Unleashing Empirical Equations with “Nonlinear Fitting” and “GUM Tree Calculator”

  • J. W. Lovell-SmithEmail author
  • P. Saunders
  • R. Feistel
Part of the following topical collections:
  1. TEMPMEKO 2016: Selected Papers of the 13th International Symposium on Temperature, Humidity, Moisture and Thermal Measurements in Industry and Science


Empirical equations having large numbers of fitted parameters, such as the international standard reference equations published by the International Association for the Properties of Water and Steam (IAPWS), which form the basis of the “Thermodynamic Equation of Seawater—2010” (TEOS-10), provide the means to calculate many quantities very accurately. The parameters of these equations are found by least-squares fitting to large bodies of measurement data. However, the usefulness of these equations is limited since uncertainties are not readily available for most of the quantities able to be calculated, the covariance of the measurement data is not considered, and further propagation of the uncertainty in the calculated result is restricted since the covariance of calculated quantities is unknown. In this paper, we present two tools developed at MSL that are particularly useful in unleashing the full power of such empirical equations. “Nonlinear Fitting” enables propagation of the covariance of the measurement data into the parameters using generalized least-squares methods. The parameter covariance then may be published along with the equations. Then, when using these large, complex equations, “GUM Tree Calculator” enables the simultaneous calculation of any derived quantity and its uncertainty, by automatic propagation of the parameter covariance into the calculated quantity. We demonstrate these tools in exploratory work to determine and propagate uncertainties associated with the IAPWS-95 parameters.


Correlation Covariance Empirical equations Generalized least squares Propagation of uncertainty 


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Measurement Standards Laboratory (MSL)Lower HuttNew Zealand
  2. 2.Leibniz Institute for Baltic Sea Research (IOW)WarnemündeGermany

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