Uncertainty of empirical correlation equations

The International Association for the Properties of Water and Steam (IAPWS) has published a set of empirical reference equations of state, forming the basis of the 2010 Thermodynamic Equation of Seawater (TEOS-10), from which all thermodynamic properties of seawater, ice, and humid air can be derived in a thermodynamically consistent manner. For each of the equations of state, the parameters have been found by simultaneously fitting equations for a range of different derived quantities using large sets of measurements of these quantities. In some cases, uncertainties in these fitted equations have been assigned based on the uncertainties of the measurement results. However, because uncertainties in the parameter values have not been determined, it is not possible to estimate the uncertainty in many of the useful quantities that can be calculated using the parameters. In this paper we demonstrate how the method of generalised least squares (GLS), in which the covariance of the input data is propagated into the values calculated by the fitted equation, and in particular into the covariance matrix of the fitted parameters, can be applied to one of the TEOS-10 equations of state, namely IAPWS-95 for fluid pure water. Using the calculated parameter covariance matrix, we provide some preliminary estimates of the uncertainties in derived quantities, namely the second and third virial coefficients for water. We recommend further investigation of the GLS method for use as a standard method for calculating and propagating the uncertainties of values computed from empirical equations.