1. Haiyan Zhang and Arthur B. Weglein
Direct non-linear inversion of multi-parameter 1D elastic media using the inverse scattering series
Haiyan Zhang and Arthur B. Weglein
M-OSRP Annual Meeting
University of Houston
May 10 –12, 2006

2. Outline Background and Motivation Theory: Inverse Scattering Series
Application on three parameter 1D elastic media
Inversion in PS space
An approach with PP data only and its numerical tests
Full non-linear elastic inversion with four components of data
Conclusions and plan
Acknowledgements

3. Background and motivation
Change in
earth properties
Non-linear
Wave response
Current Inversion methods:
Linear inversion or Born approximation
Model matching
Direct non-linear inversion method
Towards fundamentally new comprehensive and realistic target identification.

4. Derivation of the inverse series
In actual medium:
In reference medium:
Perturbation:
L-S equation:
Forward scattering Series:
Inverse scattering series:

5. Inverse Scattering Series
Linear
Non-linear

6. Three parameter elastic inversion
References:
Weglein and Stolt (1992) : introduced an elastic L-S equation and provided a specific linear inverse formalism for parameter estimation.
Matson (1997): pioneered the development and application of methods for attenuating ocean bottom and on-shore multiples.

7. Three parameter elastic inversion
In PS space
In actual medium:
In reference medium:
Perturbation:
L-S equation:
Linear inversion:
First term beyond linear: .

8. Elastic inversion: linear

9. Elastic inversion: linear
Then, in domain, we get

10. Elastic inversion: linear
Where
For 1D earth, we have

11. Elastic inversion: non-linear

12. Three parameter elastic inversion
When the acoustic work is extended to elastic case, more issues would be involved.
For example:
Data requirements: four components of data.
One more velocity, four mislocations (3 for 1D earth).

13. Three parameter elastic inversion
For towed streamer case, only PP data is available.
Can we perform non-linear elastic inversion only using PP data?
A particular non-linear approach--- only use PP data as input, to do the non-linear elastic inversion.

14. Elastic inversion: linear
Then, in domain, we get

15. Elastic inversion: non-linear

16. Non-linear elastic inversion: PP data only

17. Numerical test: PP data only
One interface model x a z

18. Numerical tests: PP data onlyx
One interface, a z
Model 1: shale (0.2 porosity) over oil sand (0.1 porosity)
Model 2: shale (0.2 porosity) over oil sand (0.2 porosity)
Model 3: shale (0.2 porosity) over oil sand (0.3 porosity)
Model 4: oil sand (0.2 porosity) over wet sand (0.2 porosity)

19. Summary of the numerical tests: PP data only
Before correction:
For one of the parameter (shear modulus) in model 3, second order approximation almost no difference, even a little bit worse than linear approximation.
After correction:
For all of the parameters of the four models, second order approximation provides improvements beyond the linear approximation.

20. Numerical tests: PP data onlyx
Model 2
shale (0.20 porosity) a
oil sand (0.20 porosity) z

21. Inversion of density
Linear+nonlinear
Linear

22. Inversion of P-bulk modulus
Linear
Linear+nonlinear

23. Inversion of Shear modulus
Linear
Linear+nonlinear

24. Numerical tests: PP data onlyx
Model 4
oil sand (0.20 porosity) a
wet sand (0.20 porosity) z

25. Inversion of density
Linear+nonlinear
Linear

26. Inversion of P-bulk modulus
Linear
Linear+nonlinear

27. Inversion of Shear modulus
Linear
Linear+nonlinear

28. Non-linear elastic inversion: all four components of data

29. Elastic inversion: linear
Then, in domain, we get

30. Non-linear elastic inversion: all four components of data
in the linear solutions equal, we have
Given , we can get the following

31. Non-linear elastic inversion: all four components of data

32. Non-linear elastic inversion: all four components of data
Through the similar way, we can get the solution for

33. Non-linear elastic inversion: all four components of data
Based on this idea, the solution for the first equation is

34. Non-linear elastic inversion: all four components of data
The solution for the second equation is

35. Non-linear elastic inversion: all four components of data
Continued

36. Non-linear elastic inversion: all four components of data
Continued

37. Non-linear elastic inversion: all four components of data
Continued
The end

38. Non-linear elastic inversion: all four components of data
The solutions for the third and fourth equations
Please see the annual report.

39. Three parameter elastic inversion
Inversion for 1D elastic earth with three parameters.
All four components of data need to be known in order to do non-linear elastic inversion.
A particular non-linear approach--- only use PP data as input, is performed and encouraging results are obtained.
A consistent method is provided to do full direct non-linear inversion using all four components of data.

40. Conclusions
A framework and algorithm for more accurate target identification have been developed.
The elastic non-linear inversion requires all four components of data.
Elastic non-linear inversion showed benefit in all the cases with PP data only.
How accurately PP data effectively synthesize PS, SP and SS data determined the degree of benefit provided by the non-linear elastic inversion.
We anticipate collecting PP, PS, SP and SS would provide further benefit. We will test this hypothesis this summer.

41. Acknowledgements
We are grateful to Robert Keys, Douglas Foster and Simon Shaw for useful comments and suggestions.
The M-OSRP sponsors are thanked for supporting this research.