paul@sep.stanford.edu

Amplitude-preserved wave-equation migration

Paul Sava & Biondo Biondi

SEP108 (pages 1-27)

paul@sep.stanford.edu

Wave-equation imaging

• Why?– Complex wavefields– Sharp velocity variation

• sub-salt

• What?– Reflectivity function of incidence angle

• Imaging• Migration Velocity Analysis (MVA)• Amplitude vs. Angle Analysis (AVA)

paul@sep.stanford.edu

Angle-Domain Common Image Gathers

• Applications– imaging

– S/G migration (Prucha et at., 1999)– shot-profile migration (Rickett, 2001)– seismic inversion (Prucha et. al., 2001)

– MVA– traveltime tomography (Clapp, 2000)– wave-equation MVA (Sava & Biondi, 2000)

– C-waves – polarity reversal (Rosales, 2001)

– AVA – wave-equation AVA (Gratwick, 2001)

paul@sep.stanford.edu

Angle-gathers vs. offset-gathers

Offset gather Angle gather

paul@sep.stanford.edu

Agenda

• ADCIG kinematics• image space• data space

• Amplitude-preserved migration

• general formulation• weighting function• COMAZ

• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT

• Applications• true-amplitude

migration• inversion• WEMVA

paul@sep.stanford.edu

Reflection scheme: global view

Source Receiver

V(x,y,z)

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Reflection scheme: local view

2h

v

h

z

tan

h

tph

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ADCIG methods

Reflection angle Offset ray-parameter

z

h

k

ktan

h

h

kp k-domain

(RTT)

h

z

tanh

tph

x-domain

(slant-stack)

paul@sep.stanford.edu

ADCIG: example

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ADCIG methods: comparison

Reflection angle Offset ray-parameter

• indirectly– function of dip

• directlyReflection

angle

• less sensitive• sensitiveInaccurate

velocity

boundaries

• data space– mixed with migration

• image space– separated from

migration

Computation

domain

paul@sep.stanford.edu

Agenda

• ADCIG kinematics• image space• data space

• Amplitude-preserved migration

• general formulation • weighting function• COMAZ

• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT

• Applications• true-amplitude

migration• inversion• WEMVA

paul@sep.stanford.edu

Spatial bandwidth

kh

kz

maxmax

z

h

k

ktan

kz

+90-90

max max

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Synthetic: ideal gather

frequency domain space domain amplitude

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Temporal bandwidth

image angle gather

data offset gather

wide frequency band

narrow frequency band

kz

kh

kz

kh

kh

kz

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Temporal bandwidth

frequency domain space domain amplitude

paul@sep.stanford.edu

RTT implementation

• Two possibilities:– push: loop over input– pull: loop over output

kh

kz kz

angle gather

offset gather

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push RTToffset-gather angle-gather

k-domain

x-domain

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pull RTToffset-gather angle-gather

k-domain

x-domain

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RTT amplitudes

paul@sep.stanford.edu

Agenda

• ADCIG kinematics• image space• data space

• Amplitude-preserved migration

• general formulation• weighting functions• COMAZ

• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT

• Applications• true-amplitude

migration• inversion• WEMVA

paul@sep.stanford.edu

Amplitude-preserving migration

• Definition: the process of recovering the amplitude of the reflectivity vector given– perfect data– infinite bandwidth– infinite aperture

paul@sep.stanford.edu

Modeling operator

0id L

L: modeling operatorA: Amplitude operatorG: Reflection operator

i0: seismic imager: reflectivityd: seismic data

paul@sep.stanford.edu

Amplitude operator

0id LA

00zr

zr

zs

zs

k

k

k

kA

Clayton & Stolt (1981)

L: modeling operatorA: amplitude operatorG: Reflection operator

i0: seismic imager: reflectivityd: seismic data

paul@sep.stanford.edu

Reflection operator

rd LAG

L: modeling operatorA: amplitude operatorG: reflection operator

i0: seismic imager: reflectivityd: seismic data

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

zrzskk

si

4

2G

ri G0

paul@sep.stanford.edu

Amplitude-preserving operator

rd )(LAG

L: modeling operatorA: amplitude operatorG: reflection operator

i0: seismic imager: reflectivityd: seismic data

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Weighting operator

)()( 0 zidk

dzi

constkz h

0 0*

i i iW LL

0i dLmodelingd i*Lmigration

paul@sep.stanford.edu

Agenda

• ADCIG kinematics• image space• data space

• Amplitude-preserved migration

• general formulation• weighting functions• COMAZ

• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT

• Applications• true-amplitude

migration• inversion• WEMVA

paul@sep.stanford.edu

Amplitude correction: the problem

frequency domain space domain amplitude

paul@sep.stanford.edu

Jacobian: general expression

1

zszrk k

s

k

ss

h

W

1

44

zszr

hm

zszr

hhp k

s

k

s

s

pks

k

s

k

s

s

pps

h

W

image space

data space

paul@sep.stanford.edu

Jacobian: 2-D, image space

1

zszrk k

s

k

ss

h

W

1

cos

1

cos

1

shk

W

cos2

10

shk W

2h

v

paul@sep.stanford.edu

Jacobian: general expression

1

zszrk k

s

k

ss

h

W

1

44

zszr

hm

zszr

hhp k

s

k

s

s

pks

k

s

k

s

s

pps

h

W

image space

data space

paul@sep.stanford.edu

Jacobian: 2-D, data space1

44

zszr

hm

zszr

hhp k

s

k

s

s

pks

k

s

k

s

s

pps

h

W

2h

v

1

)cos(

1

)cos(

1

4)cos(

1

)cos(

1

4

s

pks

s

pps hmhh

ph

W

cos

1

2

10

shk W

paul@sep.stanford.edu

Jacobian: 2-D, flat reflectors

cos2

1

shkW

cos

1

2

1

shpW

(Wapenaar et al., 1999)

paul@sep.stanford.edu

Amplitude correction: the problem

frequency domain space domain amplitude

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AVA: correct amplitudes

frequency domain space domain amplitude

paul@sep.stanford.edu

Agenda

• ADCIG kinematics• image space• data space

• Amplitude-preserved migration

• general formulation• weighting function• COMAZ

• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT

• Applications• true-amplitude

migration• inversion• WEMVA

paul@sep.stanford.edu

COMAZ: stationary-phase

view from above

2-D COMAZ

paul@sep.stanford.edu

Amplitudecomponent

Phase-shiftcomponent

COMAZ: stationary-phase correction

4sgn

02

2

2

2

2

yh

CAz

y

dk

kdi

z

h

CAz

stat e

ddkkd

A

rd statAGLA

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COMAZ: no amplitude corrections

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COMAZ: all amplitude corrections

paul@sep.stanford.edu

Agenda

• ADCIG kinematics• image space• data space

• Amplitude-preserved migration

• general formulation• weighting function• COMAZ

• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT

• Applications• true-amplitude

migration• inversion• WEMVA

paul@sep.stanford.edu

True-amplitude migration

rd LAG

rdt

t

*

*111*

L

LWAGL

L: modeling operatorA: amplitude operatorG: reflection operator

i0: seismic imager: reflectivityd: seismic data

paul@sep.stanford.edu

True-amplitude migration: COMAZ

*111 LWAG *1111 LWAGA stat

*11 LWG

OPERATORSL: modelingW: JacobianA: amplitudeAstat: stationary-phaseG: reflection

*L

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True-amplitude migration: real data

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Inversion: pseudo-unitary operators

ILL

LWL

uu

u

*

*2/1*

rd LAGInversionMigration

rd LAG

pd uL

)( 2/1 rd u AGWL

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Inversion: preconditioned regularization

pd uLp

pd u

R

L

0

rd LAG

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Wave-equation MVA

dmL

L: Wave-equation MVAm: slowness perturbationd: image perturbation

References: SEP100, SEP103, SEP105

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WEMVA: model

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WEMVA: correct amplitudes

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WEMVA: incorrect amplitudes

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Summary

• The goal– Reflectivity function of reflection angle

• The means– correct ADCIG transformations

– kinematics– amplitudes

– correct migration amplitude

paul@sep.stanford.edu

Applications

• true-amplitude migration

• seismic inversion

• AVA

• wave-equation MVA

paul@sep.stanford.edu