paul.sava@beg.utexas.edu

Time-shift imaging condition

Paul Sava & Sergey Fomel

Bureau of Economic GeologyUniversity of Texas at Austin

paul.sava@beg.utexas.edu

Imaging condition

Image

Source wavefield

Receiver wavefield

Wavefield reconstruction

Imaging sketch

S

R

Angle decomposition

Angle-dependent reflectivity

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Wavefield reconstruction

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Wavefield reconstruction

Source wavefield

Receiver wavefield

S

R

Reverse-time extrapolationKirchhoff integralWavefield extrapolation…

sssmm dDGU sss ,,,, rrrmm dDGU rrr ,,,,

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Imaging condition

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Conventional imaging condition

tUr ,m tUs ,m mR

source wavefield

receiver wavefield

image

position

time

t

zyx ,,m

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Conventional imaging condition

0,

,,,

tUR

tUtUtU sr

mm

mmm

Claerbout (1985)

position

time

t

zyx ,,m

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Conventional imaging condition

,, * mmm sr UUR

position

frequency

zyx ,,m

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distance

dept

hConventional imaging condition example

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Space-shift imaging condition

tUr ,hm tUs ,hm hm,R

source wavefield

receiver wavefield

image

position

time

t

hhh

zyx

zyx ,,

,,

h

m

space-shift

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Space-shift imaging condition

position

time

t

hhh

zyx

zyx ,,

,,

h

m

space-shift

Rickett & Sava (2002)

Biondi & Symes (2004) Sava & Fomel (2005)

0,,,

,,,,

tUR

tUtUtU sr

hmhm

hmhmhm

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Space-shift imaging condition

position

frequency

zyx hhh

zyx

,,

,,

h

m

space-shift

,,, * hmhmhm sr UUR

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distance

dept

hSpace-shift imaging condition example

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Time-shift imaging condition

tUr ,m tUs ,m ,mR

source wavefield

receiver wavefield

image

position

time

t

zyx

,,m

time-shift

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Time-shift imaging condition

position

time

t

zyx

,,m

time-shift

0,,,

,,,,

tUR

tUtUtU sr

mm

mmm

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Time-shift imaging condition

position

frequency

zyx ,,m

time-shift

isr eUUR 2* ,,, mmm

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Time-shift imaging condition example

distance

dept

h

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distance h

Imaging condition: example 1

dept

h

dept

h

dept

h

space-shift time-shift

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hdistance

Imaging condition: example 2

dept

h

dept

h

dept

h

space-shift time-shift

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Imaging condition summary

,, * mmm sr UUR

,,, * hmhmhm sr UUR

isr eUUR 2* ,,, mmmtime-shift

space-shift

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Angle decomposition

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pmp

h

srh

srm

pp2p

pp2p

ps

pr

θ,

,

m

hm

R

R

θ,

,

m

m

R

R

Angle decompositionSpace-shift Time-shift

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pmp

h

srh

srm

pp2p

pp2p

ps

pr

m

h

p

pθtan

smpθcos

Vector geometrySpace-shift Time-shift

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m

h

k

kθtan

mk

s

1θcos

Image-space angle decompositionSpace-shift Time-shift

m

h

p

pθtan

smpθcos

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Angle decomposition algorithm

θ,

θ,

,

,

m

k

kk

hm

m

hm

R

R

R

R

θ,

,

,

,

,

m

m

k

k

m

m

m

R

R

R

R

R

Space-shift Time-shift

m

h

k

kθtan

mk

s

θcos

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distance

dept

hSpace-shift angle decomposition example

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Time-shift angle decomposition example

distance

dept

h

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distance

Angle decomposition: example 1

dept

h

dept

h

dept

h

space-shift time-shift

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distance

Angle decomposition: example 2

dept

h

dept

h

dept

h

space-shift time-shift

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distance

Correct velocity

dept

h

dept

h

dept

h

space-shift time-shift

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distance

Incorrect velocity

dept

h

dept

h

dept

h

space-shift time-shift

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Angle decomposition summary

dept

h

dept

h

distance h

dept

h

dept

h

dept

h

hm,R θ,mR ,mR θ,mR

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Angle decomposition summary

dept

h

dept

h

distance h

dept

h

dept

h

dept

h

hm,R θm,R ,mR θ,mR

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Summary

• Imaging condition and angle-decomposition

– time-shifts of source/receiver wavefields

– cheaper than space-shift imaging– storage– computations

– wide applicability– migration by wavefield extrapolation– Kirchhoff migration– reverse-time migration