Fast Least SquaresFast Least Squares MigrationMigrationwith a Deblurring Filterwith a Deblurring Filter

Naoshi Aoki

Feb. 5, 2009

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OutlineOutline

• Motivation

• Theory– Deblurring filter theory– Deblurred LSM theory

• Numerical results of the deblurred LSM– Marmousi2 model test– 2D marine data test

• Conclusions

2

OutlineOutline

• Motivation

• Theories– Deblurring filter theory– Deblurred LSM theory

• Numerical results of the deblurred LSM– Marmousi2 model test– 2D marine data test

• Conclusions

3

Deblurring Migration ImageDeblurring Migration Image• Migration

• Two methods to deblur the migration image– Least Squares Migration (e.g., Nemeth et al.,1999)

– Migration Deconvolution (Hu and Schuster, 2001)

Tmigm = L d

( 1) ( ) ( ) ( ) ,k k k k Tm = m L (Lm - d)

T= L Lm

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1[ ]Tmigm = L L m

MotivationMotivation• Problems

– LSM can be more than an order of magnitude more costly than standard migration.

– MD filter is characterized by image artifacts known as MD edge artifacts.

• Solution– Use an MD image as an a priori model for a

regularized LSM.– Use an MD filter as a preconditioner for LSM.

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OutlineOutline

• Motivation

• Theories– Deblurring filter theory– Deblurred LSM theory

• Numerical results of the deblurred LSM– Marmousi2 model test– 2D marine data test

• Conclusions

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Deblurring Filter Theory (1)Deblurring Filter Theory (1)

1. Actual Migration Image

2. Reference migration image

[ ]T TL d = L L m

[ ] 'T TL d' = L L m

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Migration Model

?

Reference Migration Reference Model

Deblurring Filter Theory (2)Deblurring Filter Theory (2)

3. Find non-stationary matching operator

4. Apply the deblurring filter

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F

F

'TFL d' = mReference Migration Reference Model

TFL d mMigration Model

?

OutlineOutline

• Motivation

• Theories– Deblurring filter theory– Deblurred LSM theory

• Numerical results of the deblurred LSM– Marmousi2 model test– 2D marine data test

• Conclusions

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Deblurred LSM (DLSM) TheoryDeblurred LSM (DLSM) Theory

• DLSM is a fast LSM with a deblurring filter.• Two types of DLSM algorithms are proposed:

– Method 1: Regularized DLSM (or RDLSM)

where is , and is a regularization parameter.

– Method 2: Preconditioned DLSM (or PDLSM)

,n Tn aprig L (Lm - d) m -m

TFL d

1 .n n nm = m Fg

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aprim

OutlineOutline

• Motivation

• Theories– Deblurring filter theory– Deblurred LSM theory

• Numerical results of the deblurred LSM– Marmousi2 model test– 2D marine data test

• Conclusions

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Marmousi2 Velocity ModelMarmousi2 Velocity Model

0

3

0 15

Z (

km)

X (km)

4.51.5P wave velocity (km/s)

AnticlineStructures

Test WorkflowTest Workflow

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Migration ImageMigration Image

Velocity ModelVelocity Model

Reflectivity ModelReflectivity Model Reference Migration Reference Migration ImageImage

Reference Reference Reflectivity ModelReflectivity Model

Compute LSMCompute LSMDeblurred Migration Deblurred Migration ImageImage

Find deblurring Find deblurring operatoroperator

Data Preparation Part Deblurring Filter Part

DLSM Part

TFL d

F

Data Preparation PartData Preparation Part

Poststack KM Image

0

3

Z (

km)

6 12X (km)

Actual Reflectivity Model

0

3

Z (

km)

6 12X (km)

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Migration ImageMigration Image

Velocity ModelVelocity Model

Reflectivity ModelReflectivity Model

Deblurring Filter PartDeblurring Filter Part

Reference Migration Image

0

3

Z (

km)

6 12X (km)

Geological Reference Model

0

3

Z (

km)

6 12X (km)15

F

Reference Migration ImageReference Migration Image

Reference Reflectivity ModelReference Reflectivity Model

Find deblurring operatorFind deblurring operator

Deblurred Migration Image

0

3

Z (

km)

6 12X (km)

Actual Migration Image

0

3

Z (

km)

6 12X (km)16

F

Deblurred Migration ImageDeblurred Migration Image

Reference Migration ImageReference Migration Image

Reference Reflectivity ModelReference Reflectivity Model

Find deblurring operatorFind deblurring operatorDeblurring Filter PartDeblurring Filter Part

Method 2: PDLSM after 12 Iterations

Method 1:RDLSM after 20 Iterations

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0

3

Z (

km)

6 12X (km)

0

3

Z (

km)

6 12X (km)

Compute LSMCompute LSM

DLSM PartDLSM Part

Image ComparisonImage Comparison

0

3

Z (

km)

6 12X (km)

Method 2: PDLSM after 12 Iterations

0

3

Z (

km)

6 12X (km)

Method 1:RDLSM after 20 Iterations

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Migration Image after Deblurring Filter0

3

Z (

km)

6 12X (km)

Migration Image

0

3

Z (

km)

6 12X (km)

DLSM Residual CurvesDLSM Residual Curves

Method 2:Method 2:Preconditioned DLSMPreconditioned DLSM

1

01 30

Iteration Number

Res

idua

l

Method 1:Method 1:Regularized DLSMRegularized DLSM

1

01 30

Iteration Number

Res

idua

l

20 12

Damping parameter: Γ= 200000x0.5n-1, n=1,2,…,30

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Noise level

OutlineOutline

• Motivation

• Theories– Deblurring filter theory– Deblurred LSM theory

• Numerical results of the deblurred LSM– Marmousi2 model test– 2D marine data test

• Conclusions

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2D Poststack2D PoststackKirchhoff MigrationKirchhoff Migration

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1

1.5

Z (

km)

X (km)

1310.5

0.5

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Test WorkflowTest Workflow

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Migration ImageMigration Image

Field DataField Data

Reference Migration Reference Migration ImageImage

Reference Reference Reflectivity ModelReflectivity Model

Compute LSMCompute LSMDeblurred Migration Deblurred Migration ImageImage

Find deblurring Find deblurring operatoroperator

Standard Processing Part Deblurring Filter Part

DLSM Part

TFL d

F

Comparison of Imaging ResultsComparison of Imaging Results

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8

1

1.5

Z (

km)

X (km)

1310.5

0.5

Comparison of Images: Box AComparison of Images: Box A

0.5

0.7

Z (

km)

9.6 10.6X (km)

Migration

0.5

0.7

Z (

km)

9.6 10.6X (km)

LSM after 3 Iterations

0.5

0.7

Z (

km)

9.6 10.6X (km)

DLSM after 3 Iterations

0.5

0.7

Z (

km)

9.6 10.6X (km)

LSM after 10 Iterations

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Comparison of Images: Box BComparison of Images: Box BMigration

1

1.2

Z (

km)

11 12X (km)

LSM after 3 Iterations

1.2

Z (

km)

11 12X (km)

1

DLSM after 3 Iterations

1.2Z (

km)

11 12X (km)

1

LSM after 10 Iterations

1.2

Z (

km)

11 12X (km)

1

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OutlineOutline

• Motivation

• Theories– Deblurring filter theory– Deblurred LSM theory

• Numerical results of the deblurred LSM– Marmousi2 model test– 2D marine data test

• Conclusions

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ConclusionsConclusions

• A deblurring filter provides a fine a priori model for a regularized LSM, and can be used as an effective preconditioning filter.

• DLSM algorithms provide acceptable LSM images with 1/3 – 2/3 the cost of standard LSM.

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Continued WorksContinued Works

• 3D DLSM is tested by Wei Dai.

• An improved migration deconvolution technique is presented in my next talk.

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ThanksThanks

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