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| Titel |
Numerical Inversion with Full Estimation of Variance-Covariance Matrix |
| VerfasserIn |
Vasso Saltogianni, Stathis Stiros |
| Konferenz |
EGU General Assembly 2016
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| Medientyp |
Artikel
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| Sprache |
en
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 18 (2016) |
| Datensatznummer |
250127064
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| Publikation (Nr.) |
 EGU/EGU2016-6885.pdf |
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| Zusammenfassung |
| Various geophysical problems are described by redundant systems of highly non-linear
systems of equations with ≥3 unknown variables. Such systems are not possible to be solved
with formal algebraic techniques, and are usually solved using sampling methods (mostly
Monte Carlo-based), gradual optimization of certain of the unknown variables,
a priori fixing of the values of some variables or in the vicinity of approximate
solutions. In many cases, especially in the modeling of activated faults or of magma
sources from surface displacements, such methods lead to sub-optimal solutions
(trapped in local extrema, high uncertainties (trade-off) between certain variables, etc.)
and highly influence the understanding/ modeling of certain complex geophysical
processes.
In order to overcome these difficulties we proposed an alternative, topology-based,
deterministic, numerical approach for the inversion of such systems of equations with n
unknown variables, the TOPological INVersion (TOPINV) algorithm. TOPINV has been
inspired from traditional positioning and the geodetic theory and is based on the intersection
of spaces and the identification of clusters of points which satisfy observations equations. It is
not based on the minimization of a certain cost function and involves only forward
computations, hence avoids computational errors.
The basic concept is to assume discrete possible ranges of each variable, and from these
ranges to define a grid G in Rn space containing the true solution (discrete search
hyperspace). Each point of this hyper-grid is then tested whether it satisfies or not the
observations, given their uncertainty level. This is possible by transforming equations to
double (absolute value) inequalities using a single optimization parameter and a
trial-and-error approach. The optimal (minimal) space containing one or more solutions in the
form of one or more compact clouds (sets) of gridpoints satisfying the system of equations is
then selected, and single-point, stochastic optimal solutions are computed as the center of
gravity of these sets. A full Variance-Covariance Matrix (VCM) of each solution can be
directly computed as second statistical moment.
The overall method and the software have been tested with synthetic data (accuracy-oriented
approach) in the modeling of magma chambers in the Santorini volcano and the modeling
of double-fault earthquakes, i.e. to inversion problems with up to 18 unknowns. |
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