Seismic interferometry,the optical theorem

and a non-linear point scatterer

Kees Wapenaar Evert Slob

Roel Snieder

Society of Exploration GeophysicistsHouston, October 26, 2009

Point scatterer

Interferometry

Optical theorem

Non-linearParadox

Point scatterer

Interferometry

Optical theorem

Non-linearParadox

ModelingInversionInterferometryMigration

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.

Ax

Bx

( , , )AG 0 x

( , , )BG x 0

( , , ) ( , , ) ( , , )sB A B A B AG G G x x x x x x

( , , ) ( , , ) ( ) ( , , )sB A B AG G G x x x 0 0 x

20( )

( , , )B AG x x

( )

Ax

Bx

D

x

*( , , ) ( , , )B A B AG G x x x x

* 22( , , ) ( , , )A BD

jG G d

c

x x x x x* *( ) ( )s sA A B BG G G G

* * * *s s s sA B A B A B A BG G G G G G G G

o0

o90

o180

o90

a

b

Ax

Bx

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t (s)

�50 0 50 100 150 200 250

�0.4

�0.3

�0.2

�0.1

0

0.1

0.2

0.3

phi (degr)

t (s)

a)

a

b

-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

-50 0 50 100 150 200 250

b)-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

* 2( , , ) ( , , )A BDG G d

x x x x x

Term 1:

o0

o90

o180

o90

c

d

Ax

Bx

�0.4

�0.3

�0.2

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t (s)

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0.1

0.2

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phi (degr)

t (s)

c

d

a)-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

b)

-50 0 50 100 150 200 250

-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

Term 2:

* 2( , , ) ( , , )sA BD

G G d x x x x x

o0

o90

o180

o90

ef

Ax

Bx

�0.4

�0.3

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0.2

0.3

0.4

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2

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0.0

1

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t (s)

�50 0 50 100 150 200 250

�0.4

�0.3

�0.2

�0.1

0

0.1

0.2

0.3

phi (degr)

t (s)

e

f

a)-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

b)

-50 0 50 100 150 200 250

-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

Term 3:

* 2( , , ) ( , , )sA BD

G G d x x x x x

o0

o90

o180

o90

a

b

c

d

ef

Ax

Bx

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�0.3

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0.2

0.3

0.4

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2

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15

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0

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0.0

1

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15

0.0

2

t (s)

�50 0 50 100 150 200 250

�0.4

�0.3

�0.2

�0.1

0

0.1

0.2

0.3

phi (degr)

t (s)

a)-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

b)

-50 0 50 100 150 200 250

-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

Terms 1 + 2 + 3:

* * * 2{ }s sA B A B A BDG G G G G G d

x

�0.4 �0.3 �0.2 �0.1 0 0.1 0.2 0.3 0.4

�0.02

�0.015

�0.01

�0.005

0

0.005

0.01

0.015

0.02

t (s)-0.4 -0.3 -0.2 -0.1 0.40.0 0.1 0.2 0.3t (s)

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

�0.4

�0.3

�0.2

�0.1

00.1

0.2

0.3

0.4

�0.0

2

�0.0

15

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1

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05

0

0.0

05

0.0

1

0.0

15

0.0

2

t (s)

�50 0 50 100 150 200 250

�0.4

�0.3

�0.2

�0.1

0

0.1

0.2

0.3

phi (degr)

t (s)

a)-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

b)

-50 0 50 100 150 200 250

-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

o0

o90

o180

o90

Ax

Bx

Term 4:

* 2( , , ) ( , , )s sA BD

G G d x x x x x

g

g

h

h

i

i

�0.4 �0.3 �0.2 �0.1 0 0.1 0.2 0.3 0.4

�0.02

�0.015

�0.01

�0.005

0

0.005

0.01

0.015

0.02

t (s)-0.4 -0.3 -0.2 -0.1 0.40.0 0.1 0.2 0.3t (s)

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

�0.4 �0.3 �0.2 �0.1 0 0.1 0.2 0.3 0.4

�0.02

�0.015

�0.01

�0.005

0

0.005

0.01

0.015

0.02

t (s)-0.4 -0.3 -0.2 -0.1 0.40.0 0.1 0.2 0.3t (s)

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

Point scatterer

Interferometry

Optical theorem

Paradox

Axx

k A k

AxBx

A kBk

( , , ) ( , , ) ( , , )sB A B A B AG G G x x x x x x

4( , , ) ( , , ) ( , ) ( , , )s

B A B B A AG G f G

x x x 0 k k 0 x

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

( , )B Af k k

Axx

k A k

AxBx

Bk

* *1{ ( , ) ( , )} ( , ) ( , )

2 4A B B A A B

kf f f f d

j

k k k k k k k k

This gives:

Generalized optical theorem (Heisenberg, 1943)

A k

( , )B Af k k

Axx

k A k

AxBx

Bk

* *1{ ( , ) ( , )} ( , ) ( , )

2 4A B B A A B

kf f f f d

j

k k k k k k k k

This gives:

A k

* * 22( , ) ( , ) ( , ) ( , )B A B A A BD

jG G G G d

c

x x x x x x x x xFor comparison:

( , )B Af k k

Point scatterer

Interferometry

Optical theorem

Non-linearParadox

Axx

k A k

AxBx

* *1{ ( , ) ( , )} ( , ) ( , )

2 4A B B A A B

kf f f f d

j

k k k k k k k k

Bk A k

( , )B Af k k

Axx

k A k

AxBx

* *1{ ( , ) ( , )} ( , ) ( , )

2 4A B B A A B

kf f f f d

j

k k k k k k k kIsotropic point scatterer:

2( ) | |f k f

Bk A k

( )f

Axx

k A k

AxBx

* *1{ ( , ) ( , )} ( , ) ( , )

2 4A B B A A B

kf f f f d

j

k k k k k k k kIsotropic point scatterer:

2( ) | |f k f

Bk A k

24( ) | |

4

kf

( )

2( ) | |4

k

21 0

22 1 14

kj

1 1 1( , )regG 0 0 (van Rossum et al, 1999)

= + + + …….= + + +

(Snieder, 1999)

1 1 1 1 1 1( , ) ( , ) ( , )reg reg regG G G 0 0 0 0 0 0

Point scatterer

Interferometry

Optical theorem

Non-linearParadox

Point scatterer

Interferometry

Optical theorem

Non-linearParadox

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t (s)

�50 0 50 100 150 200 250

�0.4

�0.3

�0.2

�0.1

0

0.1

0.2

0.3

phi (degr)

t (s)

a)-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

b)

-50 0 50 100 150 200 250

-0.4

-0.3

-0.2

-0.1

0.4

0.0

0.1

0.2

0.3

t (s)

o0

o90

o180

o90

a

b

c

d

ef

Ax

Bx

Terms 1 + 2 + 3:

* * * 2{ }s sA B A B A BDG G G G G G d

x

�0.4 �0.3 �0.2 �0.1 0 0.1 0.2 0.3 0.4

�0.02

�0.015

�0.01

�0.005

0

0.005

0.01

0.015

0.02

t (s)-0.4 -0.3 -0.2 -0.1 0.40.0 0.1 0.2 0.3t (s)

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

Point scatterer

Interferometry

Optical theorem

Non-linearParadox

ModelingInversionInterferometryMigration

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

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

= + + + …….

Limiting case:Point scatterer

Resolution function for seismic migrationMiller et al., 1987; Schuster and Hu, 2000; Gelius et al., 2002; Lecomte, 2008

Migration deconvolutionYu, 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