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Migration

Migration. Intuitive. Least Squares. Green’s Theorem. Migration. ZO Migration Smear Reflections along Fat Circles. . . x x. + T. x x. o. 2-way time. x. Thickness = c*T /2. x. o. . 2. 2. ( x - x ) + y. =. x x. c/2. d ( x , ). Where did reflections

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Migration

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  1. Migration Intuitive Least Squares Green’s Theorem Migration

  2. ZO Migration Smear Reflections along Fat Circles   xx + T xx o 2-way time x Thickness = c*T /2 x o  2 2 (x-x ) + y = xx c/2 d(x , ) Where did reflections come from?

  3. ZO Migration Smear Reflections along Fat Circles  xx & Sum  1-way time x Hey, that’s our ZO migration formula d(x , )

  4. ZO Migration Smear Reflections along Circles & Sum  1-way time x In-Phase Out-of--Phase  d(x , ) xx m(x)=

  5. Migration Forward Problem: d=Lm Intuitive Least Squares T m=L d Green’s Theorem Migration

  6. i  xx’ m(x’)  x’ reflectivity d = L m e A(x,x’) i ij j j Born Forward Modeling ~ d(x) = g(x|x’)

  7. Seismic Inverse Problem 2 Soln: min || Lm-d || migration waveform inversion ‘ -1 T T m = [L L] L d T L d Given: d= Lm Find: m(x,y,z)

  8. i ZO Depth Migration (o L d) T    xx’ xx’ xx’ M(x’)       x’ x Fourier Transform reflectivity e x A(x,x’) ~ d(x, ) d(x) d(x, ) = m(x’) A(x,x’) Forward Modeling (d = Lo) ~ d(x) = g(x|x’) o(x’) ..

  9. Review  r r r xx’ ( r r ) d ( r ) M(x’)  d(r ) o Given: x reflectivity Find: dr * Soln: Soln: m( r ) (c - c )/(c + c ) r 1 2 1 2 m ( r ) d(x, ) = Smear r ò T g L d m ( r ) & Sum Data A(x,x’) • Inverse Acoustic Problem

  10. ZO Data Migration ZO Data 0 km 3 km 0 km 7 km

  11. Find: m that minimizes sum of squared residuals r = L m - d j i ij j i (r ,r) = ([Lm-d],[Lm-d]) = 2 m L Lm -2 m Ld (r ,r) = m L Lm -2m Ld-d d = 0 d d d i i dm i dm dm L Lm = Ld Recall: Lm=d Least Squares For all i Normal equations

  12. Prestack Migration

  13. d  d  x’ x’ ~ ~ W( ) W( )=1 ò ò Broadband case d(x’,  +  ) = xx’ x’x’’ A(x’’,x’) A(x,x’) 115. Diffraction Stack Migration: Prestack - -     i i * xx’ ~ e x’x’’ e m(x) = d(x’) A(x,x’) A(x’’,x’) .. Narrow band case: direct wave correlated with data

  14. r o r r r r ( ( r r r r ) ) o o ( ( r r ) ) o o o o d = L o dr o r r ò = = g g d d ( ( r r ) ) Exploding Reflector ½ Velocity V/2 Depth Time

  15. r r r r r r r ( ( ( r r r r r r ) ) ) o m m ( ( ( r r r ) ) ) Given: d = Lm r d(r) Find: inc dr d(r )dr Soln: 2 2 d + k d = 0 Exploding Reflector r r r ò ò = = = g g g d d d ( ( ( r r r ) ) ) Reflectivity Forward Acoustic Problem

  16.        j i i j i i j i Recall: (u,u) = u* u i i Recall: (v,Lu) = v* ( L u ) i ij j [ L v* ]u = j ij i = [ L* v ]* u ij i j So adjoint of L is L L* ij Dot Products and Adjoint Operators

  17. r o ( r ) Acoustic ZO Migration r r r ( r r ) d ( r ) d(r )=Lm o Given: Find: dr * Soln: Soln: m( r ) (c - c )/(c + c ) c 1 2 1 2 1 r c ò T g L d m ( r ) 2 Depth

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