Review of the Greens Theorem deghosting method Jingfeng

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

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

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

Green’s Theorem Green’s function =

Configuration for actual wavefield

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

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

Wavelet estimation algorithm 2 + 3 = Weglein and Secrest, (1990) 3 1 V =1+ 2 + 3 2 2 3

Field prediction algorithm

Field prediction algorithm

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

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

Towed streamer deghosting (Numerical tests) F. S. (0, 2) 6. 0 m M. S. 300 m c 1=1500 m/s c 2=2250 m/s

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

FSMR results (point receiver data) Primary 1 st order FSM 2 nd order FSM Using the 1 st term in the ISS FSMR series At (0, 2. 5)

FSMR results (point receiver data) At (3750, 2. 5) 1 st order FSM Using the 1 st term in the ISS FSMR series

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

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.

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

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

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.

Acknowledgements • M-OSRP sponsors and colleagues