Reciprocity and power balance for partially coupled interfaces

1. Reciprocity and power balance for partially coupled interfaces Kees Wapenaar Evert Slob Jacob Fokkema Centre for Technical Geoscience Delft University of Technology

2. Review of reciprocity theorems • Convolution type • Correlation type • Unified notation • Acoustic • Elastodynamic • Electromagnetic • Poroelastic • Seismoelectric • Review of boundary conditions • Extension of reciprocity theorems • Conclusions

3. Review of reciprocity theorems • Convolution type • Correlation type • Unified notation • Acoustic • Elastodynamic • Electromagnetic • Poroelastic • Seismoelectric • Review of boundary conditions • Extension of reciprocity theorems • Conclusions

4. A B A B ‘State A’ ‘State B’

5. A B A B ‘State A’ ‘State B’

6. V V n n State A State B Wave fields Sources Medium PA, Vk,A QA,Fk,A kA, rA PB, Vk,B QB,Fk,B kB, rB kA, rA kB, rB PB QB QA PA ‘State B’ ‘State A’

7. V V n n 0 0 0 0 PB QB QA PA ‘State B’ ‘State A’

8. V V n n PB QB QA PA ‘State B’ ‘State A’ Convolution-type reciprocity theorem: forward problems

9. Review of reciprocity theorems • Convolution type • Correlation type • Unified notation • Acoustic • Elastodynamic • Electromagnetic • Poroelastic • Seismoelectric • Review of boundary conditions • Extension of reciprocity theorems • Conclusions

10. V V n n PB QB QA PA ‘State B’ ‘State A’ Correlation-type reciprocity theorem

11. V V n n Q Q P P ‘State B’ ‘State A’ Power-flux through boundary Power dissipated in medium Power radiated by sources

12. V V n n PB QB QA PA ‘State B’ ‘State A’ Correlation-type reciprocity theorem: inverse problems

13. Review of reciprocity theorems • Convolution type • Correlation type • Unified notation • Acoustic • Elastodynamic • Electromagnetic • Poroelastic • Seismoelectric • Review of boundary conditions • Extension of reciprocity theorems • Conclusions

14. V V n n PB QB QA PA ‘State B’ ‘State A’ Convolution-type reciprocity theorem

15. Unified notation (convolution type):

16. Unified notation (convolution type):

17. Unified notation (convolution type):

18. Review of reciprocity theorems • Convolution type • Correlation type • Unified notation • Acoustic • Elastodynamic • Electromagnetic • Poroelastic • Seismoelectric • Review of boundary conditions • Extension of reciprocity theorems • Conclusions

19. Electromagnetic: Acoustic: Poroelastic: Elastodynamic: Seismoelectric:

20. V V n n PB QB QA PA ‘State B’ ‘State A’ Correlation-type reciprocity theorem

21. V V n n PB QB QA PA ‘State B’ ‘State A’

22. Unified notation (convolution type): Unified notation (correlation type):

23. Review of reciprocity theorems • Convolution type • Correlation type • Unified notation • Acoustic • Elastodynamic • Electromagnetic • Poroelastic • Seismoelectric • Review of boundary conditions • Extension of reciprocity theorems • Conclusions

24. n n V V PB QB QA PA ‘State B’ ‘State A’ Perfectly coupled interfaces: No consequences for reciprocitytheorems of convolution type and correlation type Next: consider partially coupled interfaces

25. Review of linear slip model Displacement jump:

26. Review of linear slip model of Schoenberg

27. Review of linear slip model of Pyrak-Nolte et al.

28. Review of linear slip model of Pyrak-Nolte et al. Frequency domain

29. Review of linear slip model of Pyrak-Nolte et al.

30. diagonal Schoenberg, Pyrak-Nolte et al: Nakagawa et al.: full matrix, with Generalization: Review of linear slip model

31. Arbitrary interface: Horizontal interface:

32. Generalized boundary condition:

33. Electromagnetic: Acoustic: Poroelastic: Elastodynamic: Seismoelectric:

34. Review of reciprocity theorems • Convolution type • Correlation type • Unified notation • Acoustic • Elastodynamic • Electromagnetic • Poroelastic • Seismoelectric • Review of boundary conditions • Extension of reciprocity theorems • Conclusions

35. n n V V PB QB QA PA ‘State B’ ‘State A’

36. 0 0 0 n n V V PB QB QA PA ‘State B’ ‘State A’

37. n n V V PB QB QA PA ‘State B’ ‘State A’ Convolution-type reciprocity theorem: forward problems

38. n n V V PB QB QA PA ‘State B’ ‘State A’ Correlation-type reciprocity theorem

39. Q Q P P n n V V ‘State B’ ‘State A’ Power-flux through boundary Power dissipated in medium Power radiated by sources Power dissipated by interfaces

40. n n V V PB QB QA PA ‘State B’ ‘State A’ Correlation-type reciprocity theorem: inverse problems

41. Review of reciprocity theorems • Convolution type • Correlation type • Unified notation • Acoustic • Elastodynamic • Electromagnetic • Poroelastic • Seismoelectric • Review of boundary conditions • Extension of reciprocity theorems • Conclusions

42. Unified reciprocity theorems have been formulated of the convolution and correlation type

43. Unified reciprocity theorems have been formulated of the convolution and correlation type Valid for acoustic, elastodynamic, electromagnetic, poroelastic and seismoelectric waves

44. Unified reciprocity theorems have been formulated of the convolution and correlation type Valid for acoustic, elastodynamic, electromagnetic, poroelastic and seismoelectric waves Boundary condition for imperfectly coupled interface:

45. Unified reciprocity theorems have been formulated of the convolution and correlation type Valid for acoustic, elastodynamic, electromagnetic, poroelastic and seismoelectric waves Boundary condition for imperfectly coupled interface: No effects on source-receiver reciprocity

46. Unified reciprocity theorems have been formulated of the convolution and correlation type Valid for acoustic, elastodynamic, electromagnetic, poroelastic and seismoelectric waves Boundary condition for imperfectly coupled interface: No effects on source-receiver reciprocity Imaginary part of accounts for dissipation by interfaces