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Wave-equation MVA by inversion of differential

image perturbations

Paul Sava & Biondo BiondiStanford University

SEP

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Motivation

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Wave-equation MVA (WEMVA)

• Band-limited• Multi-pathing• Resolution

• Born approximation– small anomaly

• Rytov approximation– phase unwrapping

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Wave-equation MVA (WEMVA)

• WE tomography– data space

• WE MVA– image space

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Outline

1. WEMVA overview

2. Born image perturbation

3. Differential image perturbation

4. Example

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A tomography problem

sqs LΔminΔTraveltime

MVA

Wave-equation tomography

Wave-equation MVA

q t traveltime

d

data

Rimage

L ray field wavefield wavefield

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Δss0

sii 0seW U

WEMVA: main idea

0WWWΔ

0s

0s0 eW iU

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Born approximation 1eWΔW Δs

0 i

ΔsWΔW 0 i

iei 1

ie

sR LΔ

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WEMVA: objective function

slowness perturbation

image perturbation

slownessperturbation(unknown)

Linear WEMVAoperator

imageperturbation

(known)

sRs LΔminΔ

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WEMVA: objective function

sRs LΔminΔ

Traveltime

MVA

Wave-equation tomography

Wave-equation MVA

t d R

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Fat ray: GOM example

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Outline

1. WEMVA overview

2. Born image perturbation

3. Differential image perturbation

4. Example

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“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 R0

• Velocity ratio 0RSR

RR0

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Prestack Stolt residual migration

0RRR • Image perturbation

RR0

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Born approximation

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Residual migration: the problem

Correct velocity Incorrect velocity

Zero offset image

Angle gathers

Zero offset image

Angle gathers

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Born approximation

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Outline

1. WEMVA overview

2. Born image perturbation

3. Differential image perturbation

4. Example

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Differential image perturbation

0

1

ˆ Rd

dSR

00 RRSR Image

difference

Image differential

Computed Measured

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Differential image perturbation

R

R

0

1

ˆ Rd

dSR

00 RRSR

R

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Phase perturbation

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Differential image perturbation

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Born approximation

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Example: background image

Zero offset image

Angle gathers

Background image

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Example: differential image

Zero offset image

Angle gathers

Differential image

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Example: slowness inversion

Slowness perturbation

Image perturbation

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Example: updated image

Updated slowness

Updated image

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Example: correct image

Correct slowness

Correct image

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Outline

1. WEMVA overview

2. Born image perturbation

3. Differential image perturbation

4. Example

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Field data example

• North Sea– Salt environment– Subset

– 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

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location

dep

thde

pth

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dep

th

velocity ratio velocity ratio

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1

1

1

location

dep

thde

pth

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location

dep

th

location

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location

dep

th

location

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location

dep

thde

pth

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location

dep

thde

pth

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Summary

• MVA– Wavefield extrapolation methods– Born linearization– Differential image perturbations

• Key points– Band-limited (sharp velocity contrasts)– Multi-pathing (complicated wavefields)– Resolution (frequency redundancy)

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MVA information (a)Traveltime MVA Wave-equation MVA

• Offset focusing (flat ADCIG) • Offset focusing (flat ADCIG)

z

z

xx

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MVA information (b)Traveltime MVA Wave-equation MVA

• Offset focusing (flat ADCIG) • Offset focusing (flat ADCIG)

• Spatial focusing

z

z

xx

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MVA information (c)Traveltime MVA Wave-equation MVA

• Offset focusing (flat ADCIG) • Offset focusing (flat ADCIG)

• Spatial focusing

• Frequency redundancy

low high

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low high

WEMVA cost reduction

• Full image– Offset focusing

– Spatial focusing

– Frequency

• Normal incidence image

– Spatial focusing

– “fat” rays

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Another example

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Example: correct model

Zero offset image

Angle gathers

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Example: background model

Zero offset image

Angle gathers

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Example: correct perturbation

Zero offset image

Angle gathers

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Example: differential perturbation

Zero offset image

Angle gathers

1d

dRR

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Example: perturbations comparison

Differential

Difference

Correct

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Example: differential perturbation

Zero offset image

Angle gathers

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Example: difference perturbation

Zero offset image

Angle gathers

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Example: updated model

Zero offset image

Angle gathers

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Example: correct model

Zero offset image

Angle gathers