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

This tutorial provides a comprehensive guide to understanding Green’s functions, forward modeling, reciprocity theorems, and interferometry. It covers key concepts, equations, and practical applications in geophysics.

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

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

  2. 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:

  3. Reciprocity Eqn. of Correlation Type * Free surface x 2 2 + k [ ] G(A|x) =- (x-A); B A * P(B|x) G(A|x) 2. Multiply by G(A|x) and P(B|x) and subtract 2 2 + k [ ] P(B|x) =- (x-B) * 2 2 + k P(B|x) [ ] G(A|x) =- (x-A) P(B|x) 2 2 * + k G(A|x) [ ] P(B|x) =- (x-B) * G(A|x) 2 2 2 2 - G(A|x) P(B|x) G(A|x) P(B|x) = (B-x)G(A|x) - (A-x)P(B|x) * * * [ * * * ] G(A|x) = { } G(A|x) P(B|x) P(B|x) ] [ * * ] [ * P(B|x) = P(B|x) G(A|x) G(A|x) P(B|x) G(A|x) - P(B|x) - G(A|x) 1. Helmholtz Eqns: *

  4. Reciprocity Eqn. of Correlation Type * Free surface x { } 2 2 + k [ ] G(A|x) =- (x-A); B A * P(B|x) G(A|x) 2. Multiply by G(A|x) and P(B|x) and subtract 2 2 + k [ ] P(B|x) =- (x-B) * 2 2 + k P(B|x) [ ] G(A|x) =- (x-A) P(B|x) 2 2 * + k G(A|x) [ ] P(B|x) =- (x-B) * G(A|x) 2 2 2 2 - G(A|x) P(B|x) G(A|x) P(B|x) = (B-x)G(A|x) - (A-x)P(B|x) * * * * * * G(A|x) = { } G(A|x) P(B|x) P(B|x) [ * * ] * P(B|x) = P(B|x) G(A|x) G(A|x) * * = (B-x)G(A|x) - (A-x)P(B|x) * G(A|x) P(B|x) - P(B|x) - G(A|x) - G(A|x) P(B|x) G(A|x) P(B|x) 1. Helmholtz Eqns: [ *

  5. Reciprocity Eqn. of Correlation Type - G(A|x) = G(A|B) - P(B|A) P(B|x) G(A|x) P(B|x) { } - G(A|x) = G(A|B) - P(B|A) P(B|x) G(A|x) P(B|x) n * * * { } * * * 2 3 d x d x Source line G(A|B) Free surface x B A Integration at infinity vanishes 3. Integrate over a volume 4. Gauss’s Theorem

  6. Reciprocity Eqn. of Correlation Type - G(A|x) = G(A|B) - P(B|A) P(B|x) G(A|x) P(B|x) n * * * { } 2 3 d x d x Source line Relationship between reciprocal Green’s functions G(A|B) Free surface x B A Integration at infinity vanishes 3. Integrate over a volume 4. Gauss’s Theorem { } * * * - G(A|x) = G(A|B) - G(B|A) G(B|x) G(A|x) G(B|x)

  7. Reciprocity Eqn. of Correlation Type n n r = 2i Im[G(A|B)] = 2i Im[G(A|B)] n n r iwr/c e |r| |r| iw/c G(A|x ) = ik (2a) Recall 2 2 d x d x Source line Source line n r -iwr/c n e -ik -iw/c (2b) G(B|x )* = G(B|x ) B X * * 2ik A G(B|x) G(A|x) (3) = G(A|B) - G(B|A) * 2 Neglect 1/r G(A|x ) { } (1) * * * - G(A|x) = G(A|B) - G(B|A) G(B|x) G(A|x) G(B|x) Plug (2a) and (2b) into (1)

  8. Far-Field Reciprocity Eqn. of Correlation Type n n r r ^ ^ n r k = 2i Im[G(A|B)] = 2i Im[G(A|B)] ~ n r 1 ~ A 2 2 d x d x k Source line Source line * * * * G(B|x) G(B|x) G(A|x) G(A|x) (3) (4) = G(A|B) - G(B|A) = G(A|B) - G(B|A) G(A|B) Free surface x B A

  9. Far-Field Reciprocity Eqn. of Correlation Type n n r r k = 2i Im[G(A|B)] = 2i Im[G(A|B)] ~ n r 1 ~ 2 2 d x d x k Source line Source line G(A|B) Free surface x B A * * * * G(B|x) G(B|x) G(A|x) G(A|x) (3) (4) = G(A|B) - G(B|A) = G(A|B) - G(B|A)

  10. Far-Field Reciprocity Eqn. of Correlation Type n r = 2i Im[G(A|B)] x x x 2 d x k Source line B A B A B A Virtual source G(B|x)* G(A|x) G(A|B) * * G(B|x) G(A|x) (4) = G(A|B) - G(B|A) Source redatumed from x to B

  11. Far-Field Reciprocity Eqn. of Correlation Type n r = 2i Im[G(A|B)] x x x 2 d x k Source line B A B A B A G(B|x)* G(A|x) G(A|B) * * G(B|x) G(A|x) (4) = G(A|B) - G(B|A) Source redatumed from x to B Recovering the Green’s function

  12. Summary n r = 2i Im[G(A|B)] 2 G(A|x) G(A|B) d x k Source line Free surface Free surface x x B A B A * * G(B|x) G(A|x) (4) = G(A|B) - G(B|A) Reciprocity correlation theorem, far-field, hi-freq. approx.

  13. Summary n r Green’s theorem, far-field, hi-freq. approx. = 2i Im[G(A|B)] 2 d x k Source line Inverse Fourier Transform { { * * G(B|x) G(A|x) (4) = G(A|B) - G(B|A) 0 Time Note: 2i Im[G(A|B)] = G(A|B)-G(A|B)* -g(A,t|B,0) + g(A,t|B,0) Mute negative times to get g(A,t|B,0)

  14. MATLAB Exercise |W(w)| 2 W(w) W(w)* = 2i Im[G(A|B)] Zero-Phase aurocorrelation of wavelet A B Find 2 d x Source line Given A B k * * G(A|x) G(B|x) (4) = G(A|B) - G(B|A)

  15. MATLAB Exercise n r = 2i Im[G(A|B)] Grab a trace from a shot gather 2 d x k Sum over shots x Source line Correlate trace at A with trace at B * * G(B|x) G(A|x) (4) = G(A|B) - G(B|A) function [GABT,GAB,peak]=corrsum(ntime,seismo,A,B,rick,nx) sc=zeros(1,2*ntime-1); for i=1:nx; GAx=reshape(seismo(A,i,:),1,ntime); GBx=reshape(seismo(i,B,:),1,ntime); sc=xcorr(GBx,GAx)+sc; end peak=find(max(rick)==rick); 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));

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