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

Outline Green’s Functions Forward Modeling

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

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

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

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

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

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

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

Problem with Huygen’s Principle? ZOOM + -

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

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

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

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

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

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

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

Reciprocity Eqn. of Convolution Type G(A|x) P(B|x) d x 2 = G(A|B) - P(B|A) n 1st interface n r =1 Far field G(A|x) P(B|x) d x 2 + P(B|A) = G(A|B) 2ik Freeze B and hide it 1st interface G(A|x) p(x) d x 2 =- p(A) + g(A) 2ik Remember, g(A) is a ½ space shot gather with src at B and rec at A Reflected waves { Ghost direct-Total pressure { Ghost direct-Total pressure { 1st interface B A Free surface p(x) G(A’|x) (1) 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 p(x) G(A|x) (0) p(x) = p(x) + p(x) + p(x) +… (0) (1) (2) = p(x) (0) Primary reflection

{ { Summary Green’s theorem, far-field, hi-freq. approx. G(A|x) p(x) d x 2 = p(A) 2ik n r scatt. G(A|x) p(x) d x 2 =- p(A) + g(A) 2ik Reflected waves { Scattered pressure field { obliquity 1st interface B A Free surface 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 p(x) G(A|x) (0) p(x) = p(x) + p(x) + p(x) +… (0) (1) (2) = p(x) (0) Primary reflection If A=B and p(x)=cnst. Then we reduce to intuitive ZO modeling formula