Table of contents

The publication of the Guide to the Expression of Uncertainty in Measurement (GUM), and later of its Supplement 1, can be considered to be landmarks in the field of metrology. The second of these documents recommends a general Monte Carlo method for numerically constructing the probability distribution of a measurand given the probability distributions of its input quantities. The output probability distribution can be used to estimate the fixed value of the measurand and to calculate the limits of an interval wherein that value is expected to be found with a given probability. The approach in Supplement 1 is not restricted to linear or linearized models (as is the GUM) but it is limited to a single measurand.

In this paper the theory underlying Supplement 1 is re-examined with a view to covering explicit or implicit measurement models that may include any number of output quantities. It is shown that the main elements of the theory are Bayes' theorem, the principles of probability calculus and the rules for constructing prior probability distributions. The focus is on developing an analytical expression for the joint probability distribution of all quantities involved. In practice, most times this expression will have to be integrated numerically to obtain the distribution of the output quantities, but not necessarily by using the Monte Carlo method. It is stressed that all quantities are assumed to have unique values, so their probability distributions are to be interpreted as encoding states of knowledge that are (i) logically consistent with all available information and (ii) conditional on the correctness of the measurement model and on the validity of the statistical assumptions that are used to process the measurement data. A rigorous notation emphasizes this interpretation.

The fundamental concept of the mole requires the number of entities comprising one mole, i.e. Avogadro's number, to be exactly equal to the gram-to-dalton mass ratio. If this compatibility condition is to be satisfied, the mole, the kilogram and the dalton cannot all be defined independently. This note concerns recent Metrologia publications that do, however, propose independent definitions of all three quantities: the mole by fixing the value of Avogadro's number and the kilogram by fixing the value of the Planck constant, while retaining the current carbon-12-based dalton. Adoption of these incompatible definitions would likely cause serious widespread confusion and might even result in a split in scholarly and technical communication between the quantum physics and chemistry communities. Other entirely compatible alternatives are possible: either retaining the current (inexact) carbon-12-based mole and dalton with an independently redefined kilogram or redefining the mole by fixing the value of Avogadro's number, with a compatible dalton that is exact in terms of the redefined kilogram.

We propose a novel procedure for the analysis of key comparison data. The goal of the procedure is to detect biases in the reported measurement results which are not accounted for by quoted uncertainties. A fixed effects bias model is employed which constrains the biases of some of the laboratories to zero. Only the number of these laboratories needs to be specified, not the laboratories themselves. The analysis then runs through all possible different models, each assuming zero biases for a different subset of laboratories. The results from these models are finally merged by employing a Bayesian model averaging technique. Explicit formulae are derived which allow for an easy application of the proposed approach. The procedure is illustrated by its application to data from the CCL-K1 key comparison.

Metrologists often represent the state of knowledge concerning a quantity about which scant specific information is available by a rectangular probability distribution. The end points are frequently specified by subjective judgment; therefore, they are inexactly known. If the states of knowledge about the end points may be represented by other (narrower) rectangular distributions, then the resulting probability distribution looks like a trapezoid whose sloping sides are curved. We refer to such a probability distribution as curvilinear trapezoid. Depending on the limits of rectangular distributions for the end points, the curvilinear trapezoidal distribution may be asymmetric. In a previous paper we had shown that if the mid-point of a rectangular distribution is known and the state of knowledge about the half-width may be represented by a rectangular distribution then the resulting distribution is symmetric curvilinear trapezoid. In this paper, we describe the probability density function of a curvilinear trapezoidal distribution which arises from inexactly known end points. Then we give compact analytic expressions for all moments including the expected value and the variance. Next we discuss how random numbers from such a distribution may be generated. We compare the curvilinear trapezoid which arises from inexactly known end points with the corresponding trapezoid whose sloping sides are straight. We also compare the curvilinear trapezoid which arises from inexactly known end points with the curvilinear trapezoid which arises when the mid-point is known and the state of knowledge about the half-width may be represented by a rectangular distribution. The results presented in this paper are useful in evaluating uncertainty according to the Guide to the Expression of Uncertainty in Measurement (GUM) as well as Supplement 1 to the GUM (GUM-S1).

The recent third edition of the International Vocabulary of Metrology—Basic and General Concepts and Associated Terms (VIM3) (JCGM 200:2008 (Sèvres: BIPM); also ISO/IEC Guide 99:2007 3rd edn (Geneva: ISO)) has undergone important changes, not least by adhering to ISO International Standards on terminology work (ISO 704:2000 Terminology Work—Principles and Methods; ISO 1087-1:2000 Terminology Work—Vocabulary—Part 1: Theory and Application; ISO 10241:1992 International Terminology Standards—Preparation and Layout). A recent critique (Mari 2009 Metrologia46 L11–L15)—based on Object-Oriented Analysis—centres on the meaning and relation of the two first and fundamental concepts 'quantity'Single quotation marks ('...') or bold type indicate a concept when necessary, double quotation marks ("...") a term or quotation. and the new entry 'kind-of-quantity'. This makes it timely to analyse the two concepts, their relation and their respective role in forming the generic hierarchical concept system of VIM3 from 'property' to individual quantities. It is suggested that 'kind-of-quantity' acts as a division criterionSynonyms are "criterion of subdivision", "type of characteristic(s)", see the annexe..

Leak detection is widely used nowadays in various fields such as the automotive and refrigeration industries. The leak tightness of installations charged with refrigerants must be controlled periodically by refrigerant gas detectors, qualified by refrigerant leaks called reference leaks or 'calibrated' leaks. To this day, those refrigerant leaks are not traceable to the SI units under their operating conditions. Therefore, a project involving the LNE, the CEP and the ADEME was initiated in order to develop a national standard for calibrating refrigerant reference leaks.

The method consists in measuring the accumulation of gas due to the refrigerant leak inside a closed volume at atmospheric pressure. The leak flow rate is deduced from the concentration variation, measured with a photo-acoustic spectrometer, the pressure and temperature in the accumulation volume and the capacity of the accumulation volume. The uncertainty budget thus takes into account the measurement uncertainties of these quantities. The main components of the uncertainty are due to the accumulation volume capacity and the measurement of concentration. The relative expanded uncertainty of leak flow rates between 1 g yr⁻¹ and 50 g yr⁻¹ is around 2%, depending on the value of the leak.

An experiment is described for determining the atomic mass constant by accumulating ions from an ion beam up to a weighable mass. It establishes a link between the international prototype of the kilogram, the realization of the associated SI unit and an atomic mass. The items necessary for such an experiment are a high-current ion source, an ion optical system with high transmission, a suitable ion collector and a vacuum balance. With the most recent measurement, a mass of more than 320 mg of bismuth was accumulated and its atomic mass was determined with a relative standard uncertainty of 9.4 × 10⁻⁵. Special emphasis is placed on determining the mass loss of the accumulated ions caused through sputter effects in the ion collector.

Although work on this experiment at the PTB has now been stopped, we conclude with a number of suggestions that could lead to much smaller relative uncertainty in a future experiment.

This paper analyses two possible estimators for the modulus of a complex-valued quantity. Their expectations are here recalled from a previous work and their variances and mean square errors (MSEs) are analytically derived in the case of uncorrelated, homoskedastic and normally distributed measurements of the real and imaginary parts of the considered quantity. Numerical simulations are performed in the case of (uncorrelated) heteroskedastic measurements of the real and imaginary parts. The two estimators are compared by studying the behaviour of their variances and MSEs, and some guidance is given on the best choice between the two, depending on the specific experimental context. First and second-order approximations for their variance, hence for the uncertainty associated with the estimates they produce, are also provided in a more general case, that is, for correlated and heteroskedastic measurements of the real and imaginary parts. Examples from the metrological and the physical fields are given in order to compare the estimators on actual data.

A design of an automated cryogenic current comparator (CCC) resistance ratio bridge for routine measurements in a national metrology institute or standards laboratory is described. It employs a type II CCC for use in a low loss liquid helium storage vessel and can be continuously operated from the mains power via specialized isolated supplies. All parameters including the servo control loops are digitally controlled. Noise sources in the system are analysed using the Allan deviation and it is demonstrated that non-white noise sources can be eliminated by choosing an appropriate current reversal rate.

Measurements of the refractive index of two different Lithosil samples and a sample of Suprasil 312 at cryogenic temperature and at 293 K are reported for the spectral range from 480 nm to 894 nm. Such data are useful for the design of fused silica optical components and systems destined for space missions.

The main aim of this paper is to calculate the small but significant amount of heat lost from a graphite calorimeter absorber through connecting thermistor wires during electrical calibration. Taking into account the electro-thermal interaction between the heating thermistor and its surrounding environment, a more realistic approach to the problem was developed and estimative numerical results were obtained. It was found that the wires contribute about 0.01% in extracting heat from the calorimeter core (which corresponds to a correction factor k_w^core = 0.9999). The total correction factor for heat loss through the connecting thermistor wires during the electrical calibration of the calorimeter (the total combined effect of the heater and the sensor leads due to conduction, radiation and Joule effect) was determined: k_w = 0.9989.

A second-generation standard hygrometer has been completed at the National Institute of Standards and Technology (NIST). This hygrometer measures humidity using a gravimetric method: it separates the water from the carrier gas and afterwards measures the water mass and carrier gas mass. These two measurements determine the mass ratio r (the ratio of the measured water mass to the measured dry-gas mass). The new design allows automated continuous gas collection at up to 3 L min⁻¹. This enables the hygrometer to collect larger amounts of gas and thereby measure humidity values lower than that measured by the previous NIST standard hygrometer. When operated in an optimal thermal environment (minimal thermal loads in the laboratory), the total expanded relative uncertainty (k = 2) of the gravimetric hygrometer is approximately 0.1% for atmospheric-pressure frost points higher than −35 °C (r = 250 µg g⁻¹). Below this frost point the total expanded relative uncertainty gradually increases to approximately 1% at −55 °C (r = 13 µg g⁻¹). The hygrometer has measured the humidity of gas samples produced by the NIST Hybrid Generator and the NIST Low Frost-Point Generator with dew/frost points from −35 °C to 71 °C. For both generators the differences between the humidity generated and the humidity measured by the gravimetric hygrometer are less than the combined uncertainties of the generator and the hygrometer.

This paper presents a practical ultrasonic system for near real-time imaging of spatial temperature distributions in water caused by absorption of radiation. Initial testing with radiation from a highly attenuated infrared lamp demonstrates that the system is able to map sub-millikelvin temperature changes, thus making it suitable for characterizing dose profiles of therapy-level ionizing radiation beams. The system uses a fan-beam tomographic reconstruction algorithm to invert time-of-flight data derived from ultrasonic pulses produced and detected by a circular array of transducers immersed in water. Temperature dependence of the speed of sound in water permits the conversion of these measured two-dimensional velocity distributions into temperature distributions that indicate the absorbed radiation dose. The laboratory prototype, based on a 128-element transducer array, is used to acquire temperature maps of a 230 mm × 230 mm area every 4 s with sub-millikelvin resolution in temperature and about 5 mm resolution in space. Earlier measurements with a single-channel version of this prototype suggest refinements in signal-conditioning electronics and signal-processing algorithms that would allow the present instrument to resolve temperature changes as low as a few microkelvin. Possible applications include real-time intensity profiling of radiation beams and three-dimensional characterization of the absorbed dose.

The paper introduces a distinction between true metric and parametric quantities, units and dimensions. It claims in its first part that these distinctions shed important light both on the base quantity amount of substance and VIM's definitions of kind of quantity and quantity dimension. The second part is devoted to the unit one, and it claims that this unit must be regarded as a parametric unit, but also that it is not as needed as has been thought. The third part takes for granted that both the mole and the unit one are parametric units, and it argues that, for pedagogical reasons, the mole should be exchanged for the unit one, and the parametric quantity amount of substance be renamed as 'elementary entities'.

An indirect determination of the thermodynamic temperature of the fixed point of copper was made at INRIM by measuring four cells with a Si-based and an InGaAs-based precision radiation thermometer carrying approximated thermodynamic scales realized up to the Ag point. An average value T_Cu = 1357.840 K was found with a standard uncertainty of 0.047 K. A consequent (T − T₉₀)_Cu value of 70 mK can be derived which is 18 mK higher than, but consistent with, the presently available (T − T₉₀)_Cu as elaborated by the CCT-WG4.

We have developed a blackbody comparison system to validate the fixed-point realization of the thermocouple temperature scale using the radiation thermometry scale. Platinum/palladium, type S and type B thermocouples were calibrated at the fixed points of Ag, Cu, Fe–C, Co–C and Pd (only for type B). The radiation thermometer used was an LP4 linear pyrometer operating at a central wavelength of 650 nm. To assign the Fe–C eutectic melting temperature in our laboratory, a radiometric Fe–C cell was fabricated, and its melting temperature was determined as (1154.0 ± 0.1) °C with k = 2. Two scales were compared from 962 °C to 1555 °C in the blackbody comparison system. The two scales were consistent within 0.5 °C up to 1400 °C, but the discrepancy for the type B thermocouple increased to 2.3 °C at 1554.8 °C. The thermocouple and radiometric scales realized at our laboratory were in agreement up to the freezing temperature of Pd within the measurement uncertainty.

We have measured the heat flux accompanying the melting or freezing of metal (or metalloid)–carbon eutectics, using differential scanning calorimetry (DSC) to identify appropriate binary systems for secondary thermometry fixed points. Well-known alloy systems such as Fe–C and Ni–C showed reproducible endothermic and exothermic peaks that represent melting and freezing reactions in the DSC measurement. Furthermore, a new Si–C system with a eutectic composition showed reproducible melting and freezing peaks in the DSC measurements. Based on the results by DSC, we identified the Si–SiC eutectic point as a possible eutectic fixed point. To confirm this possibility, we made a Si–SiC cell for thermocouple thermometry and measured its melting and freezing characteristics using a Pt/Pd thermocouple. The melting temperature of the Si–SiC eutectic was reproducible to within 0.02 °C (one standard deviation). From the results, we found that Si–SiC has possibility as a new eutectic fixed point at temperatures around 1400 °C. We also concluded that DSC analysis could be used to measure the reproducibility of freezing and melting reactions that are to be used as fixed points for thermometry, because it is a rapid and easy-to-use tool for characterizing the thermal behaviour of materials with only a small sample.

The measurement of layer thickness by compositional secondary ion mass spectrometry (SIMS) depth profiling is investigated for Si/Ge multilayer films using an oxygen ion beam. The original SIMS depth profiles were converted into compositional depth profiles by the relative sensitivity factors of Si and Ge derived from a Si_52.4Ge_47.6 alloy reference film. The locations of the interfaces in the Si/Ge multilayer films could be well determined by 50 at% definition where the relative composition of the constituent layer elements drops or rises to 50 at%. The layer thicknesses of Si and Ge of a test Si/Ge multilayer film were determined by the sputtering rates of Si and Ge determined from a reference Si/Ge multilayer film. Although the difference between the measured and the actual thicknesses is increased as the ion energy is increased, the layer thicknesses determined at low ion energies were very close to the actual values.

Applying the Monte Carlo method for propagation of measurement uncertainty described in the Supplement 1 to the Guide to the Expression of Uncertainty in Measurement (GUM), when the input quantities are correlated, involves the specification of a joint probability distribution for these quantities. This applies equally whether the output quantity is a scalar or a vector.

In practice, however, all that typically is available are probability distributions for the individual input quantities (their marginal distributions) and estimates of the correlations between them. Even though there are infinitely many joint distributions that are consistent with given marginal distributions and correlations, a method is needed to manufacture a particular one that may reasonably be used in practice. This paper explains how copulas may be used to this effect, illustrates their use in examples, including example H.2 from the GUM, discusses the choice of copula and provides an algorithm to delineate minimum volume coverage regions for vectorial measurands.

Johnson noise thermometers (JNT) measure the equilibrium electrical noise, proportional to thermodynamic temperature, of a sensing resistor. In the correlation method, the same resistor is connected to two amplifiers and a correlation of their outputs is performed, in order to reject amplifiers' noise. Such rejection is not perfect: the residual correlation gives a systematic error in the JNT reading. In order to set an upper limit, or to achieve a correction, for such error, a careful electrical modelling of the amplifiers and connections must be performed. Standard numerical simulation tools are inadequate for such modelling. In the literature, evaluations have been performed by the painstaking solving of analytical modelling. We propose an evaluation procedure for the JNT error due to residual correlations which blends analytical and numerical approaches, with the benefits of both: a rigorous and accurate circuit noise modelling, and a fast and flexible evaluation with a user-friendly commercial tool. The method is applied to a simple but very effective ultralow-noise amplifier employed in a working JNT.

We have developed a set of seven observational equations that include all of the physics necessary to relate the most important of the fundamental constants to the definitions of the SI kilogram and ampere. We have used these to determine the influence of alternative definitions currently under consideration for the SI kilogram and ampere on the uncertainty of three of the fundamental constants (h, e and m_u). We have also reviewed the experimental evidence for the exactness of the quantum metrology triangle resulting from experiments combining the quantum Hall effect, the Josephson effects and single-electron tunnelling.

This paper describes precision measurement methods developed at the Laboratoire national de métrologie et d'essais (LNE) for the calibration of flickermeters. Three methods are compared: one method involves calculating the rms value of each ac voltage period of the modulated signal in the time domain whereas the other methods are based on spectral analysis. The absolute standard uncertainty (1σ) is estimated to be a few parts in 10⁵ for all methods.

The basis for the description of diffuse reflecting materials is the concept of the radiance factor β, or as a quite similar depiction, the bidirectional reflectance distribution function f_r. Both characterizations use the concept of the perfectly reflecting diffuser (PRD), which reflects, by definition, the incoming radiation loss-free, completely diffuse and with Lambertian direction characteristics. The PRD is a theoretical concept only, which cannot be realized materially. Since there is no material with these characteristics, the realization is carried out with physical methods, i.e. by the measuring apparatus itself, in the context of an absolute measurement. For practical purposes, radiance factor measurements are predominantly accomplished relative to commercially available reflection standards. In the present investigation, different widely used diffuse reflection materials were measured for the first time in a multi-geometry configuration with the robot-based gonioreflectometer of the Physikalisch-Technische Bundesanstalt (PTB) in order to characterize their three-dimensional reflection behaviour. For a set of four distinct incident angles, the full hemispherical reflection indicatrix was determined at a wavelength of 550 nm.

The angle-resolved reflection data are an important reference for manufacturers, providers and users of radiometric and photometric products. This paper attempts to give users in research and industry an overview of the strongly non-Lambertian reflection behaviour of standard reflection materials, because it is a widespread false assumption that commonly used standard reflection materials have only minor deviations from the ideal specification of the PRD.

This paper presents a time transfer method based on a least-square analysis of the GPS code and carrier-phase measurements in common view (CV), constrained by TWSTFT (TW hereafter) data. Combining GPS and TWSTFT considerably increases the robustness of the remote clock comparisons thanks to the complete independence between these two techniques. An additional goal of the combination (named CV + TW) is to provide a time transfer solution, which benefits from the high short-term stability and high resolution of the GPS data and from the high accuracy of the TW data. The final solution is then calibrated by the TW data rather than by the GPS pseudoranges. As a consequence, the large day-boundary discontinuities that can exist in GPS time transfer solutions disappear in the combined solution, and are replaced by smaller discontinuities the magnitude of which depends on the noise level of the TW link. For a very noisy link, we found jumps up to 1 ns, while for the very quiet link NIST–OP, they are all lower than 200 ps.

Adaptive Monte Carlo schemes can be used to determine the number of Monte Carlo trials (the number of evaluations of the measurement model) necessary for the evaluation of uncertainty according to Supplement 1 to the GUM (GUM S1). The goal is to reach a prescribed numerical accuracy of the Monte Carlo results (the estimate, associated standard uncertainty and coverage interval endpoints) for a chosen confidence level. It is shown that simple sequential adaptive Monte Carlo schemes may not perform well in this regard and an alternative method based on a two-stage procedure due to Stein is proposed. The implementation of this two-stage scheme for GUM S1 is described, and its performance and robustness are demonstrated in terms of simulation results.

The range 3.0 K to 24.5561 K of the International Temperature Scale of 1990 (ITS-90) was realized at VSL with a ⁴He interpolating constant-volume gas thermometer (ICVGT). The standard uncertainty of the generated scale ranged from 0.32 mK at 3.3 K to 0.38 mK at 24.5561 K. The ICVGT scale was compared with the classical absolute gas thermometer scales of the National Physical Laboratory (NPL-75, 1975) and the Kamerlingh Onnes Laboratory (KOL, 1986) and was found to deviate from them by a maximum of 1.5 mK. The non-uniqueness of the generated scale was investigated by comparing it with alternative ICVGT scales (but having equal status because still satisfying all the requirements of the ITS-90) obtained from different choices of the lowest calibration temperature and interpolation function. The size of correction effects such as thermo-molecular pressure, aerostatic head pressure and dead volume was calculated and the effectiveness of the ICVGT calibration procedure in compensating for such correction effects was evaluated.

Current recommendations for evaluating uncertainty of measurement are based on the Bayesian interpretation of probability distributions as encoding the state of knowledge about the quantities to which those distributions refer. Given a measurement model that relates an output quantity to one or more input quantities, the distribution of the former is obtained by propagating those of the latter according to the axioms of probability calculus and also, if measurement data are available, by applying Bayes' theorem.

The main objective of this paper is to show that alternative ways of applying Bayes' theorem are possible, and that these alternative formulations yield the same results provided consistent use is made of measurement data and prior information. In this context the necessity of assigning non-informative priors arises often. Therefore, the second concern of the paper is to point out, by means of a specific example, that the seemingly reasonable choice of a uniform prior for a quantity about which no information is available may not conform to the accepted rules for constructing non-informative priors.

A recent paper (Baumann et al2009 Metrologia46 178–86) presents a method to evaluate the free-fall acceleration at a desired point in space, as required for the watt balance experiment. The claimed uncertainty of their absolute gravity measurements is supported by two bilateral comparisons using two absolute gravimeters of the same type. This comment discusses the case where absolute gravity measurements are traceable to a key comparison reference value. Such an approach produces a more complete uncertainty budget and reduces the risk of the results of different watt balance experiments not being compatible.

The method of uncertainty evaluation discussed in Supplement 1 to the Guide to the Expression of Uncertainty in Measurement generates a coverage interval in which the measurand is said to have a certain probability (the coverage probability) of lying. This communication contains a response to the recent claim that 'when a coverage interval summarizes the resulting state of knowledge, the coverage probability should not be interpreted as a relative frequency of successful intervals in a large series of imagined or simulated intervals' (Lira 2009 Metrologia46 616–8). First, Bernoulli's law of large numbers is used to prove that the long-run success rate of a methodology used to calculate 95% coverage intervals must be 95%. Second, the usual definition of subjective probability or 'degree of belief' is stated, and the weak law of large numbers is then used to show that this definition—and the corresponding definition of 'state of knowledge'—relies on the concept of long-run behaviour. This provides an alternative proof of the same result.

In 2009 Metrologia46 616–8 it is claimed that a coverage probability associated with an interval derived from a state-of-knowledge probability distribution should not be interpreted as a relative frequency of successful intervals in a large series of imagined or simulated intervals. In his comments, Willink affirms that this claim is untenable. This reply is limited to a summary restatement of the two examples used to support the above assertion.