1. Including headwaves in imaging and internal multiple attenuation theory Bogdan G. Nita Research Assistant Professor, Dept. of Physics University of Houston March 2004

2. Plan for this talk Brief overview of forward and inverse scattering theory Internal multiple attenuation algorithm – sub-events Headwaves Imaging with vs. Analytic example of imaging headwaves with

3. Scattering Theory Scattering theory is perturbation theory Relates differences in media to differences in wavefield

4. Scattering Theory (cont’d.) Inverse Series, V as power series in data, (  s ) m = D (2) (1) Substitute (2) into (1) and evaluate on the measurement surface, m

5. (1) Remove free-surface multiples (2) Remove internal multiples (3) Image primaries to correct spatial location (4) Invert for local earth properties Direct inversion as a series of tasks Inverse scattering series: Input:data and reference medium Output:map of earth mechanical properties and density

6. Objectives Extend the algorithms to accommodate large offset reflections, headwaves, turning waves. Expand and deepen the analysis of the mapping between seismic events described by non- perturbative methods and by the forward scattering theory. Increase the understanding of the present inverse scattering subseries methods in the presence of this type of data and seek to improve their effectiveness.

7. Past and present work Investigate the Forward Scattering Series description of headwaves –how does the series construct these events –what is the Feynman diagram representation for them –how does the scattering series deal with branch-cut singularities Examine the Inverse Scattering Series attitude towards headwaves –are they information bearing or not, –what kind of information they contain –what is the information used for (imaging, inversion). Study the present internal multiple attenuation algorithm to understand if/how the refracted multiples are processed.

8. Links with other projects Plan: Internal multiples (Innanen K) –3D –From attenuation to elimination Imaging (Shaw S) Inversion (Zhang H)

9. A multiple for which headwaves are sub-events Free surface refracted multiples (presently removed by FS demultiple algorithms as shown by Bill Dragoset on Nov. 14 Multiples Forum) Internal refracted multiples Definition of a refracted multiple

10. Internal multiple attenuation algorithm See Weglein et al (2003) Inverse Problems Topical Review pp. R61

11. Multiples are predicted by the time convolution of sub-events Time of=time of + - 2 longer time events 1 shorter time event

12. Imaging conditions in the internal multiple attenuation algorithm How is obtained? How does this imaging condition handles headwaves in the recorded data?

13. Headwaves - features Ray diagram Plane-waves do not create headwaves Impossible event

14. Wave front diagrams showing the development of the headwave

15. Internal refracted multiples Possible arrivals with headwaves as sub-events – free surface and internal refracted multiples Picture shows internal refracted multiples

16. Phase-shift migration / inversion Plane-wave decomposition Downward continuation Imaging

17. F-K constant velocity phase-shift migration Plane-wave decomposition – Fourier Transform Downward continuation – Phase-shift Imaging –

18. phase-shift migration Plane-wave decomposition – Slant-stack Downward continuation – Phase-shift Imaging – Clayton & McMechan Geophysics, 1981

19. The relationship between and__ Total travel-time is decomposed into a horizontal and a vertical time

20. Discussion Note For reflections since For headwaves there is no relationship between and

21. Events in and__

22. Analytic example: post-critical data The post critical data in a 3D acoustic experiment is is the conjugate of

23. Analytic example: downward continuation The downward continued data at any depth is

24. Analytic example: imaging the headwave The image is a delta like event at the correct depth

25. Conclusions Examine / extend the algorithms to accommodate large offset reflections, headwaves, turning waves. Imaging with places the headwaves at the correct depth Benefits: internal multiple algorithm, imaging, inversion