1. Relationship between ISS free surface multiple removal and wave-field deconvolution
Adriana C. Ramírez and Arthur B. Weglein
M-OSRP 2006 Annual Meeting, June 5 ~ June 7, 2007
M-OSRP report pages:

2. Objectives
Provide an overview of two wave theoretic free surface multiple removal methods:
Wavefield deconvolution
ISS Free surface multiple removal
Show the relationship between ISS free surface multiple removal and wave-field deconvolution.
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 Wavefield deconvolution Free surface multiple removal Analysis
Remarks
Acknowledgements

4. Wavefield deconvolution
This is a theory applied to the removal of overburden
effects (overburden refers to the medium above the receiver or measurement plane),
e.g. removal of free-surface multiples (events due to the existence of the air-water surface) and source effects (dueto a source exploiting above the location of interest).

5. Wavefield deconvolution
The method can be derived from Green’s theorem.
The choice of functions introduced in Green’s theorem is:
1) a pressure field, P,
2) a second pressure field, P’, produced by the same medium as P, but without the existence of the free surface.
The algorithm derived with this choice will remove all the free-surface
multiples and the source wavelet from the pressure field, and, it will retrieved the deconvolved wavefield at the receiver location (coincident source and receiver), (Amundsen, 1999, 2001; Holvik and Amundsen, 2005).
A similar method for wavefield deconvolution, which requires the source
wavelet, was derived by (Ziolkowski et al., 1998; Johnston and Ziolkowski, 1999).
Pressure field Green’s function
receiver
source

6. Wavefield deconvolution
The method can be derived from Green’s theorem.
The choice of functions introduced in Green’s theorem is:
1) a pressure field, P,
2) a second pressure field, P’, produced by the same medium as P, but without the existence of the free surface.
The algorithm derived with this choice will remove all the free-surface
multiples and the source wavelet from the pressure field, and, it will retrieved the deconvolved wavefield at the receiver location (coincident source and receiver), (Amundsen, 1999, 2001; Holvik and Amundsen, 2005).
A similar method for wavefield deconvolution, which requires the source
wavelet, was derived by (Ziolkowski et al., 1998; Johnston and Ziolkowski, 1999).

7. Wavefield deconvolution
The method can be derived from Green’s theorem.
The choice of functions introduced in Green’s theorem is:
1) a pressure field, P,
2) a second pressure field, P’, produced by the same medium as P, but without the existence of the free surface.
This integral equation has been derived and used for free-surface elimination by e.g. Fokkema and van den Berg (1993) and Amundsen (1999,2001).
In contrast to other solutions for Green’s theorem, the relation described by this equation is not a relation that can be readily applied to retrieve a useful result. It needs more mathematical manipulation.

8. Wavefield deconvolution
The method assumes that the pressure field is a sum of upgoing and downgoing waves,
and introduces the relations
and,
After several mathematical manipulations of P in the wavenumber domain, and introducing the fields and , a second integral equation is obtained

9. 1D Wavefield deconvolution
where denotes the horizontal coordinates for the source position. It is set to zero because the medium is laterally invariant.

10. Wavefield deconvolution
This result is a Fredholm integral equation of the first kind.
This is an equation difficult to solve and in general it is ill-conditioned.

11. Wavefield deconvolution
However, when the medium is 1D
(horizontally layered),
a much simpler solution is found,

12. 1D Wavefield deconvolution
The final result retrieves a wavefield without overburden effects, and with coincident source and receiver positions.
water
UPGOING WAVES
earth

13. 1D Wavefield deconvolution
The final result retrieves a wavefield without overburden effects, and with coincident source and receiver positions.
water
DOWNGOING WAVES
earth

14. 1D Wavefield deconvolution
The final result retrieves a wavefield without overburden effects, and with coincident source and receiver positions.
water
earth

15. 1D Wavefield deconvolution
water
earth

16. 1D Wavefield deconvolution
water
earth

17. Outline Wavefield deconvolution Free surface multiple removal Analysis
Remarks
Acknowledgements

18. Scattering Theory Scattering theory is perturbation theory
Relates differences in media to differences in wavefield

19. Scattering Theory (cont’d.)
Inverse Series, V as power series in data, (fs)m

20. Four Tasks of Direct Inversion
(1) Free surface demultiple
(2) Internal demultiple
(3) Image reflectors at depth
(4) Determine medium properties

21. Free surface multiple removal

22. Free surface multiple removal

23. Free surface multiple removal

24. FSMR theory (1) The algorithm requires: source wavelet;
Carvalho (1992), Weglein et al. (1997)
(1) The algorithm requires:
source wavelet;
Source, receiver deghosted data
(2) This method does not require any subsurface information
(3) Order by order prediction & elimination

25. Data without a free surface Data with a free surface1
Data with a free surface 1

26. Free surface demultiple algorithm
= primaries and internal multiples
= primaries, free surface multiples and internal multiples 1
Total upfield
and,

27. FSMR theory
Carvalho (1992), Weglein et al. (1997)

28. So precisely eliminates all free surface
t1 + t2
2t1
2t2
So precisely eliminates all free surface
multiples that have experienced one downward reflection
at the free surface.
The absence of low frequency (and in fact all other
frequency) plays absolutely no role in this prediction.

29. FSMR theory
water
UPGOING WAVES
earth

30. FSMR theory
water
UPGOING WAVES
earth

31. FSMR theory 1 = primaries and internal multiples
= primaries, free surface multiples and internal multiples 1
water
UPGOING WAVES
earth

32. FSMR theory 1 = primaries and internal multiples
= primaries, free surface multiples and internal multiples 1
water
UPGOING WAVES
earth

33. FSMR theory 1 = primaries and internal multiples
= primaries, free surface multiples and internal multiples 1
water
DOWNGOING WAVES
earth

34. FSMR theory = primaries and internal multiples
= primaries, free surface multiples and internal multiples
water
DOWNGOING WAVES
earth

35. FSMR theory = primaries and internal multiples
= primaries, free surface multiples and internal multiples
water
DOWNGOING WAVES
earth

36. FSMR theory
water
DOWNGOING WAVES
earth

37. FSMR theory
water
earth

38. Outline Wavefield deconvolution Free surface multiple removal Analysis
Remarks
Acknowledgements

39. 1D Wavefield deconvolution
water
earth

40. FSMR theory
water
earth

41. Wavefield deconvolution
This result is a Fredholm integral equation of the first kind.
This is an equation difficult to solve and in general it is ill-conditioned.

42. Can we use the direct wave to avoid the need of a source wavelet?
FSMR theory
Carvalho (1992), Weglein et al. (1997)
(1) This method does not require any subsurface information
(2) Order by order prediction & elimination
Can we use the direct wave to avoid
the need of a source wavelet?

43. Can we use the direct wave to avoid the need of a source wavelet?
FSMR theory
Carvalho (1992), Weglein et al. (1997)
(1) This method does not require any subsurface information
(2) Order by order prediction & elimination
Can we use the direct wave to avoid
the need of a source wavelet?

44. Outline Wavefield deconvolution Free surface multiple removal Analysis
Remarks
Acknowledgements

45. Remarks
Inverse Scattering and Wavefield Deconvolution are often seen as very different methods.
We showed that they are, in fact, very similar methods. They both deconvolve the upgoing field with the downgoing field and recover the reflectivity of the earth.
Both are subsurface independent

46. Remarks
The Inverse Scattering Free Surface Multiple removal has a series solution in multi-D.
The Wavefield Deconvolution method has a Fredholm-1 integral solution in multi-D.
It is possible that these two solutions (in Multi-D) are related.

47. Lasse Amundsen (Statoil) is also thanked for useful discussions.
Acknowledgements
I would like to thank my coauthor Arthur B. Weglein, and to acknowledge useful and invaluable discussions with Ken Matson (BP) and Rodney Johnston (BP).
I would like to acknowledge the internship opportunity I had at BP (fall/winter 2006). BP allowed me to work on a project on wavefield deconvolution, that experience triggered this work.
Lasse Amundsen (Statoil) is also thanked for useful discussions.