1. LSM Theory: Overdetermined vs Underdetermined
Workflow: Diffraction Stack vs RTM
Sensitivity: LSM sensitivity to Dv(x,z)
Dot Product Test: (Lm,d)=(m,LT d)
Examples
Summary

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

3. Motivation LSM

4. LSM=Antialiasing

5. Iterative Least Squares Migration
Overdetermined
Wrong v(x,z) will misposition reflector for src on left vs src on right. Inconsistent
Set of equations so reflector will be blurred rthat expalins all CSGs
Liability: wrong velocity model smears reflectivity
Migration image

6. Iterative Least Squares Migration Invert each shot gather
Underdetermined i
Misfit+regularization
Invert each shot gather
separately
Advantage: correct migration images prior to stacking
Penalize CSG images if they are different

7. LSM Theory: Overdetermined vs Underdetermined
Workflow: Diffraction Stack vs RTM
Sensitivity: LSM sensitivity to Dv(x,z)
Dot Product Test: (Lm,d)=(m,LT d)
Examples
Summary

8. MATLAB SD Least Squares Diffraction Stack Migration
Unlike RTM, Kirchhoff
not bothered by rabbit ears
Note: no update to smooth background c,
only hi-wavenumber m
p=p % Data without direct wave
m=adjoint(p,c) % Initial reflectivity model
c % Velocity model
for i=1:niter
p=forward(m,c) % Kirchhoff 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 - =

9. Iterative Least Squares Migration
We are now using RTM so
Both rabbit ears+ellipses -
Note: no update to smooth background so,
only hi-wavenumber ds LD D U

10. LSM Theory: Overdetermined vs Underdetermined
Workflow: Diffraction Stack vs RTM
Sensitivity: LSM sensitivity to Dv(x,z)
Dot Product Test: (Lm,d)=(m,LT d)
Examples
Summary

11. Sensitivity to V(x,z) Error

12. LSM Theory: Overdetermined vs Underdetermined
Workflow: Diffraction Stack vs RTM
Sensitivity: LSM sensitivity to Dv(x,z)
Dot Product Test: (Lm,d)=(m,LT d)
Examples
Summary

13. 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

14. LSM Theory: Overdetermined vs Underdetermined
Workflow: Diffraction Stack vs RTM
Sensitivity: LSM sensitivity to Dv(x,z)
Dot Product Test: (Lm,d)=(m,LT d)
Examples
Summary

15. 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) 16

16. Poststack 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

17. Multi-scale LSM Applied to JAPEX Data
Multi-scale (MS) LSM vs. Standard LSM
Convergence Curves
MS LSM Image
0.7
1.9
Depth (km)
2.4
4.9
X (km)
Standard LSM Image
0.7
1.9
2.4
4.9
X (km)
X10 5
3.0
0.5
Residual 40
Iteration
Multi-scale LSM
Standard LSM
20　Hz 25
I apply band-pass filter to data.
The frequency band increase with iteration 30 32 34 36 38 40 18

18. GOM Poststack Data
Poststack LSM somewhat insensitive to Dv(x,z)

19. LSM=Antialiasing

20. Prestack KM vs LSM
Prestack LSM sensitive to Dv(x,z)

21. Attenuation RTM vs LSM

22. Attenuation RTM vs LSM

23. Nonlinear LSRTM

24. Nonlinear LSRTM

25. NL Means Filter for Trim Statics
Problem: velocity error CIGs misaligned  poor stacking
Washed out
Two prestack image patches
out of phase:
Solution: Xcorr patches; recursive stacking
needs no pilot 1 A 2 B 3 4 C
shift

26. NL Means Filter for Trim Statics
Advantage: drastic improvement in feature coherency
Examples:
Disadvantage: strong migration artifacts may mislead
Simple Stacking
Future Work: 3D trim statics
Trim Statics

27. Statics+ LSRTM

28. LSM Summary 1. Prefer undetermined LSM if large Dv(xz)>>0
2. Biggest challenge: LSM sensitive Dv(x,z)
3. Q-LSM much better than LSM or RTM if Q<30
4. LSRTM cost O(20x) more than RTM
5. Multisource encoded O(LSRTM) cost = O(RTM)
6. Iterative LSM+DSO+FWI is future.