1. Migration
Intuitive
Least Squares
Migration
Green’s Theorem

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

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

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

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

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

7. ~ ~ Forward Modeling (d = Lo) ZO Depth Migration (m L d) d(x, )   e..
d(x, ) 
xx’
Fourier
Transform
A(x,x’) 
xx’  i e ~
m(x’)
d(x) ~ ~  
d(x) =  x’
g(x|x’) w 2
m(x’) x ..
m(x’)
reflectivity  x
d(x, )
A(x,x’) 
xx’ =

8. Migration m=L d Forward Problem: d=Lm Intuitive Least Squares
Japan Sea Example
Migration
m=L d T
Green’s Theorem

9. LSM Image with a Ricker Wavelet (15 Hz)
Kirchhoff Migration Image 2
Depth (km)
X (km)
LSM Image
Actual Model 2
Depth (km)
X (km)

10. 2D Poststack Data from Japan Sea
JAPEX 2D SSP marine data description: Acquired in 1974, Dominant frequency of 15 Hz. 5
TWT (s) 20
X (km) 10

11. LSM vs. Kirchhoff Migration
LSM Image
0.7
1.9
Depth (km)
2.4
4.9
X (km)
0.7
1.9
Depth (km)
2.4
4.9
X (km)
Kirchhoff Migration Image

12. Migration
Intuitive
Least Squares
Migration
Green’s Theorem

13. Summary  d(x, ) .. m(x’)  = 1. ZO migration: cos xx’ x A(x,x’)q
obliquity .. 
xx’
m(x’)  x
d(x, ) =
1. ZO migration:
cos q
Approx. reflectivity
A(x,x’)
2. ZO migration assumptions: Single scattering data
3. ZO migration matrix-vec: m=L d T ~
Compensates for
Illumination footprint and poor
illumination
4. LSM ZO migration matrix-vec: m=[L L] L d T -1
5. ZO migration smears an event along appropriate
doughnut

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

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