# Stochastic ensembles of thermodynamic potentials

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## Abstract

The provision of uncertainty estimates along with measurement results or values computed thereof is metrologically mandatory. This is in particular true for observational data related to climate change, and thermodynamic properties of geophysical substances derived thereof, such as of air, seawater or ice. The recent International Thermodynamic Equation of Seawater 2010 (TEOS-10) provides such properties in a comprehensive and highly accurate way, derived from empirical thermodynamic potentials released by the International Association for the Properties of Water and Steam (IAPWS). Currently, there are no generally recognised algorithms available for a systematic and comprehensive estimation of uncertainties for arbitrary properties derived from those potentials at arbitrary input values, based on the experimental uncertainties of the laboratory data that were used originally for the correlations during the construction process. In particular, standard formulas for the uncertainty propagation which do not account for mutual uncertainty correlations between different coefficients tend to systematically and significantly overestimate the uncertainties of derived quantities, which may lead to practically useless results. In this paper, stochastic ensembles of thermodynamic potentials, derived from randomly modified input data, are considered statistically to provide analytical formulas for the computation of the covariance matrix of the related regression coefficients, from which in turn uncertainty estimates for any derived property can be computed a posteriori. For illustration purposes, simple analytical application examples of the general formalism are briefly discussed in greater detail.

## Keywords

Uncertainty Correlation Thermodynamic potential Fundamental thermodynamic equation Covariance matrix Error propagation Thermodynamic properties Water Ice Seawater## Notes

### Acknowledgments

This paper is dedicated to Lutz Schimansky-Geier on the occasion of his 60th birthday. The author thanks Allan Harvey and Wolfgang Wagner for helpful discussions and hints on literature, as well as the anonymous reviewers for their suggestions. The paper contributes to the tasks of the SCOR/IAPSO Working Group 127 on Thermodynamics and Equation of State of Seawater.

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