1. 26 April 2002 Velocity estimation by inversion of Focusing operators: About resolution dependent parameterization and the use of the LSQR method Barbara Cox IMA Workshop: Inverse Problems and Quantification of Uncertainty

2. slide 2 26 April 2002 Outline Inversion of Focusing Operators Regularization of inversion Resolution dependent Parameterization Optimization by LSQR Synthetic example CFP method

3. slide 3 26 April 2002 Outline Inversion of Focusing Operators Regularization of inversion Resolution dependent Parameterization Optimization by LSQR Synthetic example

4. slide 4 26 April 2002 Inversion of Focusing Operators Data: one-way travel times Unknowns: slowness & exact focus point location Obtained by minimizing: Distance (km) 0481216 0 2 4 Depth (km) Distance (km) 0481216 0 2 3 Time (s) 1

5. slide 5 26 April 2002 Forward modeling by raytracing ( t i ) Optimization Fit? Y Focusing operators (data) NN Initial macro model ( s j & x p,z p ) Final macro model ( s j & x p,z p ) Inversion of Focusing Operators Distance Depth x p,z p s j=1 s j=2 s j=M

6. slide 6 26 April 2002 Forward modeling by raytracing ( t i ) Optimization Fit? Y Focusing operators (data) NN Initial macro model ( s j & x p,z p ) Final macro model ( s j & x p,z p ) Inversion of Focusing Operators Distance Depth s j=1 s j=2 s j=M x p,z p t i=1 t i=2 t i=N

7. slide 7 26 April 2002 Forward modeling by raytracing ( t i ) Optimization Fit? Y Focusing operators (data) NN Initial macro model ( s j & x p,z p ) Final macro model ( s j & x p,z p ) Inversion of Focusing Operators Solve iteratively by e.g. SVD: Assume linear relation :

8. slide 8 26 April 2002 Forward modeling by raytracing ( t i ) Optimization Fit? Y Focusing operators (data) NN Initial macro model ( s j & x p,z p ) Final macro model ( s j & x p,z p ) Inversion of Focusing Operators

9. slide 9 26 April 2002 Outline Inversion of Focusing Operators Regularization of inversion Resolution dependent Parameterization Optimization by LSQR Synthetic example

10. slide 10 26 April 2002 Regularization Parameterization: Regularizing by coarser (global) parameterization Optimization: Regularizing by e.g. resolution matrix Tomographic inverse problems are generally mixed determined Can be faced by regularization:

11. slide 11 26 April 2002 Regularization: parameterization Forward modeling by raytracing ( t i ) Optimization Fit? Y Focusing operators (data) NN Initial macro model ( s j & x p,z p ) Final macro model ( s j & x p,z p ) v1v1 v2v2 v3v3 vmvm z1z1 zkzk z2z2 v1v1 v2v2 v3v3 v4v4 Local: Global:

12. slide 12 26 April 2002 Regularization: optimization Forward modeling by raytracing ( t i ) Optimization Fit? Y Focusing operators (data) NN Initial macro model ( s j & x p,z p ) Final macro model ( s j & x p,z p ) Regularization:

13. slide 13 26 April 2002 Regularization Parameterization: Regularizing by coarser (global) parameterization Optimization: Regularizing by e.g resolution matrix Constraints resultStill over-parameterized Combine: Parameterization dependent on resolution No constraint on result, No over-parameterization

14. slide 14 26 April 2002 Outline Inversion of Focusing Operators Regularization of inversion Resolution dependent Parameterization Optimization by LSQR Synthetic example

15. slide 15 26 April 2002 N Forward modeling by ray-tracing ( t i ) Optimization Fit? Y Focusing operators (data) Final macro model ( s j & x p,z p ) N Adjustment of parameterization Calculation of resolution Initial macro model ( s j & x p,z p ) Resolution dependent Parameterization

16. slide 16 26 April 2002 Resolution dependent Parameterization Calculation of resolution 1 Resolution 0 Distance Depth Resolution in model Distance Depth Velocity model

17. slide 17 26 April 2002 Resolution dependent Parameterization Distance Adjustment of parameterization dependent on resolution Depth Resolution in model Distance Depth Velocity model Add points Remove points 0 1 0.2 0.4 0.6 0.8 Resolution Gridpoints 1M Resolution plot

18. slide 18 26 April 2002 Consequently: No constraint on result No over-parameterization The available information within the data can be completely translated to the model Resolution dependent Parameterization

19. slide 19 26 April 2002 However, Calculation of resolution or covariance requires explicit matrix inversion Explicit matrix inversion is not feasible: Optimization by iterative method: LSQR Paige & Saunders (1982) Calculate resolution during iterative optimization Zhang and McMechan (1995) Yao et al (1999) Berryman (2001) Resolution dependent Parameterization

20. slide 20 26 April 2002 Outline Inversion of Focusing Operators Regularization of inversion Resolution dependent Parameterization Optimization by LSQR Synthetic example

21. slide 21 26 April 2002 Optimization by LSQR LSQR method: Iterative SVD approximation: k iterations k basis-vectors SVDLSQR Singular Value diagonal matrix Bi-diagonal matrix If k = number of parameters then LSQR=SVD Maximum number of iterations (k) = number of parameters First k LSQR basis-vectors First k SVD eigen-vectors

22. slide 22 26 April 2002 Optimization by LSQR SVD LSQR V T k=31 U k=3 1 B k=31 VTVT USA A

23. slide 23 26 April 2002 Optimization by LSQR Largest (pseudo) singular values are obtained first SVDLSQR SV diagonal matrix Pseudo SV diagonal matrix Singular values: Bi-diagonal matrix can be converted to a pseudo singular value diagonal matrix If k = number of parameters then

24. slide 24 26 April 2002 Optimization by LSQR LSQR V T k=31 U k=3 1 B k=31 A VTBVTB UBUB SBSB SVD of B Singular values:

25. slide 25 26 April 2002 Optimization by LSQR SVDLSQR k=31 B k=12 SBSB S SVD of B Singular values:

26. slide 26 26 April 2002 Optimization by LSQR SVDLSQR diag(S B )diag(S) k=12k=3k=6k=9k=15k=12k=18k=21k=24k=27k=31 Large pseudo- singular values are solved first Singular values:

27. slide 27 26 April 2002 Calculation of resolution by LSQR SVDLSQR Calculation of resolution by means of model space matrix If k = nr of parameters (over-determined system)Calculation of covariance by means of space matrix and singular value matrix

28. slide 28 26 April 2002 SVDLSQR R k=31k=12 C Calculation of resolution by LSQR RkRk CkCk

29. slide 29 26 April 2002 SVDLSQRk=12k=31 Calculation of resolution by LSQR k=3k=6k=9k=12k=15k=18k=21k=24k=27k=31 diag(R k ) diag(C k ) diag(C) diag(R)

30. slide 30 26 April 2002 Calculation of resolution by LSQR R C The way the covariance evolves during the iterations cannot be trusted, as some parameters are not solved by the current basis-vectors Final covariance is the real covariance of the system is an indication that all parameters ARE solved, but not how WELL they are solved The way the resolution evolves during the iterations is an indication how WELL the parameters are solved Maximum iterations (=SVD)Limited number of iterations

31. slide 31 26 April 2002 LSQR Calculation of resolution by LSQR k=3 k=6k=9k=12k=15k=18k=21k=24k=27k=31 SVD diag(R k ) diag(C k ) diag(C) diag(R) Low resolution AND low covariance indicate points that are not solved yet Can be used to describe the quality of the solution quantitatively

32. slide 32 26 April 2002 Optimization by LSQR The relative resolution can be used as a criterion for adjustment of parameterization The pseudo singular values can be used to evaluate how well the system is determined The comparison between resolution and covariance can be used to evaluate which parameters are described Use of LSQR for resolution dependent parameterization: Quantitative criteria Qualitative criterion REMARK: Singular value decomposition of covariance matrix (Delphine Sinoquet) can be placed on top of this method: not expensive anymore However, don’t use covariance but resolution matrix

33. slide 33 26 April 2002 Outline Inversion of Focusing Operators Regularization of inversion Resolution dependent Parameterization Optimization by LSQR Synthetic example

34. slide 34 26 April 2002 Synthetic Example Ideal model Distance (km) 016 0 4 Depth (km) Initial model 0 2 Time (s) Focusing operators Distance (km) 016 Distance (km) 016 Modeled Foc. oper. Distance (km) 016 3500 1500 Velocity (m/s) 3500 1500 Velocity (m/s) 0 2 Time (s) 0 4 Depth (km)

35. slide 35 26 April 2002 0 0 4 16Distance Depth Velocity Resolution Resolution dependent parameterization

36. slide 36 26 April 2002 Result Ideal model Distance (km) 016 0 4 Depth (km) Data driven model Distance (km) 016 3500 1500 Velocity (m/s) 0 4 Depth (km) Distance (km) 016 0 4 Depth (km) Data driven model 3500 1500 Velocity (m/s)

37. slide 37 26 April 2002 Migration Distance (km) 016 Updated model 3500 1500 Velocity (m/s) 0 4 Depth (km) Ideal model Postupdating

38. slide 38 26 April 2002 Conclusions Resolution dependent parameterization: efficient, data dependent, minimal user interaction Resolution can be obtained in an efficient way in the LSQR algorithm Regularization of the inverse problem by means of resolution dependent parameterization The optimization can be evaluated by the LSQR algorithm, using the resolution, the ‘pseudo’ singular values and the comparison between resolution and covariance

39. slide 39 26 April 2002 Acknowledgements I would like to thank: The people of the CWP project for their Delaunay and ray-tracing software, which formed a base for the developed algorithm The sponsors of the Delphi Imaging and Characterization consortium for their support