1. Haiyan Zhang and Arthur B. Weglein
Target identification using the inverse scattering series: data requirements for the direct inversion of large-contrast, inhomogeneous elastic media
Haiyan Zhang and Arthur B. Weglein
M-OSRP Annual Meeting,
University of Houston
March 31 – April 1, 2004

2. Outline Motivation and objectives Strategy Assumptions
Briefly review of previous year’s highlight
Initial results about three parameter 2D elastic inversion
Data requirements
Computation and interpretation issues
Conclusions
Plan and acknowledgments

3. Motivation and objectives
Inversion for earth properties plays an important role in seismic exploration.
Conventional inversion typically uses linear inversion of data, D.
Objective: to provide a direct method for accurate and reliable target identification especially with large contrast, large angle target geometry.

4. Strategy Inversion of seismic data can be viewed as a series of tasks:
We isolate the inverse subseries responsible for non-linear amplitude inversion of data.
Inversion of seismic data can be viewed as a series of tasks:
Removal of free-surface multiples;
Removal of internal multiples;
Location of reflectors in space;
Target identification (parameter estimation).

5. Assumptions
All multiples have been removed from the input data (only primary reflections).
The targets have already been located in correct position.
The information of the reference medium is given.

6. Previous year’s highlight
Last year, we started with 1D acoustic multiparameter earth model (e.g. bulk modulus and density or velocity and density).
Begin with the 3D differential equations:
(1)
(2)

7. Previous year’s highlight
Then
(3)
Where

8. Previous year’s highlight
(4)
We define the data D as the measured values of the scattered wave field. Then, on the measurement surface, we have
(5)
Expand V as a series in orders of D
(6)

9. Previous year’s highlight
Substituting Eq. (6) into Eq. (5), and setting terms of equal order in the data equal, we get the equations that determine V1 ,V2 … from D and G0.

10. Previous year’s highlight For 1D acoustic earth model
(7)
(8)
Solution for first order (linear)
(9)

11. Previous year’s highlight For one interface, 1D acoustic earth model
(10)
(11)
“Linear migration-inversion”
(12)
(13)

12. (14)
(15)
Solution for second order (first term beyond linear)
(16)

13. Previous year’s highlight
1. The first 2 parameter direct non-linear inversion of 1D acoustic medium for a 2D experiment is obtained.

14. Previous year’s highlight
2. Tasks for the imaging-only and inversion-only within the series are isolated.

15. Previous year’s highlight
3. Purposeful perturbation.
When c0=c1

16. Previous year’s highlight
3. Purposeful perturbation.

17. 2D elastic inversion This year’s objective: References:
Direct non-linear inversion of 1D isotropic and inhomogeneous three parameter elastic medium for a 2D experiment is pursued.
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 multi component data.

18. 2D elastic inversion In displacement space In PS space
Elastic wave equation; perturbation; L-S equation
In PS space
Inversion in PS space

19. 2D elastic inversion In displacement space In actual medium:
In reference medium:
Perturbation:
L-S equation:
Linear inversion:
First term beyond linear: .
Notes: Operators without hats in the following talk are in the displacement space, those with hats are in PS space.

20. 2D elastic inversion In displacement space
(A.B. Weglein and R.H. Stolt, 1992.)
(17)
(18)

21. 2D elastic inversion In displacement space
(19)
Then, for 1D earth, we have,
(20)

22. 2D elastic inversion In PS space
For convenience, we change the basis from to to allow to be diagonal,
Also
Where

23. 2D elastic inversion In PS space
The operator will transform in the new basis via a transformation
Where
And

24. 2D elastic inversion In PS space In the reference medium, both
and are diagonal.

25. 2D elastic inversion In PS space In actual medium:
In reference medium:
Perturbation:
L-S equation:
Linear inversion:
First term beyond linear: .

26. On the measurement surface, we have
2D elastic inversion
In PS space
On the measurement surface, we have
Then,

27. 2D elastic inversion In PS space
For homogeneous media, no perturbation, then, there are only direct P and S waves. And they are separated. For inhomogeneous media, P and S waves will be coupled together.

28. If only P wave is incident, If only S wave is incident,
2D elastic inversion
In PS space
If only P wave is incident,
If only S wave is incident,

29. 2D elastic inversion: linear
In PS space
(21)
(22)
(23)
(24)

30. 2D elastic inversion: linear
In PS space
Where
(25)

31. 2D elastic inversion: linear
In PS space
Then, in domain, we get
Where

32. 2D elastic inversion: non-linear
In PS space
(26)

33. 2D elastic inversion: non-linear
In PS space
Since, e.g., is closely related to , direct non-linear inversion requires all components of Data.
Details of this calculation indicate that while computable this parameter choice is not favorable for interpretation and subsequent task separation.

34. 2D elastic inversion: non-linear
In PS space
The issue of how inverse series can be interpreted in terms of tasks is not new, not confined to tasks associated with primaries, and not an issue that begins with the elastic equation. We illustrate this by taking another look at the acoustic problem.

35. Interpretation in inversion
What parameters to use?
Material parameters
Free parameters

36. Interpretation in inversion
If a and b are chosen as the two material parameters,
(27)

37. Interpretation in inversion
If and are chosen as the two material parameters,
(28)

38. Interpretation in inversion
Where in domain, i.e., before the Fourier
transform over , is

39. Interpretation in inversion
The parameters that we have chosen for the elastic non-linear inversion are the generalizations of the for the acoustic case, computable but not amenable to easy interpretation and subsequent task separation. We are examining the framework to allow an interpretable elastic generalization.

40. Interpretation in inversion
Whatever the choice and convenience of parameters and free variable the requirement
for to perform direct non-linear inversion for the simplest 1D elastic interface is inescapable.

41. Interpretation in inversion
Provides a framework for that level of ambition and allows time to strategize to allow the inverse series to provide the R’s free of T’s required for this non-linear direct inversion at depth.

42. Conclusion
This talk provides a conceptual platform and analysis of issues involved in data requirements, computation and interpretation of the non-linear direct elastic inverse problem.
A detailed examination of how different choices of acoustic parameters and free parameters have a marked difference on the ability of task separated interpretation.
This analysis provides a guide and lesson for ongoing efforts at parameter inversion and structural location specific sub series for the elastic world.

43. Plan
To choose parameters for the elastic non-linear inversion problem, that are most agreeable to physical interpretation in terms of imaging and inversion tasks.

44. Acknowledgments
The M-OSRP sponsors are thanked for supporting this research.
We are grateful to Robert Keys and Douglas Foster for useful comments and suggestions.