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...
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1
Inverse problem
Jean Virieux
Year 2013-2014
1
Other inverse problems?
05/11/2013 Inversion M2PGER
From Brossier (2013)
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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 ….
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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?
05/11/2013 Inversion M2PGER 3
Please, formulate your inverse problem precisely …
Along x
Natural distance
Extension to polynomial least-square fit (Vandermonde matrix)
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4
From Brossier (2013)
Everywhere: geophysics, medical sciences, astrophysics, ocean sciences, climate simulation, signal processing, mechanics, financial market …
05/11/2013 Inversion M2PGER
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505/11/2013 Inversion M2PGER
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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
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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
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805/11/2013 Inversion M2PGER
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905/11/2013 Inversion M2PGER
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1005/11/2013 Inversion M2PGER
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1105/11/2013 Inversion M2PGER
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Non-uniqueness
Under-determined
Over-determined
Mixed-determined
05/11/2013 Inversion M2PGER 12
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1305/11/2013 Inversion M2PGER
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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
05/11/2013 Inversion M2PGER 14
cy
Model parameters
Data values
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15
Cramer methodalmost never used !
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1605/11/2013 Inversion M2PGER
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17
How to do in practice?What is inside Excel?
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1805/11/2013 Inversion M2PGER
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1905/11/2013 Inversion M2PGER
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2005/11/2013 Inversion M2PGER
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2105/11/2013 Inversion M2PGER
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2205/11/2013 Inversion M2PGER
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2305/11/2013 Inversion M2PGER
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2405/11/2013 Inversion M2PGER
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Influence of prior information
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Curve fit is influenced by the prior weight
Description of a prior model weight:guess what will bea data weight
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26Various books are useful (Menke, 1985; Tarantola, 1987)
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2705/11/2013 Inversion M2PGER
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2805/11/2013 Inversion M2PGER
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2905/11/2013 Inversion M2PGER
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30Redo the same with error (covariance matrix) on measurements
Redo
the same
by invertingx<->y
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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
05/11/2013 Inversion M2PGER 31
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3205/11/2013 Inversion M2PGER
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33
Semi-global methods
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34We are back to the « normal equations » of the linear system05/11/2013 Inversion M2PGER
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3505/11/2013 Inversion M2PGER
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3605/11/2013 Inversion M2PGER
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3705/11/2013 Inversion M2PGER
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3905/11/2013 Inversion M2PGER
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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
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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.
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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
05/11/2013 Inversion M2PGER 43
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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
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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 !
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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
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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
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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
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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
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A priori & A posteriori informationWhat is the meaning of the « final » model we provide ?
acceptable05/11/2013 50Inversion M2PGER
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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
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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)
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Sampling a posteriori distribution
Resolution estimation : spike test
Boot-Strapping
Jack-knifing
Natural Neighboring
Monte-Carlo
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Thank you …
54
Ray imprints if model description not smooth enough
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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
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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
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Sampling a posteriori distribution
Uncertainty estimation for P and S velocities using boot-strapping techniques 05/11/2013 57Inversion M2PGER
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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
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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)
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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
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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
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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
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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
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A posteriori informationWhat is the meaning of the « final » model we provide ?
acceptable05/11/2013 70Inversion M2PGER