Tutorial on Green’s Functions, Forward Modeling, Reciprocity Theorems, and Interferometry

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Tutorial on Green’s Functions, Forward Modeling, Reciprocity Theorems, and Interferometry. Reciprocity Eqn. of Correlation Type. Find:. G(A|x). G(A| B ). G( B |x). Free surface. Free surface. x. B. A. B. A. 0. Define Problem. Given:. Reciprocity Eqn. of Correlation Type. *. - PowerPoint PPT Presentation

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Tutorial on Green’s Functions, Tutorial on Green’s Functions, Forward Modeling, Forward Modeling,

Reciprocity Theorems, and Reciprocity Theorems, and InterferometryInterferometry

..

Reciprocity Eqn. of Correlation TypeReciprocity Eqn. of Correlation Type0. Define Problem0. Define Problem

Given:Given: Find:Find:

A A

G(A|G(A|BB) )

Free surface

BB

G(A|x) G(A|x)

A A B B

Free surface

G(G(BB|x) |x)

xx

Reciprocity Eqn. of Correlation TypeReciprocity Eqn. of Correlation Type1. Helmholtz Eqns: 1. Helmholtz Eqns:

2+ k

2[ ] G(A|x) =- (x-A);

2+ k

2[ ] P(B|x) =- (x-B) **

**

2 2+ k[ ] G(A|x) =- (x-A) P(B|x) P(B|x)

2 2+ k[ ] P(B|x) =- (x-B) G(A|x) G(A|x)

** **

G(A|x)P(B|x) P(B|x) - G(A|x)2

= (B-x)G(A|x) - (A-x)P(B|x) 2

**** **

2. Multiply by 2. Multiply by G(A|x)G(A|x) and and P(B|x)P(B|x) and subtract and subtract**

G(A|x)

A A B B

Free surface

P(B|x)

xx

G(A|x) = G(A|x) = P(B|x) P(B|x)P(B|x) G(A|x)G(A|x)2

{ } - - P(B|P(B|xx)) G(A|x)G(A|x)****** [ ]

P(B|P(B|xx)) = = G(A|x) G(A|x) G(A|x) P(BP(B|x|x))2

- G(A|x) - G(A|x) P(BP(B|x|x))[[ ]] ****** [ ]

Reciprocity Eqn. of Correlation TypeReciprocity Eqn. of Correlation Type1. Helmholtz Eqns: 1. Helmholtz Eqns:

2+ k

2[ ] G(A|x) =- (x-A);

2+ k

2[ ] P(B|x) =- (x-B) **

**

2 2+ k[ ] G(A|x) =- (x-A) P(B|x) P(B|x)

2 2+ k[ ] P(B|x) =- (x-B) G(A|x) G(A|x)

** **

G(A|x)P(B|x) P(B|x) - G(A|x)2

= (B-x)G(A|x) - (A-x)P(B|x) 2

**** **

2. Multiply by 2. Multiply by G(A|x)G(A|x) and and P(B|x)P(B|x) and subtract and subtract**

G(A|x) = G(A|x) = P(B|x) P(B|x)P(B|x) G(A|x)G(A|x)2

{ } - - P(B|P(B|xx)) G(A|x)G(A|x)******

P(B|P(B|xx)) = = G(A|x) G(A|x) G(A|x) P(BP(B|x|x))2

- G(A|x) - G(A|x) P(BP(B|x|x))[[ ]] ******

G(A|x)P(B|x) P(B|x) - G(A|x){ { } } = (B-x)G(A|x) - (A-x)P(B|x) ******

G(A|x)

A A B B

Free surface

P(B|x)

xx

Reciprocity Eqn. of Correlation TypeReciprocity Eqn. of Correlation Type3. Integrate over a volume3. Integrate over a volume

4. Gauss’s Theorem4. Gauss’s Theorem

Source lineSource line

G(A|x)P(B|x) P(B|x) - G(A|x) d x3

= G(A|B) - P(B|A){ }{ } ******

G(A|x)P(B|x) P(B|x) - G(A|x) d x2

= G(A|B) - P(B|A){ }{ } n** ** **

G(A|B) G(A|B)

Integration at infinity vanishesIntegration at infinity vanishesA A B B

Free surface

xx

Reciprocity Eqn. of Correlation TypeReciprocity Eqn. of Correlation Type3. Integrate over a volume3. Integrate over a volume

4. Gauss’s Theorem4. Gauss’s Theorem

Source lineSource line

G(A|x)P(B|x) P(B|x) - G(A|x) d x3

= G(A|B) - P(B|A){ }{ } ******

G(A|x)G(B|x) G(B|x) - G(A|x) d x2

= G(A|B) - G(B|A){ }{ }

n** ** **

G(A|B) G(A|B)

Integration at infinity vanishesIntegration at infinity vanishesA A B B

Free surface

xx

Relationship between reciprocal Green’s functionsRelationship between reciprocal Green’s functions

Reciprocity Eqn. of Correlation TypeReciprocity Eqn. of Correlation Type

Source lineSource line

G(A|x)G(B|x) G(B|x) - G(A|x) d x2

= G(A|B) - G(B|A){ }{ }

n** ** **= 2i Im[G(A|B)]= 2i Im[G(A|B)]

Recall Recall G(A|x ) =G(A|x ) =

|r||r|

iwr/ciwr/ceeiw/ciw/c

nn nn rr

G(BG(B|x|x )* )* = =|r||r|

-iwr/c-iwr/cee-iw/c-iw/c

nn nn rr

(1)(1)

(2a)(2a)

(2b)(2b)

Plug (2a) and (2b) into (1)Plug (2a) and (2b) into (1)

G(A|x )G(A|x )ikik

G(B|G(B|xx ) )**-ik-ik

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)r** **= 2i Im[G(A|B)]= 2i Im[G(A|B)] (3)(3)n2ik2ik

Neglect 1/rNeglect 1/r22

A XX

B

Far-Field Reciprocity Eqn. of Correlation TypeFar-Field Reciprocity Eqn. of Correlation Type

G(A|B) G(A|B)

A A B B

Free surface

xx

nn rr ~~~~ 11

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)r** **= = 2i Im[G(A| Im[G(A|BB)])] (3)(3)nkk

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)r** **= = 2i Im[G(A| Im[G(A|BB)])] (4)(4)nkk

AA

nn rr^̂ ^̂

Far-Field Reciprocity Eqn. of Correlation TypeFar-Field Reciprocity Eqn. of Correlation Type

nn rr ~~~~ 11

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)r** **= = 2i Im[G(A|B)] Im[G(A|B)] (3)(3)nkk

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)r** **= = 2i Im[G(A|B)] Im[G(A|B)] (4)(4)nkk

G(A|B) G(A|B)

A A B B

Free surface

xx

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)r** **= = 2i Im[G(A|B)] Im[G(A|B)] (4)(4)nkk

Far-Field Reciprocity Eqn. of Correlation TypeFar-Field Reciprocity Eqn. of Correlation Type

xx

B AB A

G(B|x)*G(B|x)*

xx

B AB A

G(A|x)G(A|x)

xx

B AB A

G(A|B)G(A|B)

Source redatumed from x to BSource redatumed from x to B

Virtual sourceVirtual source

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)r** **= = 2i Im[G(A|B)] Im[G(A|B)] (4)(4)nkk

Far-Field Reciprocity Eqn. of Correlation TypeFar-Field Reciprocity Eqn. of Correlation Type

Source redatumed from x to BSource redatumed from x to B

xx

B AB A

G(B|x)*G(B|x)*

xx

B AB A

G(A|x)G(A|x)

xx

B AB A

G(A|B)G(A|B)Recovering the Green’s function

SummarySummary

Reciprocity correlation theorem, far-field, hi-freq. approx.Reciprocity correlation theorem, far-field, hi-freq. approx.

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)r** **= = 2i Im[G(A|B)] Im[G(A|B)] (4)(4)nkk

G(A|x) G(A|x)

A A B B

Free surface

xx

G(A|B) G(A|B)

A A B B

Free surface

xx

SummarySummary

Green’s theorem, far-field, hi-freq. approx.Green’s theorem, far-field, hi-freq. approx.

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)r** **= = 2i Im[G(A|B)] Im[G(A|B)] (4)(4)nkk

Note: 2i Im[G(A|B)] = G(A|B)-G(A|B)*Note: 2i Im[G(A|B)] = G(A|B)-G(A|B)*Inverse FourierInverse Fourier

TransformTransform

-g(A,t|B,0) + g(A,t|B,0)-g(A,t|B,0) + g(A,t|B,0)

00

Time Time

{ { Mute negative times toMute negative times to

get g(A,t|B,0)get g(A,t|B,0)

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)** **= = 2i Im[G(A|B)] Im[G(A|B)] (4)kk

MATLAB ExerciseMATLAB Exercise

GivenGiven

AA BB

FindFindAA BB

W(W())W(W()*)* |W(|W()|)|22

Zero-Phase aurocorrelation of waveletZero-Phase aurocorrelation of wavelet

function [GABT,GAB,peak]=corrsum(ntime,seismo,A,B,rick,nx) function [GABT,GAB,peak]=corrsum(ntime,seismo,A,B,rick,nx)

sc=zeros(1,2*ntime-1);sc=zeros(1,2*ntime-1);

for i=1:nx; for i=1:nx;

GAx=reshape(seismo(A,i,:),1,ntime); GAx=reshape(seismo(A,i,:),1,ntime);

GBx=reshape(seismo(i,B,:),1,ntime); GBx=reshape(seismo(i,B,:),1,ntime);

sc=xcorr(GBx,GAx)+sc; sc=xcorr(GBx,GAx)+sc;

end end

peak=find(max(rick)==rick); peak=find(max(rick)==rick);

sc=diff(sc);[r c]=size(sc);sc=sc/max(abs(sc));GAB=sc; sc=diff(sc);[r c]=size(sc);sc=sc/max(abs(sc));GAB=sc;

s=reshape(seismo(A,B,:),1,ntime);GABT=s/max(abs(s));s=reshape(seismo(A,B,:),1,ntime);GABT=s/max(abs(s));

Source lineSource line

G(A|x)G(B|x) d x2

= G(A|B) - G(B|A)r** **= = 2i Im[G(A|B)] Im[G(A|B)] (4)nkk

MATLAB ExerciseMATLAB Exercise

Grab a trace from a shot gatherGrab a trace from a shot gather

Correlate trace at A with trace at BCorrelate trace at A with trace at B

SumSum

over over

shots xshots x

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