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Controlling the aperture in
prestack phase-shift migration
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Zero-offset Phase-shift
2 2 2
2 2 2 2 2
4 4
s m s
p p p
x v t v t
2s mz v t
2 2
21
4( , , ) ( , 0, )
x sm
k vi t
x m x mP k t P k t e
Wave equation:
Downward continuation of theupgoing wavefield(Gazdag, 1978):
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Zero-offset Phase-shift
Can also be seen as downward continuation of plane waves withdifferent propagating directions (Dubrulle, 1983)
2
22 ( )2s m
D m v tT x xv
xDkdT
m
dx
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Zero-offset Phase-shift
2
2 2
2 2
41
4
14
s m
m
s
m
D
s
mv tx x
m v
tT
m v
2 2
14
s
m
m vt
2 2
21
4
x s
m
k vt
Intercept:
( )D m
T m x x
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Extending to prestack
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Prestack phase-shift
2 2
2 21 1, ,
2 2
m s m s
D m m m m
s s
t v t vT t x x h x x h x x h
v v
2 2
2 2
1 1
1 1
2 2
m mD
s sm s m s
m m
s s
x x h x x hdTm
dx v vt v t vx x h x x h
v v
( )D m
T m x x
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Prestack phase-shift
1. 2D Fourier Transform
2. Phase shift
3. Apply imaging condition (t=0) and 1D inverse Fourier Transform
( ), 0, , ( , 0, , ) e xi t k xx m mP k t h p x t t h dtdx
, 0, , e ix mP k t h ( )
D mT m x x
2
1, , 0, , 0, , e
2
x mik xi
m m x m xp x t t h P k t h e d dk
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Controlling the aperture
How to decide the maximum slowness m?
2
max
max
21 m
s
tm
v t
When data recording time is limited: max0 t t
Levin(1984)
Mmax Is the maximum slowness for zero-offset FK migration
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Non-zero offset
Extending to non-zero offset (Ekren & Ursin, 1999):
max( , , )
D m mT t x x h t
2
max
max
max0
12
s m
m
v t tx x X
t
2
2
max 0 max
2
s
ht t
v
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Non-zero offset
max
2
s
m Av
Maximum slowness for non-zero offset:
2 2 2 2 2 24( ) 4( )s m s m
X h X hA
X h v t X h v t
2 2
2 2
1 1
1 1
2 2
m mD
s sm s m s
m m
s s
x x h x x hdTm
dx v vt v t vx x h x x h
v v
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Impulse responses
Offset=0m
Constant aperture
(x-xm)max=4000m
Variable aperture
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Impulse responses
Offset=1200m
Constant aperture
(x-xm)max=4000m
Variable aperture
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Impulse responses
Offset=1600m
Variable apertureConstant aperture
(x-xm)max=4000m
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Plot of aperture curves
Plot of Mmax-tm Plot of [x-x0]max-tm
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Marmousi
Single offset CO migration result (offset=200m)
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Marmousi
Single offset CO migration result (offset=600m)
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Marmousi
Single offset CO migration result (offset=1000m)
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