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

Thursday, December 9, 2010

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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

Thursday, December 9, 2010

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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/")

Thursday, December 9, 2010

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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

Thursday, December 9, 2010

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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

Thursday, December 9, 2010

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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

Thursday, December 9, 2010

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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

Thursday, December 9, 2010

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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

Thursday, December 9, 2010

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‣UseofsecondorderinformationinSAapproach‣TradeoffbetweenSAAandSA‣Marineacquisition

Open problems & Road ahead

Thursday, December 9, 2010

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Marine acquisition

+ =

observeddata modeleddatadobs1 + dobs

2 PH!1(q1 + q2)

Thursday, December 9, 2010

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Acknowledgements

ThisworkwasinpartfinanciallysupportedbytheNaturalSciencesandEngineeringResearchCouncilofCanadaDiscoveryGrant(22R81254)andtheCollaborativeResearchandDevelopmentGrantDNOISEII(375142‐08).ThisresearchwascarriedoutaspartoftheSINBADIIprojectwithsupportfromthefollowingorganizations:BGGroup,BP,Chevron,ConocoPhillips,Petrobras,TotalSA,andWesternGeco.

• EldadHaberandMarkSchmidtforusefulldiscussions

Thursday, December 9, 2010

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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.

Thursday, December 9, 2010