1. Review of the Green’s Theorem deghosting method
Jingfeng Zhang and Arthur B. Weglein
M-OSRP 2006 Annual Meeting, June 5 ~ June 7, 2007

2. Key points
Scattered field can be deghosted without source wavelet and vertical derivative of the wavefield
Field data test requirements: interpolation and extrapolation, high computation power
Derivation of the Green’s theorem deghosting, wavelet estimation and field prediction algorithms

3. Motivation Remove the seismic distortion due to ghosts
Angle and frequency dependent distortion, ghost notch
A pre-requisite for all of the current inverse scattering series related algorithms: FSMR, IMA, Imaging and inversion

4. Green’s Theorem
Green’s function =

5. Configuration for actual wavefield

6. Deghosting algorithm 2 1 = + 1 2 + 3 3 1 2 =
Weglein et al. (2002)

7. Wavelet estimation algorithm
is the image of
with respect to the free surface V

8. Wavelet estimation algorithm2 3 + =
Weglein and Secrest, (1990) V = 1 + 2 3 1 3 2 2 3

9. Field prediction algorithm

10. Field prediction algorithm

11. Deghosting algorithm
F.S.
Pseudo-M.S.
M.S.
Earth

12. Deghosting algorithm
F.S.
Pseudo-M.S.
M.S.
Earth

13. Towed streamer deghosting (Numerical tests)
F.S.
(0,2)
6.0m
M.S.
300m
c1=1500m/s
c2=2250m/s

14. With source wavelet
Red solid: Exact; Blue dash: deghosting result

15. FSMR results (point receiver data)
Primary
At (0,2.5)
1st order FSM
2nd order FSM
Using the 1st term in the ISS FSMR series

16. FSMR results (point receiver data)
1st order FSM
Using the 1st term in the ISS FSMR series

17. Without source wavelet
Consequences: (1) Direct wave and its ghost: remains
(2) Scattered field: no effect

18. Without source wavelet
Between FS and MS, the scattered field satisfies:
Notice there is no source term.
From similar derivation of the field prediction:
compared to
So (1) the absent of source wavelet has no effect on the prediction of
the scattered field.
(2) the term only contribute to the prediction of
the direct wave and its ghost.

19. Without source wavelet
Absent of source wavelet only cause inaccuracy in prediction of the direct wave and its ghost, while the scattered field prediction is accurate
Scattered field can be deghosted well even without source wavelet
Problems exist at places (especially for shallow water cases) where the direct wave interfere with the scattered field

20. Field data tests Surface integration: Source wavelet Receiver array
Surface data measurements: data interpolation and extrapolation
Large computation power
Source wavelet
Receiver array

21. Conclusions
Green’s theorem deghosting produce encouraging results when the source wavelet is available. Even without the source wavelet, scattered field can still be deghosted.
The surface integration requires surface data measurements and high computation power. At the same time, it provides stable results due to that integration.
Interpolation and Extrapolation, high computation power are necessary for field data tests.

22. Acknowledgements
M-OSRP sponsors and colleagues