Seismic interferometry, the optical theorem and a non-linear point scatterer Kees Wapenaar

1. Seismic interferometry, the optical theorem and a non-linear point scatterer Kees Wapenaar Evert Slob Roel Snieder Society of Exploration Geophysicists Houston, October 26, 2009

2. Interferometry Non-linear Paradox Point scatterer Optical theorem

3. Interferometry Modeling Inversion Interferometry Migration Non-linear Paradox Point scatterer Optical theorem

4. Snieder, R., K.van Wijk, M.Haney, and R.Calvert, 2008, Cancellation of spurious arrivals in Green's function extraction and the generalized optical theorem: Physical Review E, 78, 036606. Halliday, D. and A.Curtis, 2009, Generalized optical theorem for surface waves and layered media: Physical Review E, 79, 056603. van Rossum, M. C. W. and T.M. Nieuwenhuizen, 1999, Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion: Reviews of Modern Physics, 71, 313--371.

7. Term 1: a b

8. Term 2: c d

9. Term 3: f e

10. Terms 1 + 2 + 3: c a f e b d

11. Terms 1 + 2 + 3, compared with modeled G:

12. h i h i Term 4: g g

13. Terms 1 + 2 + 3 + 4, compared with modeled G:

14. Terms 1 + 2 + 3, compared with modeled G:

15. Interferometry Paradox Point scatterer Optical theorem

16. Substitute into representation for interferometry (Snieder et al., 2008, Halliday and Curtis, 2009)…..

17. This gives: Generalized optical theorem (Heisenberg, 1943)

18. This gives: For comparison:

19. Interferometry Non-linear Paradox Point scatterer Optical theorem

21. Isotropic point scatterer:

22. Isotropic point scatterer:

23. (van Rossum et al, 1999) = + + + (Snieder, 1999)

24. Interferometry Non-linear Paradox Point scatterer Optical theorem

25. Interferometry Non-linear Paradox Point scatterer Optical theorem

26. Terms 1 + 2 + 3: c a f e b d

27. Terms 1 + 2 + 3 + 4, compared with modeled G:

28. Interferometry Modeling Inversion Interferometry Migration Non-linear Paradox Point scatterer Optical theorem

29. Modeling, inversion and interferometry in scatterering media Groenenboom and Snieder, 1995; Weglein et al., 2003; Van Manen et al., 2006

30. Modeling, inversion and interferometry in scatterering media Groenenboom and Snieder, 1995; Weglein et al., 2003; Van Manen et al., 2006 Limiting case: Point scatterer

31. Resolution function for seismic migration Miller et al., 1987; Schuster and Hu, 2000; Gelius et al., 2002; Lecomte, 2008 Migration deconvolution Yu, Hu, Schuster and Estill, 2006

32. Conclusions • Born approximation is incompatible with seismic interferometry

33. Conclusions • Born approximation is incompatible with seismic interferometry • Seismic interferometry optical theorem non-linear scatterer seismic interferometry • Consequences for modeling, inversion, interferometry and migration

34. Conclusions • Born approximation is incompatible with seismic interferometry • Seismic interferometry optical theorem non-linear scatterer seismic interferometry • Consequences for modeling, inversion, interferometry and migration