Paul@sep.stanford.edu Wave-equation migration velocity analysis beyond the Born approximation Paul...

paul@sep.stanford.edu

Wave-equation migration velocity analysis

beyond the Born approximation

Paul Sava* Stanford University

Sergey Fomel UT Austin (UC Berkeley)

paul@sep.stanford.edu

Imaging=MVA+Migration

• Migration• wavefield based

• Migration velocity analysis (MVA)• traveltime based

• Compatible migration and MVA methods

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Imaging: the “big picture”

• Kirchhoff migration

• traveltime tomography

wavefronts

• wave-equation migration

• wave-equation MVA (WEMVA)

wavefields

paul@sep.stanford.edu

Agenda

Theoretical background

WEMVA methodology

Scattering

Imaging

Image perturbations

Wavefield extrapolation

Born linearization

Alternative linearizations

paul@sep.stanford.edu

Wavefields or traveltimes?

paul@sep.stanford.edu

Wavefields or traveltimes?

paul@sep.stanford.edu

Scattered wavefield

Medium perturbation

Wavefield perturbation

paul@sep.stanford.edu

Agenda

Theoretical background

WEMVA methodology

Scattering

Imaging

Image perturbations

Wavefield extrapolation

Born linearization

Alternative linearizations

paul@sep.stanford.edu

Imaging: Correct velocity

Background velocity

Migrated image

Reflectivity model

What the data tell us...What migration does...

location

depth

location

depth

depthdepth

depth

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Imaging: Incorrect velocity

Perturbed velocity

Migrated image

Reflectivity model

What the data tell us...What migration does...

location

depth

location

depth

depthdepth

depth

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Wave-equation MVA: Objective

Velocity perturbation

Image perturbation

slownessperturbation(unknown)

WEMVAoperator

imageperturbation

(known)

sLΔRminΔs

location

depth

location

depth

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– migrated images

– moveout and focusing– use amplitudes

– parabolic wave equation– multipathing

– slow

– picked traveltimes

– moveout– ignore amplitudes

– eikonal equation

– fast

Comparison of MVA methods

• Wave-equation MVA • Traveltime tomography

paul@sep.stanford.edu

Agenda

Theoretical background

WEMVA methodology

Scattering

Imaging

Image perturbations

Wavefield extrapolation

Born linearization

Alternative linearizations

paul@sep.stanford.edu

What is the image perturbation?

Focusing Flatness Residual process:• moveout• migration• focusing

slownessperturbation(unknown)

WEMVAoperator

imageperturbation

(known)

sLΔRminΔs

location

depth

angle

paul@sep.stanford.edu

Agenda

Theoretical background

WEMVA methodology

Scattering

Imaging

Image perturbations

Wavefield extrapolation

Born linearization

Alternative linearizations

paul@sep.stanford.edu

Double Square-Root Equation

Wikdz

dWz

Δsds

dkkk

0

0

ss

zzz

Fourier Finite DifferenceGeneralized Screen Propagator

Δzikz

Δzzze

W

W

Wavefield extrapolation

βΔsΔzz

0

Δzz

eW

W

βΔsΔzikΔzik0zz

paul@sep.stanford.edu

“Wave-equation” migration

z

Δzz0s

Δzz0W

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Slowness perturbation

0s Δss0

Δzz0W

z

Δzz

βΔsΔzz0 eW

paul@sep.stanford.edu

1eWΔW βΔs0

slownessperturbation

backgroundwavefield

wavefieldperturbation

ΔW

Δs

Wavefield perturbation

z

Δzz0s Δss0

paul@sep.stanford.edu

Agenda

Theoretical background

WEMVA methodology

Scattering

Imaging

Image perturbations

Wavefield extrapolation

Born linearization

Alternative linearizations

paul@sep.stanford.edu

Born approximation

iei 1

ie

Small perturbations!

Born linearization

Non-linear WEMVA

1eWΔW βΔs0

βΔsWΔW 0slowness

perturbation(unknown)

WEMVAoperator

imageperturbation

(known)

sLΔRminΔs

Unit circle

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sLΔRminΔs

Does it work?What if the perturbations are not small?

Location [km]

Depth [km

]

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Synthetic example

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Born approximation

1% 10%

paul@sep.stanford.edu

Agenda

Theoretical background

WEMVA methodology

Scattering

Imaging

Image perturbations

Wavefield extrapolation

Born linearization

Alternative linearizations

paul@sep.stanford.edu

Wavefield continuation

Wikdz

dWz βΔs

0

eW

W

Bilinear

Implicit

βΔs2

βΔs2

W

W

0

βΔs1

1

W

W

0

Explicit βΔs1W

W

0

(Born approximation)

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Exponential approximations

ξβΔs1

βΔsξ11eβΔs

0,1ξ

0ξ

1ξ

0.5ξ

Wikdz

dWz βΔs

0

eW

W

Unit circle

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1eWΔW βΔs0

A family of linearizations

ξβΔs1

βΔsξ11eβΔs

0,1ξ

βΔsξΔWWΔW 0Linear WEMVA

slownessperturbation(unknown)

WEMVAoperator

imageperturbation

(known)

sLΔRminΔs

paul@sep.stanford.edu

Improved linearizations

1% 10%40%

paul@sep.stanford.edu

Agenda

Theoretical background

WEMVA methodology

Scattering

Imaging

Image perturbations

Wavefield extrapolation

Born linearization

Alternative linearizations

paul@sep.stanford.edu

Summary

• Wave-equation MVA• wavefield-continuation• improved focusing • image space (improve the image)• interpretation guided

• Improved WEMVA• better approximations• no additional cost• further refinement