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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
