1. Theory of Multisource Crosstalk Reduction by Phase-Encoded Statics G. Schuster, X. Wang, Y. Huang, C. Boonyasiriwat King Abdullah University Science & Technology

2. Outline 1. Seismic Experiment: L m = d 1 1 2 2... N N 2. Standard vs Phase Encoded Least Squares Soln. L 1 L 2 d 1 d 2 m = vs N L + N L 1 2 1 2 [ m = [N d + N d ] ]m = [N d + N d ] 1 2 1 2 3. Theory Noise Reduction 4. Summmary and Road Ahead

3. Gulf of Mexico Seismic Survey m L m = d 1 1 2 2... N N Time (s) 6 X (km) 4 0 d Goal: Solve overdetermined System of equations for m Predicted dataObserved data Problem: Expensive, one migration/shot gather migration/shot gather Solution: Supergather migration migration

4. Brief History Multisource Phase Encoded Imaging Romero, Ghiglia, Ober, & Morton, Geophysics, (2000) Krebs, Anderson, Hinkley, Neelamani, Lee, Baumstein, Lacasse, SEG, (2009) Virieux and Operto, EAGE, (2009) Dai and Schuster, SEG, (2009) Migration Waveform Inversion and Least Squares Migration Biondi et al., SEG, (2009)

5. Outline 1. Seismic Experiment: L m = d 1 1 2 2... N N 2. Standard vs Phase Encoded Least Squares Soln. L 1 L 2 d 1 d 2 m = vs N L + N L 1 2 1 2 [ m = [N d + N d ] ]m = [N d + N d ] 1 2 1 2 3. Theory Noise Reduction 4. Summmary and Road Ahead

6. (k) 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 Problem: L is too big for IO bound hardware 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 Note: subscripts agree

7. 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) Problem: L is too big for IO bound hardware L 1 L 2d1 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 Problem: Expensive, FD solve/CSG Solution: Blend+encode Data (k)

8. Blending+Phase Encoding 2 d = N d + N d + N d d = N d + N d + N d2 1 1 3 3 PhasePhase Blending Encoding Matrix Encoded supergather L = N L + N L + N L 3 3 2 2 1 1 m[ ]m d1 L m=L m=L m=L m=1 Encoded supergather modelerd3 L m=L m=L m=L m=3d2 L m=L m=L m=L m=2 O(1/S) cost! Blending 1 e iwiw in w domain

9. (k)(k) (k) + Blended Phase-Encoded Least Squares Solution L = & d = N d + N d Given: L m= d Find: m s.t. min|| L m- d || 2 Solution: m = [ L L ] L d T TT (k+1)(k)(k)(k) or if L is too big 1 N L + N L 21 = m –  L (L m - d ) = m –  L (L m - d ) (k) + L (L m - d ) 1 1 2 2 2 1 TT [] 1 2 1 2 2 + Crosstalk + L N N (L m - d ) 2 T 1 L N N (L m - d ) L N N (L m - d ) 1 T 1 221122** (k) In general, SMALL dimension matrix (k)(k) (k) Iterations are proxy For ensemble averaging(k+1)(k)(k)(k)(k) (k) (k) (k) (k) (k) (k) (k)(k) m = m –  L ( L m - d )

10. Outline 1. Seismic Experiment: L m = d 1 1 2 2... N N 2. Standard vs Phase Encoded Least Squares Soln. L 1 L 2 d 1 d 2 m = vs N L + N L 1 2 1 2 [ m = [N d + N d ] ]m = [N d + N d ] 1 2 1 2 3. Theory Noise Reduction 4. Summary

11. Ensemble Average of Crosstalk Term With Random Time Shifts + Crosstalk: L N N (L m - d ) L N N (L m - d ) 2 T 1 1 1 T 1 221122** N N 1 2* = e e -i w  2 iwiw 1 e i w(t - t ) 1 2 <> < > > = < e i w(t - t ) 1 2 = e -.25 s (t - t ) 1 2 -2 d t 12 ~ e - s w 22 e -i w  2 Gaussian PDF Noise Crosstalk term decreases with increasing  and  :

12. Crosstalk Prediction Formula L (L m - d ) 2 T 1 1 + L (L m - d ) 1 T 2 2 e -  2 O( ) ~ X =   Pt. Scatt. Stand. Mig.Pt. Scatt.. Mig. of Supergathers.  =.05 s  =.1 s  =.01 s X Offset

13. Ensemble Average of Crosstalk Term With Random Polarity + Crosstalk: L N N (L m - d ) L N N (L m - d ) 2 T 1 1 1 T 1 221122** N N 1 2* 1 2 : <> = < > = 0 sgn(  Noise sgn(  Conclusion: Random polarity better than random time shifts Further Analysis: Variance of the crosstalk noise says that random polarity & random time shifts can be almost twice better than polarity alone <> N N 1 2* ( )2

14. 0 6.75X (km) a) Standard migration (320 CSG) b) Time static σ = 0.1 s f) SNR 0.4 Z (km) 1.48 c) Noise 0.4 Z (km) 1.48 0 6.75X (km) 0 6.75X (km) 0.4 Z (km) 1.48 0 6.75X (km) 0.4 Z (km) 1.48 0 6.75X (km) e) Noise 0.01 0.1Time static σ (s) 0.18 SNR 0 0.4 Z (km) 1.48 polarity Time static Polarity and time static Polarity+Time Statics+Location StaticsPolarity+Time StaticsPolarity  < +/- < +/-,  <  x Time Statics a) Polarity b) Noise d) Source polarity & static Key Theory+Num. Results for 320 CSG Supergather (Xin Wang, Yunsong Huang)

15. Key Results Theory of Multisource Imaging of Encoded Supergathers (Xin Wang) Sig/Noise = GI < GIN # geophones/supergather # subsupergatherss 0 6.7X (km) a) Image of 1 stack Iteration Number SNR c) Image of 50 stacks Z (km) 0 6.7 5 X (km) 06.7 X (km) b) Image of 5 stacks 1 115 d) SNR vs Iterations Observed Prediction # iterations Z (km) Bulk shift

16. Standard Migration SNR GS # geophones/CSG # CSGs SNR=... migrate SNR= d(t) = Zero-mean white noise [s(t) +n(t) ] [s(t) +n(t) ] Neglect geometric spreading Standard Migration SNR Assume: migrate + + + stack S 1 S GS G ~ ~ iterate GI Iterative Multisrc. Mig. SNR # iterations SNR= Cost ~ O(S) Cost ~ O(I)

17. Outline 1. Seismic Experiment: L m = d 1 1 2 2... N N 2. Standard vs Phase Encoded Least Squares Soln. L 1 L 2 d 1 d 2 m = vs N L + N L 1 2 1 2 [ m = [N d + N d ] ]m = [N d + N d ] 1 2 1 2 3. Theory Noise Reduction 4. Summary

18. SNR: VS3. GSGI 2. 1. Polarity+Time Statics+Location StaticsPolarity+Time StaticsPolarity  < +/- < +/-,  <  x Time Statics L 1 L 2 d 1 d 2 m = N L + N L 1 2 1 2 [ m = [N d + N d ] ]m = [N d + N d ]1 2 1 2 Summary vs L (L m - d ) 2 T 1 1 + L (L m - d ) 1 T 2 2 e -  2 O( ) ~ 4. Passive Seismic Interferometry = Multisrc Imaging Imaging

19. IO 1 1/320 Cost ~ Resolution dx 1 1 SNR~ Stnd. Mig Multsrc. LSM Stnd. Mig Multsrc. LSM Less 1 1 <1/10 Cost vs Quality Summary 1 L 1 L 2 d 1 d 2 m = N L + N L 1 2 1 2 [ m = [N d + N d ] ]m = [N d + N d ]1 2 1 2

20. SNR 0 1 Number of Iterations 300 7 The SNR of MLSM image grows as the square root of the number of iterations. SNR = GI

21. 0 6.75X (km) a) Standard migration (320 CSG) b) Time static σ = 0.1 s f) SNR 0.4 Z (km) 1.48 c) Noise 0.4 Z (km) 1.48 0 6.75X (km) 0 6.75X (km) 0.4 Z (km) 1.48 0 6.75X (km) 0.4 Z (km) 1.48 0 6.75X (km) e) Noise 0.01 0.1Time static σ (s) 0.18 SNR 0 0.4 Z (km) 1.48 polarity Time static Polarity and time static Polarity+Time Statics+Location StaticsPolarity+Time StaticsPolarity  < +/- < +/-,  <  x Time Statics a) Polarity b) Noise d) Source polarity & static Key Theory+Num. Results for 320 CSG Supergather (Xin Wang, Yunsong Huang)

22. Key Results Theory of Multisource Imaging of Encoded Supergathers (Xin Wang) Sig/Noise = GI < GIN # geophones/supergather # subsupergatherss 0 6.7X (km) a) Image of 1 stack Iteration Number SNR c) Image of 50 stacks Z (km) 0 6.7 5 X (km) 06.7 X (km) b) Image of 5 stacks 1 115 d) SNR vs Iterations Observed Prediction # iterations Z (km)

23. Key Results Theory of Multisource Imaging of Encoded Supergathers (Boonyasiriwat) Sig/Noise = GI < GIN 3.5 km Dynamic QMC Tomogram (99 CSGs/supergather) (99 CSGs/supergather) Dynamic Polarity Tomogram (1089 CSGs/supergather) 1/1000 1/300 # geophones/supergather # iterations # subsupergatherss

24. Multisource Migration: m mig =L T d Forward Model: Multisource Phase Encoded Imaging d +d =[ L +L ]m 1221 L {d { =[ L +L ](d + d ) 122 1 TT = L d +L d + 122 1 TT L d +L d L d +L d212 1 Crosstalk noise Standard migration TT m = m + (k+1)(k)

25. Polarity+Time Statics+Location StaticsPolarity+Time StaticsPolarity Relative Merits of 4 Encoding Strategies  < +/- < +/-,  <  x d1 L m=L m=L m=L m=1d3 L m=L m=L m=L m=3d2 L m=L m=L m=L m=2 Time Statics supergather Supergather #2 Supergather #3 Supergather #4 Supergather #1 Polarity+Time Statics+Location StaticsPolarity+Time StaticsPolarity  < +/- < +/-,  <  x Time Statics

26. Multisource Migration: m mig =L T d Forward Model: Phase Encoded Multisource Migration d +d =[ L +L ]m 1221 L {d { =[ L +L ](d + d ) 122 1 TT = L d +L d + 122 1 TT L d +L d L d +L d212 1 Crosstalk noise Standard migration TT m mig = L d +L d + 122 1 TT L d +L d L d +L d 212 1 T T m mig = L d +L d 122 1 m mig ++

27. Multisource Migration: m mig =L T d Forward Model: Phase Encoded Multisrce Least Squares Migration d +d =[ L +L ]m 1221 L {d { =[ L +L ](d + d ) 122 1 TT m mig = L d +L d + 122 1 TT L d +L d L d +L d212 1 Crosstalk noise Standard migration TT m = m + (k+1)(k)

28. Outline 1. Seismic Experiment: L m = d 1 1 2 2... N N 2. Standard vs Phase Encoded Least Squares Soln. L 1 L 2 d 1 d 2 m = vs N L + N L 1 2 1 2 [ m = [N d + N d ] ]m = [N d + N d ] 1 2 1 2 3. Theory Noise Reduction 4. Numerical Tests

29. RTM & FWI Problem & Possible Soln. Problem: RTM & FWI computationally costlyProblem: RTM & FWI computationally costly Solution: Multisource LSM & FWISolution: Multisource LSM & FWI Preconditioning speeds up by factor 2-3 LSM reduces crosstalk

30. Multisource Migration: m mig =L T d Forward Model: Multisource Least Squares Migration d +d =[ L +L ]m 1221 L {d { Standard migration Crosstalk term Phase encoding Kirchhoff kernel

31. Multisource Least Squares Migration Crosstalk term

32. 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 Problem: L is too big for IO bound hardware L 1 L 2d1 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 Note: subscripts agree

33. Key Results Theory of Multisource Imaging of Encoded Supergathers (Boonyasiriwat) Sig/Noise = GI < GIN 3.5 km Dynamic QMC Tomogram (99 CSGs/supergather) (99 CSGs/supergather) Dynamic Polarity Tomogram (1089 CSGs/supergather) 1/1000 1/300 # geophones/supergather # iterations # subsupergatherss