Inverse scattering terms for laterally-varying media Fang Liu, Bogdan Nita,

1. Inverse scattering terms for laterally-varying media Fang Liu, Bogdan Nita, Arthur Weglein, Kristopher Innanen M-OSRP Annual Meeting, University of Houston March 31 – April 1, 2004

2. Acknowledgments M-OSRP sponsors and members ConocoPhillips for financial support Einar Otnes of Statoil Simon Shaw Haiyan Zhang

3. Motivations • Imaging beneath sub-salt, sub-basalt and karsted sediments: a problem of high priority. • Positive depth imaging results from Shaw and Weglein (2004) encourage extension to evaluate ability to convert for vertical and lateral mis-location.

4. Questions: • Can the inverse series locate when v(x,z) produces lateral and vertical errors due to approximate velocity? • Can the concept of purposeful perturbation be extended to multi-D? • Is the “separating of tasks into imaging and inversion” logic extendable to multi-D?

5. How it works: By non-linear conversations between events, allows separate primary events to communicate and cooperate towards imaging and inversion tasks.

6. Equations to be solved:

7. Computational issues: • 4 nested integrals on the right-hand-side. If you want to invert to (x,z), another 2 nested integrals will be needed. Significant amount of computation. • What’s the best way of computing? • Direct computing is not the best choice even in 1D world. • Actually only 3 integrals are needed. For some portions, only 1 integral is needed. Appendix to the report shows the detail.

8. Progress: • Algorithm developed. • Terms in the algorithm corresponding to tasks reduced to the special case of 1D earth where the interpretation has analytic argument. • Numerical tests successful, purposeful perturbation extendable! Key features: • Constant background, one parameter (velocity). • Worked in the (km, z) domain, kept kmun-transformed. • Pre-calculated the kz inverse Fourier transform. • Currently using the parameterization kh=0.

9. Inversion results: • γ can be pre-calculated, big savings in computation. • γ and its derivation are included in the reports. • Physics behind the formula : (1) Talks happen between 2 different km components. (2) In and out logic.

10. Interpretations: 1D-1D: 1D-2D: 2D-2D:

11. Numerical results: Wavelet : First derivative of Gaussian, missing zero and high frequencies, discretely sampled data with finite aperture. Geological model: 300 m 400 m 200 m

12. Objectives: • Geological model is intended to be perfectly horizontal to the left and right, the first reflector varies smoothly in the middle. The second reflector is perfectly horizontal. • FK migration with water speed will migrate the first reflector to its correct locations. But the second reflector will be mis-located. The right half will be mis-placed more because of thicker high-velocity zone. • Can we move the second reflector in the migrated section towards its actual location? Does it migrate the left and right differently? At the same time, we don’t want to move the first reflector. • In the central part of the second reflector, there will be a false slope, can we somehow mitigate this slope?

13. A typical shot-gather Modeling algorithm : finite difference

14. α1 Stolt migration

15. Nothing to move the first reflector. Nothing to move the first reflector. Move the second reflector. Move the second reflector.

17. α1 Stolt migration

19. α1 Stolt migration

20. It is restricted inside α1

22. Right Left Plots in the next page are taken from the locations above

23. Left Amplitude Right Depth

24. Purposeful perturbation • Move reflectors only when needed • More error  algorithm corrects more • Less error  algorithm corrects less Only the second reflector is moved Inversion part of α2 is at the same location as α1 Right side perturb more Left side perturb less All the interpretation in 1D is retained! Purposeful perturbation is defined as knowledge of precisely what each term within a given task-specific sub-series is designed to accomplish.

25. Central part – maximal lateral variation Horizon picked from α1 Horizon picked from α1 +α2 Actual Horizon location Move the reflector in a reasonable way in the lateral varying portion. Extend the idea of purposeful perturbation to multi-D

26. Questions and answers: • Can the inverse series locate when v(x,z) produces lateral and vertical errors due to approximate velocity? Yes! • Can our concept of purposeful perturbation be extended to multi-D? At least the portion we are aware of. Nothing unexpected. • Is the “separating of tasks into imaging and inversion” logic extendable to multi-D? Yes!

27. Conclusions • Theory developed • Numerical tests successful • Example confirmed the potential of inverse scattering series to do imaging and inversion without sub-surface information in 2-D. Future plan • Other terms in α2, α3… • More complicated model. • Other parts of data : • Field data tests.