反演基础inversion theory课件 Tarantola

Here, the tangent linear operator On is defined via
o( mn + δm ) = o(mn) + On δm + . . .
When the notations
m = {mα} = {m1, m2, . . . , mp} o = {oi} = {o1, o2, . . . , oq} oi = oi(m1, m2, . . . , mp)
ELSA Workshop, Barcelona, 1-5 September 2008
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The Bayes-Popper approach
To better know the values of some model parameters, we use observations. Practitioners usually seek for the models implied by the data,
The maximum likelihood model is the model m maximizing fpost(m) . It is also the model minimizing S(m) . It can be obtained using a quasi-Newton algorithm,
Least-squares theory
• The model parameter manifold may be a linear space, with vectors denoted m, m , . . . , and the a priori information may have the Gaussian form
but observations can only be used to falsify possible solutions, not to create them.
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International Organization for Standardization
• The O1 = ϕ(M1) , O2 = M2, . . . are samples of the posterior volumetric probability in the observable parameter space gpost = ϕ( fpost) .
Optimization
• If the volumetric probability fpost(M) is expected to have a small number of maxima (say one, or two, or three),
• we may try to locate them by using standard optimization methods (simplex methods, gradient-based methods),
fpost(m)
=
1 ν
fprior(m) gobs( o(m) )
.
here becomes
fpost(m) = k exp( −S(m) ) ,
where the misfit function S(m) is the sum of squares
2 S(m) = (m − mprior)t Cm-1 (m − mprior) + (o(m) − oobs)t Co-1 (o(m) − oobs) .
gobs(o)ຫໍສະໝຸດ =k exp-
1 2
(o
−
oobs)t
Co-1 (o
−
oobs)
.
• The forward modeling relation becomes, with these notations,
o = o(m) .
Then, the posterior volumetric probability for the model parameters, whose general expression is
Probabilistic Formulation of Inverse Problems
Albert Tarantola, Institut de Physique du Globe de Paris
In a realistic inverse problem, there is: i) a priori information on the parameters, ii) the results of some measurements, and iii) a theoretical relation between data and parameters. The a priori information may be introduced as a probabilistic distribution. The results of the experiments are also to be introduced using probabilistic methods, because of the experimental uncertainties. While practitioners of inverse theory usually seek for some “best model”, a Bayesian point of view has to be used, where observations are used to transform the a priori distribution of models into the a posteriori distribution. This is reminiscent of the ideas introduced by Popper, excepted that instead of using observations to “falsify” physical theories, we use observations to falsify models from the prior distribution. Only when the probability distributions are very simple (usually, simplistic) one arrives at the usual optimization methods (least-squares, etc.).
• For each model Mi , solve the forward modeling problem, Oi = ϕ(Mi) .
• Give to each model Mi a probability of ‘survival’ proportional to L(Mi) = gobs( ϕ(Mi) ) .
Guide to the expression of uncertainty in measurement
When reporting the result of a measurement of a physical quantity, some quantitative indication of the result has to be given to assess its reliability and to allow comparisons to be made. The Guide to the expression of uncertainty in measurement establishes general rules for evaluating and expressing uncertainty in measurement that can be followed at many levels of accuracy and in many fields.
• The surviving models M1 , M2, . . . are sample points of the posterior volumetric probability
fpost(M)
=
1 ν
fprior(M) gobs( ϕ(M) )
L(M)
(the other models have been falsified by the observations).
fprior(m)
=
k exp
-
1 2
(m
−
mprior)t
Cm-1
(m
−
mprior)
.
• The observable parameter manifold may be a linear space, with vectors denoted o, o , . . . , and the information brought by measurements may have the Gaussian form
To estimate the posterior uncertainties: the covariance oper-
ator of the Gaussian volumetric probability that is tangent to
apply, then On is the matrix of partial derivatives
Oiα
=
∂oi ∂mα
(evaluated at point mn ).
As we have seen, the model m∞ at which the algorithm converges maximizes the posterior volumetric probability fpost(m) .