Least Squares Migration of Least Squares Migration of

Stacked SupergathersStacked Supergathers

Wei Dai and Gerard SchusterWei Dai and Gerard SchusterKAUSTKAUST

vs

RTM Problem & Possible Soln.RTM Problem & Possible Soln.

• Problem:Problem: RTM computationally costly; IO high RTM computationally costly; IO high

• Solution:Solution: Multisource LSM RTM Multisource LSM RTM

Preconditioning speeds up by factor 2-3Preconditioning speeds up by factor 2-3

Encoded LSM reduces crosstalk. Reduced comp. cost+memoryEncoded LSM reduces crosstalk. Reduced comp. cost+memory

OutlineOutline• MotivationMotivation

• Multisource LSM theoryMultisource LSM theory

• Signal-to-Noise Ratio (SNR)Signal-to-Noise Ratio (SNR)

• Numerical results Numerical results

• ConclusionsConclusions

Multisource Migration:Multisource Migration: mmmigmig=L=LTTdd

Forward Model:Forward Model:

Phase Encoded Multisource Migration Phase Encoded Multisource Migration

d +d +dd =[ =[L +L +LL ]m ]m11 222211

LL{dd{

=[=[L +L +LL ]( ](dd + + dd ) ) 11 222211

TT TT

= = L d +L d +L dL d + + 11 222211

TT TT

LL dd + +L L dd22 112211

Crosstalk noiseCrosstalk noiseStandard migrationStandard migration

TT TTmmmigmig

= = L d +L d +L dL d + + 11 222211

TT TT

LL dd + +LL dd22 112211

TT TTmmmigmig

mmmigmig

= = L d +L d +L dL d11 222211

mmmigmig

Multisource Migration:Multisource Migration: mmmigmig=L=LTTdd

Forward Model:Forward Model:

Phase Encoded Multisrce Phase Encoded Multisrce Least Squares Least Squares Migration Migration

d +d +dd =[ =[L +L +LL ]m ]m11 222211

LL{dd{

=[=[L +L +LL ]( ](dd + + dd ) ) 11 222211

TT TTmmmigmig

= = L d +L d +L dL d + + 11 222211

TT TT

LL dd + +L L dd22 112211

Crosstalk noiseCrosstalk noiseStandard migrationStandard migration

TT TT

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

OutlineOutline• MotivationMotivation

• Multisource LSM theoryMultisource LSM theory

• Signal-to-Noise Ratio (SNR)Signal-to-Noise Ratio (SNR)

• Numerical results Numerical results

• ConclusionsConclusions

Standard Migration SNR

GS# geophones/CSG# geophones/CSG

# CSGs# CSGs

SNR= ...

migrate

SNR=

d(t) =d(t) =Zero-mean white noise

[S(t) +N(t) ][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# iterations

SNR=

Cost ~ O(S)

Cost ~ O(I)

SN

R0

1 Number of Iterations 300

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

SNR = GI

Multisource LSM SummaryMultisource LSM Summary

IO 1 1/100

Cost ~

Resolution dx 1 1/2

SNR

Stnd. Mig Multsrc. LSMStnd. Mig Multsrc. LSM

GS GI

S I

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

OutlineOutline• MotivationMotivation

• Multisource LSM theoryMultisource LSM theory

• Signal-to-Noise Ratio (SNR)Signal-to-Noise Ratio (SNR)

• Numerical results Numerical results

• ConclusionsConclusions

0Z

k(m

)3

0 X (km) 16

The Marmousi2 Model

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

(k

m)

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

(k

m)

3

0 X (km) 16

Multisource KM (1 iteration)

200-source Supergather: Multisrc. KM vs LSM

Multisource KLSM (300 iterations)

1.5 x

1 x200

I=1.5S

IO 1 1/200

Cost ~

Resolution dx 1 1/2

SNR~

Stnd. Mig Multsrc. LSMStnd. Mig Multsrc. LSM

1 1.5

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

What have we empirically learned?

S=200 I=300

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

(k

m)

01.

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 (

km

)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 encodingUnconventional encoding

Conventional encoding: Polarity+Time ShiftsConventional encoding: Polarity+Time Shifts

IO 1 1/320

Cost ~

Resolution dx 1 1/2

SNR~

Stnd. Mig Multsrc. LSMStnd. Mig Multsrc. LSM

I=7

1 1/44

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

What have we empirically learned?

S=320

ConclusionsConclusions Mig vs MLSM Mig vs MLSM

1. 1.

2. Cost: 2. Cost: S S vsvs II

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

5.5. Next Step: Sensitivity analysis to starting modelNext Step: Sensitivity analysis to starting model

SNR: VSGS GI

4. Unconventional encoding: I << S4. Unconventional encoding: I << S

2. Memory 2. Memory 1 1 vsvs 1/S1/S

Back to the Future?Back to the Future?

Poststackencoded migration

DMO Prestackmigration

1980s 1980s-2010 2010?

Evolution of Migration

Poststackmigration

1960s-1970s