paul@sep.stanford.edu

Amplitude-preservingwave-equation migration

Paul Sava* Stanford University

Biondo Biondi Stanford University

paul@sep.stanford.edu

The problem

Recover reliable migration amplitudes:i.e. reflectivity as a function of incidence angle

Reflectivity description

Describe reflectivity as a function of incidence angle

Migration algorithm

Apply amplitude corrections to the migration operator

Illumination balance

Restore amplitudes in shadow zones

paul@sep.stanford.edu

Agenda

Amplitude corrections

Angle-domain common image gathers

Imaging effects

Propagation effects

Acquisition effects

Definitions

Methods

Applications

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Common image gathers

z

h

flat events

z

h

focused events

z

flat events

z

flat events

Wave-equationKirchhoff

Offset domain

Angle domain

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Angle-Domain Common Image GathersSource Receiver

V(x,y,z)

2h

v

z

h

k

ktan

h

h

kp

S-G migrationPrucha et al., 1999Sava et al., 2001

Shot-profile migrationde Bruin et al., 1990Rickett & Sava, 2001

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ADCIG: S-G migration

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Agenda

Amplitude corrections

Angle-domain common image gathers

Imaging effects

Propagation effects

Acquisition effects

Definitions

Methods

Applications

paul@sep.stanford.edu

Modeling operator

modeling

migration

0md Ldm *L

02

0*

m

mm

W

LL

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Jacobian weighting: arbitrary geometry

2h

s

1

2

)cos(

1

)cos(

1

shk

W

1

22

4)cos(

1

)cos(

1

4)cos(

1

)cos(

1

s

pk

s

ps

hm

h

hp

W

image space

data space

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Jacobian weighting: flat reflectors

cos2

12

shkW

cos

1

2

12

shpW

(Wapenaar et al., 1999)

2h

s

image space

data space

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Jacobian weighting

Idealoffset-gather

Idealangle-gather

Reflection angle

Offset ray parameter

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Jacobian weighting

Idealoffset-gather

Idealangle-gather

Reflection angle

Offset ray parameter

paul@sep.stanford.edu

Agenda

Amplitude corrections

Angle-domain common image gathers

Imaging effects

Propagation effects

Acquisition effects

Definitions

Methods

Applications

paul@sep.stanford.edu

WKBJ & scattering operators

00zr

zr

zs

zs

k

k

k

kA

Clayton & Stolt (1981)

rd

md

LAG

L

Clayton & Stolt (1981)Stolt & Benson (1986)

zrzskk

si

4

2G

paul@sep.stanford.edu

Agenda

Amplitude corrections

Angle-domain common image gathers

Imaging effects

Propagation effects

Acquisition effects

Definitions

Methods

Applications

paul@sep.stanford.edu

Stationary-phase operator

rd LSAG

4sgn

02

2

2

2

2

yh

CAz

y

dk

kdi

z

h

CAz

e

ddkkd

S

2-D

CAM

Bleistein & Handelsman (1975)

paul@sep.stanford.edu

Agenda

Amplitude corrections

Angle-domain common image gathers

Imaging effects

Propagation effects

Acquisition effects

Definitions

Methods

Applications

paul@sep.stanford.edu

Applications

Pseudo-unitary migration True-amplitude migration

2* WLL rd LSAG

-1u LWL

ILL u*u rd*

t L

*21*t LWSAGL

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Example: vertical section

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Example: vertical section

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Example: horizontal section

uncorrected

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Example: horizontal section

corrected

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Summary

• Amplitude-preserving transformation to the angle domain

• Amplitude corrections– imaging effects– propagation effects– acquisition effects