Transcript Amplitude-preserving wave

Amplitude-preserving
wave-equation migration
Paul Sava*
Biondo Biondi
Stanford University
Stanford University
paul@sep.stanford.edu
The problem
Recover reliable migration amplitudes:
i.e. reflectivity as a function of incidence angle
Reflectivity
description
Migration
algorithm
Illumination
balance
Describe reflectivity
as a function of
incidence angle
Apply amplitude
corrections to the
migration operator
Restore amplitudes in
shadow zones
paul@sep.stanford.edu
Agenda
Angle-domain common
image gathers
Definitions
Methods
Imaging effects
Amplitude corrections
Propagation effects
Acquisition effects
Applications
paul@sep.stanford.edu
Common image gathers
Kirchhoff
Wave-equation
h
h
Offset
domain
flat events
focused events
z
z
g
g
Angle
domain
flat events
flat events
z
z
paul@sep.stanford.edu
Angle-Domain Common Image Gathers
Source
Receiver
V(x,y,z)
g g
2h
g g
kh
tan g  
kz
kh
ph 

a
v
S-G migration
Prucha et al., 1999
Sava et al., 2001
Shot-profile migration
de Bruin et al., 1990
Rickett & Sava, 2001
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ADCIG: S-G migration
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Agenda
Angle-domain common
image gathers
Definitions
Methods
Imaging effects
Amplitude corrections
Propagation effects
Acquisition effects
Applications
paul@sep.stanford.edu
Modeling operator
modeling
migration
d  Lm0
*
mLd
m  L Lm 0
*
2
W m0
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Jacobian weighting: arbitrary geometry

 
1
1
s 
Wk  

h
 cos(g  a ) cos(g  a )  
1
2h
2
g g
image space
a
s
data space
2





p
1
1
2
h
 
 s 
W ph  

4s 
 cos(g  a ) cos(g  a ) 


1
1



 cos(g  a ) cos(g  a ) 
k m ph 

4s 
1
paul@sep.stanford.edu
Jacobian weighting: flat reflectors
1
Wk  cos g
h
2s
2
2h
g
g
image space
data space
s
1 1
Wp 
h
2s cos g
2
(Wapenaar et al., 1999)
paul@sep.stanford.edu
Jacobian weighting
Ideal
offset-gather
Ideal
angle-gather
Reflection
angle
Offset
ray parameter
paul@sep.stanford.edu
Jacobian weighting
Ideal
offset-gather
Ideal
angle-gather
Reflection
angle
Offset
ray parameter
paul@sep.stanford.edu
Agenda
Angle-domain common
image gathers
Definitions
Methods
Imaging effects
Amplitude corrections
Propagation effects
Acquisition effects
Applications
paul@sep.stanford.edu
WKBJ & scattering operators
d  Lm
d  LAG r
k zs k zr
A
0
0
k zs k zr
Clayton & Stolt (1981)
is
G
4k zs k zr
2
Clayton & Stolt (1981)
Stolt & Benson (1986)
paul@sep.stanford.edu
Agenda
Angle-domain common
image gathers
Definitions
Methods
Imaging effects
Amplitude corrections
Propagation effects
Acquisition effects
Applications
paul@sep.stanford.edu
Stationary-phase operator
d  LSAG r
2-D
S
z

0
 i sgn d 2 k zCA   
 dk 2  4
2

hy  

e


2 CA
d kz


d



dk h2y
Bleistein & Handelsman (1975)
CAM
paul@sep.stanford.edu
Agenda
Angle-domain common
image gathers
Definitions
Methods
Imaging effects
Amplitude corrections
Propagation effects
Acquisition effects
Applications
paul@sep.stanford.edu
Applications
LLW
*
2
Pseudo-unitary migration
L u  LW
-1
L Lu  I
*
u
d  LSAG r
True-amplitude migration
L  SAG  W L
1
*
t
2 *
Ldr
*
t
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Example: vertical section
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Example: vertical section
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Example: horizontal section
uncorrected
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Example: horizontal section
corrected
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Summary
• Amplitude-preserving transformation to the
angle domain
• Amplitude corrections
– imaging effects
– propagation effects
– acquisition effects
paul@sep.stanford.edu