yaxun seminar 1 18

8/3/2019 Yaxun Seminar 1 18

1/17

Controlling the aperture in

prestack phase-shift migration


2/17

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):


3/17


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


4/17


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


5/17

Extending to prestack


6/17

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


7/17

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


8/17

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


9/17

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


10/17

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


11/17

Impulse responses

Offset=0m

Constant aperture

(x-xm)max=4000m

Variable aperture


12/17

Impulse responses

Offset=1200m

Constant aperture

(x-xm)max=4000m

Variable aperture


13/17

Impulse responses

Offset=1600m

Variable apertureConstant aperture

(x-xm)max=4000m


14/17

Plot of aperture curves

Plot of Mmax-tm Plot of [x-x0]max-tm


15/17

Marmousi

Single offset CO migration result (offset=200m)


16/17

Marmousi



17/17

Marmousi


yaxun seminar 1 18

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