Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples

Transcription

1 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Guanghui Huang Education University of Chinese Academy of Sciences, Beijing, China Ph.D. in Computational Mathematics 9/9-7/ Thesis: Reverse time migration for inverse scattering problems Thesis supervisor: Professor Zhiming Chen Central South University, Changsha, China B.S. in Information and Computing Science 9/5-7/9 Research Interests Source-based Extended Waveform Inversion Acoustic/Electromagetic/Elastic wave inverse scattering problem Phaseless data imaging and inversion G. Huang, W. Symes and R. Nammour MSWI: Space-time /

2 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Matched Source Waveform Inversion: Space-time Extension Guanghui Huang, William W. Symes and Rami Nammour TRIP, Department of Computational and Applied Mathematics TOTAL E & P USA TRIP Annual Review Meeting April 5, 6 G. Huang, W. Symes and R. Nammour MSWI: Space-time /

3 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Outline Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples G. Huang, W. Symes and R. Nammour MSWI: Space-time /

4 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Outline Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples G. Huang, W. Symes and R. Nammour MSWI: Space-time /

5 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Seismic Inverse Problem Acoustic wave eqn: u v t u = δ(x x s)f(t). u = pressure field, v = velocity, f = input src function and x s = src position. Forward modeling operator: x r = receiver position. S[v]f = u(x, t; x s ) x=xr. Inverse Problem: Given d(x r, t; x s ), find v and f such that S[v]f = d. G. Huang, W. Symes and R. Nammour MSWI: Space-time 5 /

6 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Full Waveform Inversion FWI via least square, J FWI [v, f] = x r,x s S[v]f(x r, t; x s ) d(x r, t; x s ) dt. FWI obj is quadratic w.r.t f, but highly nonlinear and nonconvex in v. Sensitive to frequency band. Local minima: cycle skipping problem (bad init model & low frequency missing). G. Huang, W. Symes and R. Nammour MSWI: Space-time 6 /

7 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Outline Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples G. Huang, W. Symes and R. Nammour MSWI: Space-time 7 /

8 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Summary of Source-Receiver Extension FWI via src-recv extn (SEG 5, G. Huang and W. Symes) For bad v, assign different src func f(f sr (t)) for each src-recv pair (more d.o.f) G(x r, x s, t) f sr (t) = d(x r, x s, t). Fit data easily (only single trace fitting) no cycle skipping For true velocity, f(t) f sr = δ(t) focusing on t =. Minimizing non-focusing of f sr multiplied by A = tf(t). J MS [v] = Af sr dt. x s,x r See R. E. Plessix etc (), S. Luo and P. Sava (), L. Guasch and M. Warner () for similar algorithm. G. Huang, W. Symes and R. Nammour MSWI: Space-time 8 /

9 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Summary of Source-Receiver Extension Good for single arrival (equiv to traveltime tomography). Fail if strong multipathing exists. Reasons for failure: Ambiguity when fitting data from different branches; Slope of traveltime is lost (single trace fit); G(x r, x s, t) f sr (t) = d(x r, x s, t) is NOT solvable in L sense. G. Huang, W. Symes and R. Nammour MSWI: Space-time 9 /

10 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Outline Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples G. Huang, W. Symes and R. Nammour MSWI: Space-time /

11 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Motivation for New Extension If we assume G(x r, x s, t) f sr (t) = d(x r, x s, t) is solvable, it s equivalent to G(x r, x s, t) T G(x r, x s, t) f sr (t) = G(x r, x s, t) T d(x r, x s, t) NOTE: G(x r, x s, t) T d(x r, x s, t): backpropagation field (time shift of the data) in data domain (R. E. Plessix etc., ). How about backpropagation in the imaging domain? Extend the domain of f sr to the imaging domain (put src everywhere in the whole domain). G. Huang, W. Symes and R. Nammour MSWI: Space-time /

12 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Extended Modeling & Annihilator Extended Modeling: f(x, t; x s ): extended model of f(t)δ(x x s ) Extended modeling operator S f = ū: Annihilator: v ū t ū = f(x, t; x s ). A = x x s : Penalize non-focusing energy around src position x s. Do not need source function f(t)! G. Huang, W. Symes and R. Nammour MSWI: Space-time /

13 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Matched Source Waveform Inversion Extended waveform inversion: J α [v] = α S[v] f d dt + x r,x s s.t. ( S T S + αa T A) f = S T d. x,x s A f dt Key feature: data fitting via f no cycle skipping problem! G. Huang, W. Symes and R. Nammour MSWI: Space-time /

14 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Why it works (a) (b) (c) (d) Figure: (a) True model; Amplitude of the backpropagation field ( S T d) with (b) true velocity, (c) % low and (d) % high of true velocity See Y. Zhang etc. (8,9) and R. Plessix etc () for backpropagation-based waveform inversion. G. Huang, W. Symes and R. Nammour MSWI: Space-time /

15 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Relation with WRI Waveform Reconstruction Inversion (WRI, T. van Leeuwen and F. Herrmann ()): J WRI [v] = min ū + α = min f [( x,x s x r,x s v ū t ū) f(t)δ(x x s ) (ū(x, t; x s ) d(x r, t; x s )) dt. x=xr + α x,x s x r,x s f f(t)δ(x x s ) dt S f d dt. ] dt Nonlinear annihilator: A f = f(x, t; x s ) f(t)δ(x x s ). Our source focusing annihilator: A f = x x s f(x, t; x s ). G. Huang, W. Symes and R. Nammour MSWI: Space-time 5 /

16 Outline Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Transmission Configuration Diving Wave Inversion Marmousi Model BP Benchmark Model G. Huang, W. Symes and R. Nammour MSWI: Space-time 6 /

17 Numerical Implementation Forward modeling: 9-point FD method in frequency domain; Target source: zero-phased bandpassed source. Data acquisition: Receivers: fixed spread geometry on the whole surface; Source: lesser dense sampling than receivers. Inversion: Subproblem: direct solver to guarantee the accuracy of gradient; Optimization method: LBFGS with backtracking line search; Avoid inverse crime: coarse mesh grid for inversion, fine mesh grid for recording data. G. Huang, W. Symes and R. Nammour MSWI: Space-time 7 /

18 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Transmission Configuration Diving Wave Inversion Marmousi Model BP Benchmark Model G. Huang, W. Symes and R. Nammour MSWI: Space-time 8 /

19 Gaussian Model: Multipathing Figure: Transmission configuration: true model and initial model G. Huang, W. Symes and R. Nammour MSWI: Space-time 9 /

20 Simulated Data Time (s) Time (s) Trace Number 5 5 Trace Number Figure: Recorded data and simulated data with initial model at center shot x s = km G. Huang, W. Symes and R. Nammour MSWI: Space-time /

21 Inverted Results Figure: Inverted velocity by FWI and MSWI with (6,,, 8) Hz data G. Huang, W. Symes and R. Nammour MSWI: Space-time /

22 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Transmission Configuration Diving Wave Inversion Marmousi Model BP Benchmark Model G. Huang, W. Symes and R. Nammour MSWI: Space-time /

23 Diving Wave Figure: True model and ray tracing on model for shot x s =.5 km G. Huang, W. Symes and R. Nammour MSWI: Space-time /

24 Comparison of Results Figure: Top: initial model; inverted model by FWI (middle) and MSWI (bottom) using (6,7,8,9,) Hz data G. Huang, W. Symes and R. Nammour MSWI: Space-time /

25 Comparison of Results Figure: Top: initial model; inverted model by FWI (middle) and MSWI (bottom) using (5,6,7,8,9,) Hz data G. Huang, W. Symes and R. Nammour MSWI: Space-time 5 /

26 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Transmission Configuration Diving Wave Inversion Marmousi Model BP Benchmark Model G. Huang, W. Symes and R. Nammour MSWI: Space-time 6 /

27 Marmousi Model Figure: Marmousi model and D initial model G. Huang, W. Symes and R. Nammour MSWI: Space-time 7 /

28 Comparison of Results Figure: Inverted velocity by MSWI and FWI with (,5,6,7,8) Hz data G. Huang, W. Symes and R. Nammour MSWI: Space-time 8 /

29 Overview Avoiding Cycle Skipping: Model Extension MSWI: Space-time Extension Numerical Examples Transmission Configuration Diving Wave Inversion Marmousi Model BP Benchmark Model G. Huang, W. Symes and R. Nammour MSWI: Space-time 9 /

30 BP Benchmark Model Figure: BP model G. Huang, W. Symes and R. Nammour MSWI: Space-time /

31 Inverted Result Figure: Initial model and inverted model using (,,,) Hz data G. Huang, W. Symes and R. Nammour MSWI: Space-time /

32 BP Benchmark Model Figure: BP model G. Huang, W. Symes and R. Nammour MSWI: Space-time /

33 Conclusion Nonlinear extended waveform inversion can handle any kinds of waves including transmission wave and reflection without separation of data. Lower the requirement of low frequency and initial model. Source wavelet function is not required. Straightforward extension to multi-parameter inversion and elastic wave inversion. Main obstacles: Limitation to D Helmholtz eqn solver. Storage requirement of extended model f(x, xs, t) R 6 in D. G. Huang, W. Symes and R. Nammour MSWI: Space-time /

34 Acknowledgements Bertrand Duquet and Fuchun Gao Total E & P USA TRIP sponsors and members Rice University Research Computing Support Group (RCSG) All of audiences G. Huang, W. Symes and R. Nammour MSWI: Space-time /