26 april 2002 velocity estimation by inversion of focusing operators: about resolution dependent...
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26 April 2002
Velocity estimation by inversion Velocity estimation by inversion of Focusing operators:of Focusing operators:About resolution dependent About resolution dependent
parameterization and the use of the parameterization and the use of the LSQR methodLSQR method
Barbara Cox
IMA Workshop:
Inverse Problems and Quantification of Uncertainty
26 April 2002
slide 2
OutlineOutline
• Inversion of Focusing Operators
• Regularization of inversion
• Resolution dependent Parameterization
• Optimization by LSQR
• Synthetic example
CFP method
26 April 2002
slide 3
OutlineOutline
• Inversion of Focusing Operators
• Regularization of inversion
• Resolution dependent Parameterization
• Optimization by LSQR
• Synthetic example
26 April 2002
slide 4
Inversion of Focusing Inversion of Focusing OperatorsOperators
• Data: one-way travel times• Unknowns: slowness & exact focus point location• Obtained by minimizing:
Distance (km)0 4 8 12 16
0
2
4
De
pth
(km)
modelreal ttΔt
Distance (km)0 4 8 12 16
0
2
3
Tim
e (
s)
1
26 April 2002
slide 5
Forward modeling by raytracing (ti)
Optimization
Fit?
Y
Focusing operators (data)
NN
Initial macro model (sj & xp ,zp )
Final macro model (sj & xp ,zp )
Inversion of Focusing Inversion of Focusing OperatorsOperators
Distance
De
pth
xp,zp
sj=1 sj=2 sj=M
26 April 2002
slide 6
Forward modeling by raytracing (ti)
Optimization
Fit?
Y
Focusing operators (data)
NN
Initial macro model (sj & xp ,zp )
Final macro model (sj & xp ,zp )
Inversion of Focusing Inversion of Focusing OperatorsOperators
Distance
De
pth
sj=1 sj=2 sj=M
xp,zp
ti=1 ti=2 ti=N
j
i
s
t
p
i
z
t
p
i
x
t
26 April 2002
slide 7
Forward modeling by raytracing (ti)
Optimization
Fit?
Y
Focusing operators (data)
NN
Initial macro model (sj & xp ,zp )
Final macro model (sj & xp ,zp )
Inversion of Focusing Inversion of Focusing OperatorsOperators
itj
i
s
t
p
i
z
t
p
i
x
t
js px pz
mAt
•Solve iteratively by e.g. SVD:
•Assume linear relation :
TT UVSAUSVA 11 mmmtAm
kk 1
1
26 April 2002
slide 8
Forward modeling by raytracing (ti)
Optimization
Fit?
Y
Focusing operators (data)
NN
Initial macro model (sj & xp ,zp )
Final macro model (sj & xp ,zp )
Inversion of Focusing Inversion of Focusing OperatorsOperators
26 April 2002
slide 9
OutlineOutline
• Inversion of Focusing Operators
• Regularization of inversion
• Resolution dependent Parameterization
• Optimization by LSQR
• Synthetic example
26 April 2002
slide 10
RegularizationRegularization
Parameterization:Regularizing bycoarser (global)parameterization
Optimization:Regularizing by e.g.
resolution matrix
• Tomographic inverse problems are generally mixed determined
• Can be faced by regularization:
26 April 2002
slide 11
Regularization: Regularization: parameterizationparameterization
Forward modeling by raytracing (ti)
Optimization
Fit?
Y
Focusing operators (data)
NN
Initial macro model (sj & xp ,zp )
Final macro model (sj & xp ,zp )
v1v2v3
vm
z1
zk
z2 v1v2
v3v4
•Local:
•Global:
26 April 2002
slide 12
Regularization:Regularization:optimizationoptimization
Forward modeling by raytracing (ti)
Optimization
Fit?
Y
Focusing operators (data)
NN
Initial macro model (sj & xp ,zp )
Final macro model (sj & xp ,zp )
mAt •Regularization:
mm RW
1 12 2m ap m
Δt A= Δm
W Δm W
26 April 2002
slide 13
RegularizationRegularization
Parameterization:Regularizing bycoarser (global)parameterization
Optimization:Regularizing by e.g
resolution matrix
Constraints result Still over-parameterized
Combine:Parameterization dependent on
resolution
No constraint on result, No over-parameterization
26 April 2002
slide 14
OutlineOutline
• Inversion of Focusing Operators
• Regularization of inversion
• Resolution dependent Parameterization
• Optimization by LSQR
• Synthetic example
26 April 2002
slide 15
N
Forward modeling by ray-tracing (ti)
Optimization
Fit?
Y
Focusing operators (data)
Final macro model (sj & xp ,zp )
N
Adjustment of parameterizationCalculation of
resolution
Initial macro model (sj & xp ,zp )
Resolution dependent Resolution dependent ParameterizationParameterization
26 April 2002
slide 16
Resolution dependent Resolution dependent ParameterizationParameterization
• Calculation of resolution
•
• )( TVVr diag1
Re
solu
tion
0 Distance De
pth
Resolution in model
Distance De
pth
Velocity model
T
T
T
VVRUVSA
USVA
11
26 April 2002
slide 17
Resolution dependent Resolution dependent ParameterizationParameterization
Distance
• Adjustment of parameterization dependent on resolution
De
pth
Resolution in model
Distance De
pth
Velocity model
Add points
Remove points0
1
0.2
0.4
0.6
0.8
Re
solu
tion
Gridpoints 1 M
Resolution plot
26 April 2002
slide 18
Consequently:• No constraint on result• No over-parameterization• The available information within the
data can be completely translated to the model
Resolution dependent Resolution dependent ParameterizationParameterization
26 April 2002
slide 19
However, • Calculation of resolution or covariance requires explicit matrix inversion
• Explicit matrix inversion is not feasible:Optimization by iterative method: LSQR
• Paige & Saunders (1982)
• Calculate resolution during iterative optimization• Zhang and McMechan (1995)• Yao et al (1999)• Berryman (2001)
Resolution dependent Resolution dependent ParameterizationParameterization
26 April 2002
slide 20
OutlineOutline
• Inversion of Focusing Operators
• Regularization of inversion
• Resolution dependent Parameterization
• Optimization by LSQR
• Synthetic example
26 April 2002
slide 21
Optimization by LSQROptimization by LSQRLSQR method: • Iterative SVD approximation: k iterations k basis-vectors
SVD LSQRTkkkk VBUA TUSVA
Singular Value diagonal matrix
Bi-diagonal matrixkBS
Tk
Tkk
Tkkk UBBBVA
11 TUVSA 11
• If k = number of parameters then LSQR=SVD
• Maximum number of iterations (k) = number of parameters
• First k LSQR basis-vectors First k SVD eigen-vectors
26 April 2002
slide 23
Optimization by LSQROptimization by LSQR
• Largest (pseudo) singular values are obtained first
SVD LSQRTkkkk VBUA TUSVA
SV diagonal matrix Pseudo SV diagonal matrix
S
TBBBk VSUB
BS
Singular values:• Bi-diagonal matrix can be converted to a pseudo singular
value diagonal matrix
SS B• If k = number of parameters then
26 April 2002
slide 25
Optimization by LSQROptimization by LSQR
SVD LSQR
k=31
B
k=12
SBSSVD of B
Singular values:
26 April 2002
slide 26
Optimization by LSQROptimization by LSQR
SVD LSQR diag(SB)diag(S)
k=12k=3k=6k=9k=15k=12k=18k=21k=24k=27k=31
k=31
• Large pseudo- singular values are solved first
Singular values:
26 April 2002
slide 27
Calculation of resolution by Calculation of resolution by LSQRLSQR
SVD LSQR
IR M
• Calculation of resolution by means of model space matrix
TVVR IR M
IR M• If k = nr of parameters (over-determined system)
TVVSC 2 2 T
k k B B B kC V V S V V
Tk k B B kR V V V V
V
• Calculation of covariance by means of space matrix and singular value matrix
VS
26 April 2002
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SVD LSQR
R
k=31k=12
C
Calculation of resolution by Calculation of resolution by LSQRLSQR
Rk
Ck
26 April 2002
slide 29
SVD LSQR k=12 k=31
Calculation of resolution by Calculation of resolution by LSQRLSQR
k=3k=6k=9k=12k=15k=18k=21k=24k=27k=31
diag
(Rk)
diag
(Ck)
diag
(C)
diag
(R)
26 April 2002
slide 30
Calculation of resolution by Calculation of resolution by LSQRLSQR
R
C
IR M
The way the covariance evolves during the iterations cannot be trusted, as some parameters are not solved by the current basis-vectors
Final covariance is the real covariance of the system
is an indication that all parameters ARE solved, but not how WELL they are solved
The way the resolution evolves during the iterations is an indication how WELL the parameters are solved
Maximum iterations (=SVD) Limited number of iterations
26 April 2002
slide 31
LSQR
Calculation of resolution by Calculation of resolution by LSQRLSQR
k=3k=3k=6k=9k=12k=15k=18k=21k=24k=27k=31
SVD
diag
(Rk)
diag
(Ck)
diag
(C)
diag
(R) • Low resolution
AND low covariance indicate points that are not solved yet
• Can be used to describe the quality of the solution quantitatively
26 April 2002
slide 32
Optimization by LSQROptimization by LSQR
• The relative resolution can be used as a criterion for adjustment of parameterization
• The pseudo singular values can be used to evaluate how well the system is determined
• The comparison between resolution and covariance can be used to evaluate which parameters are described
Use of LSQR for resolution dependent parameterization:
Quantitative criteria
Qualitative criterion
REMARK:• Singular value decomposition
of covariance matrix (Delphine Sinoquet) can be placed on top of this method: not expensive anymore
• However, don’t use covariance but resolution matrix
26 April 2002
slide 33
OutlineOutline
• Inversion of Focusing Operators
• Regularization of inversion
• Resolution dependent Parameterization
• Optimization by LSQR
• Synthetic example
26 April 2002
slide 34
Synthetic ExampleSynthetic ExampleIdeal model Distance (km)
0 160
4
Dep
th (
km)
Initial model
0
2
Tim
e (s)
Focusing operators Distance (km)
0 16
Distance (km)
0 16
Modeled Foc. oper. Distance (km)
0 16
3500
1500
Velocity (m
/s)
3500
1500
Velocity (m
/s)
0
2
Tim
e (s)
0
4
Dep
th (
km)
26 April 2002
slide 35
00
4
16DistanceD
epth
Vel
ocity
Res
olut
ion
Resolution dependent Resolution dependent parameterizationparameterization
26 April 2002
slide 36
ResultResult
Ideal model
Distance (km)
0 160
4
Dep
th (
km)
Data driven modelDistance (km)
0 16
3500
1500
Velocity (m
/s)
0
4
Dep
th (
km)
Distance (km)
0 160
4
Dep
th (
km)
Data driven model
3500
1500
Velocity (m
/s)
0,001
0,01
0,1
1
1 2 3 4 5 6 7
update
dt(r
ms)
26 April 2002
slide 37
MigrationMigrationDistance (km)
0 16Updated model
3500
1500
Velocity (m
/s)
0
4
Dep
th (
km)
Ideal model
Postupdating
26 April 2002
slide 38
ConclusionsConclusions
• Resolution dependent parameterization: efficient,
data dependent, minimal user interaction
• Resolution can be obtained in an efficient way in the
LSQR algorithm
• Regularization of the inverse problem by means of
resolution dependent parameterization
• The optimization can be evaluated by the LSQR
algorithm, using the resolution, the ‘pseudo’ singular
values and the comparison between resolution and
covariance
26 April 2002
slide 39
AcknowledgementsAcknowledgements
I would like to thank:
• The people of the CWP project for their Delaunay and ray-tracing software, which formed a base for the developed algorithm
• The sponsors of the Delphi Imaging and Characterization consortium for their support