1 / 15

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

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. *.

wirt
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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));

More Related