Interferometric Interferometric Interpolation of 3D OBS Interpolation of 3D OBS

DataData

Weiping Cao, University of UtahWeiping Cao, University of Utah

Oct. 29 2009Oct. 29 2009

Outline

• Problems: Missing and sparse traces

• Methodology: Interferometric interpolation

• Numerical results: – 3D layered model– Anti-aliasing condition for

interferometric redatuming• Conclusions

• Problems: Missing and sparse traces

• Methodology : Interferometric interpolation

• Numerical results: – 3D layered model– Anti-aliasing condition for

interferometric redatuming• Conclusions

Outline

Motivations

Problem: Receiver interval of OBS data is (sometimes) too large

Solution: Interferometric interpolation

Water

Water

Benefits of Interferometric Benefits of Interferometric Interpolation: Interpolation:

• Accuracy (wave-equation based Accuracy (wave-equation based scheme)scheme)

• No sedimentary velocity neededNo sedimentary velocity needed

Outline

• Problems: Missing and sparse traces

• Methodology: Interferometric interpolation

• Numerical results: – 3D layered velocity model– Anti-aliasing condition for

interferometric redatuming• Conclusions

Interferometric Interpolation of OBS Data

G(B|A)Interpolated OBS

Data

Seabed

Reflectors

Ocean Surface

xB

A

G(x|A)Natural OBS

Green’s Function

Seabed

Reflectors

Ocean Surface

x

A

Go(x|B)*

Model based Green’s Function

B

Seabed

Ocean Surface

x

A

Dong S. and G. T. Schuster, 2008, Interferometric interpolation and extrapolation of sparse OBS and SSP data: UTAM 2007 annual meeting, 39 – 48.

Interferometric Interpolation of OBS Data

xx

x|Bx|A

x

x|Ax|BA|BA|B 2

*0*

0*

0

0

)()(

)()()()( d

GG

GGGG

S

2-state reciprocity equation:

xx|Ax|BA|BA|B 2**

0

00)()(2)()( dGGikGG

S

goingupgoingdown

Up-down separation, far-field approximationWater-layer

reflectionOBS reflection

Artifacts? (up-down separation, far-field approx., limited aperture, wavelet, sampling… Matching filter!

WorkflowInput Field Data

Water Layer Model

Generate GF for Water Multiples

Interpolate Missing Data

Max. Itr (MF)

Get Virtual CSG

Max Itr Intr/Extr Final CSGN

Matching Filter

N

YY

Tim

e (s

)T

ime

(s)

00

3.03.0X (km)X (km)00 4.54.5 Seabed

Ocean Surface

x

Tim

e (s

)T

ime

(s)

00

3.03.0X (km)X (km)00 4.54.5

Tim

e (s

)T

ime

(s)

00

3.03.0X (km)X (km)00 4.54.5

Input Data

Unfiltered Virtual

Filtered Virtual

Outline

• Problems: Missing and sparse traces

• Methodology: Interferometric interpolation

• Numerical results: – 3D layered velocity model– Anti-aliasing condition for

interferometric redatuming• Conclusions

Numerical Results

• 3D velocity model size: 3000 x 3000 x 1400 m3

• Source located at (10 m,10 m, 30

m)• 300 by 300 receivers dx = dy = 10 m• Sea bed is flat at a

depth of 750 m

3 km3

km

1.4 km

Source

Layered Velocity Model

Velocity (m/s)1500 2400

Sea bed

Reflector 1

Reflector 2

Synthetic DataLine y=1000m

0

5

Tim

e (s

)

0 3000X (m)

CSG in the x direction: y=1000 m, dx = 10 m

0

5

Tim

e (s

)

0 3000Y (m)

CSG in the y direction: x=1000 m, dx = 10 m

3D Interpolation

Goal: dense OBS data

• Recording interval:

10 m × 10 m

• Total number of receivers:

300 × 300 = 90, 000

Input: sparse OBS data

• Recording interval

50 m × 50 m ( =104 m )

• Total number of receivers:

60 × 60 = 3, 600

Sparse Data

0

5

Tim

e (s

)

0 3000X (m)

Line y=1000m

Decimated CSG in the X direction: Decimated CSG in the X direction: Y = 1000 m , dx = 50 m Y = 1000 m , dx = 50 m

0

5

Tim

e (s

)

0 3000Y (m)

Decimated CSG in the Y direction: Decimated CSG in the Y direction: X = 1000 m, dx = 50 m X = 1000 m, dx = 50 m

Interpolation Results: X direction

0

5

Tim

e (s

)

0 3000X (m)

Line y=1000m

Decimated CSG in the X direction: Decimated CSG in the X direction: Y =1000 m , dx = 50 mY =1000 m , dx = 50 m

0

5

Tim

e (s

)

0 3000Y (m)

Virtual dense data, dx = 10 mVirtual dense data, dx = 10 m

Tim

e (s

)T

ime

(s)

00

3.03.0X (km)X (km)00 4.54.5

Tim

e (s

)T

ime

(s)

00

3.03.0X (km)X (km)00 4.54.5

Local Matching Filter

000 ,*,, xtfxtdxtd virtualreal

0

5

Tim

e (s

)

0 3000X (m)

0

5

Tim

e (s

)

0 3000X (m)

Line y=1000m

Filtered virtual data, dx = 10 mDecimated CSG in the X direction: Y =1000 m , dx = 50 m

Interpolation Results: X direction

0

5

Tim

e (s

)

0 3000X (m)

0

5

Tim

e (s

)

0 3000X (m)

Line y=1000m

Real dense data, dx = 10 mReal dense data, dx = 10 mDecimated CSG in the X direction: Decimated CSG in the X direction: Y =1000 m , dx = 50 mY =1000 m , dx = 50 m

Interpolation Results: X direction

0

5

Tim

e (s

)

0 3000Y (m)

Line y=1000m

0

5

Tim

e (s

)

0 3000Y (m)

Virtual dense data, dx = 10 mVirtual dense data, dx = 10 mDecimated CSG in the Y direction: Decimated CSG in the Y direction: X =1000 m , dx = 50 mX =1000 m , dx = 50 m

Interpolation Results: Y direction

0

5

Tim

e (s

)

0 3000Y (m)

0

5

Tim

e (s

)

0 3000Y (m)

Line y=1000m

Virtual data after filtering, dx = 10 mVirtual data after filtering, dx = 10 mDecimated CSG in the Y direction: Decimated CSG in the Y direction: X =1000 m , dx = 50 mX =1000 m , dx = 50 m

Decimated CSG in the Y direction: Decimated CSG in the Y direction: X =1000 m , dx = 50 mX =1000 m , dx = 50 m

Interpolation Results: Y direction

0

5

Tim

e (s

)

0 3000Y (m)

0

5

Tim

e (s

)

0 3000Y (m)

Line y=1000m

Real dense data, dx = 10 mReal dense data, dx = 10 mDecimated CSG in the Y direction: Decimated CSG in the Y direction: X =1000 m , dx = 50 mX =1000 m , dx = 50 m

Interpolation Results: Y direction

1.0

3.5

Tim

e (s

)

10 2710X offset (m)

True vs. Virtual traces before Filtering

True

Virtual

Interpolation Results: Trace Comparison

1.0

3.5

Tim

e (s

)

True vs. the Virtual Traces after Filtering

True

Virtual

10 2710X offset (m)

Interpolation Results: Trace Comparison

1.0

3.5

Tim

e (s

)

True vs. the Virtual Traces after Filtering

10 2710X offset

Interpolation Results: Trace Comparison

Different Recording Spacings

0.5

1.2

Nor

mal

ized

err

or

0.20.05

Interpolation error vs. recording spacingInterpolation error vs. recording spacing

Recording spacing of input data (Recording spacing of input data (λλxxminmin))

The normalized error The normalized error ==

2

2

2

2

real

interp.real

d

dd

Outline

• Problems: Missing and sparse traces

• Methodology: Interferometric interpolation

• Numerical results: – 3D layered velocity model– Anti-aliasing condition for

interferometric redatuming• Conclusions

Interferometric redatuming equation:

Anti-aliasing Condition for Interferometric Redatuming

Phase difference between and

less than

G(A|x) G(B|x)

Axx

x

Bxx

x

Anti-aliasing condition

0

3

Tim

e (s

)

0.6 4X (km)

Remove Interf. Artifacts with the Anti-aliasing Condition

Anti-aliased Interf. ResultRegular Interf. Result

Recording interval 0.49 λ

0

3

Tim

e (s

)

0.6 4X (km)

0

3

Tim

e (s

)

0.6 4X (km)

Remove Interf. Artifacts with the Anti-aliasing ConditionAnti-aliased Interf. Result with Up-down Separation

Regular Interf. Result with Up-down Separation

Recording interval 0.97 λ

0

3

Tim

e (s

)

0.6 4X (km)

Outline

• Problems: Missing and sparse traces

• Methodology: Interferometric interpolation

• Numerical results: – 3D layered velocity model– Anti-aliasing condition for

interferometric redatuming• Conclusions

Conclusions

Encouraging results obtained for Encouraging results obtained for interpolating sparse OBS data interpolating sparse OBS data (recording spacing: )(recording spacing: )

Degraded interpolation results when the Degraded interpolation results when the recording spacing of the input sparse recording spacing of the input sparse data increasesdata increasesRemaining artifacts: Remaining artifacts:

up-down separation, anti-aliasing up-down separation, anti-aliasing conditioncondition

Acknowledgments

Thank UTAM 2008 sponsors for the support of the research.

Thank you all for your attention.