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| Titel |
Bayesian Estimation of Prior Variance in Source Term Determination |
| VerfasserIn |
Václav Šmídl, Radek Hofman |
| Konferenz |
EGU General Assembly 2015
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 17 (2015) |
| Datensatznummer |
250105965
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| Publikation (Nr.) |
 EGU/EGU2015-5563.pdf |
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| Zusammenfassung |
| The problem of determination of source term of an atmospheric release is studied. We assume
that the observations y are obtained as linear combination of the source term, x, and
source-receptor sensitivities, which can be written in matrix notation as y = Mx with source
receptor sensitivity matrix M.
Direct estimation of the source term vector x is not possible since the system is often
ill-conditioned. The solution is thus found by minimization of a cost function with
regularization terms. A typical cost function is:
C (x) = (y - M x)TR-1(y- M x) + αxTDT Dx,
(1)
where the first term minimizes the error of the measurements with covariance matrix R, and
the second term is the regularization with weight α. Various types of regularization arise for
different choices of matrix D. For example, Tikhonov regularization arises for
D in the form of identity matrix, and smoothing regularization for D in the form
of a tri-diagonal matrix (Laplacian operator). Typically, the form of matrix D is
assumed to be known, and the weight α is optimized manually by a trial and error
procedure.
In this contribution, we use the probabilistic formulation of the problem, where term
(αDTD)-1 is interpreted as a covariance matrix of the prior distribution of x. Following
the Bayesian approach, we relax the assumption of known α and D and assume
that these are unknown and estimated from the data. The general problem is not
analytically tractable and approximate estimation techniques has to be used. We
present Variational Bayesian solution of two special cases of the prior covariance
matrix.
First, the structure of D is assumed to be known and only the weight α is estimated.
Application of the Variational Bayes method to this case yields an iterative estimation
algorithm. In the first step, the usual optimization problem is solved for an estimate
of α. In the next step, the value of α is re-estimated and the procedure returns to
the first step. Positivity of the solution is guaranteed by using truncated Gaussian
priors.
In the second case, we assume that α = 1 and the matrix DTD is diagonal with all
diagonal elements unknown. The resulting estimation procedure is only minor modification
of that for the first case. This choice is known in the machine-learning community as the
automatic relevance determination prior. In effect, this prior favors sparse solutions of the
inverse problem.
Performance of both methods is evaluated on a chosen tracer experiment.
This research is supported by EEA/Norwegian Financial Mechanism under project
7F14287 STRADI. |
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