One-way acoustic reciprocity and its application in time-lapse seismic

1. One-way acoustic reciprocity and its application in time-lapse seismic By Kees Wapenaar Jacob Fokkema Menno Dillen Pieter Scherpenhuijsen Delft University of Technology Applied Earth Sciences

2. Contents • Problem statement • Review of reciprocity • Reciprocity for one-way wave fields • One-way reciprocity for time-lapse seismic • Examples • Conclusions

3. Problem statement

4. - = Monitor Difference Problem statement Reference

5. Difference Problem statement

6. Contents • Problem statement • Review of reciprocity • Reciprocity for one-way wave fields • One-way reciprocity for time-lapse seismic • Examples • Conclusions

7. Acousticreciprocity State A State B xA xA xB xB

8. Acoustic Reciprocity (continued) n V State A State B Wave fields Sources Medium PA, Vk,A QA,Fk,A kA, rA PB, Vk,B QB,Fk,B kB, rB

9. 0 0 0 0 Acoustic Reciprocity (continued) with

10. Acoustic Reciprocity (continued) with

11. Acoustic Reciprocity (continued) • Applications: • Multiple elimination: • van Borselen, Fokkema and van den Berg, Geophysics 1996 • Time-lapse seismic: • Fokkema and van den Berg, 1996

12. Contents • Problem statement • Review of reciprocity • Reciprocity for one-way wave fields • One-way reciprocity for time-lapse seismic • Examples • Conclusions

13. One-way wave fields or : is the square root operator

14. c1,r1 c2,r2 hence One-way acoustic Reciprocity(horizontally layered medium) c1,r1 c2,r2

15. One-way acoustic Reciprocity(arbitrary inhomogeneous medium) State A State B xA xA xB xB

16. Flux-normalized one-way wave fields Power flux: Note:

17. c1,r1 c2,r2 hence One-way acoustic Reciprocity(horizontally layered medium) c1,r1 c2,r2

18. One-way acoustic Reciprocity(arbitrary inhomogeneous medium) State A State B xA xA xB xB

19. One-way Acoustic Reciprocity (continued) V State A State B Wave fields Sources Medium PA, Vk,A SA,Sk,A kA, rA PB, Vk,B QB,Fk,B kB, rB

20. One-way Acoustic Reciprocity (continued) and

21. One-way Acoustic Reciprocity (cont’d) • Applications • Representations of scattered data in inhomogeneous dissipative media: • Generalised Bremmer series expansion • Generalised primary representation • Inverse wave field extrapolation in inhomogeneous dissipative media • Multiple elimination • Time-lapse seismic

22. Contents • Problem statement • Review of reciprocity • Reciprocity for one-way wave fields • One-way reciprocity for time-lapse seismic • Examples • Conclusions

23. Time-lapse seismic S State A (reference wave field): State B (monitoring wave field):

24. Time-lapse seismic

25. Evaluation of boundary integral A B A B S S A B Fokkema and van den Berg (1996) Dillen (2000, Ph.d thesis) Wapenaar et al. (2000, SEG)

26. Evaluation of boundary integral A B A B S S A B Fokkema and van den Berg (1996) Dillen (2000, Ph.d thesis) Wapenaar et al. (2000, SEG)

27. Contents • Problem statement • Review of reciprocity • Reciprocity for one-way wave fields • One-way reciprocity for time-lapse seismic • Examples • Conclusions

28. Numerical example S Scherpenhuijsen (2000, M.Sc. Thesis)

29. - = Upgoing wave fields at surface

30. - = Boundary integrals below reservoir

31. Difference data Numerical example (continued)

32. Boundary integral Numerical example (continued)

33. Contents • Problem statement • Review of reciprocity • Reciprocity for one-way wave fields • One-way reciprocity for time-lapse seismic • Examples • Conclusions

34. Conclusions • Reciprocity theorems formulate a relation between wave fields, sources and medium parameters in two states. • Usual (i.e. pressure normalized) one-way wave fields don’t obey reciprocity. • Flux-normalized one-way wave fields do obey reciprocity.

35. Conclusions • Applications: • Representations of scattered data in inhomogeneous dissipative media • Inverse wave field extrapolation in inhomogeneous dissipative media • Multiple elimination • Time-lapse seismic