Least Squares Migration of Least Squares Migration of
Stacked SupergathersStacked Supergathers
Wei Dai and Gerard SchusterWei Dai and Gerard SchusterKAUSTKAUST
vs
RTM Problem & Possible Soln.RTM Problem & Possible Soln.
• Problem:Problem: RTM computationally costly; IO high RTM computationally costly; IO high
• Solution:Solution: Multisource LSM RTM Multisource LSM RTM
Preconditioning speeds up by factor 2-3Preconditioning speeds up by factor 2-3
Encoded LSM reduces crosstalk. Reduced comp. cost+memoryEncoded LSM reduces crosstalk. Reduced comp. cost+memory
OutlineOutline• MotivationMotivation
• Multisource LSM theoryMultisource LSM theory
• Signal-to-Noise Ratio (SNR)Signal-to-Noise Ratio (SNR)
• Numerical results Numerical results
• ConclusionsConclusions
Multisource Migration:Multisource Migration: mmmigmig=L=LTTdd
Forward Model:Forward Model:
Phase Encoded Multisource Migration Phase Encoded Multisource Migration
d +d +dd =[ =[L +L +LL ]m ]m11 222211
LL{dd{
=[=[L +L +LL ]( ](dd + + dd ) ) 11 222211
TT TT
= = L d +L d +L dL d + + 11 222211
TT TT
LL dd + +L L dd22 112211
Crosstalk noiseCrosstalk noiseStandard migrationStandard migration
TT TTmmmigmig
= = L d +L d +L dL d + + 11 222211
TT TT
LL dd + +LL dd22 112211
TT TTmmmigmig
mmmigmig
= = L d +L d +L dL d11 222211
mmmigmig
Multisource Migration:Multisource Migration: mmmigmig=L=LTTdd
Forward Model:Forward Model:
Phase Encoded Multisrce Phase Encoded Multisrce Least Squares Least Squares Migration Migration
d +d +dd =[ =[L +L +LL ]m ]m11 222211
LL{dd{
=[=[L +L +LL ]( ](dd + + dd ) ) 11 222211
TT TTmmmigmig
= = L d +L d +L dL d + + 11 222211
TT TT
LL dd + +L L dd22 112211
Crosstalk noiseCrosstalk noiseStandard migrationStandard migration
TT TT
m = m +(k+1) (k)
OutlineOutline• MotivationMotivation
• Multisource LSM theoryMultisource LSM theory
• Signal-to-Noise Ratio (SNR)Signal-to-Noise Ratio (SNR)
• Numerical results Numerical results
• ConclusionsConclusions
Standard Migration SNR
GS# geophones/CSG# geophones/CSG
# CSGs# CSGs
SNR= ...
migrate
SNR=
d(t) =d(t) =Zero-mean white noise
[S(t) +N(t) ][S(t) +N(t) ] Neglect geometric spreading
Standard Migration SNR
Standard Migration SNR
Assume:
migrate+++
stack
S1
SGS G~~
iterate
GI
Iterative Multisrc. Mig. SNR
# iterations# iterations
SNR=
Cost ~ O(S)
Cost ~ O(I)
SN
R0
1 Number of Iterations 300
7The SNR of MLSM image grows as the square root of the number of iterations.
SNR = GI
Multisource LSM SummaryMultisource LSM Summary
IO 1 1/100
Cost ~
Resolution dx 1 1/2
SNR
Stnd. Mig Multsrc. LSMStnd. Mig Multsrc. LSM
GS GI
S I
Cost vs Quality: Can I<<S?Cost vs Quality: Can I<<S?
OutlineOutline• MotivationMotivation
• Multisource LSM theoryMultisource LSM theory
• Signal-to-Noise Ratio (SNR)Signal-to-Noise Ratio (SNR)
• Numerical results Numerical results
• ConclusionsConclusions
0Z
k(m
)3
0 X (km) 16
The Marmousi2 Model
The area in the white box is used for SNR calculation.
200 CSGs.
Born Approximation
Conventional Encoding: Static Time Shift & Polarity Statics
0 X (km) 16
0Z
k(m
)3
0Z
(k
m)
3
0 X (km) 16
Conventional Source: KM vs LSM (50 iterations)
Conventional KM
50x
1x
Conventional KLSM
0 X (km) 16
0Z
k(m
)3
0Z
(k
m)
3
0 X (km) 16
Multisource KM (1 iteration)
200-source Supergather: Multisrc. KM vs LSM
Multisource KLSM (300 iterations)
1.5 x
1 x200
I=1.5S
IO 1 1/200
Cost ~
Resolution dx 1 1/2
SNR~
Stnd. Mig Multsrc. LSMStnd. Mig Multsrc. LSM
1 1.5
Cost vs Quality: Can I<<S?Cost vs Quality: Can I<<S?
What have we empirically learned?
S=200 I=300
SEG/EAGE Salt Reflectivity Model
• Use constant velocity model with c = 2.67 km/s
• Center frequency of source wavelet f = 20 Hz
• 320 shot gathers, Born approximation
Z
(k
m)
01.
4
0 X (km) 6
• Encoding: Dynamic time, polarity statics + wavelet shaping
• Center frequency of source wavelet f = 20 Hz
• 320 shot gathers, Born approximation
0 X (km) 6
0Z
k(m
)1.
40
Z (
km
)1.
4
0 X (km) 6
Standard Phase Shift Migration (320 CSGs)
Standard Phase Shift Migration vs MLSM (Yunsong Huang)
Multisource PLSM (320 blended CSGs, 7 iterations)
1 x
1 x
44
Single-source PSLSM(Yunsong Huang)
Mod
el E
rror
1.0
0.30 50Iteration Number
Unconventional encodingUnconventional encoding
Conventional encoding: Polarity+Time ShiftsConventional encoding: Polarity+Time Shifts
IO 1 1/320
Cost ~
Resolution dx 1 1/2
SNR~
Stnd. Mig Multsrc. LSMStnd. Mig Multsrc. LSM
I=7
1 1/44
Cost vs Quality: Can I<<S? Yes.Cost vs Quality: Can I<<S? Yes.
What have we empirically learned?
S=320
ConclusionsConclusions Mig vs MLSM Mig vs MLSM
1. 1.
2. Cost: 2. Cost: S S vsvs II
3. Caveat: Mig. & Modeling were adjoints 3. Caveat: Mig. & Modeling were adjoints of one another. LSM sensitive starting model of one another. LSM sensitive starting model
5.5. Next Step: Sensitivity analysis to starting modelNext Step: Sensitivity analysis to starting model
SNR: VSGS GI
4. Unconventional encoding: I << S4. Unconventional encoding: I << S
2. Memory 2. Memory 1 1 vsvs 1/S1/S
Back to the Future?Back to the Future?
Poststackencoded migration
DMO Prestackmigration
1980s 1980s-2010 2010?
Evolution of Migration
Poststackmigration
1960s-1970s