1. Reverse Time Migration=
Generalized Diffraction Migration
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.

2. Outline 1. RTM = GDM 2. Implications Superresolution Filtering
Target Oriented RTM
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.
Fast LSM
Diffraction Selective
Perfect Migration Operators

3. = dot product data with hyperbola
Generalized Diff. Migration
Reverse Time Migration
Trial image pt x
Calc. Green’s Func. By FD solves
d(r) = m(x) * *
G(s|x)
G(x|r) [ ] w
,r,s
Generalized Kirchhoff kernel
Convolution of G(s|x) with G(x|r)
= dot product data with hyperbola
Direct wave
Backpropagated
traces
T=0
QED: RTM can now enjoy:
Anti-aliasing filter
Obliquity factor
Angle Gathers
UD Separation
Decomplexify back&forward
felds according 2 taste
Etc. etc. x s r
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.
Expensive to store Calc. Green’s Func. By FD solves

4. = dot product data with hyperbola
Most Kirchhoff Tricks for Kirchhoff Migration can be Implemented for RTM
Generalized Kirch. Migration
Reverse Time Migration
Trial image pt x
Calc. Green’s Func. By FD solves
d(r) = m(x) * *
G(s|x)
G(x|r) [ ] w
,r,s
Generalized Kirchhoff kernel
Convolution of G(s|x) with G(x|r)
= dot product data with hyperbola
Direct wave
Backpropagated
traces
T=0
QED: RTM can now enjoy:
Anti-aliasing filter
Obliquity factor
Angle Gathers
UD Separation
Decomplexify back&forward
felds according 2 taste
Etc. etc. x s r
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.
Expensive to store Calc. Green’s Func. By FD solves

5. Outline 1. RTM = GDM 2. Implications Superresolution Filtering
Target Oriented RTM
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.
Fast LSM
Diffraction Selective
Perfect Migration Operators

6. 1. Low-Fold Stack vs Superstack 2. Poor Resolution vs Superresolution
Resolution of KM vs GDM
Kirchhoff Mig vs GDM
Multiples
time
Multiples
Primary
1. Low-Fold Stack vs Superstack
2. Poor Resolution vs Superresolution
3. Caution: RTM sensitive to mig. vel. errors

7. Rayleigh Resolution
time L
migrate
Dx = 0.25lz/L

8. This is highest fruit on the tree..who dares pick it?
Is Superresolution by RTM Achievable?
Tucson, Arizona Test
This is highest fruit on the tree..who dares pick it?
60 m
~Kirchhoff Mig.
Poststack Migration
~Scattered RTM
Can RTM achieve superresolution via scattering? Test in Arizona suggests 3x improvement in spatial resolution if RTM is done right. Sources were excited in mine and seismograms recorded at surface. These seismograms were migrated by the EXACT RTM migration operator (Green’s functions were recorded so we used these to exactly RTM migrate data..no velocity model needed!). Results show 3x improvement in spatial resolution of RTM scattered image compared to ~KM. The ~KM was achieved by muting out all but first arrival in Green’s functions before we formed focusing kernel. See next slide for muted Green’s functions. Above should be resolution goal we might all try to achieve,,,above shows the highest fruit on the tree..who dares pick it? Not only can we achieve better resolution but above suggests we can possibly cut aperture width by half.
(Hanafy et al., 2008)

9. Can Scatterers Beat the Resolution Limit?
Recorded Green’s functions G(s|x) divided into:
Shot gathers with direct arrivals only
Shot gathers with scattered arrivals only

10. Outline 1. RTM = GDM 2. Implications Superresolution
Filtering: 1st Arrival Filtering
Target Oriented RTM
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.
Fast LSM
Diffraction Selective
Perfect Migration Operators

11. S S S Phase Shift, Beam, Kirchhoff Migrations
are Special Cases of True RTM S
1. RTM:
de (x) =
[G(s|x)G(x|g)]* d(s|g)
Frechet Derivative ds
s,g S
[{ } { } ]* d(s|g) =
G(s|x) +
G(s|x)
G(x|g) +
G(x|g)
s,g S {
G(s|x) } * ~
G(x|g)
d(s|g)
True RTM
s,g
First Arrival Filter
& U p+Down filter
First Arrival Filter
Early Arrival Filter
Super-wide Angle Phase
Shift Migration
Single Arrival Kirchhoff
w/o high-freq. appox
Multiple Arrival Kirchhoff
w/o high-freq. appox
(or Super beam migration) 11

12. Efficient RT Migration Operators
SALT
FD only in
expanding box

13. Standard RTM vs Early Arrival RTM
Example (Min Zhou, 2003)
Standard FD Wavefront G(s|x)
Early Arrival FD Wavefront G(s|x)
Standard RTM vs Early Arrival RTM 13

14. Efficient RT Migration Operators
Standard FD km
1.5 km
Wavefront FD

15. FD/ Wavefront FD Cost 45 FD/ Wavefront FD Cost 5 # Gridpts along side
# Gridpts along side

16. Wavefront Migration Image
Model
1.5 km
Wavefront Migration Image
1.5 km/s
2.2 km/s
1.8 km/s km

17. Reverse Time Migration
1.5 km
1.5 km/s
2.2 km/s
1.8 km/s
Wavefront Migration Image
1.5 km km

18. Outline 1. RTM = GDM 2. Implications Superresolution
Filtering: 1st Arrival Filtering
Target Oriented RTM
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.
Fast LSM
Diffraction Selective
Perfect Migration Operators

19. Filtering of Wave Equation
Migration Operators
Truncation: anti-aliasing
SALT
SALT

20. Filtering of Wave Equation
Migration Operators
Slant stack
SALT

21. Filtering of Wave Equation Migration Operators
COG Mig. Op.
Filtered COG Mig. Op.
0 s
1.0 s
Z=70 m
0 s
1.0 s
Time (s)
Z=270 m
0 s
1.0 s
Z=1190 m
0 km
4.5 km
X (km)

22. Outline 1. RTM = GDM 2. Implications Superresolution
Filtering: 1st Arrival Filtering
Target Oriented RTM
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.
Fast LSM
Diffraction Selective
Perfect Migration Operators

23. Standard Reverse Time Redatuming
Special case: 10 Shot Gathers at the Surface, 3 Receivers at Depth
Procedure: Compute 10 FD Solves, one for each shot at z=0
Cost = 10 FD Solves
to get G(x|x)
Datum

24. Target Oriented Reverse Time Redatuming
Special case: 10 Shot Gathers at the Surface, 3 Receivers at Depth
Trick: By Reciprocity G(x|x)=G(x|x)
Procedure: Compute 3 FD Solves, one for each shot at z=datum
Cost = 3 FD Solves
to get G(x|x)
Datum
Benefit: Several orders
magnitude less expensive

25. 3D Synthetic Data (Dong)W E
Kirchhoff Migration
Depth (Km)
Redatum + KM
3.5
Offset (km)
1.24
2.0
Offset (km)
3.5
A slice of 3D SEG/EAGE model at x=2.0 km

26. 3D Field Data Test New Datum km/s OBC geometry: 50,000 shots
Z (km)
8.0
y (km)
6.0
x (km) 12
Interval velocity model
km/s
5.5
1.5
OBC geometry:
50,000 shots
180 receivers per shot
Datum depth:
1.5 km
RVSP Green’s functions:
5,000 shots
New Datum

27. 3D Field Data Test y (km) 4.5 Time (s) 6.0 Original CSG y (km) 4.5
4.5
Time (s)
6.0
Original CSG
y (km)
4.5
Time (s)
6.0
Redatumed CSG

28. 3D Field Data Test KM of redatumed data Z (km) 8 y (km) x (km) 58
y (km) 5
x (km) 12
Z (km) 8
y (km) 5
x (km) 12
KM of original data
KM of RTD data

29. 3D Field Data Test ( Inline No. 61 ) KM of original data
KM of RTD data
X (km) 12
Z (km)
8.0
X (km) 12
Z (km)
8.0

30. 3D Field Data Test ( Crossline No. 41 ) KM of original data
KM of RTD data
Y (km)
5.0
Z (km)
8.0
Y (km)
5.0
Z (km)
8.0

31. 3D Field Data Test ( Crossline No. 61 ) KM of original data
KM of RTD data
Y (km)
5.0
Z (km)
8.0
Y (km)
5.0
Z (km)
8.0

32. 3D Field Data Test ( Depth 2.0 km ) KM of original data KM of RTD data
X (km) 12
Y (km)
5.0
X (km) 12
Y (km)
5.0

33. 3D Field Data Test ( Depth 2.5 km ) KM of original data KM of RTD data
X (km) 12
Y (km)
5.0
X (km) 12
Y (km)
5.0

34. 3D Field Data Test ( Depth 4.0 km ) KM of original data KM of RTD data
X (km) 12
Y (km)
5.0
X (km) 12
Y (km)
5.0

35. Computational Costs RTM (CPU-hours) RTD Speed up 3D field data test
5,000,000 (estimated)
52,000
100

36. Outline 1. RTM = GDM 2. Implications Superresolution
Filtering: 1st Arrival Filtering
Target Oriented RTM
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.
Fast LSM
Diffraction Selective
Perfect Migration Operators

37. Motivation (Ge Zhan) Problem Solution
Kirchhoff (diffraction-stack) migration is efficient
but with a high-frequency approximation.
WEM method (RTM) is accurate but computationally
intensive compared to KM.
Conventional RTM suffers from imaging artifacts.
Solution
Compressed generalized diffraction-stack migration (GDM) .
Wavelet compression of Green’s functions (10x or more).
Least squares algorithm.

38. Theory G(s|x)G(x|g) (5 dimensions) Migration Operator r s x
Size = nx*nz*ns*ng*nt = 645*150*323*176*1001*4 = 20 TB
Too big to store.
2D Wavelet Transform
appropriate threshold
10x compression

39. Theory Can Scatterers Beat the Resolution Limit ? Green’s Function
trace

40. Numerical Results 323 shots 176 geophones peak freq = 13 Hz
SEG/EAGE Salt Model
1.5
2.5
3.5
4.5
km/s
X (km)
Z (km) 15 3
323 shots
176 geophones
peak freq = 13 Hz
dx = 24.4 m
dg = 24.4 m
ds = 48.8 m
nsamples = 1001
dt = s
X (km)
Z (km) 15 3
Zoom View

41. Numerical Results Wavelet Transform Compression 200 MB 20 MB
Calculated GF
Reconstructed GF
Time (s)
Trace# 4
1.5
401 1
Trace Comparison
101
201
301
Time (s)
Trace # 4
401 1 1
401
Trace #
200 MB
20 MB

42. Numerical Results Early-arrivals Trace# 401 1 Multiples Time (s)
401 1

43. Numerical Results X (km) Z (km) 15 3 (a) GDM using Early-arrivals15 3
(a) GDM using Early-arrivals
X (km) 15
(b) GDM using Full Wavefield
X (km)
Z (km) 15 3
(c) GDM using Multiples
X (km) 15
(d) Optimal Stack of (a) and (c)

44. Outline 1. RTM = GDM 2. Implications Superresolution
Filtering: 1st Arrival Filtering
Target Oriented RTM
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.
Fast LSM
Diffraction Selective
Perfect Migration Operators

45. IMPLICATION #2
Exact Migration Operators from VSP
SALT
g(s|x)

46. IMPLICATION #2 * Exact Migration Operators from VSP * g(s|x) g(r|x)
SALT

47. Exxon RVSP Data X 0 km 0.2 km Direct Reflections Multiples Focusing
Z = .18 km
0 km X
0.2 km
Focusing
Operator
g(x|r)
g(s|x)

48. X X Exxon RVSP Data 0 km 0.2 km 0 km 0.2 km
Prim Refl. Kirchhoff Operator
Interbed Multiple Refl. Kirchhoff Operator
Prim Refl. Focusing Operator
0.2 s
0.28 s
0 km X
0.2 km
Interbed Multiple Refl. Focusing Operator
0.31 s
0.37 s X
0 km
0.2 km