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

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

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

Least Squares Migration m (k+1) = m (k) +  L  d (k) T m = [L T L] -1 L T d Geom. Spreading: 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

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),

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

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

RTM vs LSM X (km) Reverse Time Migration Plane-Wave LSM X (km)

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

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)

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

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

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

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

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

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

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

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

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

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)

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

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

0 Z (km) X (km) True Reflectivity Acoustic LSRTM 0 X (km) 2 Viscoacoustic LSRTM Z (km) X (km) 2 Q Model Q=20 Q=20000

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

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

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 ) TT [] In general, huge dimension matrix Linear Optimization = Least Squares Migration Non-Linear Optimization = Full Waveform Inversion

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

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

Shots per supergather gain Computational gain Conventional migration: Sensitivity to input noise level SNR=10dB SNR=30dB SNR=20dB

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

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

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

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