1. Remarks on Green’s Theorem for seismic interferometry
Adriana C. Ramírez and Arthur B. Weglein
M-OSRP 2006 Annual Meeting, June 5 ~ June 7, 2007
M-OSRP report pages:

2. In this talk we would like to…
Provide an overview of a broad set of seismic applications that recognize Green’s theorem as their starting point.
Show that Green’s theorem presents a platform and unifying principle for the field of seismic interferometry.
Explain artifacts and spurious multiples (in certain interferometry approaches) as fully anticipated errors and as violations of the theory.
Provide a systematic approach to understanding, comparing and improving upon many current concepts, approximations and compromises.
The attention given by the energy industry and the literature to methods dealing with wavefield retrieval, or seismic interferometry, and its applications to different seismic exploration problems, has brought about a renewed interest in Green’s theorem.

3. Outline Motivation Green’s theorem Seismic interferometry
Standard seismic interferometry
Direct wave seismic interferometry
Dirichlet seismic interferometry
Other applications of Green’s theorem
Remarks
Acknowledgements

7. Free Surface

8. `
Free Surface ?

9. Outline Motivation Green’s theorem Seismic interferometry
Standard seismic interferometry
Direct wave seismic interferometry
Dirichlet seismic interferometry
Other applications of Green’s theorem
Remarks
Acknowledgements

10. Green’s theorem
Green’s theorem relates a surface integral of two scalar functions u and , and their normal derivatives with a volume integral of the same functions and their Laplacians,
where x is a three dimensional vector (x1, x2, x3) characterizing the volume V enclosed by the surface S, and n is the unit vector normal to this surface.
The functions u(x) and v(x) can be any pair of scalar functions that have normal derivatives at the surface S and Laplacians in V.
The importance of this
theorem lays in its generality. Green’s theorem has been extended to displacements by Betti (1872) and elastodynamic fields by Rayleigh (1873).

11. Green’s theorem Introduce P=u and G+=v into :
We use Green’s theorem to derive an integral representation of the pressure field P, assumed to satisfy the inhomogeneous Helmholtz equation,
The causal Green’s function for the Helmholtz operator is given by
The importance of this
theorem lays in its generality. Green’s theorem has been extended to displacements by Betti (1872) and elastodynamic fields by Rayleigh (1873).
Introduce P=u
and G+=v into :

12. Green’s Theorem
Two cases will be considered:
and

13. Green’s Theorem
Two cases will be considered:
and

14. Green’s Theorem
Two cases will be considered:
and

15. Configuration
First case:

16. Green’s Theorem
Consider and within the volume.
First case:

17. Let’s go back to our starting point
The pressure field, P, is assumed to satisfy the inhomogeneous Helmholtz equation,
The causal Green’s function for the Helmholtz operator is given by
The importance of this
theorem lays in its generality. Green’s theorem has been extended to displacements by Betti (1872) and elastodynamic fields by Rayleigh (1873).
Since and ,
we can use and in Green’s theorem…

18. Let’s go back to our staring point
Introduce, P,
and
into Green’s theorem,
The importance of this
theorem lays in its generality. Green’s theorem has been extended to displacements by Betti (1872) and elastodynamic fields by Rayleigh (1873).
is the integration variable

19. Configuration
This form of Green’s theorem is used as the starting point for
common approaches to seismic interferometry.

20. Configuration
This form of Green’s theorem is used as the starting point for
common approaches to seismic interferometry.

21. Outline Motivation Green’s theorem Seismic interferometry
Standard seismic interferometry
Direct wave seismic interferometry
Dirichlet seismic interferometry
Other applications of Green’s theorem
Remarks
Acknowledgements

23. Free Surface

24. Using a zero-pressure surface

25. Configuration with a free surface
First solution:
Use an approximation
Using two measured wavefields to construct new data introduces an extra factor of the source wavelet multiplied to the synthesized data.
This form of Green’s theorem is used as the starting point for
common approaches to seismic interferometry.

26. Configuration with a free surface
First solution:
Use an approximation
This is a high frequency
and one-way wave approximation
Using two measured wavefields to construct new data introduces an extra factor of the source wavelet multiplied to the synthesized data.

27. Configuration with a free surface
First solution:
Use an approximation
Using two measured wavefields to construct new data introduces an extra factor of the source wavelet multiplied to the synthesized data.

28. Seismic interferometry
 Free surface 
(Wapenaar and Fokkema, Geophysics, 2006)
The process of using two measured wavefields to construct new data introduces an extra factor of the source wavelet multiplied to the synthesized data.

29. Seismic interferometry
(Wapenaar and Fokkema, Geophysics, 2006)
 Free surface 

30. Seismic interferometry
(Wapenaar and Fokkema, Geophysics, 2006)
 Free surface
The waves are assumed to travel everywhere.

31. Seismic interferometry
(Wapenaar and Fokkema, Geophysics, 2006)
 Free surface

32. Seismic interferometry
(Wapenaar and Fokkema, Geophysics, 2006)
Using two measured wavefields to construct new data introduces an extra factor of the source wavelet multiplied to the synthesized data.
Using two recorded wavefields, constrains the reconstructed wavefield to locations where actual sources or receivers exist, and requires two approximations to the exact theory.

33. Seismic interferometry
(Wapenaar and Fokkema, Geophysics, 2006)
This is a (double) compromised form of Green’s theorem and gives rise to spurious multiples (artifacts).
The normal derivative information required by Green’s theorem cancels these artifacts by using differences in sign that identify opposite directions of the wavefield.
The directionality information is part of the wavefield’s normal derivative.

34. Outline Motivation Green’s theorem Seismic interferometry
Standard seismic interferometry
Direct wave seismic interferometry
Dirichlet seismic interferometry
Other applications of Green’s theorem
Remarks
Acknowledgements

35. Free Surface

36. Free Surface

37. Free Surface
-Amplitude
-Phase
-Directionality

38. Reference Green’s function
Explain more , predict more
More effective
More realism
Pushing the boundaries
Whenever you can use physics to predict…
Don´t give adaptive more responsability

39. Anticausal reference Green’s function
Explain more , predict more
More effective
More realism
Pushing the boundaries
Whenever you can use physics to predict…
Don´t give adaptive more responsability

40. Configuration

41. Let’s go back to our starting point
Introduce, P,
and
into Green’s theorem,
The importance of this
theorem lays in its generality. Green’s theorem has been extended to displacements by Betti (1872) and elastodynamic fields by Rayleigh (1873).
is the integration variable

42. Configuration with a free surface
This form of Green’s theorem is used as the starting point for
Direct wave seismic interferometry.

43. Configuration with a free surface
(Ramirez, Hokstad and Otnes, EAGE, 2007)
This form of Green’s theorem is used as the starting point for
Direct wave seismic interferometry.

44. Configuration with a free surface
First solution:
Use an approximation
 Free surface 
(Ramirez,Hokstad and Otnes, EAGE, 2007)
This form of Green’s theorem is used as the starting point for
Direct wave seismic interferometry.

45. Configuration with a free surface
First solution:
Use an approximation
 Free surface 
(Ramirez,Hokstad and Otnes, EAGE, 2007)
This form of Green’s theorem is used as the starting point for
Direct wave seismic interferometry.

46. Direct wave seismic interferometry
First solution:
Use an approximation
(Ramirez,Hokstad and Otnes, EAGE, 2007)

47. Direct wave seismic interferometry
(Ramirez, Hokstad and Otnes, EAGE, 2007)
 Free surface 

48. Direct wave seismic interferometry
(Ramirez, Hokstad and Otnes, EAGE, 2007) 

49. Direct wave seismic interferometry
(Ramirez, Hokstad and Otnes, EAGE, 2007) 

50. Direct wave seismic interferometry
(Ramirez, Hokstad and Otnes, EAGE, 2007)

51. Direct wave seismic interferometry
(Ramirez, Hokstad and Otnes, EAGE, 2007)
The method uses an analytic anticausal Green’s function and only makes one approximation.
The output is approximately equal to the scattered field and the imaginary part of the direct wave.
This result is more accurate than the standard seismic interferometry method.

52. Direct wave seismic interferometry
(Ramirez, Hokstad and Otnes, EAGE, 2007)
The wavefield can be reconstructed anywhere between the measurement surface and the free surface (it is not constrained to positions where sources and/or receivers exist).
This allows for applications like extrapolation and regularization.

53. Outline Motivation Green’s theorem Seismic interferometry
Standard seismic interferometry
Direct wave seismic interferometry
Dirichlet seismic interferometry
Other applications of Green’s theorem
Remarks
Acknowledgements

54. Free Surface
Zero pressure surface for the second source

55. -Phase -Directionality -Amplitude Free Surface
Zero pressure surface for the second source

56. Dirichlet Green’s function
2 zero-pressure surfaces
Explain more , predict more
More effective
More realism
Pushing the boundaries
Whenever you can use physics to predict…
Don´t give adaptive more responsability
Weglein and Devaney (1992), Tan (1992),
Osen et al. (1998), Tan (1999), Weglein et al. (2000).

57. Configuration

58. Let’s go back to our starting point
Introduce, P,
and
into Green’s theorem,
The importance of this
theorem lays in its generality. Green’s theorem has been extended to displacements by Betti (1872) and elastodynamic fields by Rayleigh (1873).
is the integration variable

59. Configuration with Dirichlet boundary conditions
 Free surface 
A Green’s function that vanishes on both the free and measurement surfaces eliminates the data requirement of the normal derivative in previous forms of Green’s theorem.
(Weglein and Devaney, 1992; Tan, 1992;Osen et al., 1998).

60. Configuration with Dirichlet boundary conditions
 Free surface 
Weglein and Devaney (1992),
Tan (1992),
Osen et al. (1998),
Weglein et al. (2000),
Zhang and Weglein (2006).
The algorithm described by this equation doesn’t require the normal derivative of the pressure field.

61. Dirichlet seismic interferometry
 Free surface 

62. Dirichlet seismic interferometry

63. Dirichlet seismic interferometry

64. Dirichlet seismic interferometry
The method uses an analytic Green’s function with two-surface Dirichlet boundary conditions.
The method requires knowledge of the wavelet to be exact.

65. Dirichlet seismic interferometry
If an estimate of the wavelet is available, or obtainable, we can easily remove the error on the left hand side, by adding a factor of
This will obtain an exact equation and the only limitation for a perfect output would be due to aperture limitations in the recorded pressure data. The rest of the ingredients required by Green’s function would be fulfilled analytically.

66. Outline Motivation Green’s theorem Seismic interferometry
Standard seismic interferometry
Direct wave seismic interferometry
Dirichlet seismic interferometry
Other applications of Green’s theorem
Remarks
Acknowledgements

67. Outline Motivation Green’s theorem Seismic interferometry
Standard seismic interferometry
Direct wave seismic interferometry
Dirichlet seismic interferometry
Other applications of Green’s theorem
Remarks
Acknowledgements

68. Outline Motivation Green’s theorem Green’s Theorem
Standard seismic interferometry
Direct wave seismic interferometry
Dirichlet seismic interferometry
Other applications of Green’s theorem
Remarks
Acknowledgements

69. Outline Motivation Green’s theorem Green’s Theorem
Standard Green’s theorem
Direct wave seismic interferometry
Dirichlet seismic interferometry
Other applications of Green’s theorem
Remarks
Acknowledgements

70. Outline Motivation Green’s theorem Green’s Theorem
Standard Green’s theorem
Direct wave Green’s theorem
Dirichlet seismic interferometry
Other applications of Green’s theorem
Remarks
Acknowledgements

71. Outline Motivation Green’s theorem Green’s Theorem
Standard Green’s theorem
Direct wave Green’s theorem
Dirichlet Green’s theorem
Other applications of Green’s theorem
Remarks
Acknowledgements

72. Removing the direct wave with Green’s theorem
Use the pressure field and the reference Green’s function (Causal) in Green’s theorem
This form of Green’s theorem retrieves the scattered field
 Free surface 

73. Wavelet estimation with Green’s theorem
Use the pressure field and the reference Green’s function (Causal) in Green’s theorem
 Free surface 

74. Wavelet estimation with Green’s theorem (Dirichlet boundary conditions)
Use the pressure field and the reference Green’s function (Causal) in Green’s theorem
 Free surface 

75. … Wavefield retrieval Imaging Deghosting Up-Down separation
The list of application in seismic physics that uses Green’s theorem includes:
Wavefield retrieval
Imaging
Deghosting
Up-Down separation
Wavefield Deconvolution
Free-Surface multiple removal
Wavelet estimation, etc.

76. Outline Motivation Green’s theorem Green’s Theorem
Standard Green’s theorem
Direct wave Green’s theorem
Dirichlet Green’s theorem
Other applications of Green’s theorem
Remarks
Acknowledgements

77. Remarks
The attention given by the energy industry and the literature to methods dealing with wavefield retrieval, or seismic interferometry, and its applications to different seismic exploration problems, has brought about a renewed interest in Green’s theorem.
Explain more , predict more
More effective
More realism
Pushing the boundaries
Whenever you can use physics to predict…
Don´t give adaptive more responsability

78. Today, Green’s theorem was used to:
1) show that Green’s theorem provides a platform and unifying principle for the field of seismic interferometry;
2) explain artifacts and spurious multiples (in certain interferometry approaches) as fully anticipated errors and as violations of the theory,
3) provide a systematic approach to understanding, comparing and improving upon many current concepts, approximations and compromises.
Explain more , predict more
More effective
More realism
Pushing the boundaries
Whenever you can use physics to predict…
Don´t give adaptive more responsability

79. Acknowledgements

80. Acknowledgements

81. Acknowledgements
I would like to thank my coauthor Arthur B. Weglein, and to acknowledge useful and invaluable discussions with Ketil Hokstad (Statoil) and Einar Otnes (Statoil).
Simon A. Shaw (ConocoPhillips) is thanked for sharing insights on boundary conditions.

85. Seismic interferometry and deghosting
Medium parameters for both fields are only equal at the boundary and within the volume.
Medium parameters for both fields are equal everywhere
Seismic interferometry and deghosting
Seismic interferometry
Wavelet estimation
both sources are strictly inside S
only the observation point lies within V
only the source lies within V

86. Functional relation (imaging)
Medium parameters for both fields are only equal at the boundary and within the volume.
Medium parameters for both fields are equal everywhere
Functional relation (imaging)
Seismic interferometry
both sources are strictly inside S
only the observation point lies within V
only the source lies within V