INVERSION 2013 M2PGER · LEAST SQUARES METHOD m G G G d G G m G d m E m E m d G m d G m t t est t t t 0 1 0 0 0 0 0 0 0 0 ( ) ( ) ( ) ( ) L 2 norm locates the minimum of E normal

1

Inverse problem

Jean Virieux

Year 2013-2014

1

Other inverse problems?

05/11/2013 Inversion M2PGER

From Brossier (2013)

Least-square method

Sum of vertical distances between data points and expected y values from the unknown line y=ax+b should be minimum: find a and b?

05/11/2013 Inversion M2PGER 2

From Excel

It is an inversion ….

X or Y ? Very specific formation (x is

supposed to be perfectly know while y is the measurement)

Minimisation of distance along x? Minimisation of distance

perpendicular to the line?


Please, formulate your inverse problem precisely …

Along x

Natural distance

Extension to polynomial least-square fit (Vandermonde matrix)

4

From Brossier (2013)

Everywhere: geophysics, medical sciences, astrophysics, ocean sciences, climate simulation, signal processing, mechanics, financial market …



Inversion vs Assimilation Common features

Need data (and uncertainties) Need model (and prior uncertainties) Need an updating procedure (optimization)

Main differences Inversion

The initial state is assumed to be knownThe observation and the model solution are

time-independent Assimilation

The initial state is part of the solutionThe observation and the model solution are

time-dependent 605/11/2013 Inversion M2PGER

Forward versus inverse « Determine and characterize the causes of a

(physical) phenomenon from (observed) effectsand consequences »

Forward problem: natural and easy as samecause(s) give(s) same consequences

– a well-posed problem

Inverse problem: not natural and complexe as a same fact can have different distinc origins

– an ill-posed problem






Non-uniqueness

Under-determined

Over-determined

Mixed-determined



Basics of linear algebra Why you have learnt linear algebra during

undergraduate studies! Consider the linear system where x is

an unknow vector and y is a data vector. The matrix A (called an operator when no discretization) is the model relation


cy

Model parameters

Data values

15

Cramer methodalmost never used !



17

How to do in practice?What is inside Excel?









Influence of prior information


Curve fit is influenced by the prior weight

Description of a prior model weight:guess what will bea data weight

26Various books are useful (Menke, 1985; Tarantola, 1987)





30Redo the same with error (covariance matrix) on measurements

Redo

the same

by invertingx<->y


SUM UP: LEAST SQUARES METHOD

dGGGm

dGmGGmmE

mGdmGdmE

ttest

tt

t

0

1

00

000

00

0)()()()(

L2 norm

locates the minimum of E

normal equations

if exists 1

00

GG t

Least-squares estimation

Operator on data will derive a new model : this is called

the generalized inverse

tt GGG 01

00

gG0

G0 is a N by M matrice

is a M by M matrice 1

00

GG t

Under-determination M > N

Over-determination N > MMixed-determination – seismic tomography



33

Semi-global methods


34We are back to the « normal equations » of the linear system05/11/2013 Inversion M2PGER







LINEAR INVERSE PROBLEM

1 10 0

0 0

u G t m G dt G u d G m

Updating slowness perturbation values from time residuals

Formally one can write

with the forward problem

Existence, Uniqueness, Stability, RobustnessDiscretisation

Identifiability

of the model

Small errors propagates

Outliers effects

NON-UNIQUENESS & NON-STABILITY : ILL-POSED PROBLEM

REGULARISATION : ILL-POSED -> WELL-POSED

05/11/2013 41Inversion M2PGER

LEAST-SQUARES SOLUTIONS

AtDT=AtA DM•The linear system can be recast into a least-square system, which means a system of normal equations. The resolution of this system gives the solution. DM=(AtA)-1AtDT•The system is both under-determined and over-determined depending on the considered zone (and tne number of rays going through.


LEAST SQUARES METHOD

dGGGm

dGmGGmmE

mGdmGdmE

ttest

tt

t

0

1

00

000

00

0)()()()(

L2 norm

locates the minimum of E

normal equations

if exists 1

00

GG t

Least-squares estimation

Operator on data will derive a new model : this is called

the generalized inverse

tt GGG 01

00

gG0

G0 is a N by M matrice

is a M by M matrice 1

00

GG t

Under-determination M > N

Over-determination N > MMixed-determination – seismic tomography


Maximum Likelihood method One assume a gaussian distribution of data

Joint distribution could be written

)()(

21exp)( 0

10 mGdCmGddp d

Where G0m is the data mean and Cd is the data covariance matrice: this method is very similar to the least squares method

)()()()()()( 01

0100 mGdCmGdmEmGdmGdmE dt

)()()( 01

02 mGdWmGdmE d

Even without knowing the matrice Cd, we may consider data weight Wdthrough


SVD analysis for stability and uniqueness

SVD decomposition :

U : (N x N) orthogonal Ut=U-1

V : (M x M) orthogonal Vt=V-1

: (N x M) diagonal matrice Null space for i=0

tVUG 0

UtU=I and VtV=I (not the inverse !)

][

][

0

0

UUU

VVV

p

p

tpp

p VVUUG 000 000

Vp and V0 determine the uniqueness while Up and U0 determine the existence of the solution

tppp

tppp

UVG

VUG11

0

0

Up and Vp have now inverses !


Solution, model & data resolution

RmmVVmVUUVmGGdGm tp

tppp

tpppest 1

01

01

0 )(The solution is

where Model resolution matrice : if V0=0 then R=VVt=I tppVVR

NddUUmGd tppestest 0

dUUN tppwhere Data resolution matrice : if U0=0 then N=UUt=I

importance matriceGoodness of resolution

SPREAD(R)=

SPREAD(N)=

Spreading functions

2

2

IN

IR

Good tools for quality estimation


PRIOR INFORMATION Hard bounds

Prior model

is the damping parameter controlling the importance of the model mp

Gaussian distribution

Model smoothness

Penalty approach

add additional relations between model parameters (new lines)

)()()()()( 005 pmt

pdt mmWmmmGdWmGdmE

With Wd data weighting and Wm model weighting

tmd

tg

pmt

pdt

GCGCGG

mmCmmmGdCmGdmE

011

01

00

10

104 )()()()()(

)()()()()( 003 pt

pt mmmmmGdmGdmE

BmA i Seismic velocity should be positive


UNCERTAINTY ESTIMATION Least squares solution

Model covariance : uncertainty in the data

curvature of the error function

Sampling the error function around the estimated model often this has to be done numerically

dGdGGGm gttest 00

100

1

2

22

100

2

20000

21cov

cov

covcov

estmmdest

tdest

dd

gtd

ggtgest

mEm

GGm

IC

GCGGdGm

Uncorrelated data


A posteriori model covariance matrice True a posteriori distribution

Tangent gaussian distribution

S diagonal matrice eigenvalues

U orthogonal matrice eigenvectors

Error ellipsoidal could be estimated

WARNING : formal estimation related to the gaussian distribution hypothesis

If one can decompose this matrice


A priori & A posteriori informationWhat is the meaning of the « final » model we provide ?

acceptable05/11/2013 50Inversion M2PGER

Flow chart

true ray tracing

data residual

sensitivity matrice

model update

new modelmmmdGm

mgG

ddd

mgdmd

synobs

syn

obs

10

0

)(

collected data

starting modelloop

Calculate for formal uncertainty estimation

small model variation or small errors exit

22

mE


LSQR method The LSQR method is a conjugate gradient method developped by Paige & Saunders

Good numerical behaviour for ill-conditioned matrices

When compared to an SVD exact solution, LSQR gives main components of the solution while SVD requires the entire set of eigenvectors

Fast convergence and minimal norm solution (zero components in the null space if any)


Sampling a posteriori distribution

Resolution estimation : spike test

Boot-Strapping

Jack-knifing

Natural Neighboring

Monte-Carlo


Thank you …

54

Ray imprints if model description not smooth enough


Discrete Model Spacecube

kjikji huzyxu ,,,,),,(

m

m

m

nn

m

n

n

cubekji

cube rayonkjikji

rayon cubekjikji

uu

uu

ut

ut

ut

ut

tt

tt

uutrst

dlhudlhurst

1

2

1

1

1

1

1

1

2

1

,,

,,,,,,,,

...

),(

),(00

Slowness perturbation description

0t G u

Matrice of sensitivity or of partial derivatives

Discretisation of the medium fats the ray

Sensitivity matrice is a sparse matrice


Another error function pmpmg mmCmmCmE ()(

21)( 2/12/1

))(())(( 2/12/1 mgdCmgdC dd

Scalar product on D x M

mCmgC

mCdC

mCmgC

mCdC

mEm

d

pm

d

m

d

pm

dg 2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/1 )()(21)(

We must minimize 2

2/1

2/1

2/1

2/1 )(21

mCmgC

mCdC

m

d

pm

d

which is related to the possible following factorisation

2/10

2/1

2/10

2/11

01

0m

d

t

m

dmd C

GCC

GCCGCG

t

m

d VUC

GC

2/10

2/1

SVD decomposition if possible : please note that this is a sparse matrice good for tomography


Sampling a posteriori distribution

Uncertainty estimation for P and S velocities using boot-strapping techniques 05/11/2013 57Inversion M2PGER

Steepest descent methods )()( 1 kk mEmE

kk

kk

kkkk

k

kkk

DmE

mEmEd

dEE

d

mEdmEmEtmE

)(

)()(

)()())((

2

12

1

0

Gradient method

Conjugate gradient

Newton

Quasi-Newton

Gauss-Newton is Quasi-Newton for L2 norm

quadratic approximation of E


Tomographic descent 2

2/1

2/1

2/1

2/1 )(21

mCmgC

mCdC

m

d

pm

dMinimisation of this vector

2/1

2/1

m

kdk C

GCAIf one computes

then

)())((

02/1

2/1

km

kdtkk

tk mmC

dmgCAmAA

Gaussian error distribution of data and of a posteriori model

Easy implementation once Gk has been computed

Extension using Sech transformation (reducing outliers effects while keeping L2 norm simplicity)








THE Cm-1/2 MATRICE

)exp(2

ji

ij

xxc

Shape independent of

Values depend on

SATURATION

The matrice Cm has a band diagonal shape

- is the standard error (same for all nodes)

is the correlation length

n=nx.ny.nz=104 Cm=USUt (Lanzos decomposition)

tm UUSC 2/12/1


Analysis of coefficients

Values independent of n (n>5000)

Values are only related to and

2/12/1 ~0 m

nm

n CC

Typical sizes 200x200x50

deduced from 20x20x5 (few minutes)

Strategy of libraries of Cm-1/2 for

various and =

Other coefficients could be deduced

R: Cm-1/2 sparse matrice


An example

=0.8

v=100 km/s

x=100 km

t=100 s

=0.1

Ray imprints

Same numerical grid for all simulations (either 100x100 or 400x400)

Same results at the limit of numerical precision related to the estimation of the sensitivity matrice


Illustration of selection {,v}

= 5 km and v= 3 km/s

Error function analysis will give us optimal values of a priori standard error and correlation length (2D analysis)

v influence

influence


A posteriori informationWhat is the meaning of the « final » model we provide ?

acceptable05/11/2013 70Inversion M2PGER

Documents

INVERSION 2013 M2PGER · LEAST SQUARES METHOD m G G G d G G m G d m E m E m d G m d G m t t est t t t 0 1 0 0 0 0 0 0 0 0 ( ) ( ) ( ) ( ) L 2 norm locates the minimum of E normal