Transcript Amplitude-preserved wave

Amplitude-preserved
wave-equation migration
Paul Sava & Biondo Biondi
SEP108 (pages 1-27)
[email protected]
Wave-equation imaging
• Why?
– Complex wavefields
– Sharp velocity variation
• sub-salt
• What?
– Reflectivity function of incidence angle
• Imaging
• Migration Velocity Analysis (MVA)
• Amplitude vs. Angle Analysis (AVA)
[email protected]
Angle-Domain Common Image Gathers
• Applications
– imaging
– S/G migration (Prucha et at., 1999)
– shot-profile migration (Rickett, 2001)
– seismic inversion (Prucha et. al., 2001)
– MVA
– traveltime tomography (Clapp, 2000)
– wave-equation MVA (Sava & Biondi, 2000)
– C-waves
– polarity reversal (Rosales, 2001)
– AVA
– wave-equation AVA (Gratwick, 2001)
[email protected]
Angle-gathers vs. offset-gathers
Offset gather
Angle gather
[email protected]
Agenda
• ADCIG kinematics
• image space
• data space
• ADCIG amplitudes
• spatial bandwidth
• temporal bandwidth
• RTT
• Amplitude-preserved
migration
• general formulation
• weighting function
• COMAZ
• Applications
• true-amplitude
migration
• inversion
• WEMVA
[email protected]
Reflection scheme: global view
Source
Receiver
V(x,y,z)
g g
[email protected]
Reflection scheme: local view
2h
a
g g
z
tan g  
h
t
ph 
h
v
[email protected]
ADCIG methods
Reflection angle
z
tan g  
h
kh
tan g  
kz
Offset ray-parameter
x-domain
(slant-stack)
k-domain
(RTT)
t
ph 
h
ph 
kh

[email protected]
ADCIG: example
[email protected]
ADCIG methods: comparison
Reflection angle
Computation
domain
• image space
– separated from
migration
Reflection
angle
• directly
Inaccurate
velocity
boundaries
• sensitive
Offset ray-parameter
• data space
– mixed with migration
• indirectly
– function of dip
• less sensitive
[email protected]
Agenda
• ADCIG kinematics
• image space
• data space
• ADCIG amplitudes
• spatial bandwidth
• temporal bandwidth
• RTT
• Amplitude-preserved
migration
• general formulation
• weighting function
• COMAZ
• Applications
• true-amplitude
migration
• inversion
• WEMVA
[email protected]
Spatial bandwidth
kh
tan g  
kz
kz
kz
g
kh
-90
 gmax
+ gmax
+90
gmax
+gmax
[email protected]
Synthetic: ideal gather
frequency domain
space domain
amplitude
[email protected]
Temporal bandwidth

kz
kh
image
kz
kh
data
kz
g
kh
offset
gather
angle
gather
wide frequency band
narrow frequency band
[email protected]
Temporal bandwidth
frequency domain
space domain
amplitude
[email protected]
RTT implementation
• Two possibilities:
– push: loop over input
– pull: loop over output
kz
offset
gather
kz
kh
angle
gather
g
[email protected]
push RTT
offset-gather
angle-gather
x-domain
k-domain
[email protected]
pull RTT
offset-gather
angle-gather
x-domain
k-domain
[email protected]
RTT amplitudes
[email protected]
Agenda
• ADCIG kinematics
• image space
• data space
• ADCIG amplitudes
• spatial bandwidth
• temporal bandwidth
• RTT
• Amplitude-preserved
migration
• general formulation
• weighting functions
• COMAZ
• Applications
• true-amplitude
migration
• inversion
• WEMVA
[email protected]
Amplitude-preserving migration
• Definition: the process of recovering the
amplitude of the reflectivity vector given
– perfect data
– infinite bandwidth
– infinite aperture
[email protected]
Modeling operator
d  Li0
i0: seismic image
r: reflectivity
d: seismic data
L: modeling operator
A: Amplitude operator
G: Reflection operator
[email protected]
Amplitude operator
d  LAi0
k zs k zr
A
0
0
k zs k zr
Clayton & Stolt (1981)
i0: seismic image
r: reflectivity
d: seismic data
L: modeling operator
A: amplitude operator
G: Reflection operator
[email protected]
Reflection operator
d  LAG r
i0  Gr
is
G
4k zs k zr
2
Clayton & Stolt (1981)
Stolt & Benson (1986)
i0: seismic image
r: reflectivity
d: seismic data
L: modeling operator
A: amplitude operator
G: reflection operator
[email protected]
Amplitude-preserving operator
d  (LAG )r
i0: seismic image
r: reflectivity
d: seismic data
L: modeling operator
A: amplitude operator
G: reflection operator
[email protected]
i  L*d
Weighting operator
d  Li
0
0
i  L*Li  Wi
migration
d
i( z ) 
dk z
modeling
0
i0 ( z )
k h  const
[email protected]
Agenda
• ADCIG kinematics
• image space
• data space
• ADCIG amplitudes
• spatial bandwidth
• temporal bandwidth
• RTT
• Amplitude-preserved
migration
• general formulation
• weighting functions
• COMAZ
• Applications
• true-amplitude
migration
• inversion
• WEMVA
[email protected]
Amplitude correction: the problem
frequency domain
space domain
amplitude
[email protected]
Jacobian: general expression
 
1



ph  ph  s s   km  ph  s s 


Wph   s 
+  +  s 



 k
4
s
k
k
4

s



zs  
 zr
 zr k zs 
data space
  s s 

Wkh   s
+
  k zr k zs 
1
image space
[email protected]
Jacobian: 2-D, image space
 

1
1

Wkh  s
+




cos
g

a
cos
g
+
a

 
a  0  Wk 
h
  s s 

Wkh   s
+
  k zr k zs 
1
1
1
cos g
2s
2h
g g
a
v
[email protected]
Jacobian: general expression
 
1



ph  ph  s s   km  ph  s s 


Wph   s 
+  +  s 



 k
4
s
k
k
4

s



zs  
 zr
 zr k zs 
data space
  s s 

Wkh   s
+
  k zr k zs 
1
image space
[email protected]
Jacobian: 2-D, data space
 
1



ph  ph  s s   km  ph  s s 


Wph   s 
+  +  s 



 k
4
s
k
k
4

s



zs  
 zr
 zr k zs 
 
1



  km  ph 

p  p 
1
1
1
1

 +  s 

Wph   s  h h 
+


4s  cos(g  a ) cos(g + a )  
4s  cos(g  a ) cos(g + a ) 

1 1
a  0  Wkh 
2s cos g
2h
g g
a
v
[email protected]
Jacobian: 2-D, flat reflectors
1
Wkh  cos g
2s
1 1
Wp h 
2s cos g
(Wapenaar et al., 1999)
[email protected]
Amplitude correction: the problem
frequency domain
space domain
amplitude
[email protected]
AVA: correct amplitudes
frequency domain
space domain
amplitude
[email protected]
Agenda
• ADCIG kinematics
• image space
• data space
• ADCIG amplitudes
• spatial bandwidth
• temporal bandwidth
• RTT
• Amplitude-preserved
migration
• general formulation
• weighting function
• COMAZ
• Applications
• true-amplitude
migration
• inversion
• WEMVA
[email protected]
COMAZ: stationary-phase
2-D
COMAZ
view from above
[email protected]
COMAZ: stationary-phase correction
d  LA stat AGr
A stat 
z

0
Phase-shift
component
 i sgn d 2 k zCA  
 dk 2  4
2

hy  

e


2 CA
d kz


d 

2
dk hy
Amplitude
component
[email protected]
COMAZ: no amplitude corrections
[email protected]
COMAZ: all amplitude corrections
[email protected]
Agenda
• ADCIG kinematics
• image space
• data space
• ADCIG amplitudes
• spatial bandwidth
• temporal bandwidth
• RTT
• Amplitude-preserved
migration
• general formulation
• weighting function
• COMAZ
• Applications
• true-amplitude
migration
• inversion
• WEMVA
[email protected]
True-amplitude migration
d  LAG r
1
1
1 *
L G A W L
*
t
Ld r
*
t
i0: seismic image
r: reflectivity
d: seismic data
L: modeling operator
A: amplitude operator
G: reflection operator
[email protected]
True-amplitude migration: COMAZ
L*
G 1W 1L*
G 1A 1W 1L*
1
A stat
G 1A 1W 1L*
OPERATORS
L: modeling
W: Jacobian
A: amplitude
Astat: stationary-phase
G: reflection
[email protected]
True-amplitude migration: real data
[email protected]
Inversion: pseudo-unitary operators
Migration
Inversion
d  LAG r
d  LAG r
L W
*
u
1/ 2
*
L
d  Lu ( W AGr )
1/ 2
L Lu  I
*
u
d  Lu p
[email protected]
Inversion: preconditioned regularization
d  LAG r
d  Lu p
d  Lu p
0  Rp
[email protected]
Wave-equation MVA
Lm  d
L: Wave-equation MVA
m: slowness perturbation
d: image perturbation
References: SEP100, SEP103, SEP105
[email protected]
WEMVA: model
[email protected]
WEMVA: correct amplitudes
[email protected]
WEMVA: incorrect amplitudes
[email protected]
Summary
• The goal
– Reflectivity function of reflection angle
• The means
– correct ADCIG transformations
– kinematics
– amplitudes
– correct migration amplitude
[email protected]
Applications
•
•
•
•
true-amplitude migration
seismic inversion
AVA
wave-equation MVA
[email protected]
[email protected]