Table of contents

Quantities that lie within inexactly prescribed intervals arise frequently. A formula for the maximum entropy probability density function (PDF) that describes the state of knowledge of such a quantity is given in Supplement 1 to the Guide to the Expression of Uncertainty in Measurement. However, this formula applies only to the case for which the limits of the interval are known to lie within subintervals of equal lengths centred on the best estimates of the limits. In this paper, a PDF is derived for a quantity that is not so restricted.

When constructing a coverage interval from the probability density function that describes the state of knowledge about a measurand, it seems reasonable to expect that the long-run success rate of that interval will be about equal to the stipulated coverage probability. Through a specific example, the validity of this criterion is examined.

Ten years after the last comparison of the World Radiometric Reference (WRR) to the Système International (SI) radiometric scale and in respect of the recent introduction of a quality management system for the maintenance and dissemination of WRR, the need for a third comparison became apparent. In this third comparison, the two scales are related through two separate radiometers representing WRR and two independent realizations of SI by cryogenic radiometers at the Bundesamt für Metrologie (METAS) in Wabern, Switzerland, and at the National Physical Laboratory in Teddington, UK. The final results have confirmed the previously stated agreement between WRR and SI scales to better than 0.03% ± 0.14%.

We describe two methods for evaluating the dead time of a time-to-amplitude converter (TAC). The dead time is obtained by measuring either the corresponding time interval in an oscilloscope trace or the relation between the single count rate and the coincidence count rate. Values for the TAC dead time are obtained in the range from 3.4 µs to 14.3 µs for the two methods with respective standard uncertainties of 2.9 × 10⁻⁸ s and 3.3 × 10⁻⁹ s. The TAC dead time is applied to the calibration of coincidence-counting measurements of optical transmission and photon-heralding efficiency.

The paper deals with the uncertainty of differential measurements, obtained from the subtraction of a pair of absolute measurements. It is shown that if the same sensor is used to perform both measurements, a model of the sensor will reveal a correlation component between the uncertainty of each absolute measurement, reducing the uncertainty on its subtraction. The procedure followed is based on the Gauss–Markov estimation method, showing that differential measurement uncertainty vanishes when the gradient to be measured is zero. If the two absolute measurements are to be performed using different sensors, a calibration by comparison between them will result in a similar uncertainty reduction. Finally, a simulated example based on commercially available thermistor data is included.

This paper describes methods for reducing the statistical uncertainty in measurements made by noise thermometers using digital cross-correlators and, in particular, for thermometers using pseudo-random noise for the reference signal. First, a discrete-frequency expression for the correlation bandwidth for conventional noise thermometers is derived. It is shown how an alternative frequency-domain computation can be used to eliminate the spectral response of the correlator and increase the correlation bandwidth. The corresponding expressions for the uncertainty in the measurement of pseudo-random noise in the presence of uncorrelated thermal noise are then derived. The measurement uncertainty in this case is less than that for true thermal-noise measurements. For pseudo-random sources generating a frequency comb, an additional small reduction in uncertainty is possible, but at the cost of increasing the thermometer's sensitivity to non-linearity errors. A procedure is described for allocating integration times to further reduce the total uncertainty in temperature measurements. Finally, an important systematic error arising from the calculation of ratios of statistical variables is described.

When spectral ultraviolet (UV) irradiance measurements are carried out, the signal rendered by the spectroradiometer should be proportional to the cosine of the angle between the incident radiation and the normal to the entrance optics. The deviation from this ideal response is referred to as cosine error. The cosine error effect on solar irradiance measurements performed by spectroradiometer systems can be accounted for by using a correcting factor B. If the cosine response is reasonably good, the factor B can be considered to be 1, such that the irradiance estimates are not affected. However, the uncertainty of B is not zero and it can be under certain conditions the greatest uncertainty source affecting solar spectral UV irradiance measurements. In this paper, we evaluated the B uncertainty of a state-of-the-art UV diffuser under different radiance conditions. We applied a Monte Carlo-based uncertainty propagation technique that allowed us to estimate an upper limit for the cosine error influence. We found that irrespective of the wavelength and the solar zenith angle, the standard uncertainty of B is about 2.5% in the case of high aerosol load. Under low aerosol and cloudless conditions, the standard uncertainty increased with the solar zenith angle, reaching a maximum of 3.5% at about 55° and decreasing thereafter. We conclude that the cosine response of even modern entrance optics is still a significant uncertainty source affecting ground-based spectral UV irradiance measurements.

This paper presents an approach for making inferences about the intercept and slope of a linear regression model when both variables are subject to measurement errors. The approach is based on the principle of fiducial inference. A procedure is presented for computing uncertainty regions for the intercept and slope that can be used to assess agreement between two instruments. Computer codes for performing these calculations, written using open-source software, are listed.

A large air-filled parallel-plate extrapolation chamber with thin graphite front and back electrodes is used as a primary standard measuring device for low-energy interstitial brachytherapy sources from which the unit of air-kerma strength or reference air-kerma rate can be derived. The chamber is suitable for low-energy photons with energies up to 40 keV. The underlying principle is that the air-kerma rate at a given point is proportional to the increment of ionization per increment of chamber volume at chamber depths greater than the range of secondary electrons originating from the electrode. The fundamentals for evaluating the extrapolation curves are presented as well as a detailed description of the present set-up of the PTB large-volume extrapolation chamber (GROVEX). Comparisons between the GROVEX and other primary standards for air-kerma and reference air-kerma rate are presented.

A Josephson voltage standard (JVS) from the BIPM, incorporating a PTB 10 V programmable array, has been compared with a JVS from the LNE, operating with a conventional hysteretic array. The metrological characteristics of both JVSs were carefully studied. We report on the experiments performed for this comparison and on the uncertainty components. The final comparison result is (U_PJVS − U_JVS)/U_JVS = −3.2 × 10⁻¹¹ with a relative total combined standard uncertainty of 6.1 × 10⁻¹¹ (k = 1).

Johnson noise thermometry comes with many imperfection effects that arise primarily from electromagnetic interference, the non-linearity of the electronic system, the transmission line error, etc. Current Johnson noise thermometers (JNTs) operate according to the principle of correlation by which interference noises would be rejected completely. Unfortunately, correlators of JNTs are observed to perform imperfectly in the elimination of these imperfection effects. This paper presents the direct imperfection effect measurements of the JNT of the National Institute of Metrology at the melting point of gallium. The result of these measurements implied that the imperfection effect of correlation acted as an equivalent input noise equally distributed on both probes of the JNT. A relation was derived in this paper to treat this equivalent input noise as a systematic error to Johnson noise thermometry. As a result, the systematic error arising from the measured imperfection effect was corrected from the measured thermodynamic temperature of the melting point of gallium.

The idea of using a 'robust' method of data analysis seems popular. Many scientists will regard such a method as being superior to one for which no such property is claimed. But what actually is a 'robust' method? This paper discusses what is meant by robustness in statistical analysis and, by implication, in general data analysis. It is motivated by recent debate surrounding the estimation of the Newtonian constant G from a set of measurements.

The artificial ear specified in IEC 60318-1 is used for the measurement of headphones and has been designed to present an acoustic load equivalent to that of normal human ears. In this respect it is specified in terms of an acoustical impedance, and modelled by a lumped parameter approach. However, this has some inherent frequency limitations and becomes less valid as the acoustic wavelength approaches the characteristic dimensions within the device. In addition, when sound propagates through structures such as narrow tubes, annular slits or over sharp corners, noticeable thermal and viscous effects take place causing further departure from the lumped parameter model. A new numerical model has therefore been developed, which gives proper consideration to the aforementioned effects. Both kinds of losses can be simulated by means of the LMS Virtual Lab acoustic software which facilitates finite and boundary element modelling of the whole artificial ear. A full 3D model of the artificial ear has therefore been developed based on key dimensional data found in IEC 60318-1. The model has been used to calculate the acoustical impedance, and the results compared with the corresponding data determined from the lumped parameter model. The numerical simulation of the artificial ear has been shown to provide realistic results, and is a powerful tool for developing a detailed understanding of the device. It is also proving valuable in the revision of IEC 60318-1 that is currently in progress.

The raison d'être for this paper is to outline mathematical tools for evaluating uncertainties of atomic weights from those of isotopic abundances despite the absence of information on isotopic abundance correlations—a task usually deemed impossible for elements with more than three stable isotopes.

Evaluation of measurement uncertainty is considered when the value of the measurand depends on the continuous variable time. A concept of dynamic measurement uncertainty is introduced by generalizing the GUM approach. The concept is applied to linear and time-invariant systems which are often appropriate to model dynamic measurements. Digital filtering is proposed for estimating the time-dependent value of the measurand and the design of an appropriate FIR filter is described. Dynamic uncertainty evaluation is then carried out for this analysis and conditions are specified for its proper use. The approach is illustrated for the particular example of a second-order model. It is shown in terms of simulations that the proposed analysis yields significantly improved results when compared with the sometimes applied quasi-static treatment.

A careful study was performed to evaluate and compare uncertainties associated with the measurement of luminescence from chrysene by using fluorimetry in solution and by using solid surface room-temperature phosphorimetry (SSRTP). Chrysene was chosen as the luminescent analyte (measurand) for this study because it is a substance of environmental interest and also because it presents strong natural fluorescence and phosphorescence that is easily induced by the external heavy atom effect. The most common approach for uncertainty calculation is the one indicated in the Guide to the Expression of Uncertainty in Measurement which is recognized worldwide. The most relevant sources of uncertainty were identified. The expanded uncertainty (k = 2; 95%) for SSRTP was 3.27 × 10⁻⁶ mol L⁻¹ (3.7 ng of chrysene) with relevant contributions from the internal reproducibility, preparation of solutions and from the analytical curve. This value is equivalent to 28% of the reference analyte value. For fluorimetry, an acceptable value for the expanded uncertainty was achieved by preparing solutions by weighing masses of analyte and solvent. In this case, the huge contribution from the preparation of solutions by adjusting volume is minimized and the uncertainty magnitude was 2.12 × 10⁻⁹ mol L⁻¹ (21.2% of the chrysene reference concentration). In this case, relevant contributions were from the repeatability, the internal reproducibility and from the analytical curve. The preparation of solutions by weighing does not have a relevant impact in improving the quality of the signal measurements by SSRTP.

The stability of an electron pump composed of three junctions and on-chip chromium resistors at the ends is investigated as a function of applied frequencies on the gate electrodes. For the first time current steps have been obtained with frequencies f_SET as high as 100 MHz. Moreover in the complete studied frequency range we show that the nature (white) and the level of the noise are independent of pumping speed within the noise floor. By generating a single-electron current close to 16.02 pA and measuring over 7 h, a relative type A uncertainty has been found of 3.9 parts in 10⁶. Although the experimental set-up described in this paper does not allow one to measure accurately an absolute value of the current, an alternative set-up including an external current source is proposed for investigating the eventual deviation of the current from the quantization level, e · f_SET. As a matter of fact this set-up proves to be similar to the quantum metrological triangle experiment (QMT). We have demonstrated that the electron R-pump is a plausible candidate for closing the QMT experiment with a type A uncertainty level of 10⁻⁶.