Iterative Migration Deconvolution (IMD) Iterative Migration Deconvolution (IMD) with Migration Green’s Functions as with Migration Green’s Functions as

PreconditionersPreconditioners

Naoshi Aoki

Feb. 5, 2009

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OutlineOutline• Introduction

• Theory– Inexpensive IMD

• Numerical results– 2D model IMD test– 3D model IMD test

• When should we use IMD and LSM ?

• Conclusions

2

OutlineOutline• Introduction

• Theory– Inexpensive IMD

• Numerical results– 2D model IMD test– 3D model IMD test

• When should we use IMD and LSM ?

• 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)

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Tmigm = L d T= L Lm

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

1[ ]Tmigm = L L m

Grid Reflectivity Model

Migration Green’s Function or MGFMigration Green’s Function or MGF

• is known as the migration Green’s function.

• It is an impulse response of migration operator.

• MGF variation depends on:– acquisition geometry,– and velocity distribution.

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Z (

km)

X (km)

TW

T (

s)

Synthetic Data

MGFs

[ ]TL L

OutlineOutline• Introduction

• Theory– Inexpensive IMD

• Numerical results– 2D model IMD test– 3D model IMD test

• When should we use IMD and LSM ?

• Conclusions

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Inexpensive IMD TheoryInexpensive IMD Theory

( 1) ( ) ( ) ( )[ ] [ ] ,k k k T T kmig m m L L L L m -m

Expensive IMD

Inexpensive IMD with Preconditioned MGFs

( 1) ( ) ( ) ( )ˆ ˆ ˆ ,k k k T kmig m m G Gm -m

where and represent the k+1 and k th models, is a step length, and is expensive MGF.

where represents a preconditioned normal matrix that contains the preconditioned MGF in each subsection, denotes amplitude compensated migration image.

[ ]TL L( )k

( 1)km ( )km

1ˆ ( )T Tmig xx

m L L L d

1ˆ ( ) [ ]T TxxG L L L L

Expensive and Inexpensive MGFsExpensive and Inexpensive MGFs

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[ ]TL L 1ˆ ( ) [ ]T TxxG L L L L

OutlineOutline• Introduction

• Theory– Inexpensive IMD

• Numerical results– 2D model IMD test– 3D model IMD test

• When should we use IMD and LSM ?

• Conclusions

9

Test WorkflowTest Workflow

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Migration ImageMigration Image

Reflectivity ModelReflectivity Model

Compute MGFsCompute MGFs

Point Scatterer ModelPoint Scatterer Model

Compute IMDCompute IMD

Data Preparation Part MGF Computation Part

IMD Computation Part

Compare with LSMCompare with LSM

Data Preparation PartData Preparation Part

0 1.8

0

1.8

X (km)

Z (

km)

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2D Stick Model

0 1.8

0

1.8

X (km)

Z (

km)

Prestack Migration

Migration ImageMigration Image

Reflectivity ModelReflectivity Model

0 1.8

0

1.8

X (km)

Z (

km)

Scatterers

0 1.8

0

1.8

X (km)

Z (

km)

MGFs

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MGF Computation PartMGF Computation PartCompute MGFsCompute MGFs

Point Scatterer ModelPoint Scatterer Model

0 1.8

0

1.8

X (km)

Z (

km)

IMD Imageafter 43 Iterations

130 1.8

0

1.8

X (km)

Z (

km)

Prestack Migration

IMD Computation PartIMD Computation PartCompute IMDCompute IMD

0 1.8

0

1.8

X (km)

Z (

km)

0 1.8

0

1.8

X (km)

Z (

km)

IMD Imageafter 43 Iterations

LSM Imageafter 30 Iterations

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Compute IMDCompute IMD

Compare with LSMCompare with LSM

IMD vs LSMIMD vs LSM

Model ResidualModel Residual

7

5

Res

idua

l

1 4330

Iteration number 15

IMD Noise level

Computational CostsComputational Costs

Process Computational Costs(CPU seconds)

1 Migration

54 MGFs

43 IMD Iterations

30 LSM Iterations

9

640

55

860

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Expensive and Inexpensive MGFsExpensive and Inexpensive MGFs

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Compute MGFs One by One Compute MGFs at Once

Clean MGFs without interference A possible problem is interference from other MGFs.

OutlineOutline• Introduction

• Theory– Inexpensive IMD

• Numerical results– 2D model IMD test– 3D model IMD test

• When should we use IMD and LSM ?

• Conclusions

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3D Model Test3D Model Test

Model Model Description• Model size:

– 1.8 x 1.8 x 1.8 km

• U shape reflectivity anomaly

• Cross-spread geometry– Source : 16 shots, 100 m int.– Receiver : 16 receivers , 100 m int.

Depth (m) Reflectivity

250 1

500 -1

750 1

1000 -1

1250 1

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0

20

2

0

2X (km) Y (km)

Z (

km)

● Source● Receiver

Test WorkflowTest Workflow

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Migration ImageMigration Image

Reflectivity ModelReflectivity Model

Compute MGFsCompute MGFs

Point Scatterer ModelPoint Scatterer Model

Compute IMDCompute IMD

Data Preparation Part MGF Computation Part

IMD Computation Part

Compare with LSMCompare with LSM

Data PreparationData Preparation

0

1.80 1.8

Z (

km)

Y (

km)

X (km)

Prestack MigrationY = 1 km

0

1.80 1.8

X (km)

Prestack MigrationZ = 750 m

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Migration ImageMigration Image

Reflectivity ModelReflectivity Model

MGF Computation PartMGF Computation Part

0

1.80 1.8

Z (

km)

Y (

km)

X (km)

MGF ImageY = 1 km

0

1.80 1.8

X (km)

MGF ImageZ = 750 m

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Compute MGFsCompute MGFs

Point Scatterer ModelPoint Scatterer Model

IMD Computation PartIMD Computation Part

IMD Image after 30 Iterations

Y = 1000 m

Z = 750 m

Z (

km)

X (km)0 1.8

Y (

km)

0

1.80 1.8

0

1.823

PrestackMigration

Z (

km)

X (km)0 1.8

Y (

km)

0

1.80 1.8

0

1.8

Compute IMDCompute IMD

IMD vs Prestack MigrationIMD vs Prestack Migration

750 m

0

1.8 0 1.8

Y (

km)

0 1.8 0 1.8 0 1.8 0 1.8X (km)

Z = 250 m 1000 m 1250 m 1500 m

1.80 1.8

Y (

km)

0 1.8 0 1.8 0 1.8 0 1.8X (km)

0

IMD after 30 Iterations

Prestack Migration

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IMD vs LSMIMD vs LSM

IMD image after 30 Iterations

0

1.80 1.8

Z (

km)

Y = 1000 m

X (km)0 1.8

Z = 750 m

Y (

km)

Z (

km)

X (km)0 1.8

Y (

km)

0

1.8

0

1.80 1.8

0

1.8

LSM Image after 30 Iterations

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Compute IMDCompute IMD

Compare with LSMCompare with LSM

IMD vs LSMIMD vs LSM

750 m

0

1.8 0 1.8

Y (

km)

0 1.8 0 1.8 0 1.8 0 1.8X (km)

Z = 250 m 1000 m 1250 m 1500 m

1.8 0 1.8

Y (

km)

0 1.8 0 1.8 0 1.8 0 1.8X (km)

0

IMD Images after 30 Iterations

LSM Images after 30 Iterations

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Model ResidualModel Residual

84

76

Res

idua

l

1 30

Iteration number 27

IMD Noise level

Computational CostsComputational Costs

Process Computational Costs(CPU seconds)

1 Migration

486 MGFs

30 IMD Iterations

30 LSM Iterations

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190

25500

65400

15400

Why Is IMD So Slow?Why Is IMD So Slow?

• Computational cost of IMD is 6 times higher than that of LSM because:– the cross-spread geometry has a large MGF

variation,– convolution / cross-correlation is used in the

space domain.

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OutlineOutline• Introduction

• Theory– Inexpensive IMD

• Numerical results– 2D model IMD test– 3D model IMD test

• When should we use IMD and LSM ?

• Conclusions

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Difference Between LSM and IMDDifference Between LSM and IMD• Both methods minimize the misfit with the data.

LSM IMD

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( 1) ( ) ( ) ( )ˆ ˆ ˆk k k T kmig m m G Gm m- ( 1) ( ) ( ) ( )k k k T k m m L Lm - d

d

( )km ( )km

ˆ migm

When Should We Use IMD and LSM ?When Should We Use IMD and LSM ?

IMDLarger amount of 5-D data

provides smaller MGF variation.

LSMSmaller amount of 5-D data

provides larger MGF variation.

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d d

Suitable ApplicationSuitable Application

IMD LSM

Large amount of data

Coarse acquisition geometry

Complicated geology

Target oriented deblurring

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OutlineOutline• Introduction

• Theory– Inexpensive IMD

• Numerical results– 2D model IMD test– 3D model IMD test

• When should we use IMD and LSM ?

• Conclusions

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ConclusionsConclusions• Inexpensive IMD method with preconditioned

MGF is developed.

• 2D IMD achieves a quality almost equal to that from LSM with cheaper computational cost.

• 3D IMD test suggests that IMD quality and cost depend on required MGF density, and investigation of the required MGF density is important.

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Continued WorkContinued Work

• An IMD test on PEMEX RTM image is presented by Qiong Wu.

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ThanksThanks

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