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

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

Outline • Green’s Functions • Forward Modeling

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

{ 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

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

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

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

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

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

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

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

Problem with Huygen’s Principle? ZOOM + -

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 +

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

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

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:

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: *

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

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

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

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

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