Stochastic ensembles of thermodynamic potentials

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.

Appendix: Selected elements of matrix calculus used in the text

The mathematical product of two matrices A, B is defined as the matrix C that has as its elements the scalar products of the row vector of the first factor with the column vector of the second, for example,

$$ {\mathbf{AB}} \equiv \left( {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {b_{11} } & {b_{12} } \\ {b_{21} } & {b_{22} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {a_{11} b_{11} + a_{12} b_{21} } & {a_{11} b_{12} + a_{12} b_{22} } \\ {a_{21} b_{11} + a_{22} b_{21} } & {a_{21} b_{12} + a_{22} b_{22} } \\ \end{array} } \right) \equiv {\mathbf{C}}. $$

The number of columns of the first factor must equal the number of rows of the second factor. Transposition is indicated by the superscript T and obeys the equation C ^T = B ^T A ^T:

$$ {\mathbf{B}}^{\text{T}} {\mathbf{A}}^{\text{T}} \equiv \left( {\begin{array}{*{20}c} {b_{11} } & {b_{21} } \\ {b_{12} } & {b_{22} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {a_{11} } & {a_{21} } \\ {a_{12} } & {a_{22} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {a_{11} b_{11} + a_{12} b_{21} } & {a_{21} b_{11} + a_{22} b_{21} } \\ {a_{11} b_{12} + a_{12} b_{22} } & {a_{21} b_{12} + a_{22} b_{22} } \\ \end{array} } \right) \equiv {\mathbf{C}}^{\text{T}} . $$

Vectors may be represented as matrices having just one column and n rows. Depending on the sequence of the factors, the result of a product of two vectors a, b is either a (1×1) matrix, namely the usual scalar product that is written in vector notation as $\left( {{\mathbf{ab}}} \right) = \left( {{\mathbf{ba}}} \right)$, and in matrix notation in the form,

$$ {\mathbf{a}}^{\text{T}} {\mathbf{b}} \equiv \left( {\begin{array}{*{20}c} {a_{1} } & {a_{2} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {b_{1} } \\ {b_{2} } \\ \end{array} } \right) = \left( {a_{1} b_{1} + a_{2} b_{2} } \right), $$

or a (n×n) matrix, the dyadic product, written in vector notation as $\left( {{\mathbf{a}} \otimes {\mathbf{b}}} \right)$, and in matrix notation in the form,

$$ {\mathbf{ab}}^{\text{T}} \equiv \left( {\begin{array}{*{20}c} {a_{1} } \\ {a_{2} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {b_{1} } & {b_{2} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {a_{1} b_{1} } & {a_{1} b_{2} } \\ {a_{2} b_{1} } & {a_{2} b_{2} } \\ \end{array} } \right). $$

In particular, the quadratic form of a dyadic product can be written as a product of scalar products, in matrix notation, ${\mathbf{x}}^{\text{T}} \left( {{\mathbf{ab}}^{\text{T}} } \right){\mathbf{y}} = \left( {{\mathbf{x}}^{\text{T}} {\mathbf{a}}} \right)\left( {{\mathbf{b}}^{\text{T}} {\mathbf{y}}} \right)$, or in vector notation, ${\mathbf{x}}\left( {{\mathbf{a}} \otimes {\mathbf{b}}} \right){\mathbf{y}} = \left( {{\mathbf{xa}}} \right)\left( {{\mathbf{by}}} \right)$.

The inverse of a transpose equals the transpose of the inverse, $\left( {{\mathbf{A}}^{\text{T}} } \right)^{ - 1} = \left( {{\mathbf{A}}^{ - 1} } \right)^{\text{T}}$. A matrix S is symmetric if it equals its transpose, ${\mathbf{S}}^{\text{T}} = {\mathbf{S}}$. The inverse of a symmetric matrix is symmetric. If a non-zero vector e obeys the equation ${\mathbf{Ae}} = \lambda {\mathbf{e}}$, e is termed an eigenvector and λ an eigenvalue of the quadratic matrix A. Eigenvalues of symmetric real matrices are real. The rank of a quadratic matrix is the number of its non-zero eigenvalues. Rayleigh’s theorem states that for any column vector x the quadratic form x ^T Sx satisfies the inequality $\lambda_{\min } \left( {\mathbf{x}}^{\text{T}} {\mathbf{x}} \right) \le {\mathbf{x}}^{\text{T}} {\mathbf{Sx}} \le \lambda_{\max } \left( {{\mathbf{x}}^{\text{T}} {\mathbf{x}}} \right)$ if λ _min, λ _max are the minimum and the maximum eigenvalue, respectively, of the symmetric matrix S.