1. Zero-Offset Data d = L o ò r ) ( g = d dr r ) ( g = d
(aka poststack data)
d = L o o r ) ( ò g = d dr o r ) ( g = d
V/2
Depth
Time

2. 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

3. Seismic Inverse Problem
Given: d = Lm
Find: m(x,y,z)
Soln: min || Lm-d || 2
Waveform inversion
(non-linear)
Least squares migration
(linear)
m = [L L] L d T -1
Migration
L d T

4. Migration Intuitive: Modeling Least Squares Poststack Mig Prestack
Green’s Theorem

5. Smear Reflections along Fat Circles
ZO Migration
Smear Reflections along Fat Circles  xx
+ T o
Exploding Reflector assumption:
Let c = v/2, or ½ actual velocity so
Exploding reflector time=2-way time
2-way time
(x-x ) + y 2 c  xx = x
Where did reflections
come from?
Thickness = c*T /2 o x
d(x , )  xx

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

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

8. ZO Data
Migration
ZO Data
0 km
3 km
0 km km

9.  m(x) =  d (g, ) ZO Migration: Smear Trace Sample over Circle g xg
Loop over data
for ixtrace=1:ntrace;
for ixs=istart:iend;
for izs=1:nz;
r = sqrt((ixtrace*dx-ixs*dx)^2+(izs*dx)^2);
time = 1 + round( r/c/dt );
mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time);
end;
Loop over x in model
Loop over z in model
Smear over
circle
Traveltime

10.  m(x) =  d (g, ) ZO Migration: Smear Trace Sample over Circle g xg
Loop over data
for ixtrace=1:ntrace;
for ixs=istart:iend;
for izs=1:nz;
r = sqrt((ixtrace*dx-ixs*dx)^2+(izs*dx)^2);
time = 1 + round( r/c/dt );
mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time);
end;
Loop over x in model
Loop over z in model
Traveltime

11. ZO Migration: Sum Trace Samples along migration hyperbola into m(x)
(x,z)
(x’,z’)
Loop over x in model
Loop over z in model
for ixtrace=1:ntrace;
for ixs=istart:iend;
for izs=1:nz;
r = sqrt((ixtrace*dx-ixs*dx)^2+(izs*dx)^2);
time = 1 + round( r/c/dt );
mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time);
end;
Loop over data
Sum samples along hyperbola

12. ZO Diffraction Stack Migration
d (g, ) xg
m(x) =  g 
Trial image pt x
traces x g
It is often thought that RTM does not enjoy filtering tricks of KM such as U+D separation, obliquity factor, angle gather separation, anti-aliasing filter, etc. This is not true as shown above. The RTM formula is shown above in traditional form: apply adjoint Green’s function to data and backpropagate data, then zero-lag correlation with source field. Rearranging brackets gives different interpretation: RTM is just like KM in the sense that you apply a dot product of the hyperbolas to the data to get migration image. In this case the hyperbolas conatin all the scattering events and the Green’s functions are computed by FD solves rather than ray tracing.

13. 2D dot product of migration
ZO Diffraction Stack Migration
d (g, ) xg
m(x) =  g 
Trial image pt x
traces
2D dot product of migration
Operator and d(g,t) x g
It is often thought that RTM does not enjoy filtering tricks of KM such as U+D separation, obliquity factor, angle gather separation, anti-aliasing filter, etc. This is not true as shown above. The RTM formula is shown above in traditional form: apply adjoint Green’s function to data and backpropagate data, then zero-lag correlation with source field. Rearranging brackets gives different interpretation: RTM is just like KM in the sense that you apply a dot product of the hyperbolas to the data to get migration image. In this case the hyperbolas conatin all the scattering events and the Green’s functions are computed by FD solves rather than ray tracing.
Migration Image

14. ZO Diffraction Stack Migration: C(x,z)
d (g, ) xg
m(x) =  g 
Trial image pt x
Ray tracing
It is often thought that RTM does not enjoy filtering tricks of KM such as U+D separation, obliquity factor, angle gather separation, anti-aliasing filter, etc. This is not true as shown above. The RTM formula is shown above in traditional form: apply adjoint Green’s function to data and backpropagate data, then zero-lag correlation with source field. Rearranging brackets gives different interpretation: RTM is just like KM in the sense that you apply a dot product of the hyperbolas to the data to get migration image. In this case the hyperbolas conatin all the scattering events and the Green’s functions are computed by FD solves rather than ray tracing.

15. 3D ZO Diffraction Stack Migration
d (g, ) xg
m(x) =  g 
Trial image pt x
Impulse Response of Mig. Op.
It is often thought that RTM does not enjoy filtering tricks of KM such as U+D separation, obliquity factor, angle gather separation, anti-aliasing filter, etc. This is not true as shown above. The RTM formula is shown above in traditional form: apply adjoint Green’s function to data and backpropagate data, then zero-lag correlation with source field. Rearranging brackets gives different interpretation: RTM is just like KM in the sense that you apply a dot product of the hyperbolas to the data to get migration image. In this case the hyperbolas conatin all the scattering events and the Green’s functions are computed by FD solves rather than ray tracing.

16. Migration Intuitive: Modeling Least Squares Poststack Prestack
Green’s Theorem

17. 3D Prestack Diffraction Stack Migration
Motivation: ZO only good if no lateral vel change
It is often thought that RTM does not enjoy filtering tricks of KM such as U+D separation, obliquity factor, angle gather separation, anti-aliasing filter, etc. This is not true as shown above. The RTM formula is shown above in traditional form: apply adjoint Green’s function to data and backpropagate data, then zero-lag correlation with source field. Rearranging brackets gives different interpretation: RTM is just like KM in the sense that you apply a dot product of the hyperbolas to the data to get migration image. In this case the hyperbolas conatin all the scattering events and the Green’s functions are computed by FD solves rather than ray tracing. s g x

18.  m(x) = 3D Prestack Diffraction Stack Migration = d(x’,  +  ) s g xsx
s,g xg
m(x) =
Trial image pt x
It is often thought that RTM does not enjoy filtering tricks of KM such as U+D separation, obliquity factor, angle gather separation, anti-aliasing filter, etc. This is not true as shown above. The RTM formula is shown above in traditional form: apply adjoint Green’s function to data and backpropagate data, then zero-lag correlation with source field. Rearranging brackets gives different interpretation: RTM is just like KM in the sense that you apply a dot product of the hyperbolas to the data to get migration image. In this case the hyperbolas conatin all the scattering events and the Green’s functions are computed by FD solves rather than ray tracing. s g x

19. Prestack Migration Question: Why Prestack when poststack
migration seems good enough?
Answer: Stacking to get stacked section assumes
layered medium assumption.
Solution: Migrate shot gathers so no layer
assumption needed. This is prestack
migration.

20. Diffraction Stack Migration: Prestack
Down time
Up time
T(s,g) =  sx xg +
Where is scatterer? s x g  sx  xg 
s,g
d(s,g, )  sx xg +
Narrow band case: direct wave correlated with data

21. Diffraction Stack Modeling: Prestack
115.
Diffraction Stack Modeling: Prestack
m = L d T
d = L m     i i ~
W( ) ~ e
m(x) ~ sx  x xg e
d(s,g) = w 2
A(s,x)
A(g,x)

22. Diffraction Stack Migration: Prestack
115.
Diffraction Stack Migration: Prestack
m = L d T -  -
d(s,g,  +  ) sx xg    d ò i i ~
W( ) ~ * e sx 
s,g xg e w 2
m(x) = ~
d(s,g)
A(s,x)
A(g,x)
Broadband case
W( )=1 ~ ..
A(s,x) 
s,g
A(x,g) =
m(x)
Narrow band case: direct wave correlated with data

23. MATLAB Inefficient Prestack Migration
Data Loops
Model Loops
for isx=1:nx % Loop over shot
for igx=1:nx % Loop over receivers
for ix=1:nx % Loop over model x
for iz=1:nx % Loop over model z
t=timer(ix,iz,isx)+timer(ix,iz,igx)
sample=gather(isx,igx,t) % Shot gather has 2 time derivatives
mig(ix,iz)=mig(ix,iz)+sample
end

24. MATLAB Prestack Migration

25. Poststack vs Prestack Migration

26. Poststack vs Prestack Migration

27. Prestack Migration 1. No 1D assumption about velocity model
2. More sensitive to velocity model errors
compared to poststack migration
3. More than 100 times slower than poststack
migration
4. More sensitive to velocity model than time
migration

28. Summary m(x) =    m(x) = 3D ZO Diffraction Stack Migration d (g, )
Migration Motivation: diffractions, dipping layers,
conflicting dips, out-of-plane reflections
3D ZO Diffraction Stack Migration
d (g, ) xg
m(x) =  g 
Trial image pt x
3D Prestack Diffraction Stack Migration  =
d(x’,  +  ) sx
s,g xg
m(x) =
Trial image pt x

29. Migration Intuitive: Modeling Least Squares Poststack Prestack
Green’s Theorem

30. Iterative Least Squares Migration
Step 1:
Step 2:
Step 3:
Step 4:

31. MATLAB SD Least Squares Migration
p=p % Data without direct wave
m=adjoint(p,c) % Initial reflectivity model
c % Velocity model
for i=1:niter
p=forward(m,c) % predicted data
alpha=step(p,p0,c,m) % step length
dP=p-p % data residual
dm =adjoint(dP,c) % migrate residual
m = m –alpha*dm % Update model
end

32. 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*

33. Dot Product Test with CG code
Actual
model
Predicted
model
Actual
data
Predicted
data
d Lm =(d,Lm) = (Lm,d) = m L d T T T
d=forward(m,c)
m=adjoint(d,c)
d d = T m T m
All migration codes should pass the dot product test

34. Migration r = L L L Least Squares: Intuitive: Modeling Examples
Footprint
Intuitive: Modeling
Poststack
Prestack
Migration
r = L L L T . . . 1 .
Green’s Theorem . .
Migration butterfly

35. Migration = Blurred r m = L d d = L r T but Migrated Section Modeling
Data

36. True Reflectivity Model r
Migration = Blurred r T
m = L
but
d = L r
L r
Migrated Section
Migration Image m =
True Reflectivity Model r

37. r L = m Migration Deconvolution = Reflectivity Migration image T
Migration Green’s function

38. L r L = m = Migration Deconvolution ] [ Migration Deconvolution -1 1 T

39. Migration Deconvolution-1 ] [ r = m = 1

40. L r = m = Migration Deconvolution ] [ -1 1 T
Assume Local v(z) Approximation

41. Migration Noise Problems
Note: Artifacts stronger near surface. Why?
Footprint
Migration noise and artifacts
Depth (km)
Weak illumination
Irregular acquisition geometry
Limited recording aperture
Footprint and aliasing
Some footprint caused by
Irregular acquisition geometry
Limited recording aperture
aliasing and illumination loss in migration imaging
3.5

42. Wave Equation Migration Before LSM
Depth (km) 10
X (km) 20

43. Wave Equation Migration after MD X (km)
Depth (km) 10
X (km) 20

44. Acquisition Footprint (Geophone Aliasing)
Coarse Kirchhoff Migration Image (15 Hz) 2
Depth (km)
X (km)
LSM Image (15 Hz)
Actual Model 2
Depth (km)
X (km)
Note: Artifacts stronger near surface. Why?

45. Standard Kirchhoff Image vs LSM Image
Kirchhoff Migration Image (15 Hz) 2
Depth (km)
X (km)
LSM Image (15 Hz)
Actual Model 2
Depth (km)
X (km)

46. Migration Least Squares: Intuitive: Modeling Examples Poststack
Footprint
Intuitive: Modeling
Poststack
Prestack
Migration
Green’s Theorem

47. 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) 48

48. 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

49. Kirchhoff MD

50. Kirchhoff LSM

51. Migration Summary m(x) =    m(x) = d (g, )
Migration Motivation: diffractions, dipping layers,
conflicting dips, out-of-plane reflections
Full Waveform Inversion
m = m - aL dm
(k+1)
(k) T
Least Squares Migration
m = m - aL dm
(k+1)
(k) T
3D ZO Diffraction Stack Migration
d (g, ) xg
m(x) =  g 
Trial image pt x
3D Prestack Diffraction Stack Migration  =
d(x’,  +  ) sx
s,g xg
m(x) =
Trial image pt x

52. ZO 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

53. 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

54. Kirchhoff MD

55. Kirchhoff MD

56. Kirchhoff MD

57. 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