Overview of Multisource Phase

Encoded Seismic Inversion

Wei Dai, Ge Zhan, and Gerard SchusterKAUST

Outline1. Seismic Experiment:

L m = d

L m = d1 1

L m = d2 2...N N

2. Standard vs Phase Encoded Least Squares Soln.

L1

L2

d 1

d 2m = vs N L + N L1 21 2[ ]m = [N d + N d ]1 21 2

3. Theory Noise Reduction 4. Summmary and Road Ahead

Gulf of Mexico Seismic SurveyL m = d

Time (s)

4

0

d

Goal: Solve overdeterminedSystem of equations for m

Predicted data Observed data

m(x,y,z)

Common Shot Gather

Streamer Reel

Streamer Cables

4 km

Details of Lm = d

Time (s)

6 X (km)

4

0

1 d

G(s|x)G(x|g)m(x)dx = d(g|s)

Reflectivityor velocity

model

Predicted data = Born approximationSolve wave eqn. to get G’s

m

Outline1. Seismic Experiment:

L m = d

L m = d1 1

L m = d2 2...N N

2. Standard vs Phase Encoded Least Squares Soln.

L1

L2

d 1

d 2m = vs N L + N L1 21 2[ ]m = [N d + N d ]1 21 2

3. Theory Noise Reduction 4. Summmary and Road Ahead

Conventional Least Squares Solution: L= & d =

Given: Lm=dFind: m s.t. min||Lm-d||2

Solution: m = [L L] L d T T-1

m = m – a L (Lm - d) T(k+1) (k) (k)(k)

or if L is too big

L1

L2

d 1

d 2

= m – a L (L m - d ) (k)

+ L (L m - d ) 1 1 2 2 21

TT[ ]

In general, hugedimension matrix

Conventional Least Squares Solution: L= & d =

Given: Lm=dFind: m s.t. min||Lm-d||2

Solution: m = [L L] L d T T-1

m = m – a L (Lm - d) T(k+1) (k) (k)(k)

or if L is too big

L1

L2

d 1

d 2

= m – a L (L m - d ) (k)

+ L (L m - d ) 1 1 2 2 21

TT[ ]

In general, hugedimension matrix

Note: subscripts agree

Conventional Least Squares Solution: L= & d =

Given: Lm=dFind: m s.t. min||Lm-d||2

Solution: m = [L L] L d T T-1

m = m – a L (Lm - d) T(k+1) (k) (k)(k)

L1

L2

d 1

d 2

= m – a L (L m - d ) (k)

+ L (L m - d ) 1 1 2 2 21

TT[ ]

In general, hugedimension matrix

Problem: Each prediction is a FD solveSolution: Blend+encode Data

Blending+Phase Encoding

2 d = N d + N d + N d211 33

PhasePhaseBlending

Encoding MatrixSupergather

L = N L + N L + N L3 32 21 1m [ ]m

d 1L m=1

Encoded supergather modeler

d 3L m=3d 2L m=2

O(1/S) cost!

Blending

Blended Phase-Encoded Least Squares Solution L = & d = N d + N d

Given: Lm=dFind: m s.t. min||Lm-d||2

Solution: m = [L L] L d T T-1

m = m – a L (Lm - d) T(k+1) (k) (k)(k)

or if L is too big

1N L + N L2 1

= m – a L (L m - d ) (k)

+ L (L m - d ) 1 1 2 2 21

TT[ ]

1 2 1 2 2

In general, SMALLdimension matrix

+ Crosstalk+ L (L m - d ) 2

T

11 + L (L m - d ) 1

T

22

Iterations are proxyFor ensemble averaging

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, SEG, (2009)

Outline1. Seismic Experiment:

L m = d

L m = d1 1

L m = d2 2...N N

2. Standard vs Phase Encoded Least Squares Soln.

L1

L2

d 1

d 2m = vs N L + N L1 21 2[ ]m = [N d + N d ]1 21 2

3. Theory + Numerical Results 4. Summmary and Road Ahead

SEG/EAGE Salt Reflectivity Model

• Use constant velocity model with c = 2.67 km/s • Center frequency of source wavelet f = 20 Hz• 320 shot gathers, Born approximation

Z

(km

)

0

1.4

0 X (km) 6

• Encoding: Dynamic time, polarity statics + wavelet shaping• Center frequency of source wavelet f = 20 Hz• 320 shot gathers, Born approximation

0 X (km) 6

0Z

k(m

)1.

40

Z (k

m)

1.4

0 X (km) 6

Standard Phase Shift Migration (320 CSGs)

Standard Phase Shift Migration vs MLSM (Yunsong Huang)

Multisource PLSM (320 blended CSGs, 7 iterations)

1 x

1 x

44

Single-source PSLSM(Yunsong Huang)

Mod

el E

rror

1.0

0.30 50Iteration Number

Unconventional encoding

Conventional encoding: Polarity+Time Shifts

Multi-Source Waveform Inversion Strategy(Ge Zhan)

Generate multisource field data with known time shift

Generate synthetic multisource data with known time shift from estimated

velocity model

Multisource deblurring filter

Using multiscale, multisource CG to update the velocity model with

regularization

Initial velocity model

144 shot gathers

3D SEG Overthrust Model(1089 CSGs)

15 km

3.5 km

15 km

3.5 km

Dynamic QMC Tomogram (99 CSGs/supergather)

Static QMC Tomogram(99 CSGs/supergather)

15 km

Dynamic Polarity Tomogram(1089 CSGs/supergather)

Numerical Results

1000x

300x

300x

Outline1. Seismic Experiment:

L m = d

L m = d1 1

L m = d2 2...N N

2. Standard vs Phase Encoded Least Squares Soln.

L1

L2

d 1

d 2m = vs N L + N L1 21 2[ ]m = [N d + N d ]1 21 2

3. Theory + Numerical Results 4. Summmary and Road Ahead

Multisource Migration: mmig=LTd

Forward Model:

Multisource Least Squares Migration

d +d =[L +L ]m1 221

L{d{Standard migration

Crosstalk term

Phase encoding

Kirchhoff kernel

34

Multisource Least Squares Migration Crosstalk term

Crosstalk Prediction FormulaL (L m - d ) 2

T

11 + L (L m - d ) 1

T

22 e-s w2 2

O( )~X =

X

s .01 1.0

Standard Migration SNR

GS# geophones/CSG

# CSGs

SNR= ...migrate

SNR=

d(t) = Zero-mean white noise [S(t) +N(t) ] Neglect geometric spreading

Standard Migration SNR

Standard Migration SNR

Assume:

migrate+++

stack

S1

SGS G~~

iterate

GI

Iterative Multisrc. Mig. SNR

# iterations

SNR=

Cost ~ O(S)

Cost ~ O(I)

SNR

0

1 Number of Iterations 300

7The SNR of MLSM image grows as the square root of the number of iterations.

SNR = GI

IO 1 1/320

Cost ~

Resolution dx 1 1

SNR~

Stnd. Mig Multsrc. LSM

Less 1

1 <1/44

Cost vs Quality

Summary

1

L1

L2

d 1

d 2m = N L + N L

1 21 2[ ]m = [N d + N d ]

1 21 2

Multisource FWI Summary(We need faster migration algorithms & better velocity models)

Future: Multisource MVA, Interpolation, Field Data, Migration Filtering, LSM

Issues: Optimal encoding strategies, datacompression, loss of information.

Summary(We need faster migration algorithms & better velocity models)

IO 1 vs 1/20 or better

Cost 1 vs 1/20 or better

Resolution dx 1 vs 1

Sig/MultsSig ?

Stnd. FWI Multsrc. FWI

Multisource Migration: mmig=LTd

Forward Model:

Multisource Least Squares Migration

d +d =[L +L ]m1 221

L{d{Standard migration

Crosstalk term

Phase encoding

Kirchhoff kernel

34

Multisource Least Squares Migration Crosstalk term

Numerical Result of Multi-source Super stacking Reflectivity model

5.9X (km)0

Z (k

m)

1. 40

KM of 320 Single Source CSG

5.9X (km)0

Z (k

m)

1. 40

Narrowed Spectrum Wavelet

0.5time (s)0

Am

plitu

de

- 0. 3

0.4

Signal

FT of Wavelet

0.5Frequency (Hz)0

04.

5

50

Dominant frequency

(Xin Wang)

Numerical Result of Multi-source Super stacking KM of 320 Shots Supergather w/o

PE

5.9X (km)0

Z (k

m)

1. 40

-0.05

040

00

0.05

KM of 3000 Stacking Supergather

5.9X (km)0

Z (k

m)

1. 40

320 × 3000

0

KM of 320 Shots Supergather with PE

5.9X (km)

Z (k

m)

1. 40

Gaussian Distribution

0.05-0.05

05 0 320

Signal + Noise Singal + Noise

Singal + Noise

(Xin Wang)

Numerical Result of Multi-source Super stacking Noise

= Σ Σ Γ(g,x,s)* D0 (g|s)sg + R Σ Σ Σ Γ (g,x,s)* D0 (g|s’)

sg s≠s’

= Signal + Noise − Signal

= < N (g,s) N (g,s’)* > if s≠s’ R = e-2ω σ2 2Crosstalk damping coefficientR (σ) / R (σ0) = e 2ω (σ0 - σ )2 2 2

(Xin Wang)

0Z

k(m

)3

0 X (km) 16

The Marmousi2 Model(Wei Dai)

The area in the white box is used for SNR calculation.

200 CSGs.

Born Approximation

Conventional Encoding: Static Time Shift & Polarity Statics

0 X (km) 16

0Z

k(m

)3

0Z

(km

)3

0 X (km) 16

Conventional Source: KM vs LSM (50 iterations)Conventional KM

50x

1x

Conventional KLSM

0 X (km) 16

0Z

k(m

)3

0Z

(km

)3

0 X (km) 16

Multisource KM (1 iteration)

200-source Supergather: Multisrc. KM vs LSM

Multisource KLSM (300 iterations)

1 x200

Outline

1. Migration Problem and Encoded Migration

2. Standard vs Monte Carlo Least Squares Soln.

3. Numerical Results: Kirchhoff, Phase Shift, RTM

4. Summary

L1

L2

d 1

d 2m = vs N L + N L1 21 2[ ]m = [N d + N d ]1 21 2

SEG/EAGE Salt Reflectivity Model

• Use constant velocity model with c = 2.67 km/s • Center frequency of source wavelet f = 20 Hz• 320 shot gathers, Born approximation

Z

(km

)

0

1.4

0 X (km) 6

• Encoding: Dynamic time, polarity statics + wavelet shaping• Center frequency of source wavelet f = 20 Hz• 320 shot gathers, Born approximation

0 X (km) 6

0Z

k(m

)1.

40

Z (k

m)

1.4

0 X (km) 6

Standard Phase Shift Migration (320 CSGs)

Standard Phase Shift Migration vs MLSM (Yunsong Huang)

Multisource PLSM (320 blended CSGs, 7 iterations)

1 x

1 x

44

Single-source PSLSM(Yunsong Huang)

Mod

el E

rror

1.0

0.30 50Iteration Number

Unconventional encoding

Conventional encoding: Polarity+Time Shifts

Outline

1. Migration Problem and Encoded Migration

2. Standard vs Monte Carlo Least Squares Soln.

3. Numerical Results: Kirchhoff, Phase Shift, RTM

4. Summary

L1

L2

d 1

d 2m = vs N L + N L1 21 2[ ]m = [N d + N d ]1 21 2

3D SEG Overthrust Model(1089 CSGs, Chaiwoot)

15 km

3.5 km

15 km

3.5 km

Dynamic QMC Tomogram (99 CSGs/supergather)

Static QMC Tomogram(99 CSGs/supergather)

15 km

Dynamic Polarity Tomogram(1089 CSGs/supergather)

Numerical Results(Chaiwoot Boonyasiriwat)

1000x

300x

300x

IO 1 1/320

Cost ~

Resolution dx 1 1/2

SNR~

Stnd. Mig Multsrc. LSM

I=7

1 1/44

Cost vs Quality: Can I<<S? Yes.

What have we empirically learned?

S=320

Outline

1. Migration Problem and Encoded Migration

2. Standard vs Monte Carlo Least Squares Soln.

3. Numerical Results

4. S/N Ratio

L1

L2

d 1

d 2m = vs N L + N L1 21 2[ ]m = [N d + N d ]1 21 2

Standard Migration SNR

GS# geophones/CSG

# CSGs

SNR= ...migrate

SNR=

d(t) = Zero-mean white noise [S(t) +N(t) ] Neglect geometric spreading

Standard Migration SNR

Standard Migration SNR

Assume:

migrate+++

stack

S1

SGS G~~

iterate

GI

Iterative Multisrc. Mig. SNR

# iterations

SNR=

Cost ~ O(S)

Cost ~ O(I)

SNR

0

1 Number of Iterations 300

7The SNR of MLSM image grows as the square root of the number of iterations.

SNR = GI

Summary

IO 1 1/100

Cost ~

Resolution dx 1 1/2

SNR

Stnd. Mig Multsrc. LSM

GS GI

S I

Cost vs Quality: Can I<<S?

L1

L2

d 1

d 2m = vs N L + N L1 21 2[ ]m = [N d + N d ]1 21 2

Outline• Motivation• Multisource LSM theory• Signal-to-Noise Ratio (SNR)• Numerical results • Conclusions

Conclusions Mig vs MLSM

1.

2. Cost: S vs I

3. Caveat: Mig. & Modeling were adjoints of one another. LSM sensitive starting model

5. Next Step: Sensitivity analysis to starting model

SNR: VSGS GI

4. Unconventional encoding: I << S

2. Memory 1 vs 1/S

Back to the Future?

Poststackencoded migration

DMO Prestackmigration

1980s 1980s-2010 2010?

Evolution of Migration

Poststackmigration

1960s-1970s

1980

Multisource SeismicImaging

vs

copper

VLIW

Superscalar

RISC

1970 1990 2010

1

100

100000

10

1000

10000

Aluminum

Year202020001980

Spee

d

CPU Speed vs Year

Multisource Migration: mmig=LTd

Forward Model:

Multisource Phase Encoded Imaging

d +d =[L +L ]m1 221

L{d{

=[L +L ](d + d ) 1 221

T T

= L d +L d + 1 221

T T

L d +L d2 121

Crosstalk noiseStandard migration

T T

m = m +(k+1) (k)

FWI Problem & Possible Soln.• Problem: FWI computationally costly

• Solution: Multisource Encoded FWI Preconditioning speeds up by factor 2-3

Iterative encoding reduces crosstalk

Outline

1. Migration Problem and Encoded Migration

2. Standard vs Monte Carlo Least Squares Soln.

3. Numerical Results: Kirchhoff, Phase Shift, RTM

4. Summary

L1

L2

d 1

d 2m = vs N L + N L1 21 2[ ]m = [N d + N d ]1 21 2