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

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

Gulf of Mexico Seismic Survey m L m = d 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

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)

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

(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 ) TT [] In general, huge dimension matrix Note: subscripts agree

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 ) TT [] In general, huge dimension matrix Problem: Expensive, FD solve/CSG Solution: Blend+encode Data (k)

Blending+Phase Encoding 2 d = N d + N d + N d d = N d + N d + N d PhasePhase Blending Encoding Matrix Encoded supergather L = N L + N L + N L 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

(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 ) TT [] Crosstalk + L N N (L m - d ) 2 T 1 L N N (L m - d ) L N N (L m - d ) 1 T ** (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 )

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

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

Crosstalk Prediction Formula L (L m - d ) 2 T 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

Ensemble Average of Crosstalk Term With Random Polarity + Crosstalk: L N N (L m - d ) L N N (L m - d ) 2 T T ** 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

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) X (km) X (km) 0.4 Z (km) X (km) 0.4 Z (km) X (km) e) Noise Time static σ (s) 0.18 SNR 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)

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) X (km) 06.7 X (km) b) Image of 5 stacks d) SNR vs Iterations Observed Prediction # iterations Z (km) Bulk shift

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)

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

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

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 [ m = [N d + N d ] ]m = [N d + N d ]

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

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) X (km) X (km) 0.4 Z (km) X (km) 0.4 Z (km) X (km) e) Noise Time static σ (s) 0.18 SNR 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)

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) X (km) 06.7 X (km) b) Image of 5 stacks d) SNR vs Iterations Observed Prediction # iterations Z (km)

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

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 ) TT = L d +L d TT L d +L d L d +L d212 1 Crosstalk noise Standard migration TT m = m + (k+1)(k)

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

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 ) TT = L d +L d TT L d +L d L d +L d212 1 Crosstalk noise Standard migration TT m mig = L d +L d TT L d +L d L d +L d T T m mig = L d +L d m mig ++

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 ) TT m mig = L d +L d TT L d +L d L d +L d212 1 Crosstalk noise Standard migration TT m = m + (k+1)(k)

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

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

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

Multisource Least Squares Migration Crosstalk term

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 ) TT [] In general, huge dimension matrix Note: subscripts agree

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