Seismic interferometry, the optical theorem and a non-linear point scatterer Kees Wapenaar Evert Slob Roel Snieder Society of Exploration Geophysicists Houston, October 26, 2009

Point scatterer Interferometry Optical theorem Non-linear Paradox

Point scatterer Interferometry Optical theorem Non-linear Paradox Modeling Inversion Interferometry Migration

Snieder, R., K.van Wijk, M.Haney, and R.Calvert, 2008, Cancellation of spurious arrivals in Green's function extraction and the generalized optical theorem: Physical Review E, 78, Halliday, D. and A.Curtis, 2009, Generalized optical theorem for surface waves and layered media: Physical Review E, 79, van Rossum, M. C. W. and T.M. Nieuwenhuizen, 1999, Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion: Reviews of Modern Physics, 71,

a b Term 1:

c d Term 2:

ef Term 3:

a b c d ef Terms :

Terms , compared with modeled G:

Term 4: g g h h i i

Terms , compared with modeled G:

Terms , compared with modeled G:

Point scatterer Interferometry Optical theorem Paradox

Substitute into representation for interferometry (Snieder et al., 2008, Halliday and Curtis, 2009)…..

This gives: Generalized optical theorem (Heisenberg, 1943)

This gives: For comparison:

Point scatterer Interferometry Optical theorem Non-linear Paradox

Isotropic point scatterer:

(van Rossum et al, 1999) =+++ (Snieder, 1999)

Point scatterer Interferometry Optical theorem Non-linear Paradox

Point scatterer Interferometry Optical theorem Non-linear Paradox

a b c d ef Terms :

Terms , compared with modeled G:

Point scatterer Interferometry Optical theorem Non-linear Paradox Modeling Inversion Interferometry Migration

Modeling, inversion and interferometry in scatterering media Groenenboom and Snieder, 1995; Weglein et al., 2003; Van Manen et al., 2006

Modeling, inversion and interferometry in scatterering media Groenenboom and Snieder, 1995; Weglein et al., 2003; Van Manen et al., 2006 Limiting case: Point scatterer

Resolution function for seismic migration Miller et al., 1987; Schuster and Hu, 2000; Gelius et al., 2002; Lecomte, 2008 Migration deconvolution Yu, Hu, Schuster and Estill, 2006

Born approximation is incompatible with seismic interferometry Conclusions

Born approximation is incompatible with seismic interferometry Seismic interferometry optical theorem non-linear scatterer seismic interferometry Consequences for modeling, inversion, interferometry and migration Conclusions

Born approximation is incompatible with seismic interferometry Seismic interferometry optical theorem non-linear scatterer seismic interferometry Consequences for modeling, inversion, interferometry and migration Conclusions