1. On acoustic reciprocity theorems and the construction of transmission response from reflection data
Bogdan G. Nita
*University of Houston
M-OSRP Annual Meeting
20-21 April, 2005
University of Houston

2. Outline Motivation One-way wavefield decomposition
Reciprocity theorems
Reconstructing the phase from amplitude information
Minimum phase condition
Summary and Conclusions

3. Outline Motivation One-way wavefield decomposition
Reciprocity theorems
Reconstructing the phase from amplitude information
Minimum phase condition
Summary and Conclusions

4. Motivation Virtual source – Shell Seismic interferometry
Deep earth seismology
Model type independent imaging

5. Inverse scattering imaging
Inverse scattering imaging subseries method has shown tremendous value for 1D and 2D acoustic media (Shaw, Liu)
H. Zhang leads the efforts to identify the subseries for imaging in a 1D elastic medium
Model type independent method

6. Internal multiple attenuation subseries
= G0
= D
The attenuation algorithm requires three reflection data sets to build up an internal multiple
= Imaged Data

7. Leading order imaging sub-series
= V1
Linear
2nd Order +
3rd Order
A subseries of the inverse series +
4th Order + + …

8. Data requirements for model type independent imaging
Reflection data
Transmission data

9. Methods for obtaining transmission data
Measure/record it (e.g. VSP)
Determine it from reflection data using reciprocity theorems
Inverse scattering series constructs the transmission response order by order from reflection data

10. Outline Motivation One-way wavefield decomposition
Reciprocity theorems
Reconstructing the phase from amplitude information
Minimum phase condition
Summary and Conclusions

11. Seismic experiment FS
At any depth, the total wavefield has an up-going and a down-going component

12. Two way wavefield reciprocityFS
Acoustic response does not change if the source and receiver are interchanged

13. Two way wavefield reciprocityFS
Acoustic response does not change if the source and receiver are interchanged

14. Why do we need one-way wavefields
Migration
Deghosting
To be able to define reflection and transmission responses

15. One-way wavefields Reciprocity is not obvious for one way wavefields
One way wavefield decomposition is not unique

16. Up-down wavefield decomposition
Pressure normalized one-way wavefields
Widely used
Do not satisfy the reciprocity theorem
Flux normalized one-way wavefields
Satisfy the reciprocity theorem
M.V. De Hoop 1996, Wapenaar 2004, 2005

17. Pressure normalized up-down decomposition
Acoustic pressure
Particle velocity

18. Pressure normalized up-down decomposition
1D medium
Continuity of P and Vz at the interface
Reciprocity is not satisfied!

19. Flux normalized up-down decomposition
Acoustic pressure
Particle velocity

20. Flux normalized up-down decomposition
1D medium
Continuity of P and Vz at the interface
Reciprocity is satisfied!

21. Medium dependence
The one-way wavefield decompositions only depend on the medium where the data is collected

22. Conclusions: one-way wavefield decomposition
Decomposition is not unique
Pressure normalized one-way wavefields do not satisfy reciprocity
Flux-normalized one-way wavefields satisfy reciprocity
The two decompositions only depend on the medium where the data is collected

23. Outline Motivation One-way wavefield decomposition
Reciprocity theorems
Reconstructing the phase from amplitude information
Minimum phase condition
Summary and Conclusions

24. Reciprocity theorems Two-way wavefields One way wavefields
Convolution type
Correlation type
One way wavefields
Fokkema and van den Berg 1990

25. One-way wavefield theorem of the correlation type
Independent acoustic states
The region between and is source free
Valid only for lossless media with evanescent waves neglected

26. Transmission from reflection
Use the one way reciprocity of the correlation type
Same experiments and
Substitute into the one-way reciprocity theorem of correlation type and divide by the source wavelet

27. Transmission from reflection
relation between the amplitude of reflection data and that of transmission data
all the phase information is lost and there is no unique way of recovering it
phase reconstruction requires one additional relation which is sometimes provided by the minimum phase condition
minimum phase property for a wavefield depends on the medium that the wave propagates through
for general 3D acoustic and elastic media the wavefield usually has mixed phase

28. Conclusions for reciprocity theorems
One way reciprocity theorem of correlation type provides a relation between the amplitude of the reflection data and that of the transmission data
To recover the phase one needs one additional relation which is sometimes provided by the minimum phase condition

29. Outline Motivation One-way wavefield decomposition
Reciprocity theorems
Reconstructing the phase from amplitude information
Minimum phase condition
Summary and Conclusions

30. A real signal
For arbitrary functions there is no connection between X and Y

31. Causal signals Causal Causal Causal
Is analytic in the upper half complex plane
Causal
Causal
Related through Hilbert transforms

32. Causal signals
A causal signal can be fully reconstructed from its frequency domain real or imaginary parts

33. Amplitude and phase relations

34. Amplitude and phase relations
When F contains no zeroes in the upper complex-frequency half plane

35. Amplitude and phase relations
F is analytic and has no zeros implies is analytic and hence its real and imaginary parts are related through Hilbert transforms
phase is constructed from amplitude

36. Amplitude and phase relations
F is analytic in the upper complex-frequency half plane - Causality
F has no zeroes in the upper complex-frequency half plane – Minimum phase condition

37. Conclusions: Reconstructing the phase from amplitude information
The phase can be reconstructed from amplitude information only if the signal is
Causal
Satisfies the minimum phase condition

38. Outline Motivation One-way wavefield decomposition
Reciprocity theorems
Reconstructing the phase from amplitude information
Minimum phase condition
Summary and Conclusions

39. Minimum phase condition
A signal is minimum phase if it has no zeroes in the upper complex-frequency half plane
The inverse has no poles hence it is analytic
Zeroes create phase-shifts
Passing beneath a zero causes a phase-shift of
Minimum phase-shift
Complex frequency plane

40. Minimum phase condition in time domain
Eisner (1984)
Output energy
the output energy of a minimum phase signal integrated up to time T is greater than that of a non-minimum phase signal with the same frequency-domain magnitude
Hence a minimum phase signal has more energy concentrated at earlier times than any other signal sharing its spectrum

41. Minimum phase reflectors
A minimum phase reflector has the property of reflecting the acoustic energy faster than any non-minimum phase reflector
In a minimum phase medium the perfect velocity transfer condition is satisfied: the wave that enters the medium and the one that exits it have the same propagation speed
This holds for normal incident intramodal reflection – more general situations (e.g. converted waves) are presently under investigation

42. Outline Motivation One-way wavefield decomposition
Reciprocity theorems
Reconstructing the phase from amplitude information
Minimum phase condition
Summary and Conclusions

43. Summary and conclusions
Model type independent ISS imaging requires both reflection and transmission data
One-way reciprocity theorem of the correlation type relates amplitude of the reflection data and transmission data
To recover the phase an additional condition – minimum phase condition – is necessary
Seismic arrivals are presently under investigation to determine their phase properties

44. Acknowledgements Co-author: Arthur B. Weglein.
Support: M-OSRP sponsors.
Collaboration with Gary Pavlis and Chengliang Fan, Indiana University