Randomized Full Waveform Inversion

Transcription

1 Consortium 2010 Randomized Full Waveform Inversion Peyman P. Moghaddam SLIM University of British Columbia

2 Motivation Cost of the FWI is propor?onal to the number of shots and it requires hundreds of RTM (Reverse Time Migra?on). Dimensionality reduc?on with compressive sensing aims at compressing the data volume in inversion. Stochas?c op?miza?on provides bejer solu?on to randomized inversion problem than conven?onal op?miza?on.

3 Overview Full Waveform Inversion (FWI) Simultaneous Source Experiment Randomized FWI Stochastic Optimization Methods Examples Conclusion Future Plans

4 Full Waveform Inversion Mathema?cally FWI can be formed as min σ J(σ) = 1 2 d Du(σ) 2 subject to F(u, σ) = 0 d : data D : detection operator σ : sloweness F(u, σ) = (ω 2 σ )u + q = 0 u(σ) : wavefield ω : angular frequency q : source 2 : Laplacian (Ben Hadj Ali 08)

5 Full Waveform Inversion Gradient of the cost func?on with respect to slowness is defined as, J(σ)/ σ = R{( u/ σ) H D T [d Du(σ)]} R : real part (.) H : Hermitian with u/ σ defined as, u/ σ = 2ω 2 (ω 2 σ ) 1 σu = K K : de-migration operator, linearized Born (Plessix 2009)

6 Conventional Optimization Limited memory BFGS (Plessix 2009) σ k+1 = σ k τh k J(σ k ) H k : inverse Hessian τ : line search value Precondi?oned Gradient method (Ravaut 2004) σ k+1 = σ k τdiag(k T K + ɛi) 1 J(σ k ) Conjugate Gradient method (Virieux 2009)

7 Simultaneous Source Experiment 0.1 Shot time (s) Shot Shot offset (m) 0.2 time (s) time (s) time (s) offset (m) offset (m) Super-Shot offset (m) (Krebs 2009)

8 updates read Simultaneous Source Experiment σ k+1 = σ k τ J(σ k, Q) Q : all sources with J(σ k, Q) 1 i=n s N s J(σ k, Q i ) i=1 Q i : a simultaneous source ( ) i=n 1 s E J(σ k, Q i ) N s i=1 J(σ k, Q) E(.) : expectation

9 Randomized FWI σ σ 0 initial model {J(σ, Q i ), σ J(σ, Q i )} new randomized super-shot While σ J(σ) ɛ σ update model with J(σ, Q i ), σ J(σ, Q i ) {J(σ, Q i ), σ J(σ, Q i )} new randomized super-shot end (Moghaddam 2010, Krebs 2009)

10 Stochastic Optimization Approaches Stochastic Gradient Descent σ k+1 = σ k τ J(σ k, d k ) Integrated Stochastic Gradient Descent (isgd) J(σ k ) σ k+1 = σ k η k J(σ k ) with is averaging on the past numbers of gradients with weights, k i=k m eα[i (k m)] J(σ i, d i ) J(σ k )= k i=k m eα[i (k m)] (Moghaddam 2010)

11 Stochastic Optimization Limited Memory BFGS (quasi Newton methods), σ k+1 = σ k η k H k J(σ k, d k ) with updates: H k+1 = V T k H k V k + ρ k s k s T k s k = σ k+1 σ k y k = J(σ k+1 ) J(σ k ) V k = I ρ k y k s T k On line Limited Memory BFGS H 0 = m i=1 s T k i y k i y T k i y k i (Schraudolph 2007)

12 Stochastic Optimization Regular Limited Memory BFSG (quasi Newton methods), min H subject to H k+1 H k F H T k+1 = H k+1, H T k+1y k = s k s k = σ k+1 σ k y k = J(σ k+1 ) J(σ k ) Integrated Limited Memory BFSG (ilbfgs), min H H k+1 k H k F subject to H T k+1 = H k+1, H T k+1y k = s k

13 Examples (Marmoussi Model) depth (m) offset (m) 113 shots with 40 (m) spacing, 249 receivers with 20 (m) spacing, WAZ survey with 5 (km) max. aperture, Ricker source with 10 Hz central frequency, 3.6 second recording?me with.9 (ms)?me sampling. 1.5

14 Examples (Ini?al Model) depth (m) offset (m)

15 Examples (Inverted Model) depth (m) offset (m) inverted model acer 18 itera?ons of LBFGS, 113 sequen?al shots, 50 frequency components has been used from 5 to 33 Hz with.55 Hz resolu?on 1.5

16 Randomized FWI (Inverted Model) depth (m) offset (m) Stochas?c Gradient Descent, SNR= 4.65 db, 1.5 SNR = 20 log 10 ( δm δ m 2 δm 2 ) 16 Randomized simultaneous shots, 4 frequencies, 40?mes speed up

17 Randomized FWI (Inverted Model) depth (m) online LBFGS, SNR= 7.17 db offset (m) Randomized simultaneous shots, 4 frequencies, 40?mes speed up

18 Randomized FWI (Inverted Model) depth (m) ilbfgs, SNR= 9.10dB offset (m) Randomized simultaneous shots, 4 frequencies, 40?mes speed up

19 Randomized FWI (Inverted Model) depth (m) isgd, SNR= 10.85dB offset (m) Randomized simultaneous shots, 4 frequencies, 40?mes speed up

20 Comparison depth (m) offset (m) Inversion for all the shots, 1 week on the 32 CPU cluster offset (m) isgd, SNR= 10.85dB 8 hours on the 32 CPU cluster 16 Randomized simultaneous shots, 4 frequencies, 40?mes speed up

21 Comparison 12 ISGD SGD 10 8 SNR (db) iteration Comparison between conven?onal gradient descent and stochas?c gradient descent.

22 Examples (Marmoussi Model) 900 shots with 10 (m) spacing, 900 receivers with 10 (m) spacing, WAZ survey with 5 (km) max. aperture, Ricker source with 10 Hz central frequency, 3.6 second recording?me with.9 (ms)?me sampling.

23 Examples (Ini?al Model)

24 Randomized FWI (Inverted Model) isgd method, 1 Randomized simultaneous shots, 900?mes speed up!

25 Conclusion Super shot experiment combined with stochas?c op?miza?on methods produce promising results for solu?on for FWI Randomized FWI greatly increases the performance of the FWI. Dimensionality reduc?on algorithms, open possibility of replacing migra?on with FWI with no extra cost.

26 Future Plans Further inves?ga?on on the choice of random frequency and super shot Stochas?c op?miza?on strategies for FWI, improved ilbfgs, Natural gradient Regulariza?on for the FWI Solving the uniqueness problem, exploi?ng the mul? scale nature of the FWI

27 Acknowledgements The authors would like to thank Yogi Erlangga and Tim Lin for their Helmoltz operator. The authors would like to thank Eldad Haber for discussion on stochastic optimization and regularized inverse problem. 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 ( ). This research was carried out as part of the SINBAD II project with support from the following organizations: BG Group, BP, Chevron, ConocoPhillips, Petrobras, Total SA, and WesternGeco.