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Amplitude-preserved wave-equation migration
Paul Sava & Biondo Biondi
SEP108 (pages 1-27)
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)
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)
Agenda
• ADCIG kinematics• image space• data space
• Amplitude-preserved migration
• general formulation• weighting function• COMAZ
• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT
• Applications• true-amplitude
migration• inversion• WEMVA
ADCIG methods
Reflection angle Offset ray-parameter
z
h
k
ktan
h
h
kp k-domain
(RTT)
h
z
tanh
tph
x-domain
(slant-stack)
ADCIG methods: comparison
Reflection angle Offset ray-parameter
• indirectly– function of dip
• directlyReflection
angle
• less sensitive• sensitiveInaccurate
velocity
boundaries
• data space– mixed with migration
• image space– separated from
migration
Computation
domain
Agenda
• ADCIG kinematics• image space• data space
• Amplitude-preserved migration
• general formulation • weighting function• COMAZ
• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT
• Applications• true-amplitude
migration• inversion• WEMVA
Temporal bandwidth
image angle gather
data offset gather
wide frequency band
narrow frequency band
kz
kh
kz
kh
kh
kz
RTT implementation
• Two possibilities:– push: loop over input– pull: loop over output
kh
kz kz
angle gather
offset gather
Agenda
• ADCIG kinematics• image space• data space
• Amplitude-preserved migration
• general formulation• weighting functions• COMAZ
• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT
• Applications• true-amplitude
migration• inversion• WEMVA
Amplitude-preserving migration
• Definition: the process of recovering the amplitude of the reflectivity vector given– perfect data– infinite bandwidth– infinite aperture
Modeling operator
0id L
L: modeling operatorA: Amplitude operatorG: Reflection operator
i0: seismic imager: reflectivityd: seismic data
Amplitude operator
0id LA
00zr
zr
zs
zs
k
k
k
kA
Clayton & Stolt (1981)
L: modeling operatorA: amplitude operatorG: Reflection operator
i0: seismic imager: reflectivityd: seismic data
Reflection operator
rd LAG
L: modeling operatorA: amplitude operatorG: reflection operator
i0: seismic imager: reflectivityd: seismic data
Clayton & Stolt (1981)Stolt & Benson (1986)
zrzskk
si
4
2G
ri G0
Amplitude-preserving operator
rd )(LAG
L: modeling operatorA: amplitude operatorG: reflection operator
i0: seismic imager: reflectivityd: seismic data
Weighting operator
)()( 0 zidk
dzi
constkz h
0 0*
i i iW LL
0i dLmodelingd i*Lmigration
Agenda
• ADCIG kinematics• image space• data space
• Amplitude-preserved migration
• general formulation• weighting functions• COMAZ
• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT
• Applications• true-amplitude
migration• inversion• WEMVA
Jacobian: general expression
1
zszrk k
s
k
ss
h
W
1
44
zszr
hm
zszr
hhp k
s
k
s
s
pks
k
s
k
s
s
pps
h
W
image space
data space
Jacobian: 2-D, image space
1
zszrk k
s
k
ss
h
W
1
cos
1
cos
1
shk
W
cos2
10
shk W
2h
v
Jacobian: general expression
1
zszrk k
s
k
ss
h
W
1
44
zszr
hm
zszr
hhp k
s
k
s
s
pks
k
s
k
s
s
pps
h
W
image space
data space
Jacobian: 2-D, data space1
44
zszr
hm
zszr
hhp k
s
k
s
s
pks
k
s
k
s
s
pps
h
W
2h
v
1
)cos(
1
)cos(
1
4)cos(
1
)cos(
1
4
s
pks
s
pps hmhh
ph
W
cos
1
2
10
shk W
Agenda
• ADCIG kinematics• image space• data space
• Amplitude-preserved migration
• general formulation• weighting function• COMAZ
• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT
• Applications• true-amplitude
migration• inversion• WEMVA
Amplitudecomponent
Phase-shiftcomponent
COMAZ: stationary-phase correction
4sgn
02
2
2
2
2
yh
CAz
y
dk
kdi
z
h
CAz
stat e
ddkkd
A
rd statAGLA
Agenda
• ADCIG kinematics• image space• data space
• Amplitude-preserved migration
• general formulation• weighting function• COMAZ
• ADCIG amplitudes• spatial bandwidth• temporal bandwidth• RTT
• Applications• true-amplitude
migration• inversion• WEMVA
True-amplitude migration
rd LAG
rdt
t
*
*111*
L
LWAGL
L: modeling operatorA: amplitude operatorG: reflection operator
i0: seismic imager: reflectivityd: seismic data
True-amplitude migration: COMAZ
*111 LWAG *1111 LWAGA stat
*11 LWG
OPERATORSL: modelingW: JacobianA: amplitudeAstat: stationary-phaseG: reflection
*L
Inversion: pseudo-unitary operators
ILL
LWL
uu
u
*
*2/1*
rd LAGInversionMigration
rd LAG
pd uL
)( 2/1 rd u AGWL
Wave-equation MVA
dmL
L: Wave-equation MVAm: slowness perturbationd: image perturbation
References: SEP100, SEP103, SEP105
Summary
• The goal– Reflectivity function of reflection angle
• The means– correct ADCIG transformations
– kinematics– amplitudes
– correct migration amplitude
Applications
• true-amplitude migration
• seismic inversion
• AVA
• wave-equation MVA