26 april 2002 velocity estimation by inversion of focusing operators: about resolution dependent...

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






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 )


Distance

De

pth

xp,zp

sj=1 sj=2 sj=M

26 April 2002

slide 6


Optimization

Fit?

Y


NN




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


Optimization

Fit?

Y


NN




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


Optimization

Fit?

Y


NN




26 April 2002

slide 9

OutlineOutline






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


Optimization

Fit?

Y


NN



v1v2v3

vm

z1

zk

z2 v1v2

v3v4

•Local:

•Global:

26 April 2002

slide 12

Regularization:Regularization:optimizationoptimization


Optimization

Fit?

Y


NN



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






26 April 2002

slide 15

N

Forward modeling by ray-tracing (ti)

Optimization

Fit?

Y



N

Adjustment of parameterizationCalculation of

resolution


Resolution dependent Resolution dependent ParameterizationParameterization

26 April 2002

slide 16


• 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


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


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)


26 April 2002

slide 20

OutlineOutline






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


SVD LSQR

k=31

B

k=12

SBSSVD of B

Singular values:

26 April 2002

slide 26


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

slide 28

SVD LSQR

R

k=31k=12

C


Rk

Ck

26 April 2002

slide 29

SVD LSQR k=12 k=31


k=3k=6k=9k=12k=15k=18k=21k=24k=27k=31

diag

(Rk)

diag

(Ck)

diag

(C)

diag

(R)

26 April 2002

slide 30


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


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


• 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






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

26 april 2002 velocity estimation by inversion of focusing operators: about resolution dependent...

Documents

lsqrsynthetic exampleslide

zp slide

bidiagonal matrix

iterative method

zp initial macro model

asingular values

iterative optimizationzhang

number of parametersslide