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Elastodynamic reciprocity theorems for time-lapse seismic methods
Kees WapenaarMenno DillenJacob Fokkema
Centre for Technical GeoscienceDelft University of Technology
Review of reciprocity theorems and seismic applications
Kees WapenaarMenno DillenJacob Fokkema
Centre for Technical GeoscienceDelft University of Technology
‘State A’
‘State B’
A
A
B
B
),(),( tptp ABBA xx
‘State A’
‘State B’
A
A
B
B
),(),( ABBA PP xx
‘State A’
‘State B’
),(),( ABBA PP xx
‘State A’
‘State B’
),(),( ABBA PP xx
‘State A’
‘State B’
V
kBAkBkAABBA nPVVPPP xxx 2,, d}{),(),(
Vn
State A State BPA, Vk,A
QA,Fk,A
A, A
PB, Vk,B
QB,Fk,B
B, B
Wave fields
Sources
Medium
‘State B’‘State A’
QAQB
PA
PB
V n
V
V n
V
A, A B, B
‘State B’‘State A’
QAQB
PA
PB
V
BABkAkBkAkBA
V
BkAkBA
V
kBAkBkA
PQVFFVQP
VVPPjnPVVP
x
xx
3,,,,
3,,
2,,
d}{
d}{d}{ 0 0
0 0
)( Axx )( Bxx
V n
V
V n
V
‘State B’‘State A’
QAQB
PA
PB
V
BABkAkBkAkBA
V
BkAkBA
V
kBAkBkA
PQVFFVQP
VVPPjnPVVP
x
xx
3,,,,
3,,
2,,
d}{
d}{d}{
V n
V
V n
V
‘State B’‘State A’
V
BijAijBiAiBiAiBijAij
V
BiAiijklBklAij
V
jBijAiBiAij
THVFFVHT
VVsTTjnTVVT
x
xx
3,,,,,,,,
3,,,,
2,,,,
d}{
d}{d}{
V n
V
V n
V
1872, Betti: Elastostatic fields
1873, Rayleigh: Acoustic time-harmonic fields
1959, Knopoff and Gangi: Elastodynamic fields in inhomogeneous anisotropic media
Some history
1966, De Hoop: visco-elastic anisotropic media
1993, Fokkema and van den Berg: applications in seismic processing
(source: Arntsen and Carcione, Geophysics, 2000)
‘State A’
‘State B’
V
kBAkBkAABBA nPVVPPP xxx 2,, d}{),(),(
Vn
‘State B’
V
kBAkBkAABBA nPVVPPP xxx 2,, d}{),(),(
‘State A’
‘State A’ ‘State B’
Multiple elimination (van Borselen et al., 1996)
V
kBAkBkAABBA nPVVPPP xxx 2,, d}{),(),(
‘State A’ ‘State B’
Multiple elimination (van Borselen et al., 1996)
‘State B’‘State A’
V n
V
V n
V
V
jBijAiBiAijABiAiBAiBi nTVVTVfVf xxx 2,,,,,,,, d}{),(),(
Aif ,
AiV ,
BiV ,Bif ,
Surface wave suppression (van Zanen et al., 2000)
‘State A’ ‘State B’
V
jBijAiBiAijABiAiBAiBi nTVVTVfVf xxx 2,,,,,,,, d}{),(),(
Surface wave suppression (van Zanen et al., 2000)
‘State A’ ‘State B’
Surface wave suppression (van Zanen et al., 2000)
‘State B’‘State A’
QAQB
PA
PB
V
BABkAkBkAkBA
V
BkAkBA
V
kBAkBkA
PQVFFVQP
VVPPjnPVVP
x
xx
3,,,,
3,,
2,,
d}{
d}{d}{
V n
V
V n
V
Time-lapse seismic (Dillen et al, 2000)
‘State B’‘State A’
QBPB
V
BABkAkBkAkBA
V
BkAkBA
V
kBAkBkA
PQVFFVQP
VVPPjnPVVP
x
xx
3,,,,
3,,
2,,
d}{
d}{d}{
V n
V
Time-lapse seismic (Dillen et al, 2000)
V
BABkAkBkAkBA
V
BkAkBA
V
kBAkBkA
PQVFFVQP
VVPPjnPVVP
x
xx
3,,,,
3,,
2,,
d}{
d}{d}{
‘State A’ ‘State B’
Time-lapse seismic (Dillen et al, 2000)
State A State B
xA xA
xB xB
),x(),x( ABBA PP
Acoustic reciprocity for one-way wave fields
V
BABABABA
V
HBABA n
x
x
3
23
d}{
d}{
PSPSSPSP
PPPP
V
BABkAkBkAkBA
V
kBAkBkA
PQVFFVQP
nPVVP
x
x
3,,,,
2,,
d}{
d}{
Acoustic reciprocity for full (two-way) wave fields
Acoustic reciprocity for one-way wave fields
A BA B
03x
13x
23x
33x
03x
13x
23x
33x
HRBSA x)x,x()x,x( 2d PP
HRBSA x)x,x()x,x( 2dPP
Fokkema and van den Berg (1996) Dillen (2000, Ph.d thesis) Wapenaar et al. (2000, SEG)
B A
Application in time-lapse seismic
A BA B
03x
13x
23x
33x
03x
13x
23x
33x
HRBSA x)x,x()x,x( 2d PP
HRBSA x)x,x()x,x( 2dPP
Fokkema and van den Berg (1996) Dillen (2000, Ph.d thesis) Wapenaar et al. (2000, SEG)
B A
Application in time-lapse seismic
Difference data
Application in time-lapse seismic
Boundary integral
Application in time-lapse seismic
Conclusions
Source - receiver reciprocity is a special caseof general reciprocity
General reciprocity theorems formulate a relationbetween wave fields, sources and medium parameters in two different states
- Acoustic and elastodynamic
- Two-way and one-way
Applications:
- Multiple elimination- Surface wave suppression- Time-lapse seismic- Etc.
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