Wavefield Reconstruction Inversion (WRI) a new take on wave-equation based inversion Felix J. Herrmann

Transcription

1 Wavefield Reconstruction Inversion (WRI) a new take on wave-equation based inversion Felix J. Herrmann SLIM University of British Columbia van Leeuwen, T and Herrmann, F J (2013). Mitigating local minima in full- waveform inversion by expanding the search space. Geophysical Journal International. van Leeuwen, T and Herrmann, F J (2013). A penalty method for PDE- constrained optimization. Submitted for publication van Leeuwen, T and Herrmann, F J (2013). US Provisional Patent Application No. 61/815,533. A Penalty Method for PDE- Constrained Optimization with Applications to Wave- Equation Based Seismic Inversion

2 Wavefield Reconstruction Inversion (WRI) a new take on wave-equation based inversion Ernie Esser, Tristan van Leeuwen*, and Bas Peters * Mathematical Institute at Utrecht University SLIM University of British Columbia van Leeuwen, T and Herrmann, F J (2013). Mitigating local minima in full- waveform inversion by expanding the search space. Geophysical Journal International. van Leeuwen, T and Herrmann, F J (2013). A penalty method for PDE- constrained optimization. Submitted for publication van Leeuwen, T and Herrmann, F J (2013). US Provisional Patent Application No. 61/815,533. A Penalty Method for PDE- Constrained Optimization with Applications to Wave- Equation Based Seismic Inversion

3 Motivation Full- waveform inversion is plagued with local minima Derive an alternative extended formulation less prone to local minima computationally feasible relaxes the physics while staying solidly grounded

4 Waveform inversion Retrieve the medium parameters from partial measurements of the solution of the wave- equation:a(m)u i = q i q i P i u i m

5 [Tarantola, 84; Pratt, 98; Haber, 00; Plessix, 06] Waveform inversion Adjoint- state/reduced- space methods: Optimize over earth models to minimize the misfit between observed and simulated data while solving the wave equation exactly for each earth model. Full- space or all- at- once methods: Optimize over earth models & wavefields jointly to minimize the misfit between observed and simulated data subject to wavefields that satisfy the wave equation.

6 Waveform inversion Both approaches assume flawless wave physics i.e., known physics known source A(m)u i = q i unknown wavefield holds exactly for each source i differ on insisting wave equations to hold for each iteration different unknowns: m m & u

7 [Richter, 81] Equation error approach If we know the wavefields everywhere, we solve for m from A(m)u i = q i via min m ka(m)p 1 i d i q i k 2 2 cf. min kp ia(m) 1 q i d i k 2 2 m The challenge is to reconstruct wavefields from partial measurements...

8 [van Leeuwen & FJH, 2013] WRI Wavefield Reconstruction Inversion For m fixed, reconstruct wavefields by jointly fitting observed shots P u i d i and wave- equations A(m)u i q i via least- squares solutions of the data- augmented wave- equation min u i Pi A(m) u i di q i 2 2 followed by fixing u i and solving min m ka(m)u i q i k 2 2

9 wave- equation x wavefield = source versus ( ) ( wave- equation sampling operator x wavefield = ) source data

10 observed data initial data data- augmented solution

11 wavefield in true model data- augmented

12 wavefield in true model data- augmented

13 wavefield in true model data- augmented

14 wavefield in true model data- augmented

15 wavefield in true model data- augmented

16 wavefield in true model data- augmented

17 wavefield in true model data- augmented

18 wavefield in true model data- augmented

19 wavefield in true model data- augmented

20 wavefield in true model data- augmented

21 wavefield in true model data- augmented

22 wavefield in true model data- augmented

23 wavefield in true model data- augmented

24 wavefield in true model data- augmented

25 wavefield in true model data- augmented

26 wavefield in true model data- augmented

27 [Heinkenschloss, 98, Haber, 00] PDE-constrained optimization all-at-once full-space approach simulated data simulated wavefield min m,u MX i=1 kp i u i d i k 2 2 s.t. A i (m)u i = q i observed data source Helmholtz equation avoids having to solve the PDE explicitly sparse (GN) Hessian requires storing all variables (m,u) does not scale to industry- scale seismic problems

28 [Tarantola 84; Pratt, 98; Plessix, 06] Adjoint-state/reduced-space formulation Elimination of the constraint leads for all sources to MX min m red (m) = i=1 kp i A i (m) 1 q i d i k 2 2 no need to store all wavefields (block- elimination) suitable for black- box optimization (e.g., l- BFGS) need to solve forward & adjoint PDEs very non- linear in earth model (m) dense (GN) Hessian, involves additional PDE solves paints you in a corner by insisting on the physics...

29 [Bertsekas, 96; Wright, 00; van Leeuwen & FJH, 13] WRI penalty formulation Instead of eliminating, we add constraints as penalties i.e., min m,u (m, u) = MX i=1 kp u i d i k A i (m)u i q i k 2 2 coincides with original problem when "1

30 [Aravkin & van Leeuwen, 12; van Leeuwen & FJH, 13] Variable projection Solve data- augmented wave equation for each source P i A i (m) u i, di q i Define reduced objective with proxy wavefields (m) = (m, ū )=kp ū dk A(m)ū qk 2 2

31 [van Leeuwen & FJH, 13] Wavefield Reconstruction Inversion WRI method for each source i Conventional method for each source i solve P i A i (m) u,i di q i solve solve A(m)u i = q i A(m) v i = P i (P i u i d i ) g = g + 2! 2 diag(ū i, ) (A(m)ū i, q i ) g = g +! 2 diag(u i ) v i m = m end g correlation proxy wavefield & PDE residual end m = m g correlation wavefield & data residual

32 Wavefield Reconstruction Inversion no need to store all the fields (u) no adjoint solves sparse approximation of GN Hessian for small less non- linear in m need to solve overdetermined PDEs...

33 Diving wave example true model and wavefield 0 z [m] x [m] 0 z [m] x [m]

34 Wavefields in homogeneous background FWI WRI 2 2 data x [m] r x [m] r 0 0 wavefield z [m] 500 z [m] x [m] x [m] model update z [m] x [m] z [m] x [m]

35 Local minima single shot, single frequency data for linear velocity profile v(z) =v 0 + z, misfit as function of (v 0, ) misfit v0 [m/s] constant λ = 10 3 λ = 10 6 λ = 10 9 reduced misfit α [1/s] slope λ = 10 3 λ = 10 6 λ = 10 9 reduced

36 Connections

37 [Aria Abubakar et. al. 09] Related work Contrast- source formulation combined objective is similar but does not eliminate wavefields via variable projection requires storage of wavefields for all sources

38 Extended modelling The penalty formulation min m,u P u d A(m)u q 2 2 can be interpreted as min m misfit( m)+annihilator( m) with m =(m, u) For a physically plausible model we have annihilator( m) =0 [Symes, personal communication]

39 Warping The overdetermined WE is a way of warping 1 1 Re(u) 0 Re(u) x [m] r x [m] r 0 0 z [m] 500 z [m] x [m] x [m] [Baek 13, Ma 13]

40 WRI vs. FWI Larger # of degrees of freedom more convex m u m

41 [van Leeuwen & FJH, 13] Wavefield Reconstruction Inversion WRI method for each source i Conventional method for each source i solve P i A i (m) u,i di q i solve solve A(m)u i = q i A(m) v i = P i (P i u i d i ) g = g + 2! 2 diag(ū i, ) (A(m)ū i, q i ) g = g +! 2 diag(u i ) v i H GN = H GN + 2! 4 diag(u i ) diag(u i ) m = m g m = m H 1 GN g end diagonal = end dense & pseudo Hessian too expsensive

42 Example BG Compass model Low frequencies missing, 24 frequency batches (15 iterations each) {5 6},{6 7},...,{28 29} Hertz. Each interval contains 5 frequencies. 103 sources/receivers w/ 55m sample interval Inaccurate initial model

43 True & initial model z [m] True model x [m] Velocity [m/s] z [m] Initial model x [m] Velocity [m/s]

44 FWI vs WRI z [m] Result FWI x [m] Velocity [m/s] z [m] Result WRI, = x [m] Velocity [m/s]

45 Gradients 0 First update FWI 200 z [m] x [m] Velocity [m/s] 0 First update WRI, =1 200 z [m] x [m] Velocity [m/s]

46 Cross sections 0 x =2063.1[m] 0 x =3443.1[m] 0 x =4305.6[m] z [m] z [m] z [m] True model WRI start Velocity [m/s] Velocity [m/s] Velocity [m/s]

47 Relative model errors difference with true model, cycle 1 1 difference with true model, cycle m est m true 2 / m0 m true m est m true 2 / m0 m true Iteration nr Iteration nr.

48 Objective function value 10 5 Objective WRI, cycle 1 Objective WRI, cycle Data fit Data fit PDE fit PDE fit Data fit increases at some iterates

49 Data fit Imaginary part, source in middle of domain Receiver number 5 0 Real part, source in middle of domain Receiver number Observed data Data from wave equation in start model Data from data augmented wave eqution in start model

50 Ernie Esser, Tristan van Leeuwen, Aleksandr Y. Aravkin, and Felix J. Herrmann, A scaled gradient projection method for total variation regularized full waveform inversion Bas Peters and Felix J. Herrmann, A sparse reduced Hessian approximation for multi-parameter Wavefield Reconstruction Inversion Extensions Total-variation regularization via scaled gradient projections & bound constraints Multi-parameter case via sparse approximate Gauss- Newton scaling

51 BP model number of sources: 126 (starting 1000m in from boundary) number of receivers: 299 frequency range: 3-20Hz in overlapping batches of 2 maximum number of outer iterations per frequency batch: 25 maximum number of inner iterations for convex subproblems: 2000 known Ricker wavelet sources with 15Hz peak frequency two simultaneous shots with Gaussian weights w/ redraws no added noise

52 True and initial velocity

53 Results w/ TV After one cycle through the frequencies After two cycles through the frequencies

54 Results w/o TV After one cycle through the frequencies After two cycles through the frequencies

55 Conclusions New alternating method for wave- equation based inversion: same extended search space as in all- at- once but with memory & CPU requirements as in adjoint- state approach no adjoints & sparse GN- Hessian approximation less susceptible to local minima due to data fit sparse GN Hessians bilinear Challenge: Stationary points are not necessary global minima

56 Acknowledgements Thank you for your attention! 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, BGP, CGG, Chevron, ConocoPhillips, ION, Petrobras, PGS, Statoil, Subsalt Ltd, Total SA, WesternGeco, and Woodside.