1. paul@sep.stanford.edu Wave-equation migration velocity analysis beyond the Born approximation Paul Sava* Stanford University Sergey Fomel UT Austin (UC Berkeley)

2. paul@sep.stanford.edu Imaging=MVA+Migration Migration wavefield based Migration velocity analysis (MVA) traveltime based Compatible migration and MVA methods

3. paul@sep.stanford.edu Imaging: the “big picture” Kirchhoff migration traveltime tomography wavefronts wave-equation migration wave-equation MVA (WEMVA) wavefields

4. paul@sep.stanford.edu Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

5. paul@sep.stanford.edu Wavefields or traveltimes?

6. paul@sep.stanford.edu Wavefields or traveltimes?

7. paul@sep.stanford.edu Scattered wavefield Medium perturbation Wavefield perturbation

8. paul@sep.stanford.edu Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

9. paul@sep.stanford.edu Imaging: Correct velocity Background velocity Migrated image Reflectivity model What the data tell us...What migration does... location depth location depth

10. paul@sep.stanford.edu Imaging: Incorrect velocity Perturbed velocity Migrated image Reflectivity model What the data tell us...What migration does... location depth location depth

11. paul@sep.stanford.edu Wave-equation MVA: Objective Velocity perturbation Image perturbation slowness perturbation (unknown) WEMVA operator image perturbation (known) location depth location depth

12. paul@sep.stanford.edu –migrated images –moveout and focusing –use amplitudes –parabolic wave equation –multipathing –slow –picked traveltimes –moveout –ignore amplitudes –eikonal equation –fast Comparison of MVA methods Wave-equation MVATraveltime tomography

13. paul@sep.stanford.edu Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

14. paul@sep.stanford.edu What is the image perturbation? FocusingFlatness Residual process: moveout migration focusing slowness perturbation (unknown) WEMVA operator image perturbation (known) location depth angle

15. paul@sep.stanford.edu Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

16. paul@sep.stanford.edu Double Square-Root Equation Fourier Finite Difference Generalized Screen Propagator Wavefield extrapolation

17. paul@sep.stanford.edu “Wave-equation” migration

18. paul@sep.stanford.edu Slowness perturbation

19. paul@sep.stanford.edu slowness perturbation background wavefield perturbation Wavefield perturbation

20. paul@sep.stanford.edu Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

21. paul@sep.stanford.edu Born approximation Small perturbations! Born linearization Non-linear WEMVA slowness perturbation (unknown) WEMVA operator image perturbation (known) Unit circle

22. paul@sep.stanford.edu Does it work? What if the perturbations are not small? Location [km] Depth [km]

23. paul@sep.stanford.edu Synthetic example

24. paul@sep.stanford.edu Born approximation 1%10%

25. paul@sep.stanford.edu Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

26. paul@sep.stanford.edu Wavefield continuation Bilinear Implicit Explicit(Born approximation)

27. paul@sep.stanford.edu Exponential approximations Unit circle

28. paul@sep.stanford.edu A family of linearizations Linear WEMVA slowness perturbation (unknown) WEMVA operator image perturbation (known)

29. paul@sep.stanford.edu Improved linearizations 1%10% 40%

30. paul@sep.stanford.edu Agenda Theoretical background WEMVA methodology Scattering Imaging Image perturbations Wavefield extrapolation Born linearization Alternative linearizations

31. paul@sep.stanford.edu Summary Wave-equation MVA wavefield-continuation improved focusing image space (improve the image) interpretation guided Improved WEMVA better approximations no additional cost further refinement