1. Making the Most from the Least (Squares Migration)
G. Dutta, Y. Huang, W. Dai, X. Wang, and Gerard Schuster
KAUST
Standard Migration
Least Squares Migration

2. Least Squares Migration:
Outline
Least Squares Migration:
Examples of LSM:
Problems with LSM: Cost and V(x,z) Sensitivity
Multisource LSM: Gulf of Mexico Data
Viscoacoustic LSM: Marmousi & GOM data
Summary and Road Ahead

3. Migration Problems = L Solution: Least squares migration Given: d = Lm
x-z
x-y
Problem: mmig=LTd
defocusing
aliasing
Solution: Least squares migration
= L
Modeling
operator d
Given: d = Lm
Find: min ||Lm - d || 2 m
predicted
observed
Soln: m(k+1) = m(k) + a L Dd(k) T

4. Migration Problems Migration Algorithm Soln: m(k+1) = m(k) + a L Dd(k)
m = : select initial reflectivity model and smooth v
for L x z v
d = L m : compute initial data (only direct wave)
time x
L m
Dd = L m -dobs : compute initial residual
time x =
dobs
L m Dd -
For k=1:N
Dm = migrate(L,Dd) : migrate residual traces Dd x Dd  z
time
m = m – a Dm : update reflectivity model m
d = Lm : update predicted data d x z
time
d = Lm
Soln: m(k+1) = m(k) + a L Dd(k) T
Dd = Lm - dobs : update residual Dd
time
end

5. Least Squares Migration
m(k+1) = m(k) + a L Dd(k) T
m = [LTL]-1LT d
[w(t) w(t)]-1w(t) w(t)
Source Decon:
Geom. Spreading:
r4 r2 r2
1/r
Inconsistent events
Anti-aliasing:
migrate
model
Aliasing
artifacts

6. Least Squares Migration
Standard Migration
(176x176)
Least Squares Migration
(176x176)
1 km
1 km
1 km

7. Least Squares Migration
Standard Migration
(176x176)
Least Squares Migration
(176x176)

8. Brief History of Least Squares Migration
Linearized Inversion
Lailly (1983), Tarantola (1984)
Least Squares Migration
Cole & Karrenbach (1992), GTS (1993), Nemeth (1996)
Nemeth et al (1999), Duquet et al (2000), Sacchi et al (2006)
Guitton et al (2006),
Multisource Migration
Romero et al. (2000)
Multisource Least Squares Migration
Tang & Biondi (2009), Dai & GTS (2009), Dai (2011, 2012),
Zhang et al. (2013), Dai et al. (2013), Dutta et al (2014)

9. Least Squares Migration:
Outline
Least Squares Migration:
Examples of LSM:
Problems with LSM: Cost and V(x,z) Sensitivity
Multisource LSM: Gulf of Mexico Data
Viscoacoustic LSM: Marmousi & GOM data
Summary and Road Ahead

10. Acquisition Footprint Mitigation
5 sail lines
200 receivers/shot
45 shot gathers
Standard Migration
LSM 10
Y (km)
X (km)
X (km)

11. RTM vs LSM Reverse Time Migration Plane-Wave LSM 0.8 Z (km) 1.2
X (km)
Plane-Wave LSM
0.8
1.2
Z (km)
X (km)

12. Least Squares Migration:
Outline
Least Squares Migration:
Examples of LSM:
Problems with LSM: Cost and V(x,z) Sensitivity
Multisource LSM: Gulf of Mexico Data
Viscoacoustic LSM: Marmousi & GOM data
Summary and Road Ahead

13. Problem #1 with LSM Problem: High Sensitivity to Inaccurate V(x,y,z)
Partial Solutions:
a) Statics corrections
LSM
LSM+Statics
LSM
CSG1
CSG2
b) Iterative LSM+MVA
RTM+Traveltime Tomo
RTM+MVA
Sanzong Zhang (2014)

14. Problem #2 with LSM Problem: LSM Cost >10x than RTM
Solution: Migrate Blended Supergathers

15. Standard Migration vs Multisource LSM
Romero, Ghiglia, Ober, & Morton, Geophysics, (2000)
Given: d1 and d2
Given: d1 + d2
Find: m
1 RTM per
shot gather
Find: m
1 RTM to migrate many
shot gathers
Soln:
m(k+1) = m(k) + a (L1 + L2)(d1+d2) T
Soln: m=L1 d1 + L2 d2 T
= m(k) + a[L1 d1 + L2 d2 T
+ L1 d2 + L2 d1 ]
Iteratively encode data so
L1T d2 = 0 and L2T d1 = 0
Benefit: 1/10 reduced cost+memory

16. (304 blended shot gathers)
Multisource LSM
(304 blended shot gathers)
Z (km)
1.48
a) Original
b) Standard Migration
6.75
X (km)
c) Standard Migration with 1/8 subsampled shots
Z (km)
1.48 38 76
152
304
9.4
5.4 1
Shots per supergather
Computational gain
Conventional migration:
SNR=30dB
Comp. Gain
X (km)
6.75
6.75
X (km)
d) 304 shots/gather
26 iterations

17. Outline Least Squares Migration: Examples of LSM:
Problems with LSM: Cost and V(x,z) Sensitivity
Multisource LSM: 3D SEG Salt Model
Viscoacoustic LSM: Marmousi & GOM data
Summary and Road Ahead

18. SEG/EAGE Model+Marine Data
(Yunsong Huang)
sources in total
40 m
100 m
16 swaths,
50% overlap
256 sources
a swath
6 km
3.7 km
20 m
16 cables
13.4 km

19. Numerical Results 8 x gain in computational efficiency
(Yunsong Huang)
6.7 km
True reflectivities
3.7 km
Conventional migration
3.7 km
13.4 km
256 shots/super-gather, 16 iterations
8 x gain in computational
efficiency
The multisource result is of slightly higher resolution than the conventional shot-gather migration; indistinguishable in this size on screen.

20. Outline Least Squares Migration: Examples of LSM:
Problems with LSM: Cost and V(x,z) Sensitivity
Multisource LSM: Gulf of Mexico Data
Viscoacoustic LSM: Marmousi & GOM data
Summary and Road Ahead

21. Plane-wave LSRTM of 2D GOM Data
Model size: 16 x 2.5 km.
Source freq: 25 hz
Shots: 515
Cable: 6km
Receivers: 480
km/s
2.1
Z (km)
2.5
1.5 16
X (km)

22. Conventional GOM RTM (cost: 1)
(Wei Dai)
Z (km)
2.5
Plane-wave LSRTM (cost: 12)
Encoded Plane-wave LSRTM (cost: 0.4)
Plane-wave RTM (cost: 0.2)
Z (km)
2.5 16
X (km)

23. LSM RTM Conventional GOM RTM (cost: 1)
(Wei Dai)
LSM
Z (km)
RTM
2.5
Encoded Plane-wave LSRTM (cost: 0.4)
Plane-wave LSRTM (cost: 12)
Plane-wave RTM (cost: 0.2)
Z (km)
2.5 16
X (km)

24. Outline Least Squares Migration: Examples of LSM:
Problems with LSM: Cost and V(x,z) Sensitivity
Multisource LSM: 3D SEG Salt Model
Viscoacoustic LSM: Marmousi & GOM data
Summary and Road Ahead

25. Viscoacoustic Least Squares Migration
m(k+1) = m(k) + a L Dd(k) T
L = viscoacoustic wave equation

26. True Reflectivity Q Model Acoustic LSRTM Viscoacoustic LSRTM Q=20000
Z (km)
Z (km)
Q=20000
Q=20
X (km)
X (km)
Acoustic LSRTM
Viscoacoustic LSRTM
Z (km)
X (km)
X (km)

27. Post Processing LSM Standard Migration Least Squares Migration
12 km
12 km
12 km
12 km
12 km
Preconditioned LSM
12 km

28. Road Ahead Summary
1. LSM Benefits: Anti-aliasing, better resolution, focusing
2. Cost: MLSM ~ RTM, MLSM has better resolution
3. Sensitivity: Quality LSM = RTM if inaccurate v(x,y,z)
4. Viscoacoustic LSM: Required if Q<25?
Broken LSM: Multiples. Quality degrades below 2 km?
Collect 4:1 data?
6. Road Ahead: Iterative MVA+MLSM+Statics

29. Data Misfit Function e=||D-d||2 = D2 + d2 – 2Re(D*d)
Difference between predicted and observed traces.
Match phase and amplitudes
e=||D-d||2 = D2 + d2 – 2Re(D*d)
e=||D*d||2 = 2Re(D*d) = |D||d|cos(F-f)
Correlation between predicted and observed traces.
Match phase, no need to match amplitudes
Zhang et al. (2013) Dutta et al. (2014)

30. Model Misfit Function e=1/2||mLT Lm-mmig||2 e=1/2||mmig m||2
Difference between predicted and observed migration
Match phase and amplitudes
e=1/2||mLT Lm-mmig||2
e=1/2||mmig m||2
Correlation between predicted and observed traces.
Match phase, no need to match amplitudes

31. Examples: Data vs Phase Misfit