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Time-shift imaging condition
Paul Sava & Sergey Fomel
Bureau of Economic GeologyUniversity of Texas at Austin
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Imaging condition
Image
Source wavefield
Receiver wavefield
Wavefield reconstruction
Imaging sketch
S
R
Angle decomposition
Angle-dependent reflectivity
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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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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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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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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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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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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
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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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