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

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

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

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

12. ‘State B’ ‘State A’ V n PB QB QA PA
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. ‘State B’ ‘State A’ V n PB QB QA PA
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. Acoustic:
Poroelastic:
Elastodynamic:
Seismoelectric:
Electromagnetic:

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

21. V 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. ‘State B’ ‘State A’ n n V V PB QB QA PA 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. Review of linear slip model
Schoenberg, Pyrak-Nolte et al:
diagonal
Nakagawa et al.:
full matrix, with
Generalization:

31. Horizontal interface:
Arbitrary interface:

32. Generalized boundary condition:

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

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

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

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

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

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