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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. Interferometry. Non-linear. Paradox. Point scatterer. Optical theorem. Interferometry. Modeling

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Seismic interferometry, the optical theorem and a non-linear point scatterer Kees Wapenaar

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Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

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


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Interferometry

Non-linear

Paradox

Point

scatterer

Optical

theorem


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Interferometry

Modeling

Inversion

Interferometry

Migration

Non-linear

Paradox

Point

scatterer

Optical

theorem


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

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, 036606.

Halliday, D. and A.Curtis, 2009,

Generalized optical theorem for surface waves and

layered media:

Physical Review E, 79, 056603.

van Rossum, M. C. W. and T.M. Nieuwenhuizen, 1999,

Multiple scattering of classical waves: microscopy,

mesoscopy, and diffusion:

Reviews of Modern Physics, 71, 313--371.


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Term 1:

a

b


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Term 2:

c

d


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Term 3:

f

e


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Terms 1 + 2 + 3:

c

a

f

e

b

d


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Terms 1 + 2 + 3, compared with modeled G:


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

h

i

h

i

Term 4:

g

g


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Terms 1 + 2 + 3 + 4, compared with modeled G:


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Terms 1 + 2 + 3, compared with modeled G:


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Interferometry

Paradox

Point

scatterer

Optical

theorem


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Substitute into representation for interferometry

(Snieder et al., 2008, Halliday and Curtis, 2009)…..


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

This gives:

Generalized optical theorem (Heisenberg, 1943)


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

This gives:

For comparison:


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Interferometry

Non-linear

Paradox

Point

scatterer

Optical

theorem


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Isotropic point scatterer:


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Isotropic point scatterer:


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

(van Rossum et al, 1999)

=

+

+

+

(Snieder, 1999)


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Interferometry

Non-linear

Paradox

Point

scatterer

Optical

theorem


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Interferometry

Non-linear

Paradox

Point

scatterer

Optical

theorem


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Terms 1 + 2 + 3:

c

a

f

e

b

d


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Terms 1 + 2 + 3 + 4, compared with modeled G:


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Interferometry

Modeling

Inversion

Interferometry

Migration

Non-linear

Paradox

Point

scatterer

Optical

theorem


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Modeling, inversion and interferometry in scatterering media

Groenenboom and Snieder, 1995; Weglein et al., 2003;

Van Manen et al., 2006


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

Modeling, inversion and interferometry in scatterering media

Groenenboom and Snieder, 1995; Weglein et al., 2003;

Van Manen et al., 2006

Limiting case:

Point scatterer


Seismic interferometry the optical theorem and a non linear point scatterer kees wapenaar

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


Conclusions

Conclusions

  • Born approximation is incompatible with seismic interferometry


Conclusions1

Conclusions

  • Born approximation is incompatible with seismic interferometry

  • Seismic interferometry optical theorem

    non-linear scatterer seismic interferometry

  • Consequences for modeling, inversion, interferometry and migration


Conclusions2

Conclusions

  • Born approximation is incompatible with seismic interferometry

  • Seismic interferometry optical theorem

    non-linear scatterer seismic interferometry

  • Consequences for modeling, inversion, interferometry and migration


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