A new approximation for the full waveform inversion (FWI) that does not require any forward modeling to estimate the gradient, making the routine cheaper, faster and with competitive resolution. It also opens the possibility for a post-stack approximation. Project presented at the GeoConvention 2017 in Calgary, AB, Canada: http://www.geoconvention.com/uploads/2017abstracts/039_GC2017_FWI_without_tears.pdf

1. www.crewes.org
Supervisors:
Dr. Larry Lines
Dr. Robert Ferguson
Presenter:
Marcelo Guarido de Andrade

2. Outline
• Introduction
• Gradient approximation
•Impedance inversion
•Forward modeling-free (pre-stack)
•Forward modeling-free (post-stack)
• All results
• Conclusions
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3. Introduction
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4. Introduction
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Least Squares

5. Introduction
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C 𝑚 ≡ ∆𝑑 𝑚
2Minimize
Least Squares

6. Introduction
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C 𝑚 ≡ ∆𝑑 𝑚
2Minimize
Least Squares
𝒅 𝒐𝒃𝒔 − 𝒅 𝒔𝒚𝒏

7. Introduction
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C 𝑚 ≡ ∆𝑑 𝑚
2Minimize
Least Squares
𝒅 𝒐𝒃𝒔 − 𝒅 𝒔𝒚𝒏

8. m0

9. m0

10. m0
∆dm

11. m0
∆dm
g0

12. m0
∆dm
g0
α0

13. m0
∆dm
g0
m1
α0

14. m0
∆dm
g0
m1
α0

15. 𝒎(𝒏+𝟏) = 𝒎(𝒏) − 𝜶(𝒏) 𝒈(𝒏)
m0
∆dm
g0
m1
α0

16. Introduction
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The final update is assumed to be the closest one to the true model

17. Introduction
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Velocity(m/s)
Velocity(m/s)
The final update is assumed to be the closest one to the true model

18. Gradient Approximation:
Impedance Inversion
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19. Gradient Approximation: Impedance Inversion
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• Gradient -> reflection coefficients or impedance?
• Impedance inversion
• FWI as processing tools
• I – impedance inversion
• S – stacking
• M – migration

20. Gradient Approximation: Impedance Inversion
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• Pratt et al. (1990):
• Step length:
Forward modeling over a small
perturbation of the model

21. Gradient Approximation: Impedance Inversion
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22. Gradient Approximation: Impedance Inversion
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23. Gradient Approximation: Impedance Inversion
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24. Gradient Approximation: Impedance Inversion
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25. Gradient Approximation:
Forward Modeling-Free (Pre-Stack)
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26. Gradient Approximation: Forward Modeling-Free (Pre-Stack)
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• Linear operators
Current Model
And:
Step length requires 2 forward modeling
No forward modeling is required
to estimate the gradient
Separate the terms

27. Gradient Approximation: Forward Modeling-Free (Pre-Stack)
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28. Gradient Approximation: Forward Modeling-Free (Pre-Stack)
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29. Gradient Approximation: Forward Modeling-Free (Pre-Stack)
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30. Gradient Approximation: Forward Modeling-Free (Pre-Stack)
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31. Gradient Approximation:
Forward Modeling-Free (Post-Stack)
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32. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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• Seismic processing workflow
• Commute the order of the Migration and Stacking processes

33. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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• Seismic processing workflow
• Commute the order of the Migration and Stacking processes

34. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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• Seismic processing workflow
• Commute the order of the Migration and Stacking processes
Theoretically, same thing
Pre-stack migration
Post-stack migration

35. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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Input: Stacked observed data

36. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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37. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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38. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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39. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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40. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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41. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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Simple exponential of trace integration as impedance inversion

42. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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43. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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44. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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Velocity analysis output as initial model

45. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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46. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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47. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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2nd pass of the FWI processing tool

48. Gradient Approximation: Forward Modeling-Free (Post-Stack)
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• Improves initial model
• Better on the shallow
• 2nd pass -> good choice
• Fast
• Simple
• Cheap

49. All Results
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50. All Results
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Standard approximation with BLIMP

51. All Results
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Pre-stack approximation

52. All Results
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Post-stack approximation

53. All Results
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Simple exponential of trace integration as impedance inversion

54. Conclusions
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55. Conclusions
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• Impedance inversion of the gradient is reasonable
• Forward modeling-free gradient is good enough and stable
• No source estimation for gradient
• Two forward modeling runs is required to estimate step length
• Possibility for a post-stack approximation
• Trace integration has higher resolution on simpler models
• Requires reasonable good initial model
• Promising results

56. Acknowledgments
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• NSERC
• CREWES Sponsors
• Students and staff