1. Prestack Migration
Intuitive
Least Squares
Migration
Green’s Theorem

2. 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

3.  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

4. Outline Prestack DS Migration Theory RTM RTM vs Poststack vs Prestack
MATLAB Code

5. 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.

6. 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

7. 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)

8. 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

9. Outline Prestack DS Migration Theory RTM RTM vs Poststack vs Prestack
MATLAB Code

10. Prestack RTM vs One-way Wave Equation Migration

11. Prestack RTM vs One-way Wave Equation Migration

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 Reverse Time Migration
d (g, ) xg
m(x) =  g 
Trial image pt x
traces x x
~Scattered RTM 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.
Super-resolution

15. ZO Reverse Time Migration
d (g, ) xg
m(x) =  g 
Trial image pt x
traces x x
~Scattered RTM 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.
Super-resolution

16. Prestack RTM vs One-way Wave Equation Migration

17. Outline Prestack DS Migration Theory RTM RTM vs Poststack vs Prestack
MATLAB Code

18. Shortest Traveltime or Shortest Raypath Maximum Energy Traveltimes
Types of Traveltimes
Shortest Traveltime or Shortest Raypath
Maximum Energy Traveltimes
Shortest path ray
Shortest traveltime
ray
Maximum energy ray

19. Poststack vs Prestack Migration

20. Poststack vs Prestack Migration

21. RRTM vs KM Migration

22. RRTM vs KM Migration

23. Prestack RTM vs One-Way Mig.

24. Prestack RTM vs One-Way Mig.

25. Outline Prestack DS Migration Theory RTM RTM vs Poststack vs Prestack
DS MATLAB Code

26. MATLAB Prestack Migration

27. MATLAB Inefficient Prestack Migration
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

28. MATLAB Prestack Migration

29. Prestack Migration 1. No assumption about velocity model
2. More sensitive to velocity model errors
compared to poststack migration
3. More than 10 – 10 times slower than
poststack migration 2 6
4. More sensitive to velocity model than time
migration

30. Poststack vs Prestack Migration

31. Poststack vs Prestack Migration

32. ZO Reverse Time Migration
d (g, ) xg
m(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.

33. This is highest fruit on the tree..whom dare pick it?
Is Superresolution by RTM Achievable?
Tucson, Arizona Test
This is highest fruit on the tree..whom dare 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)

34. This is highest fruit on the tree..whom dare pick it?
Is Superresolution by RTM Achievable?
Tucson, Arizona Test
This is highest fruit on the tree..whom dare 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, TLE)