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: 15-39

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.

Outline Wavefield deconvolution Free surface multiple removal Analysis Remarks Acknowledgements

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).

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

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).

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.

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

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

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.

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

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

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

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

1D Wavefield deconvolution water earth

1D Wavefield deconvolution water earth

Outline Wavefield deconvolution Free surface multiple removal Analysis Remarks Acknowledgements

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

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

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

Free surface multiple removal

Free surface multiple removal

Free surface multiple removal

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

Data without a free surface Data with a free surface 1 Data with a free surface 1

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

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

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.

FSMR theory water UPGOING WAVES earth

FSMR theory water UPGOING WAVES earth

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

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

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

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

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

FSMR theory water DOWNGOING WAVES earth

FSMR theory water earth

Outline Wavefield deconvolution Free surface multiple removal Analysis Remarks Acknowledgements

1D Wavefield deconvolution water earth

FSMR theory water earth

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.

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?

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?

Outline Wavefield deconvolution Free surface multiple removal Analysis Remarks Acknowledgements

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

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.

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.