paul@sep.stanford.edu

Analytical image perturbations for wave-equation

migration velocity analysis

Paul Sava & Biondo BiondiStanford University

paul@sep.stanford.edu

Wave-equation MVA (WEMVA)

• Wavefield-based MVA method

• Closely related to– Wave-equation migration– Wave-equation tomography

• Benefits– Finite-frequency– Multipathing– Hi resolution

paul@sep.stanford.edu

A tomography problem

sqs LΔminΔTraveltime

tomography/MVA

Wave-equation tomography

Wave-equation MVA

q t traveltime

d

data

Rimage

L ray field wavefield wavefield

paul@sep.stanford.edu

Outline

1. WEMVA review

2. Image perturbation

3. Field data example

paul@sep.stanford.edu

Δss1

Δss1eW fU

WEMVA: main idea 1eWΔW Δs

1 f

1s

1s1 eW fU

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Born approximation 1eWΔW Δs

1 f

ΔsWΔW 1 f

iei 1

ie

sR LΔ

paul@sep.stanford.edu

WEMVA: objective function

slowness perturbation

image perturbation

slownessperturbation(unknown)

Linear WEMVAoperator

imageperturbation

(known)

sRs LΔminΔ

paul@sep.stanford.edu

Slowness backprojection

slowness perturbation

image perturbation

slowness perturbation

image perturbation

Rs Δ*L

paul@sep.stanford.edu

MVA informationTraveltime MVA Wave-equation MVA

• Offset focusing (flat gathers) • Offset focusing (flat gathers)

• Spatial focusing

• Frequency redundancy

z

z

xx

paul@sep.stanford.edu

Outline

1. WEMVA review

2. Image perturbation

3. Field data example

paul@sep.stanford.edu

“Data” estimate

Traveltime

MVA

Wave-equation tomography

Wave-equation MVA

t d Rray

tracing

data

modeling

residual

migration

sRs LΔminΔ

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Prestack Stolt residual migration

• Background image R1

• Velocity ratio ),( 1 RSR

1RRR • Image perturbation

R

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Incorrect velocityCorrect velocity

Zero offset image

Angle gathers

Synthetic model

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Residual migration: the problem

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Differential image perturbation

1d

dRR

1RRR Image

difference

Image differential

Computed Measured

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Background image

Zero offset image

Angle gathers

Background image

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Differential image

Differential image

Zero offset image

Angle gathers

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Image to slowness perturbation

Slowness perturbation

Image perturbation

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Image comparison

Updated slownessCorrect slowness

Zero offset image

slowness

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Outline

1. WEMVA review

2. Image perturbation

3. Field data example

paul@sep.stanford.edu

Field data example

• North Sea– Salt environment

– One non-linear iteration• Migration (background image)

• Residual migration (image perturbation)

• Slowness inversion (slowness perturbation)

• Slowness update (updated slowness)

• Re-migration (updated image)

location

dep

th

paul@sep.stanford.edu

location

dep

thde

pth

Zero offset image

Angle gathers

Background slowness

Background image

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dep

th

velocity ratio velocity ratio

Semblance Angle-gathers

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1

1

1

location

dep

th

Zero offset image

Background image

location

“Ratio” map

1d

dRR

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location

dep

th

location

Zero offset image Zero offset image

Background image

Image perturbation

paul@sep.stanford.edu

location

dep

th

location

Zero offset image

Image perturbation

Slowness perturbation

sRs LΔminΔ

paul@sep.stanford.edu

location

dep

thde

pth

Zero offset image

Angle gathers

Background slowness

Background image

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location

dep

thde

pth

Zero offset image

Angle gathers

Updated slowness

Updated image

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dep

thde

pth

location

Angle gathers

“Correct” slowness

Zero offset image

“Correct” image

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Summary

• Wave-equation MVA– Finite frequency– Multipathing– Hi resolution– Image space objective function

• Image perturbation– From prestack Stolt residual migration– Differential method– Compliant with the Born approximation