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. Point scatterer Interferometry Optical theorem Non-linear Paradox

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

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. a b Term 1:

8. c d Term 2:

9. ef Term 3:

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

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

12. Term 4: g g h h i i

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

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

15. Point scatterer Interferometry Optical theorem Paradox

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. Point scatterer Interferometry Optical theorem Non-linear Paradox

21. Isotropic point scatterer:

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

24. Point scatterer Interferometry Optical theorem Non-linear Paradox

25. Point scatterer Interferometry Optical theorem Non-linear Paradox

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

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

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

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. Born approximation is incompatible with seismic interferometry Conclusions

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

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