1. Haiyan Zhang and Arthur B. Weglein
The inverse scattering series for tasks associated with primaries: Depth imaging and direct non-linear inversion of 1D variable velocity and density acoustic media
Haiyan Zhang and Arthur B. Weglein
M-OSRP annual meeting
University of Houston
April 20 –21, 2005

2. Marine Seismic Exploration Geometry
Air
Free surface (F.S.)
Water
Earth

3. Objectives of Seismic Exploration
Where is the location of the medium changes?
(Imaging or Migration)
What are the medium changes? (Inversion or Target identification)

4. Events in the recorded data
Event categories
Did not touch the earth
(Direct wave)
Traveled through the earth
Up from source or,
Down to receiver
(Ghosts)
Down from source and up to receiver
More than one
upward reflections
(Multiples)
One upward reflection
(Primaries)
Reflected by F.S.
(F.S. multiples)
Not reflected by F.S.
(Internal multiples)

5. M-osrp projects Wavelet estimation; Deghosting;
Data interpolation and extrapolation;
Free surface multiple removal;
Internal multiple removal;
Location of reflectors in space (imaging)
Target identification (inversion/parameter estimation)

6. Stages within strategy for primaries
(1) 1D acoustic medium with one parameter (e.g., velocity);
(2) 2D acoustic medium with one parameter (e.g., velocity);
(3) 1D acoustic medium with two parameters (velocity and density), one propagation velocity, and one shot record of PP data, and
(4) 1D elastic earth with three parameters ( P and S velocity and density), two wave speeds, for P and S waves, and four (PP, PS, SP, and SS) shot records collected.

7. Target identification using the inverse scattering series:
direct non-linear inversion
Current methods:
Linear inversion or Born approximation
My method:
direct non-linear inversion

8. What’s non-linear? Why non-linear?
Intrinsic non-linear

9. Intrinsic non-linear a in terms of R R in terms of a
The 1D plane-wave normal incidence acoustic example. 1 R
R in terms of a
a in terms of R

10. Non-linear by choice Depth Imaging: Adequate information: Linear
Inadequate information:
Non-linear t

11. Outline Derivation of the inverse series
Solution for first order and second order
Numerical tests
Conclusions
Acknowledgements

12. Derivation of the inverse series
In actual medium:
In reference medium:
Perturbation:
L-S equation:
Forward scattering (Born) Series:

13. Inverse Scattering Series
Linear
Non-linear

14. Two parameter 2D acoustic inversion
1D acoustic two parameter earth model
(bulk modulus and density or velocity and density)

15. Two parameter 2D acoustic inversion
The 3D differential equations:
Then
Where

16. Two parameter 2D acoustic inversion

17. Two parameter 2D acoustic inversion For 1D acoustic earth model
Solution for first order (linear)

18. Two parameter 2D acoustic inversion
Relationship of is shown in the fig.1. z
Fig.1

19. Two parameter 2D acoustic inversion
Solution for second order (first term beyond linear)

20. Two parameter 2D acoustic inversion
1. The first 2 parameter direct non-linear inversion of 1D acoustic medium for a 2D experiment is obtained.

21. Two parameter 2D acoustic inversion
2. Tasks for the imaging-only and inversion-only within the series are isolated.

22. Two parameter 2D acoustic inversion
3. Purposeful perturbation.

23. Two parameter 2D acoustic inversion
4. Leakage.

24. One parameter 2D acoustic inversion
(velocity)
Solution for second order (first term beyond linear)

25. Two parameter 2D acoustic inversion
(velocity and density)
Solution for second order (first term beyond linear)

26. Two parameter 2D acoustic inversion
1. Leakage and special parameter

27. Two parameter 2D acoustic inversion
2. Purposeful perturbation

28. Numerical test x
One interface model a z
Fig. 2

29. Numerical test
Choose two different angles to solve for a1 and b1 and then a2 and b2 .

30. exact value of a=0.292 critical angle=61.90
θ1 ≠ θ2
(Bulk modulus)
exact value of a= critical angle=61.90

31. exact value of b=0.09, critical angle=61.90
(density)
exact value of b=0.09, critical angle=61.90

32. exact value of critical angle=61.90
(P-wave impedance)
exact value of critical angle=61.90

33. exact value of critical angle=61.90
(P-wave velocity)
exact value of critical angle=61.90

34. Two parameter 2D acoustic inversion
1D acoustic medium with two parameters (bulk modulus and density or velocity and density)
Only primaries involved
Solution for first order and second order
Numerical results

35. Conclusion
I developed a framework and algorithms for more accurate target identification.
In this task I have illustrated these concepts for a two parameter 1D acoustic medium.
Numerical tests showed that including terms beyond linear in earth property identification sub series provides added value.
In the next talk, I’ll generalize it to the elastic (solid earth problem) case using three parameters.

36. Acknowledgements
The M-OSRP sponsors are thanked for supporting this research.
We are grateful to Robert Keys (ExxonMobil) and Douglas Foster ( ConocoPhillips ) for useful comments and suggestions.