P. P., and F. J. Herrmann, 2010, Randomized full-waveform inversion: a dimenstionality-reduction approach: SEG Technical Program Expanded Abstracts, 29, 977–982.Herrmann, F. J., Y. A. Erlangga, and T. Lin, 2009, Compressive simultaneous full-waveform simulation: Geophysics, 74, A35.Plessix, R.-E., 2006, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications: Geophysical Journal International, 167, 495–503.Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, 1259–1266.Symes, W., 2010, Source synthesis for waveform inversion: SEG Expanded Abstracts, 29, 1018–1022.
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