Main text This report is the final report for key comparison CCAUV.W-K2. This key comparison covers primary free-field standards for sound in water at frequencies between 250 Hz and 500 kHz. This project is one of the key comparisons organised under the auspices of the Consultative Committee on Acoustics, Ultrasound and Vibration (CCAUV) of the CIPM. This report has the status of a Final Report and has been submitted to the Key Comparison Database (KCDB). In the report, the results of participants are presented with the Key Comparison Reference Values and Degrees of Equivalence. The results are calculated according to the procedures agreed after consideration of the Draft A1 and A2 reports, and the Draft B report has been approved by the CCAUV. All participants have had the opportunity to give final agreement on the contents and amendments have been made to account for their comments. In many respects, the comparison has been a success with good agreement achieved over an extended lower frequency range compared to the previous CCAUV.W-K1 comparison, the lower frequency limit for CCAUV.W-K2 being extended down by two octaves to 250 Hz. The generally more difficult frequency range from 100 kHz to 500 kHz has also shown very good agreement between the participants. However, in the range 60 kHz to 100 kHz the agreement was not as good, with three participants exhibiting some discrepant results. To reach the main text of this paper, click on Final Report . Note that this text is that which appears in Appendix B of the BIPM key comparison database https://www.bipm.org/kcdb/ . The final report has been peer-reviewed and approved for publication by the CCAUV, according to the provisions of the CIPM Mutual Recognition Arrangement (CIPM MRA).

The European Metrology Network for Mathematics and Statistics (MATHMET) is creating a Quality Management System (QMS) to ensure that research outputs in the forms of data, software and guidelines are fit-for-purpose, achieve a sufficient level of quality, and are consistent with the aims of National Measurement Institutes to provide quality-assured and trusted outputs. The essential components of the QMS for all three forms of research output are discussed. On-line, interactive risk assessment tools that guide a user through the process of assigning an integrity level for the research outputs of data and software, are described. Examples of case studies that have been used to demonstrate the QMS are indicated.

The systems and control theory was developed since before the 19th century and provides the necessary mathematics for the analysis of dynamic measurement uncertainty, which is a new concept needed for the development of the future technology. The knowledge of the theory of systems and control is based on disciplines such as mathematics, statistics and signal processing. The purpose of the present study is to compile this technical knowledge to develop methods for the metrological reliability of dynamic quantities through uncertainty analysis and, as a complement, to unify the concepts with the measurement science.

A new method for performance validation of surface texture parameter calculation software is introduced, focussing on field surface texture parameters. Surface height functions are defined mathematically, either using Fourier series or polynomials, and are then input into the surface texture parameter definitions to obtain mathematical parameter values. A series of user-adjustable parametric surface functions are defined that correspond to each surface texture parameter, enabling users to create a variety of surfaces to assess their software whilst still retaining mathematical traceability. This method is expanded to include complex surface textures. Chebyshev polynomials are used to perform numerical calculations of surface texture parameters for a selection of polynomial surface functions. Mathematical reference parameter values are calculated for a series of fifteen predefined surfaces and ten parametric surfaces to assess the performance of the software under test for a given dataset resolution. Assessment of the number of significant figures of the software-obtained values that agree with the reference values is used as a performance metric that enables comparison between different third-party software applications for a given dataset resolution. An assessment of the sampling methods used to create discrete datasets of a mathematical surface function for use with numerical third-party software is performed. Two implementations of surface height sampling are used to create datasets that are input into four third-party surface texture parameter calculation software packages, and the results compared, showing a significant variation in the performance metric values for different sampling methods.

Abstract A new method for performance validation of surface texture parameter calculation software is introduced, focussing on field surface texture parameters. Surface height functions are defined mathematically, either using Fourier series or polynomials, and are then input into the surface texture parameter definitions to obtain mathematical parameter values. A series of user-adjustable parametric surface functions are defined that correspond to each surface texture parameter, enabling users to create a variety of surfaces to assess their software whilst still retaining mathematical traceability. This method is expanded to include complex surface textures. Chebyshev polynomials are used to perform numerical calculations of surface texture parameters for a selection of polynomial surface functions. Mathematical reference parameter values are calculated for a series of fifteen predefined surfaces and ten parametric surfaces to assess the performance of the software under test for a given dataset resolution. Assessment of the number of significant figures of the software-obtained values that agree with the reference values is used as a performance metric that enables comparison between different third-party software applications for a given dataset resolution. An assessment of the sampling methods used to create discrete datasets of a mathematical surface function for use with numerical third-party software is performed. Two implementations of surface height sampling are used to create datasets that are input into four third-party surface texture parameter calculation software packages, and the results compared, showing a significant variation in the performance metric values for different sampling methods.

By using a simple model to relate the electromotive force drift rate of Pt–Rh thermoelements to dS/dc, i.e. the sensitivity of the Seebeck coefficient, S, to rhodium mass fraction, c, the composition of the optimal pair of Pt–Rh wires that minimizes thermoelectric drift can be determined. The model has been applied to four multi-wire thermocouples each comprising 5 or 7 Pt–Rh wires of different composition. Two thermocouples were exposed to a temperature of around 1324 °C, one thermocouple to around 1492 °C, i.e. the melting points of the Co–C and Pd–C high temperature fixed points, respectively, and one thermocouple to a series of temperatures between 1315 °C and 1450 °C. The duration of exposure at each temperature was several thousand hours. By performing repeated calibrations in situ with the appropriate fixed point during the high temperature exposure, the drift performance has been quantified with high accuracy, entirely free from errors associated with thermoelectric homogeneity. By combining these results it is concluded that the Pt-40%Rh versus Pt-6%Rh is the most stable at the temperatures investigated. A preliminary reference function was determined and is presented.

Software for the evaluation of areal surface texture function parameters is described. Definitions of the parameters, expressed in terms of the inverse areal material ratio function, are provided along with details of the numerical algorithms employed in the software to implement calculations to evaluate approximations to the parameters according to those definitions. Results obtained using the software to process a number of data sets representing different surfaces are compared with those returned by proprietary software for surface texture measurement. Differences in the results, arising from different choices being made when implementing the steps in the parameter evaluation process, are discussed.

Recent advances in primary acoustic gas thermometry (AGT) have revealed significant differences between temperature measurements using the International Temperature Scale of 1990, \(T_{90}\), and thermodynamic temperature, T. In 2015, we published estimates of the differences \((T-T_{90})\) from 118 K to 303 K, which showed interesting behavior in the region around the triple point of water, \(T_\mathrm{TPW}=273.16\) K. In that work, the \(T_{90}\) measurements below \(T_\mathrm{TPW}\) used a different ensemble of capsule standard platinum resistance thermometers (SPRTs) than the \(T_{90}\) measurements above \(T_\mathrm{TPW}\). In this work, we extend our earlier measurements using the same ensemble of SPRTs above and below \(T_\mathrm{TPW}\), enabling a deeper analysis of the slope \(\mathrm{d}(T-T_{90})/\mathrm{d}T\) around \(T_\mathrm{TPW}\). In this article, we present the results of seven AGT isotherms in the temperature range 258 K to 323 K. The derived values of \((T-T_{90})\) have exceptionally low uncertainties and are in good agreement with our previous data and other AGT results. We present the values \((T-T_{90})\) alongside our previous estimates, with the resistance ratios W(T) from two SPRTs which have been used across the full range 118 K to 323 K. Additionally, our measurements show discontinuities in \(\mathrm{d}(T-T_{90})/\mathrm{d}T\) at \(T_\mathrm{TPW}\) which are consistent with the slope discontinuity in the SPRT deviation functions. Since this discontinuity is by definition non-unique, and can take a range of values including zero, we suggest that mathematical representations of \((T-T_{90})\), such as those in the mise en pratique for the kelvin (Fellmuth et al. in Philos Trans R Soc A 374:20150037, 2016. doi: 10.1098/rsta.2015.0037), should have continuity of \(\mathrm{d}(T-T_{90})/\mathrm{d}T\) at \(T_\mathrm{TPW}\).

Using exceptionally accurate measurements of the speed of sound in argon, we have made estimates of the difference between thermodynamic temperature, T, and the temperature estimated using the International Temperature Scale of 1990, T-90, in the range 118K to 303 K. Thermodynamic temperature was estimated using the technique of relative primary acoustic thermometry in the NPL-Cranfield combined microwave and acoustic resonator. Our values of (T - T-90) agree well with most recent estimates, but because we have taken data at closely spaced temperature intervals, the data reveal previously unseen detail. Most strikingly, we see undulations in (T - T-90) below 273.16 K, and the discontinuity in the slope of (T - T-90) at 273.16K appears to have the opposite sign to that previously reported.

The determination of thermal diffusivity is at the heart of modern materials characterisation. The evaluation of the associated uncertainty is difficult because the determination is performed in an indirect way, in the sense that the thermal diffusivity cannot be measured directly. The well-known GUM uncertainty framework does not provide a reliable evaluation of measurement uncertainty for such inverse problems, because in that framework the underlying measurement model is supposed to be a direct relationship between the measurand (the quantity intended to be measured) and the input quantities on which the measurand depends. This paper is concerned with the development of a Bayesian approach to evaluate the measurement uncertainty associated with thermal diffusivity. A Bayesian model is first developed for a single thermogram and is then extended to the case of several thermograms obtained under repeatability and reproducibility conditions. This multi-thermogram based model is able to take into consideration a large set of influencing quantities that occur during the measurements and yields a more reliable uncertainty evaluation than the one obtained from a single thermogram. Different aspects of the Bayesian model are discussed, including the sensitivity to the choice of the prior distribution, the Metropolis-Hastings algorithm used for the inference and the convergence of the Markov chains.

A turbine flow meter indicates the volume of fluid flowing through the device per unit of time. Such a flow meter is commonly calibrated at a few known flow rates over its measurement range. A calibration curve relating the pulse factor of the meter to the flow rate is then fitted to calibration data using an ordinary least squares approach. This approach does not consider prior knowledge that may exist about the flow meter or the calibration procedure. A Bayesian analysis enables prior knowledge to be taken into account. A Bayesian inference results in a posterior distribution for the unknown parameters of the calibration curve that may be seen as the most comprehensive uncertainty information about these unknowns. This paper investigates for a flow meter calibration problem the effects of appreciating prior knowledge on values of the calibration curve and their associated uncertainties. It presents the results of a Bayesian analysis and compares them to those obtained by an ordinary least squares approach.

The 'Guide to the Expression of Uncertainty in Measurement' (GUM) requires that the way a measurement uncertainty is expressed should be transferable. It should be possible to use directly the uncertainty evaluated for one measurement as a component in evaluating the uncertainty for another measurement that depends on the first. Although the method for uncertainty evaluation described in the GUM meets this requirement of transferability, it is less clear how this requirement is to be achieved when GUM Supplement 1 is applied. That Supplement uses a Monte Carlo method to provide a sample composed of many values drawn randomly from the probability distribution for the measurand. Such a sample does not constitute a convenient way of communicating knowledge about the measurand. In this paper consideration is given to obtaining a more compact summary of such a sample that preserves information about the measurand contained in the sample and can be used in a subsequent uncertainty evaluation. In particular, a coverage interval for the measurand that corresponds to a given coverage probability is often required. If the measurand is characterized by a probability distribution that is not close to being Gaussian, sufficient information has to be conveyed to enable such a coverage interval to be computed reliably.A quantile function in the form of an extended lambda distribution can provide adequate approximations in a number of cases. This distribution is defined by a fixed number of adjustable parameters determined, for example, by matching the moments of the distribution to those calculated in terms of the sample of values. In this paper, alternative flexible models for the quantile function and methods for determining a quantile function from a sample of values are proposed for meeting the above needs.

The 'Guide to the Expression of Uncertainty in Measurement' (GUM) provides a framework and procedure for evaluating and expressing measurement uncertainty. The procedure has two main limitations. Firstly, the way a coverage interval is constructed to contain values of the measurand with a stipulated coverage probability is approximate. Secondly, insufficient guidance is given for the multivariate case in which there is more than one measurand. In order to address these limitations, two specific guidance documents (or 'Supplements to the GUM') on, respectively, a Monte Carlo method for uncertainty evaluation (Supplement 1) and extensions to any number of measurands (Supplement 2) have been published. A further document on developing and using measurement models in the context of uncertainty evaluation (Supplement 3) is also planned, but not considered in this paper.An overview is given of these guidance documents. In particular, a Monte Carlo method, which is the focus of Supplements 1 and 2, is described as a numerical approach to implement the 'propagation of distributions' formulated using the 'change of variables formula'. Although applying a Monte Carlo method is conceptually straightforward, some of the practical aspects of using the method are considered, such as the choice of the number of trials and ensuring an implementation is memory-efficient. General comments about the implications of using the method in measurement and calibration services, such as the need to achieve transferability of measurement results, are made.

The use of a combined microwave and acoustic resonator to determine the Boltzmann constant, k(B), permits several checks on the internal consistency of the data. Using measurements in argon gas in the NPL-Cranfield quasispherical copper resonator (NPLC-2), we describe four distinct types of internal consistency check. Firstly, we estimate k(B) using six distinct acoustic resonances varying in frequency from 3.55 kHz to 21.77 kHz. We thus span a wide range of systematic corrections, most notably in the effect of the thermal boundary layer (TBL), which varies strongly with mode. Secondly, the same theory which predicts the TBL corrections to the acoustic resonance frequencies also predicts the widths of the resonances. By comparing the measured and theoretically-expected widths we can place limits on the effect of any un-modeled physics. Thirdly, the equivalent radius of the resonator (similar to 62.03 mm) is inferred from analysis of 8 TM microwave resonances and the spread of the radius values inferred from each mode is a measure of how well the resonator has been modeled. Finally, the microwave data can be used to check the inferred density of gas within the resonator. Based on measurements of the dielectric permittivity of the argon gas, pressure discrepancies greater than +/-6 Pa can be detected at all pressures up to 700 kPa. Taken together, these four checks improve confidence in the final estimate for k(B) and restrict the types of systematic error which may affect the result.

Corrections for the so-called self-heating effect in measurements with resistance thermometers are made by extrapolating the resistances (or bridge readings) measured at two or more currents to obtain the value at 0 mA. Such corrections are of particular significance when using standard platinum resistance thermometers with the lowest uncertainties. The purpose of this paper is to establish how to optimize the measurements so as to achieve the minimum uncertainty in the correction for a given total measurement time and maximum allowed measuring current. The assumptions are that the self-heating effect is proportional to the square of the current, and that the statistical uncertainties vary inversely with the current and with the square root of the measurement time. Under these assumptions it can be shown, using analytical or Monte Carlo methods, that there is no benefit in using more than two measuring currents to determine the correction, and that the uncertainty is minimized when the lower current is approximately 0.5 times the higher current. Specifically, it is shown that the optimum solution is to employ two measuring currents in the ratio 1 :2 and allocate the measurement times at the lower and higher currents in the ratio 8 :1. Under these conditions the uncertainty in the resistance extrapolated to zero current is equal to the uncertainty at the higher current and root 2 times the uncertainty at the lower current. It is expected that the techniques used here could be applied in other extrapolations, such as for isotherms in the various forms of gas thermometry.

In this paper we give the results of four methods of calculating uncertainty associated with a mass calibration problem, three based on different implementations - the first and second order law of propagation of uncertainty and the Monte Carlo method - of the general methodology described by the Guide to the Expression of Uncertainty in Measurement, the fourth based on a Bayesian formulation. The nonlinearities present in the model for the calibration problem means that the first order approach can be an unreliable method for evaluating uncertainties, relative to the other three approaches.

The Comite international des poids et mesures (CIPM) has projected a major revision of the International System of Units (SI) in which all of the base units will be defined by fixing the values of fundamental constants of nature. In preparation for this we have carried out a new, low-uncertainty determination of the Boltzmann constant, k(B), in terms of which the SI unit of temperature, the kelvin, can be re-defined. We have evaluated k(B) from exceptionally accurate measurements of the speed of sound in argon gas which can be related directly to the mean molecular kinetic energy, 3/2k(B)T. Our new estimate is k(B) = 1.380 651 56 (98) x 10(-23) J K-1 with a relative standard uncertainty u(R) = 0.71 x 10(-6).