1. Migration MigrationIntuitive Least Squares Green’s Theorem

2. 2-way time (x-x ) + y 22 c/2 xxxxxxxx =  xxxxxxxx + T o ZO Migration ZO Migration Smear Reflections along Fat Circles Smear Reflections along Fat Circlesx x d(x, )  xxxxxxxx Thickness = c*T /2 o Where did reflections come from?

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

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

5. Migration MigrationIntuitive Least Squares Green’s Theorem Forward Problem: d=Lm m=L dm=L dm=L dm=L dT

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

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

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

9. Review Inverse Acoustic ProblemInverse Acoustic Problem Find: m( r ) (c - c )/(c + c ) m( r ) (c - c )/(c + c )1122 Given: d(r ) rm  )( Soln: L d L d T Soln:  o rdrr)()(  g rm  )(  dr * M(x’) reflectivity  x d(x, ) A(x,x’)  xx’ = Smear & Sum Data

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

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

12. Prestack Migration

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

14. Exploding Reflector ½ Velocity Depth Time V/2  oo rorr)()(  g  rd  )(  dr o  oo rorr)()( g  rd  )(  r o d = L o

15.  rorr)()(  g  rd  )(  dr ExplodingReflector Forward Acoustic Problem Given: d + k d = 0 22 Find: d(r) Soln:  rmrr)()(  g  rd  )(  d(r )dr inc  rmrr)()( g  rd  )(  r d = L m Reflectivity

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

17. Acoustic ZO Migration Depth c 2c1 Find: m( r ) (c - c )/(c + c ) m( r ) (c - c )/(c + c )1122 Given: d(r )=Lm ro  )( Soln: L d L d T Soln:  o rdrr)()(  g rm  )(  dr *