1 / 17

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

thanh
Download Presentation

Migration

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. 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

More Related