tutorial on green’s functions, forward modeling, reciprocity theorems, and interferometry
DESCRIPTION
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 PresentationTRANSCRIPT
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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
..
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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
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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))[[ ]] ****** [ ]
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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
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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
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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
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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
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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^̂ ^̂
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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
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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
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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
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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
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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)
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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
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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