Amplitude-preservingwave-equation migration
Paul Sava* Stanford University
Biondo Biondi Stanford University
The problem
Recover reliable migration amplitudes:i.e. reflectivity as a function of incidence angle
Reflectivity description
Describe reflectivity as a function of incidence angle
Migration algorithm
Apply amplitude corrections to the migration operator
Illumination balance
Restore amplitudes in shadow zones
Agenda
Amplitude corrections
Angle-domain common image gathers
Imaging effects
Propagation effects
Acquisition effects
Definitions
Methods
Applications
Common image gathers
z
h
flat events
z
h
focused events
z
flat events
z
flat events
Wave-equationKirchhoff
Offset domain
Angle domain
Angle-Domain Common Image GathersSource Receiver
V(x,y,z)
2h
v
z
h
k
ktan
h
h
kp
S-G migrationPrucha et al., 1999Sava et al., 2001
Shot-profile migrationde Bruin et al., 1990Rickett & Sava, 2001
Agenda
Amplitude corrections
Angle-domain common image gathers
Imaging effects
Propagation effects
Acquisition effects
Definitions
Methods
Applications
Jacobian weighting: arbitrary geometry
2h
s
1
2
)cos(
1
)cos(
1
shk
W
1
22
4)cos(
1
)cos(
1
4)cos(
1
)cos(
1
s
pk
s
ps
hm
h
hp
W
image space
data space
Jacobian weighting: flat reflectors
cos2
12
shkW
cos
1
2
12
shpW
(Wapenaar et al., 1999)
2h
s
image space
data space
Jacobian weighting
Idealoffset-gather
Idealangle-gather
Reflection angle
Offset ray parameter
Jacobian weighting
Idealoffset-gather
Idealangle-gather
Reflection angle
Offset ray parameter
Agenda
Amplitude corrections
Angle-domain common image gathers
Imaging effects
Propagation effects
Acquisition effects
Definitions
Methods
Applications
WKBJ & scattering operators
00zr
zr
zs
zs
k
k
k
kA
Clayton & Stolt (1981)
rd
md
LAG
L
Clayton & Stolt (1981)Stolt & Benson (1986)
zrzskk
si
4
2G
Agenda
Amplitude corrections
Angle-domain common image gathers
Imaging effects
Propagation effects
Acquisition effects
Definitions
Methods
Applications
Stationary-phase operator
rd LSAG
4sgn
02
2
2
2
2
yh
CAz
y
dk
kdi
z
h
CAz
e
ddkkd
S
2-D
CAM
Bleistein & Handelsman (1975)
Agenda
Amplitude corrections
Angle-domain common image gathers
Imaging effects
Propagation effects
Acquisition effects
Definitions
Methods
Applications
Applications
Pseudo-unitary migration True-amplitude migration
2* WLL rd LSAG
-1u LWL
ILL u*u rd*
t L
*21*t LWSAGL
Summary
• Amplitude-preserving transformation to the angle domain
• Amplitude corrections– imaging effects– propagation effects– acquisition effects