Simultaneous estimation of wavefields & medium parameters

Transcription

1 Simultaneous estimation of wavefields & medium parameters reduced-space versus full-space waveform inversion Bas Peters, Felix J. Herrmann Workshop W- 2, The limit of FWI in subsurface parameter recovery. SEG 85TH annual meeting, Friday, 23 October SLIM University of British Columbia

2 Motivation Full- Waveform Inversion (FWI) works well when a good start model and/or low- frequency data is available. m 2 kph(m) q dk 2 2 = 2 kd pred(m) d obs k 2 2 H(m) 2 C N N discrete PDE m 2 R N medium parameters P 2 R m N selects field at receivers 2 u 2 C N d 2 C m q 2 C N field observed data source

3 Motivation Full- Waveform Inversion (FWI) works well when a good start model and/or low- frequency data is available. m 2 kph(m) q dk 2 2 = 2 kd pred(m) d obs k 2 2 If iterative solvers in frequency domain are used, we cannot compute: H(m) q = u Instead we obtain: û = H(m) (q + r u )=H(m) r u + u 3

4 Motivation Full- Waveform Inversion (FWI) works well when a good start model and/or low- frequency data is available. m 2 kph(m) q dk 2 2 = 2 kd pred(m) d obs k 2 2 If iterative solvers in frequency domain are used, we cannot compute: H(m) q = u Instead we obtain: residual û = H(m) (q + r u )=H(m) r u + u 4 error

5 Example Illustration of what happens when PDE s are solved inaccurately 3-5Hz good start model full- offset source & receiver array noise free data 5

6 0 500 True velocity model z [m] x [m] 6

7 0 500 Initial velocity model z [m] x [m] 7

8 FWI using very accurate iterative solver & LBFGS estimated model z [m] x [m] 8

9 FWI using accurate iterative solver & LBFGS estimated model z [m] x [m] 9

10 FWI using less accurate iterative solver & LBFGS estimated model z [m] x [m] 0

11 FWI using inaccurate accurate iterative solver & LBFGS estimated model z [m] x [m]

12 FWI using very inaccurate accurate iterative solver &LBFGS estimated model z [m] x [m] 2

13 Inexact PDE solves reduced- space: (includes FWI) error in objective function value error in gradient error in Hessian [A Tarantola, 984; E Haber et al., 2000; I Epanomeritakis et al., 2008] error in medium parameter update storage as low as two fields at a time dense reduced- Hessian requires extra safeguards/accuracy control [T. van Leeuwen & F.J. Herrmann, 204] 3

14 Goal This talk is about deriving an algorithm which: allows for inexact solutions of linear systems enjoys similar parallelism and memory requirements as FWI 4

15 Data, PDE s and constraints For the true medium parameters and true fields we know that: H(m)u = q & P u = d Many ways to use these equations to form: objectives, constraints algorithms Adjoint- state based FWI is just one algorithm. 5

16 Data, PDE s and constraints objective m,u 2 kp u dk2 2 s.t. H(m)u = q constraint 6

17 Data, PDE s and constraints m,u 2 kp u dk2 2 s.t. H(m)u = q objective m,u 2 kh(m)u qk2 2 s.t. P u = d constraint 7

18 Data, PDE s and constraints m,u 2 kp u dk2 2 s.t. H(m)u = q m,u 2 kh(m)u qk2 2 s.t. P u = d m 2 kph(m) q dk 2 2 = 2 kd pred(m) d obs k 2 2 8

19 Data, PDE s and constraints m,u 2 kp u dk2 2 s.t. H(m)u = q m,u 2 kh(m)u qk2 2 s.t. P u = d m 2 kph(m) q dk 2 2 = 2 kd pred(m) d obs k 2 2 m,u kh(m)u qk2 2 s.t. kp u dk 2 2 apple 9

20 Data, PDE s and constraints m,u 2 kp u dk2 2 s.t. H(m)u = q m,u 2 kh(m)u qk2 2 s.t. P u = d m 2 kph(m) q dk 2 2 = 2 kd pred(m) d obs k 2 2 m,u kh(m)u qk2 2 s.t. kp u dk 2 2 apple 20

21 Multi-experiment structure: P u d H(m)u q 0 P P 2 Pk C A 0 u u 2 C A 0 d d 2 B. A 0 H H 2 Hk C A 0 u u 2 C A 0 q q 2 B. A u k d k u k q k k = n src n freq k N field parameters 2

22 A quadratic-penalty based full space method m,u kh(m)u qk2 2 s.t. kp u dk 2 2 apple - relation is known [W. Gander, 980; A. Bjork, 996] m,u 2 kp u dk kh(m)u qk2 2 22

23 A quadratic-penalty based full space method m,u 2 kp u dk kh(m)u qk2 2 Newton s method: P P + 2 H H r 2 u,m P (P u d)+ 2 H Hu q r 2 m,u 2 G mg m u m = 2 G m Hu q updates for for working subset of fields & medium parameters 23

24 A quadratic-penalty based full space method P P + 2 H H r 2 u,m P (P u d)+ 2 H Hu q r 2 m,u 2 G mg m u m = 2 G m Hu q update fields & medium parameters simultaneously function value, gradient, Hessian evaluation is ~free & exact sparse Hessian theory allows for inexact updates computations requires storage of working subset of fields + working memory (gradients, Hessian & update) 24 update computation is challenging

25 A quadratic-penalty based full space method Approximate: block diagonal & positive (semi) definite P P + 2 H H 0 P (P u d)+ 2 H Hu q 0 2 G mg m u m = 2 G m Hu q give up some of Newton s method properties update computation intrinsically parallel per field no need to form off- diagonal blocks philosophy: more & cheaper iterations 25

26 Memory requirements save fields for the working subset of frequencies & sources can be distributed over multiple nodes feasible? need parallel computing simultaneous sources (redrawing is possible) small frequency batches 26

27 Algorithm 0. construct initial guess for medium and for each field while not converged do. form Hessian and gradient // form (~free) 2. ignore the blocks // approximate 3. find & each in parallel // solve 4. find steplength using linesearch // evaluate (~free) 5. & // update model and fields end δm 2 u,m m = m + αδm m ϕ, ϕ δu i α 2 m,u u = u + αδu u i 27 Algorithm 2 field-medium parameter uncoupled Newton for Quadratic Penalty form.

28 FWI using very inaccurate accurate iterative solver &LBFGS estimated model z [m] x [m] 28

29 Full-space method of this talk using very inaccurate accurate iterative solver estimated model z [m] x [m] 29

30 Related work [E. Haber & U.M. Ascher, 200 ; G. Biros & O. Ghattas, 2005 ; Grote et. al., 20] The presented algorithm is a quadratic- penalty version of Lagrangian- based all- at- once algorithms: m,u 2 kp u dk2 2 s.t. H(m)u = q L(m, u, v) = 2 kp u dk2 2 + v H(m)u q G solve (inexactly) at every iteration: Newton- KKT G m G P P H u A H v + P (P u d) A G H 0 v Hu q 30

31 Related work [E. Haber & U.M. Ascher, 200 ; G. Biros & O. Ghattas, 2005 ; Grote et. al., 20] Lagrangian based full- space methods also store the multipliers + corresponding gradient & Hessian blocks. no intrinsic parallel structure G m G P P H u A H v + P (P u d) A G H 0 v Hu q * higher order terms number of field variables: 2 n src n freq n grid 3

32 Inexact PDE solves full-space vs reduced-space reduced- space (FWI): error in objective function value error in gradient error in Hessian error in medium parameter update full- space (this talk): objective function value always exact gradient always exact Hessian always exact globally convergent inexact Newton methods [S.C. Eisenstat & H.F. Walker, 994] 32

33 Full vs Reduced-space Hessian, gradient & function evaluation Hessian, gradient & function evaluation Reduced-space solve PDE s inexact Full-space ~free exact Hessian dense sparse memory for fields working memory 2 fields per parallel process gradient & update direction working subset of simultaneous source fields in memory update (can be directions distributed & over gradients nodes) in memory 33 ~free = sparse matrix-vector products

34 Conclusions Constructed a quadratic- penalty based full- space method which: updates fields & medium parameters simultaneously main computations are intrinsically parallel suitable for frequency domain waveform inversion with iterative solvers con: need to store working subset of simultaneous source fields but, less storage needed compared to Lagrangian full- space methods 34

35 Acknowledgements Thanks to our sponsors This work was financially supported by SINBAD Consortium members BG Group, BGP, CGG, Chevron, ConocoPhillips, DownUnder GeoSolutions, Hess, Petrobras, PGS, Schlumberger, Statoil, Sub Salt Solutions and Woodside; and by the Natural Sciences and Engineering Research Council of Canada via NSERC Collaborative Research and Development Grant DNOISEII (CRDPJ ).

36 References. Eisenstat, Stanley C., and Homer F. Walker. "Globally convergent inexact Newton methods." SIAM Journal on Optimization 4.2 (994): Tristan van Leeuwen and Felix J. Herrmann, frequency- domain seismic inversion with controlled sloppiness, SIAM Journal on Scientific Computing, 36 (204), pp. S92 S B Peters, FJ Herrmann, T van Leeuwen. Wave- equation Based Inversion with the Penalty Method- Adjoint- state Versus Wavefield- reconstruction Inversion. 76th EAGE Conference, M.J. Grote, J. Huber, and O. Schenk, Interior point methods for the inverse medium problem on massively parallel architectures, Procedia Computer Science, 4 (20), pp Proceedings of the International Conference on Computational Science, {ICCS} Eldad Haber, Uri M Ascher, and Doug Oldenburg, On optimization techniques for solving nonlinear inverse problems, Inverse Problems, 6 (2000), pp E Haber and U M Ascher, Preconditioned all- at- once methods for large, sparse parameter estimation problems, Inverse Problems, 7 (200), p I Epanomeritakis, V Akcelik, O Ghattas, and J Bielak. A Newton- CG method for large- scale three- dimensional elastic full- waveform seismic inversion. Inverse Problems, 24(3):03405, June George Biros and Omar Ghattas, Parallel lagrange newton krylov schur methods for pde- constrained optimization. part i: The krylov schur solver, SIAM Journal on Scientific Computing, 27 (2005), pp R.E. Kleinman and P.M.van den Berg, A modified gradient method for two- dimensional problems in tomography, Journal of Computational and Applied Mathematics, 42 (992), pp

37 References 0. Ake Bjork, Numerical methods for least squares problems. siam, Walter Gander, Least squares with a quadratic constraint, Numerische Mathematik, 36 (980), pp