1. Rigorous Derivation of LSM (& FWI)

2. Outline LSM Theory Extrapolation: U(x)=Sg G(x|g)*d(g|s)
m(x)= Zero-Lag Correlation U(x) D(x)
m(x)=Dot product G(g\x)G(x|s) & d(g|s)
LSM Physical Meaning: Wavepaths
LSM Code

3. 1. 2.

4. 2a.
gradient
Gradient
Frechet Derivative
gradient
(i)
Slowness
update

5. Special Cases: Prestack Migrationx
Prestack migration of the residual s x g
More traces better resolution
Dx=Poststack trace at farthest offset from x
Dz=Poststack trace at nearest offset from x
Migration ellipse: reflection smeared
along ellipse. Scatterer anywhere along
ellipse

6. Special Cases: Poststack Migration

7. Outline LSM Theory Extrapolation: U(x)=Sg G(x|g)*d(g|s)
m(x)= Zero-Lag Correlation U(x) D(x)
m(x)=Dot product G(g\x)G(x|s) & d(g|s)
LSM Physical Meaning: Wavepaths
LSM Code

8. Extrapolation: U(x)=Sg Sw G(x|g)D(g|s)*
= Sg g(x,t|g) D(g,-t|s)
Sg g(x,t|g) d(g,t|s)
Sg g(x,t|g) d(g,-t|s) 7 8 9 10 6
Time (s) -7 -8 -9
-10 -6
Time (s)
7 s
8 s
backpropagated
wave
Forward modeled
wave
8.5 s
9.2 s
-7 s

9. Outline LSM Theory Extrapolation: U(x)=Sg G(x|g)*d(g|s)
m(x)= Zero-Lag Correlation U(x) D(x)
m(x)=Dot product G(g\x)G(x|s) & d(g|s)
LSM Physical Meaning: Wavepaths
LSM Code

10. m(x) = Zero-lag xcorr D(x,t)U(x,t)
Migration: m(x) = Sg Sw G(x|s)[G(x|g)D(g|s)*]
= Sg Sw G(x|s)[G(x|g)*D(g|s)]*
= Sg g(x,t|s) [g(x,-t|g) d(g,t|s)]
m(x) = Zero-lag xcorr D(x,t)U(x,t)
Sg g(x,t|g) d(g,t|s)
Sg g(x,t|g) d(g,-t|s) 7 8 9 10 6
Time (s) -7 -8 -9
-10 -6
Time (s)
7 s
8 s
backpropagated
wave
Forward modeled
wave
8.5 s
9.2 s
-7 s
What would happen if you applied
extrapolation operator above free surface?

11. m(x) = Zero-lag xcorr D(x,t)U(x,t)
Migration: m(x) = Sg Sw G(x|s)[G(x|g)D(g|s)*]
= Sg Sw G(x|s)[G(x|g)*D(g|s)]*
g(x,t|s)ref [g(x,-t|g)dir d(g,t|s)]
m(x) = Zero-lag xcorr D(x,t)U(x,t)
Sg g(x,t|g) d(g,t|s)
Sg g(x,t|g) d(g,-t|s) 7 8 9 10 6
Time (s) -7 -8 -9
-10 -6
Time (s)
7 s
8 s
backpropagated
wave
Forward modeled
wave
8.5 s
9.2 s
-7 s
What would happen if you applied
extrapolation operator above free surface?

12. Backprojection data and zero-lag xcorrel with direct field
Physical Meaning: RTM
Backprojection data and zero-lag
xcorrel with direct field s x g
D R
10 s
9 s
8 s
7 s

13. Outline LSM Theory Extrapolation: U(x)=Sg G(x|g)*d(g|s)
m(x)= Zero-Lag Correlation U(x) D(x)
m(x)=Dot product G(g\x)G(x|s) & d(g|s)
LSM Physical Meaning: Wavepaths
LSM Code

14. Alternative Physical Meaning: GDM
m(x)=Dot product between observed
and predicted hyperbolas s g xo x
Write a 6-line code for FWI using GDM approach Given the forward modeling subroutine p=Forw(m)

15. Alternative Physical Meaning: GDM
m(x)=Dot product between observed
And predicted hyperbolas s g
Small shift in trial image point x away from trial
image pt xo leads to enormous decrease in correlation xo x
Why superresolution?

16. d Change in data w/r change in model

17. Outline LSM Theory Extrapolation: U(x)=Sg G(x|g)*d(g|s)
m(x)= Zero-Lag Correlation U(x) D(x)
m(x)=Dot product G(g\x)G(x|s) & d(g|s)
LSM Physical Meaning: Wavepaths
LSM Code

18. Ellipse Wavepath s g x
Only reflection
good for migration (smear all along ellipse),
so this is why we have smooth background
Assume G = Gdir where G(x|s)dir=eiwtsx and DP=reflection
Assume G(x|xs)G(x|xg)= Gdir (x|xs)G(x|xg)dir
Where is DP smeared? How Thick is it?

19. Ellipse Wavepath s g x
Only reflection
Ellipse update high wavenumber
portion of model & is good for
LSM.
good for migration (smear all along ellipse),
so this is why we have smooth background
To create the ellipse simply mute everything but the reflection
arrival for a single trace and migrate it in a smooth medium
to get the above ellipse.

20. Cigar Wavepath Assume G = Gdir where G(x|s)dir=eiwtsx
Cigars update low wavenumber
portion of model & is good for
FWI. It is bad for LSM.
That is why we kill direct wave
For LSM.
reflection
diving wave s g x
bad for migration (smear all along ellipse),
so this is why we have smooth background
Assume G = Gdir where G(x|s)dir=eiwtsx
Assume G(x|xs)G(x|xg)= Gdir (x|xs)G(x|xg)dir
Where is DP smeared? How Thick is it?

21. Cigar Wavepath
Cigars update low wavenumber
portion of model & is good for
FWI. It is bad for LSM.
That is why we kill direct wave
For LSM.
reflection
diving wave s g x
bad for migration (smear all along ellipse),
so this is why we have smooth background
To create the cigar simply mute everything but the direct
arrival for a single trace and migrate it in a smooth medium
to get the above cigar.

22. Rabbit Ear Wavepath
2. Exploding reflector r(x,z)d(x,t)=Gx|s)refl
Rabbit Ear update low wavenumber
portion of model & is good for
FWI. It is bad for LSM and is a reason
We use smooth velocity model with LSM. s x g x*
Not good for migration (smear all along wavepath),
so this is why we have smooth background
To create the cigar simply mute everything but the reflection
arrival for a single trace & do following:
1. Migrate d(x,t) r(x,z)
3. Migrate d(g,-t)G(g,t|x) G(x,t|s)refl
2. Exploding reflector r(x,z)d(x,t)=g(x,t|s)refl
and forward model exploding reflector
d(x,t)
g(x,t|s)refl
r(x,z) x x

23. Rabbit Ear Wavepath s g x x* Rabbit Ear update low wavenumber
portion of model & is good for
FWI. It is bad for LSM and is a reason
We use smooth velocity model with LSM. s x g x*
Not good for migration (smear all along wavepath),
so this is why we have smooth background

24. Outline LSM Theory Extrapolation: U(x)=Sg G(x|g)*d(g|s)
m(x)= Zero-Lag Correlation U(x) D(x)
m(x)=Dot product G(g\x)G(x|s) & d(g|s)
LSM Physical Meaning: Wavepaths
LSM Code

25. FWI Code -

26. FWI Code
T=5
T=4
T=3
T=6 -
T=7
T=6
T=5