The International System of Units, SI, in use since 1946, was formally established in October 1960 by the eleventh Conférence Générale des Poids et Mesures, CGPM, with its resolution 12. In the past years, several changes have been made to the system. The 26th Conference, on 16th November 2018, adopted a revised SI, due to come into force on 20 May 2019. This revision was by far the most radical in the history of the SI. In this paper, I review the system from its origin to the present time, discuss the needs that suggested, and the conditions that allowed such an epochal change, and present the SI of the third millennium.

There is much confusion on the topic of uncertainty of measurement. Yet, measurement uncertainty is both a pivotal concept in measurement theory and, above all, a basic requisite in practice, from the physics laboratory measuring some exotic property of Nature to the shop floor. The purpose of this paper is to give the author's view on measurement uncertainty in an unambiguous way, thus privileging clarity over diplomacy, for which he apologizes once and for all. Accordingly, the scope of the paper is to discuss the fundamental metrological concepts and associated terms, as given in the International Vocabulary of Metrology, VIM, in the light of their relevance to the topic of uncertainty, as treated in the Guide to the expression of uncertainty in measurement, GUM. In this scheme, the focus is on the concepts of error and uncertainty and on their intimate connection, often masked by misunderstanding when not buried under the misconception that they are opposite and competing concepts. It will be shown that probability theory is the correct framework in which error and uncertainty are reconciled in a convenient and rigorous way. The author is convener of the Joint Committee for Guides in Metrology (JCGM) Working Group 1 (GUM). The opinion expressed in this paper does not necessarily represent the view of this Working Group.

Absolute gravity measurements are based on the reconstruction of the free-falling motion of a test body in vacuum. In this paper, two large disturbing effects are studied, namely the non-gravitational accelerations originated by rotation and translation of the flying body. Their contribution to the uncertainty of the free-fall acceleration is evaluated using the Monte Carlo method as proposed in Supplement 1 to the GUM. The analysis is specifically applied to the IMGC-02 absolute gravimeter, but can be easily extended to other instruments, including cold-atom gravimeters currently under development.

Absolute ballistic gravimeters can measure the free-fall acceleration with an uncertainty of few parts in 10(9). Typically, the vertical trajectory of a test body subjected to the gravity field is tracked using interferometric methods, and a mathematical model of the motion is fitted to the time-position coordinates in a least-squares adjustment. In this paper, we describe a non-linear regression analysis applied to the IMGC-02 transportable gravimeter, developed at the Istituto Nazionale di Ricerca Metrologica (INRIM). We show how the method yields an accurate estimate of the free-fall acceleration avoiding measurement of the vertical gradient.

The forthcoming Supplement 1 to the GUM: Numerical methods for the propagation of distributions proposes the use of Monte Carlo simulation for uncertainty evaluation. Here we apply a modified implementation of the proposed Monte Carlo Simulation to construct intervals of confidence for a complex-valued measurand. In particular, we analyze the so-called three-voltage method for impedance calibration, which relates complex-valued impedances to voltage moduli measurements. We compare and discuss the results obtained with those given by application of Bootstrap Resampling on the same model.

The techniques used for generating multiples and submultiples of the kilogram are reviewed, and their historical evolution is outlined. Emphasis is given to estimation of the values of the measurands, in connection with the need for prior knowledge about them. This need has deep motivation and implies the introduction of constraints on their values. The alternatives of deterministic constraints, leading to the Lagrangian multipliers method, and uncertain constraints, leading to the minimum variance estimator, are discussed, as well as the mathematical relationship between the two methods.

Despite the greater attention given in recent years to covariances and correlations in metrology, they still seem to be undervalued. Although in the recent ISO Guide to the expression of uncertainty in measurement [1] covariances between input estimates are treated to some extent, no mention is made of their propagation to output estimates in the multivariate case. Undoubtedly, the topic is considered difficult, probably because of the matrix formalism often used in its treatment. The present communication describes a simple approach, which is based on partial derivatives and thus leads to a varianceand covariancepropagation law analogous to, but more general than, the well-known law of propagation of variances. The law described is used in a simple example showing that correlations between output estimates may reach large values even when input quantities are uncorrelated. No matrix notation is used: for a matrix approach, see, for example, [2].

To evaluate uncertainty in mass measurements with accuracy and in accordance with recent international documents, a general model is developed, which takes account of the various contributing quantities in a multivariate context. On this basis, the variance-covariance matrix of the in-vacuo mass differences is constructed in its general form and tailored for application to some of the most commonly adopted weighing methods. The usual assumption of equal-variance, uncorrelated observations is shown to be inappropriate for mass comparisons.

Comparison calibration designs are rank insufficient to permit a purely batch estimation of parameters, and require the input of some a priori knowledge. The estimator commonly used, which is based on Restrained Least Squares, is shown to be inappropriate because of its inability to take account of uncertainty in the a priori knowledge. A model in which the parameters represent the state of a dynamic stochastic system is proposed together with the appropriate recursive estimator, namely the Kalman filter-predictor. This estimator is less affected than Restrained Least Squares by errors in the a priori knowledge. Its application in mass comparisons in conjunction with a fully recursive approach, i.e. with use of all the available a priori knowledge, is discussed.

An intercomparison of standards of mass of 50 g and 10 g was carried out between the national standards laboratories of four European countries during the period May 1988 to July 1989. The transfer standards used were carefully selected by the pilot laboratory (NPL) and the comparison scheme was chosen to minimise the influence of any instability in their mass. The results obtained show good agreement to within the uncertainties of measurement of the participants, the widest variation in mass value between any two participants being 8 μg at 50 g and 3 μg at 10 g.

The use of Linear, Ordinary Least-Squares Estimation in mass metrology leads to singular normal equations. In the classic approach the singularity is avoided by using the method of the Lagrange Multipliers. A Gauss-Markov approach is suggested instead and its effects on the estimates and on their covariance matrix are discussed. A practical application is presented.