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: 15-39

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.

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

Free Surface

` Free Surface ?

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

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).

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 :

Green’s Theorem Two cases will be considered: and

Green’s Theorem Two cases will be considered: and

Green’s Theorem Two cases will be considered: and

Configuration First case:

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

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…

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

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

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

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

Free Surface

Using a zero-pressure surface

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.

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.

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.

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.

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

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

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

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.

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.

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

Free Surface

Free Surface

Free Surface -Amplitude -Phase -Directionality

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

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

Configuration

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

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

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.

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.

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.

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

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

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

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

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

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.

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.

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

Free Surface Zero pressure surface for the second source

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

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).

Configuration

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

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).

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.

Dirichlet seismic interferometry  Free surface 

Dirichlet seismic interferometry 

Dirichlet seismic interferometry

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.

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.

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

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

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

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

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

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

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 

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

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 

… 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.

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

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

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

Acknowledgements

Acknowledgements

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.

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

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