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 reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

3. Problem statement

4. -= Reference Monitor Difference

5. Problem statement Difference

6. Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

7. Acoustic reciprocity State AState B xAxA xAxA xBxB xBxB

8. Acoustic Reciprocity (continued) V State AState B P A, V k,A Q A,F k,A  A,  A P B, V k,B Q B,F k,B  B,  B Wave fields Sources Medium n

9. with 00 00 Acoustic Reciprocity (continued)

10. with Acoustic Reciprocity (continued)

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

12. Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

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

14. One-way acoustic Reciprocity (horizontally layered medium) hence c1,c2,c1,c2, c1,c2,c1,c2,

15. One-way acoustic Reciprocity (arbitrary inhomogeneous medium) State AState B xAxA xAxA xBxB xBxB

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

17. One-way acoustic Reciprocity (horizontally layered medium) hence c1,c2,c1,c2, c1,c2,c1,c2,

18. One-way acoustic Reciprocity (arbitrary inhomogeneous medium) State AState B xAxA xAxA xBxB xBxB

19. One-way Acoustic Reciprocity (continued) V State AState B P A, V k,A S A,S k,A  A,  A P B, V k,B Q B,F k,B  B,  B Wave fields Sources Medium

20. and One-way Acoustic Reciprocity (continued)

21. 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 One-way Acoustic Reciprocity (cont’d)

22. Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

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

24. Time-lapse seismic

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

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

27. Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

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

29. Upgoing wave fields at surface -=

30. Boundary integrals below reservoir -=

31. Numerical example (continued) Difference data

32. Numerical example (continued) Boundary integral

33. Contents Problem statement Review of reciprocityReview of reciprocity Reciprocity for one-way wave fieldsReciprocity for one-way wave fields One-way reciprocity for time-lapse seismicOne-way reciprocity for time-lapse seismic ExamplesExamples ConclusionsConclusions

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