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Tutorial on Green’s Theorem and Forward Modeling, SRME, Interferometry, and Migration

Learn about Green's Functions, Huygen’s Principle, SRME, Interferometry, and Migration in seismic data analysis. Understand the principles and applications through detailed explanations and examples.

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Tutorial on Green’s Theorem and Forward Modeling, SRME, Interferometry, and Migration

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  1. Tutorial on Green’s Theorem and Forward Modeling, SRME, Interferometry, and Migration ..

  2. Outline • Green’s Functions • Forward Modeling

  3. G(x,t|x ,t )= s s { Listening time Listening & source coords. 1 t=|r|/c (t-|r|/c) = 0 |r| ottherwise Geometrical Spreading r t=1 t=2 t=3 2 2 2 |r| = x + y + z Propagation time = c t=4 c T 4 What is a Green’s Function? Answer: Point Src. Response of Medium Answer: Impulse-Point Src. Response of Medium

  4. { 1 t-t =|r|/c (t -|r|/c) = s |r| |r| |r| |r| G(x,t|x ,0 ) = s 0 ottherwise Fourier derivative ~ iwr/c e G(x|x ) = s ~ d iwr/c e G(x|x ) = iw/c s 2 + O(1/r ) dr n n r iwr/c e G(x|x ) = iw/c s 2 + O(1/r ) What is a Green’s Function? Answer: Point Src. Response of Medium Answer: Impulse-Point Src. Response of Medium Far field approx k

  5. Wave in Heterogeneous Medium i(wt - wt) G = Ae satisfies except at origin Geometrical speading w i i i i i i Time taken along ray segment .. 1 2 G c G = 2 2 3 4 (2) kr = (kc ) r /c = wt r Valid at high w and smooth media

  6. Outline • Green’s Functions • Forward Modeling • SRME • Acausal GF, Stationarity, Reciprocity • Interferometry • Migration

  7. P(x,T ) 5 T 4 What is Huygen’s Principle? Answer: Every pt. on a wavefront is a secondary src. pt. Common tangent kinematically defines next wavefront. T 5

  8. P(x’,T ) s 5 G(x,t|x’ ,T ) s 5 G(x,t|x ,T ) s 5 s P(x,T ) 5 What is Huygen-Fresnel Principle? Answer: Every pt. on a wavefront is a secondary src. pt. Next wavefront is sum of weighted Green’s functions. T 5

  9. G(x,t|x ,T ) s 5 P(x,T ) 5 x t s s What is Huygen-Fresnel Principle? Answer: Every pt. on a wavefront is a secondary src. pt. Next wavefront is sum of weighted Green’s functions. T 5

  10. Problem with Huygen’s Principle? G(x,t|x ,T ) s 5 P(x,T ) 5 x t s s T 5

  11. Problem with Huygen’s Principle? ZOOM G(x,t|x ,T ) s 5 P(x,T ) 5 x t s s T 5

  12. Problem with Huygen’s Principle? ZOOM + -

  13. Problem with Huygen’s Principle? ZOOM P(x’,T ) P(x’,T ) + 5 5 - -G(x,t|x’,T ) G(x,t|x’ ,T ) s s 5 5 + dn - = PdG/dn +

  14. P(x ) - G(x|x) d d dn dn Green’s Theorem Forward Modeling } 2 d x G(x|x) P(x ) P(x ) = }

  15. Outline • Green’s Functions • Forward Modeling • SRME • Acausal GF, Stationarity, Reciprocity • Interferometry • Migration

  16. Reciprocity Eqn. of Convolution Type Find: G(A|x) G(A|B) G(B|x) Free surface Free surface x B A B A 0. Define Problem Given:

  17. Reciprocity Eqn. of Convolution Type * * { } 2 2 + k [ ] G(A|x) =- (x-A); * 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) - G(A|x) - P(B|x) - G(A|x) P(B|x) G(A|x) P(B|x) 1. Helmholtz Eqns: *

  18. Reciprocity Eqn. of Convolution 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 * * * { } * * * 3 2 d x d x Source line 3. Integrate over a volume 4. Gauss’s Theorem G(A|B) Free surface x B A Integration at infinity vanishes

  19. Reciprocity Eqn. of Convolution 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 { } Hence, the two integrands above are opposite and equal in far-field hi-freq. approximation 3 2 d x d x Layer interface If boundary is far away the major contribution is specular reflection |A-x| |B-x| A B (B-x) n - (A-x) n = x n 3. Integrate over a volume 4. Gauss’s Theorem

  20. Reciprocity Eqn. of Convolution Type n 2 P(B|x) G(A|x) = G(A|B) - P(B|A) 1st interface n r =1 2ik P(B|x) G(A|x) + P(B|A) = G(A|B) Freeze B and hide it 2ik p(x) G(A|x) =- p(A) + g(A) 2 2 2 d x d x d x { { { Far field Reflected waves Ghost direct-Total pressure Ghost direct-Total pressure Free surface p(x) (1) G(A’|x) B A p(x) (0) G(A|x) p(x) = p(x) + p(x) + p(x) +… (0) (1) (2) = p(x) (0) Primary reflection 1st interface Remember, g(A) is a ½ space shot gather with src at B and rec at A 1st interface Dashed yellow ray is just a primary and ghost reflection off of sea-floor. .Higher-order multiples will also contribute..but we assume they are weak

  21. Summary Green’s theorem, far-field, hi-freq. approx. scatt. p(x) G(A|x) = p(A) 2ik obliquity n r 2ik p(x) G(A|x) =- p(A) + g(A) 2 2 d x d x { { Reflected waves Scattered pressure field Free surface B A p(x) (0) G(A|x) p(x) = p(x) + p(x) + p(x) +… (0) (1) (2) If A=B and p(x)=cnst. Then we reduce to intuitive ZO modeling formula = p(x) (0) Primary reflection 1st interface Dashed yellow ray is just a primary and ghost reflection off of sea-floor. .Higher-order multiples will also contribute..but we assume they are weak

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