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.

http://iopscience.iop.org/article/10.1088/0026-1394/53/4/1079/pdf

R. Ludwig, Methoden der Fehler- und Ausgleichsrechnung. VIII + 259 S. m. 29 Abb. Braunschweig 1969. Friedr. Vieweg & Sohn. [Preis brosch. DM 16,80] Berlin 1969. Deutscher Verlag der Wissenschaften. Preis brosch. M 16,80

Uncertainty of empirical correlation equations

Evaluation and calibration in quantitative thin-layer chromatography

Zum Abschatzen der in brenngeschnittenen Blechproben in der Brennschneidzone vorhandenen Eigenspannungen wurden je vier Proben aus Armco-Eisen und aus Stahl St 60-1, die man mit verschiedenen Schneidbedingungen hergestellt hatte, nach dem Biegepfeilmesverfahren untersucht. Fur die umfangreiche numerische Auswertung der Meswerte stand eine programmgesteuerte Ziffernrechenanlage zur Verfugung. Zur Deutung der ermittelten Randspannungen wurden Hartemessungen herangezogen. Fur die Stahlproben lies sich ein Einflus der Heizflammenlange und der Schneidgeschwindigkeit auf die Spannungen in der Brennschneidflache nachweisen. Auserdem lieferte die Untersuchung fur jede Probe die durch Bearbeitung der Brennschneidflache hervorgerufenen Spannungsumlagerungen fur unterschiedliche Abarbeitungstiefen.

Bestimmung der durch Brennschneiden von Stahlblechen verursachten Eigenspannungen

Positioning technology can be organized by the underlying geometric principles such as triangulation, dead reckoning, or presence detection which can be further subdivided. Alternatively, the algorithms can be organized along observable variables (time, time difference, angle-of-arrival, angle-of-emission, signal strength, acceleration, rotation, etc.). This chapter provides basic algorithms for positioning organized along the underlying geometric principles. After explaining the basic algorithms, a description of real-world approaches is given, organized by the employed sensor technology and observable variables.

Basic Positioning Techniques

The linear observation equation is usually expressed as

$$ {\user2{l}} = {\user2{Ax}} + {\user2{e}} $$

(61)

where the non-random design matrix $ {\user2{A}} \in R^{m \times n} , $ the vector of unknown parameters $ {\user2{x}} \in R^{n \times 1} , $ the vector of measurements $ {\user2{l}} \in R^{m \times 1} $and contaminated by random error vector $ {\user2{e}} $ with zero mean and variance–covariance matrix $ \sigma_{0}^{2} {\user2{P}}^{ - 1} , $ where P is the weight matrix and $ \sigma_{0}^{2} $ is the variance of unit weight If the coefficient matrix A of the observation equation possesses very large condition number, the observation equation is ill-conditioned, which is defined as ill-posed problems by Hadamard (1932) In geodesy ill-posed problems are frequently encountered in satellite gravimetry due to downward continuation, or in geodetic date procession due to the colinearity among parameters that are to be estimated Most useful and necessary adjustment algorithms for data processing are outlined in the second part of this chapter The adjustment algorithms discussed here include least squares adjustment, sequential application of least squares adjustment via accumulation, sequential least squares adjustment, conditional least squares adjustment, a sequential application of conditional least squares adjustment, block-wise least squares adjustment, a sequential application of block-wise least squares adjustment, a special application of block-wise least squares adjustment for code-phase combination, an equivalent algorithm to form the eliminated observation equation system and the algorithm to diagonalize the normal equation and equivalent observation equation A priori constrained adjustment and filtering are discussed for solving the rank deficient problems After a general discussion on the a priori parameter constraints, a special case of the so-called a priori datum method is given A quasi-stable datum method is also discussed A summary is given at the end of this part of the chapter

R. Ludwig, Methoden der Fehler- und Ausgleichsrechnung. VIII + 259 S. m. 29 Abb. Braunschweig 1969. Friedr. Vieweg & Sohn. [Preis brosch. DM 16,80] Berlin 1969. Deutscher Verlag der Wissenschaften. Preis brosch. M 16,80

Citations

Uncertainty of empirical correlation equations

Evaluation and calibration in quantitative thin-layer chromatography

Bestimmung der durch Brennschneiden von Stahlblechen verursachten Eigenspannungen

Basic Positioning Techniques

Regularization and Adjustment

Related Papers (1)

Methoden der Fehler- und Ausgleichsrechnung