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

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

3. x-y x-z Problem: m mig =L T d Migration Problems Soln: m (k+1) = m (k) +  L  d (k) T Solution: Least squares migration Given: d = Lm predictedobserved = L = L Modeling operatord Find: min ||Lm - d || Find: min ||Lm - d ||2m defocusing aliasing

4. Least Squares Migration m (k+1) = m (k) +  L  d (k) T m = [L T L] -1 L T d Geom. Spreading: 1 -1 1 1 r 4 r 2 r 2 Anti-aliasing: [w(t) w(t)] -1 w(t) w(t) Source Decon: 1/r 1/r Aliasing artifacts migrate model Inconsistent events

5. Brief History of Least Squares Migration Romero et al. (2000) Tang & Biondi (2009), Dai & GTS (2009), Dai (2011, 2012), Zhang et al. (2013), Dai et al. (2013), Dutta et al (2014) Multisource Migration Multisource Least Squares Migration Lailly (1983), Tarantola (1984) Linearized Inversion 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),

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

7. Acquisition Footprint Mitigation 0 10 X (km) 0 10 Standard Migration LSM X (km) 5 sail lines 200 receivers/shot 45 shot gathers

8. RTM vs LSM 6.3 9.9 X (km) 0.8 1.2 Reverse Time Migration 0.8 1.2 Plane-Wave LSM 6.3 9.9 X (km)

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

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

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

12. Standard Migration vs Multisource LSM Given: d 1 and d 2 Find: m Soln: m=L 1 d 1 + L 2 d 2 TT Given: d 1 + d 2 Find: m = m (k) +  [L 1 d 1 + L 2 d 2 TT + L 1 d 2 + L 2 d 1 TT Soln: m (k+1) = m (k) +  (L 1 + L 2 )(d 1 +d 2 ) T Romero, Ghiglia, Ober, & Morton, Geophysics, (2000) Iteratively encode data so L 1 T d 2 = 0 and L 2 T d 1 = 0 1 RTM to migrate many shot gathers 1 RTM per shot gather ] Benefit: 1/10 reduced cost+memory

13. 0 6.75 X (km) 0 Z (km) 1.48 a) Original b) Standard Migration Multisource LSM (304 blended shot gathers) 0 6.75 X (km) c) Standard Migration with 1/8 subsampled shots 0 Z (km) 1.48 0 6.75 X (km) d) 304 shots/gather 26 iterations 3876152304 9.4 5.4 1 Shots per supergather Computational gain Conventional migration: SNR=30dB

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

15. a swath 16 16 swaths, 50% overlap 16 cables 100 m 6 6 km 40 40 m 256 256 sources 20 m 4096 sources in total SEG/EAGE Model+Marine Data (Yunsong Huang) 13.4 km 3.7 km

16. Numerical Results (Yunsong Huang) 6.7 km True reflectivities 3.7 km Conventional migration 13.4 km 25616 256 shots/super-gather, 16 iterations 8 x gain in computational efficiency 3.7 km

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

18. Plane-wave LSRTM of 2D GOM Data 0X (km) 16 0 Z (km) 2.5 2.1 1.5 km/s Model size: 16 x 2.5 km.Source freq: 25 hz Shots: 515Cable: 6km Receivers: 480

19. 0X (km) 16 0 Z (km) 2.5 Conventional GOM RTM (cost: 1) (Wei Dai) Z (km) 2.5 Plane-wave RTM (cost: 0.2) Plane-wave LSRTM (cost: 12) Encoded Plane-wave LSRTM (cost: 0.4) 0

20. 0X (km) 16 0 Z (km) 2.5 Z (km) 2.5 Plane-wave RTM (cost: 0.2) Plane-wave LSRTM (cost: 12) Encoded Plane-wave LSRTM (cost: 0.4) 0 RTM LSM Conventional GOM RTM (cost: 1) (Wei Dai)

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

22. Viscoacoustic Least Squares Migration m (k+1) = m (k) +  L  d (k) T L = viscoacoustic wave equation

23. 0 Z (km) 1.5 0 X (km) 2 1.0 -1.0 True Reflectivity Acoustic LSRTM 0 X (km) 2 Viscoacoustic LSRTM 1.0 -1.0 0 Z (km) 1.5 0 X (km) 2 Q Model Q=20 Q=20000

24. Road Ahead Summary 3. Sensitivity: Quality LSM = RTM if inaccurate v(x,y,z) 1. LSM Benefits: Anti-aliasing, better resolution, focusing 5.Broken LSM: Multiples. Quality degrades below 2 km? Collect 4:1 data? Collect 4:1 data? 2. Cost: MLSM ~ RTM, MLSM has better resolution 4. Viscoacoustic LSM: Required if Q<25? 6. Road Ahead: Iterative MVA+MLSM+Statics

25. IO 1 ~1/36 Cost Resolution dx 1 ~double Migration SNR Stnd. Mig Multsrc. LSM Stnd. Mig Multsrc. LSM ~1 1.1-8 Cost vs Quality: Can I<<S? Yes. What have we empirically learned about MLSM? 1

26. Conventional Least Squares Solution: L= & d = Given: Lm=d Find: m s.t. min||Lm-d|| 2 Solution: m = [L L] L d TT m = m –  L (Lm - d) T(k+1)(k)(k)(k) or if L is too big L 1 L 2 d 1 d 2 = m –  L (L m - d ) = m –  L (L m - d ) (k) + L (L m - d ) 1 1 2 2 2 1 TT [] In general, huge dimension matrix Linear Optimization = Least Squares Migration Non-Linear Optimization = Full Waveform Inversion

27. Numerical Results 1. Land and Marine MLSM 2. Multiscale Waveform Inversion 3. Multisource Waveform Inversion

28. Numerical Results 1. Land and Marine MLSM 2. Multiscale Waveform Inversion 3. Multisource Waveform Inversion

29. 3876152304 9.4 8.0 6.6 5.4 3.8 1 Shots per supergather gain Computational gain Conventional migration: Sensitivity to input noise level SNR=10dB SNR=30dB SNR=20dB

30. Data Misfit Function  =||D-d|| 2 = D 2 + d 2 – 2Re(D * d)  =||D*d|| 2 = 2Re(D * d) = |D||d|cos(  -  ) Correlation between predicted and observed traces. Match phase, no need to match amplitudes Difference between predicted and observed traces. Match phase and amplitudes

31. Model Misfit Function  =1/2||mL T Lm-m mig || 2  =1/2||m mig m|| 2 Correlation between predicted and observed traces. Match phase, no need to match amplitudes Difference between predicted and observed migration Match phase and amplitudes

32. Data Misfit Function  =||D-d|| 2 = D 2 + d 2 – 2Re(D * d)  =||D*d|| 2 = 2Re(D * d) = |D||d|cos(  -  ) Correlation between predicted and observed traces. Match phase, no need to match amplitudes Zhang et al. (2013) Dutta et al. (2014) Difference between predicted and observed traces. Match phase and amplitudes

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