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The medical use of radionuclides depends on the accurate measurement of activity (Bq) for regulatory compliance, patient safety, and effective treatment or image quality. In turn, these measurements rely on the realization of primary standards of activity by national metrology institutes, with uncertainties that are fit for purpose. This article reviews the current status of primary standards of activity for radionuclides used in medical imaging and therapy applications. Results from international key comparisons carried out through the International Bureau of Weights and Measures transfer instruments (SIR and SIRTI) are used to verify that standards for a variety of radionuclides are consistent and conform with practitioners’ expectations.

The main motivation for this research is the growing awareness of the impact of climate change and the increasing relevance of the United Nations Sustainable Development Goals, aiming to contribute to the measurement of quantities like precipitation and rate of rainfall. This knowledge is widely used in hydrology, climatology and meteorology, providing data and information applied in modelling, pattern definition and recognition, and forecasting. This work is concerned with estimating the average areal rainfall in a stipulated region from rainfall intensity observations made at measurement stations within that region. It focuses on three straightforward estimation approaches: the arithmetic mean method, the Thiessen polygon method and the isohyetal method. The evaluation of the associated measurement uncertainty, for which the law of propagation of uncertainty and a Monte Carlo method as described in guidance documents from the Joint Committee for Guides in Metrology are applied, is the main consideration. The approaches described may be readily applied by practitioners. A comparison of results from applying these methods to a simple example is made. Such results are required for conformity assessment and support in urban management and water resources management worldwide.

A criticism levelled at the Guide to the Expression of Uncertainty in Measurement (GUM) is that it is based on a mixture of frequentist and Bayesian thinking. In particular, the GUM's Type A (statistical) uncertainty evaluations are frequentist, whereas the Type B evaluations, using state-of-knowledge distributions, are Bayesian. In contrast, making the GUM fully Bayesian implies, among other things, that a conventional objective Bayesian approach to Type A uncertainty evaluation for a number n of observations leads to the impractical consequence that n must be at least equal to 4, thus presenting a difficulty for many metrologists. This paper presents a Bayesian analysis of Type A uncertainty evaluation that applies for all , as in the frequentist analysis in the current GUM. The analysis is based on assuming that the observations are drawn from a normal distribution (as in the conventional objective Bayesian analysis), but uses an informative prior based on lower and upper bounds for the standard deviation of the sampling distribution for the quantity under consideration. The main outcome of the analysis is a closed-form mathematical expression for the factor by which the standard deviation of the mean observation should be multiplied to calculate the required standard uncertainty. Metrological examples are used to illustrate the approach, which is straightforward to apply using a formula or look-up table.

The eutectic alloys rhenium–carbon, platinum–carbon and cobalt–carbon have been proposed as reference standards for thermometry, with temperature and uncertainty values specified within the mise en pratique of the definition of the kelvin. These alloys have been investigated in a collaboration of eleven national measurement institutes and laboratories. Published results reported the point-of-inflection in the melting curve with extremely low uncertainties. However, to be considered as standards it is necessary to stipulate what phenomenon a temperature value has been ascribed to; specifically, this should be a thermodynamic state. Therefore, the data have been further evaluated and the equilibrium liquidus temperatures determined based on a consideration of limits and assuming a rectangular probability distribution. The values are: for rhenium–carbon 2747.91 ± 0.44 K, for platinum–carbon 2011.50 ± 0.22 K and for cobalt–carbon 1597.48 ± 0.14 K, with uncertainties at approximately a 95% coverage probability. It is proposed that these values could be used as the basis of thermodynamic temperature measurement at high temperatures (above 1300 K).

A computer model of a patient-specific clinical (177)Lu-DOTATATE therapy dosimetry system is constructed and used for investigating the variability of renal absorbed dose and biologically effective dose (BED) estimates. As patient models, three anthropomorphic computer phantoms coupled to a pharmacokinetic model of (177)Lu-DOTATATE are used. Aspects included in the dosimetry-process model are the gamma-camera calibration via measurement of the system sensitivity, selection of imaging time points, generation of mass-density maps from CT, SPECT imaging, volume-of-interest delineation, calculation of absorbed-dose rate via a combination of local energy deposition for electrons and Monte Carlo simulations of photons, curve fitting and integration to absorbed dose and BED. By introducing variabilities in these steps the combined uncertainty in the output quantity is determined. The importance of different sources of uncertainty is assessed by observing the decrease in standard deviation when removing a particular source. The obtained absorbed dose and BED standard deviations are approximately 6% and slightly higher if considering the root mean square error. The most important sources of variability are the compensation for partial volume effects via a recovery coefficient and the gamma-camera calibration via the system sensitivity.

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.

International Standard ISO 17025:2005 states that the degree of rigour needed in an estimation of uncertainty of measurement depends on several factors including the requirements of the customer. So, generally to deliver a measurement result acceptable to a customer requires a degree of assurance in its quality. We investigate the extent to which such assurance can be given in applying the Guide to the expression of uncertainty in measurement (GUM). For many practical cases, a measurement result incorporating an evaluated uncertainty that is correct to one significant decimal digit would be acceptable. Accordingly, attention is given in published GUM Supplements and in the GUM revision process to quantifying the numerical precision of an uncertainty statement provided by following those documents. Any such quantification is naturally relative to the adequacy of the measurement model and the knowledge used about the input quantities in that model. For general univariate and multivariate measurement models, we emphasize the use of a Monte Carlo method, as recommended in GUM Supplements 1 and 2. One use of this method is as a benchmark in terms of which measurement results provided by the GUM can be assessed in any particular instance. For measurement models that are linear in the input quantities and for which the probability distributions for those quantities are independent, we indicate the use of other approaches such as convolution methods based on the fast Fourier transform and Chebyshev polynomials as benchmarks.

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.

Circular dichroism (CD) spectroscopy of proteins is widely used to measure protein secondary structure, and to detect changes in secondary and higher orders of structure, for applications in research and in the quality control of protein products such as biopharmaceuticals. However, objective comparison of spectra is challenging because of a limited quantitative understanding of the sources of error in the measurement. Statistical methods can be used for comparisons, but do not provide a mechanism for dealing with systematic, as well as random, errors. Here we present a measurement model for CD spectroscopy of proteins, incorporating the principal sources of uncertainty, and use the model in conjunction with experimental data to derive an uncertainty budget. We show how this approach could be used in practice for the objective comparison of spectra, and discuss the benefits and limitations of this strategy.

According to the CIPM Mutual Recognition Arrangement, a calibration and measurement capability (CMC) uncertainty for a laboratory offering a particular calibration or measurement service is normally expressed at a \SI{95}{\percent} level of confidence. When laboratories' CMC claims are unsupported by the relevant key comparison (KC), one option is for modified values to be assigned to their CMC uncertainties. In many cases when CMCs apply to a continuous interval of values such as mass concentration or wavelength, there is no directly relevant KC that can be used in support since KCs are carried out only for selected discrete values of the quantity concerned. Under realistic assumption, we developed a method that is applicable in such an instance and for which the reported CMC uncertainties are amplified so that they are underpinned by the results of the KC. The amplification factors depend on the laboratories' degrees of equivalence (DoEs) for these discrete values, and judiciously adjusted to achieve consistency with the key comparison reference values. The method is based on the patterns in the individual behaviour of the DoEs of the participating laboratories for the discrete values, implying the presence of correlation associated with the DoE values. It applies when the weighted mean of some or all of the measured values reported by the participating laboratories in the KC is used to obtain the KC reference value.

The Joint Committee for Guides in Metrology, Working Group 1, JCGM-WG1, is currently revising the 'Guide to the Expression of Uncertainty in Measurement'. In this communication, the motivation for undertaking such a revision is given and the main changes with respect to the current, 2008 edition are outlined.

The concern is with key comparisons for which the participating laboratories provide measured data at each of a number of prescribed values of an auxiliary parameter, such as wavelength or frequency. Key comparison reference curves provide a continuous counterpart of the key comparison reference values (KCRVs) currently determined independently from the measured data relating to each value of the auxiliary parameter. They have the potential to provide a smoother realization of KCRVs across the interval of values of the auxiliary parameter and thus avoid the anomalous erratic sequences of KCRVs, associated standard uncertainties and degrees of equivalence that can be obtained otherwise. Key comparison reference curves may also help in assuring the consistency of statements of calibration and measurement capabilities with the results of key comparisons in cases where the former relate to a continuous interval of values of an auxiliary parameter and the latter to discrete values within the interval.

Data analysis of a regional metrology organization (RMO) when linked to an International Committee of Weights and Measures key comparison (KCs) is discussed. One of the purposes of an RMO KC is to check the appropriateness of participants - results from a statistical point of view. For this purpose, a statistical test is useful. In this study, we discuss the derivation of a unilateral Degree of Equivalence (DoE) as a test statistic, and apply the test to dummy data. The unilateral DoEs given by the method in this study can be significantly different from those given by previous studies. These differences can be ascribed to the use of different statistical models. A Consultative Committee (CC) should choose an appropriate statistical model for the analysis in accordance with its purpose in implementing an RMO KC.

We consider a probabilistic model of the measurement process, based on identifying two main sub-processes, named observation and restitution. Observation constitutes the transformations involved in producing the observable output. Restitution constitutes the determination of the measurand (the quantity measured) from the observable output, and includes data processing. After providing a probabilistic representation of the observation sub-process, we derive appropriate formulae for addressing restitution and describing the overall measurement process. The model allows the treatment in probabilistic terms of both the random and systematic effects that influence the measurement process, and may prove particularly useful in the formulation phase of uncertainty evaluation. We also discuss the different ways in which the measurand can be characterized by a probability distribution, and demonstrate the application of the approach to the analysis of risk in conformance testing.

The problem is addressed of determining numerical approximations to the area under a curve specified by arbitrarily spaced data. A formulation of this problem is given in which the data are used to model the curve as a piecewise polynomial, each piece having the same degree. That piecewise function is integrated to provide an approximation to the area. A corresponding compound quadrature rule is derived. Different degrees of polynomial give rise to different orders of quadrature rule. The widely used trapezoidal rule is a special case, as is the Gill–Miller rule. The remainder of the paper is concerned with evaluating the measurement uncertainties associated with the approximations to the area obtained by the use of these rules in the case where the data ordinates correspond to measured values having stated associated uncertainties. The case in which there is correlation associated with the measured values, frequently arising when the measured values are obtained using the same measuring instrument, is also treated. A statistical test is used to select a suitable polynomial degree.

The requirement to aggregate measurement data, in the case where each measured value corresponds to nominally the same quantity, is important throughout metrology. Correlation associated with measured values is shown to arise in a number of measurement problems, with examples taken from the areas of interlaboratory comparisons, including key comparisons, the calibration of measuring systems and instruments, wireless sensor networks and prediction on the basis of different models. Consideration is given to the effect that correlation has on the determination of an aggregated estimate of the measured quantity, and on testing the consistency of measurement data. Approaches to quantifying the correlation associated with measured values are presented, and a formulation is given of the problem of determining the largest consistent subset of data having associated correlation. Results for three measurement problems are given, concerned with an interlaboratory comparison of noise in a coaxial line, the calibration of a thermometer and the analysis of data arising from a wireless sensor network.

Suppose a single stable travelling standard is circulated around the national metrology institutes (NMIs) participating in a key comparison. Consider the set of data consisting of a measurement result, comprising a measured value and the associated standard uncertainty, provided independently by each such NMI. Each measured value is the corresponding NMI's best estimate of a single stipulated property of the standard. The weighted mean (WM) of the measured values can be formed, the weights being proportional to the reciprocals of the squared standard uncertainties. If this WM is consistent with the measured values according to a statistical test, it can be accepted as a key comparison reference value for the comparison. Otherwise, the WM of a largest consistent subset (LCS) can be determined. The LCS contains as many as possible of those results of participating NMIs that are consistent with the WM of that subset. An efficient approach for determining the LCS having smallest chi-squared value is described, and applied to length, temperature and ionizing radiation comparisons.

The performance of underwater electroacoustic transducers often depends on water temperature and, for accurate calibrations, it is necessary to take account of this influence on measurement data. Doing so is particularly important for open-water calibration facilities where the environmental conditions cannot be controlled and seasonal variations in temperature can contribute significant measurement uncertainty. This paper describes the characterization of the sensitivity of underwater electroacoustic transducers in terms of their variation with water temperature. A model containing adjustable parameters is developed for providing frequency-dependent temperature coefficients whose use enables measurement data to be corrected for temperature. Estimates of these coefficients are determined by applying the model to the time history of measurement data obtained over a period of several years. The influence of seasonal temperature variation can then be separated from the slow temporal drift in the transducer sensitivity. The uncertainties associated with the corrected sensitivity values are evaluated.

A number of Joint Committees of the Bureau International des Poids et Mesures and other international organizations carry out particular tasks of common interest. The Joint Committee for Guides in Metrology (JCGM) has amongst its tasks the promotion of the 'Guide to the Expression of Uncertainty in Measurement' (GUM), the preparation of further documents for its broad application, and revision and promotion of the use of the 'International Vocabulary of Basic and General Terms in Metrology'. This paper summarizes the documents relating to the GUM planned by JCGM.

The Guide to the Expression of Uncertainty in Measurement (GUM) is the internationally accepted master document for the evaluation of uncertainty. It contains a procedure that is suitable for many, but not all, uncertainty evaluation problems met in practice. This procedure constitutes an approximation to the general solution of the Markov formula, which infers the probability density function (PDF) for the output quantities (measurands) from the model of the measurement and the PDFs for the input quantities. This paper shows that a Monte Carlo method is an effective and versatile tool for determining the PDF for the measurands. This method provides a consistent Bayesian approach to the evaluation of uncertainty. Although in principle straightforward, some care is required in representing and validating the results obtained using the method. The paper provides guidance on optimizing the approach, identifies some pitfalls and indicates means for validating the results.

The CCPR K1-a key comparison of spectral irradiance (from 250 nm to 2500 nm) was carried out to meet the requirements of the Mutual Recognition Arrangement by 13 participating national metrology institutes. Because of the fragile nature of the tungsten halogen lamps used as comparison artefacts the comparison was arranged as a star comparison with many more lamps than participants. The National Physical Laboratory (UK) piloted the comparison and, by measuring all lamps, provided a link between participants' measurements. The comparison was analysed using a model-based method, which ensured that all participants, including the pilot, were treated equitably. This paper presents the comparison philosophy, methodology, analysis and results.

Obtaining confidence in a measured value requires a quantitative statement of its quality, which in turn necessitates the evaluation of the uncertainty associated with the value. The basis for the value and the associated uncertainty is traceability of measurement, involving the relationship of relevant quantities to national or international standards through an unbroken chain of measurement comparisons. Each comparison involves calibration of a standard at one level in the chain using a standard at a higher level. Global economy considerations mean that this basis also requires the national measurement institutes to carry out comparative assessment of the degree of equivalence of national standards through their participation in key comparisons. The evaluation of uncertainty of measurement is founded on the use of models of measurement for each stage of the chain and at the highest level to interrelate national standards. Basic aspects of uncertainty evaluation are covered in this paper, and forms for the above types of model considered, with attention given to least squares as a basis for calibration curves (and certain other types of calibration) and also for key comparison data evaluation.

Several studies highlight the need for appropriate statistical and probabilistic tools to analyse the data provided by the participants in an interlaboratory comparison. In some temperature comparisons, where the measurand is a physical state, independent realizations of the same physical state are acquired in each participating institute, which should be considered as belonging to a single super-population. This paper introduces the use of a probabilistic tool, a mixture of probability distributions, to represent the overall population in such a temperature comparison. This super-population is defined by combining the local populations in given proportions. The mixture density function identifies the total data variability, and the key comparison reference value has a natural definition as the expectation value of this probability density.

Two-way satellite time and frequency transfer (TWSTFT) has been used operationally by the international time and frequency community for several years. This paper describes analysis techniques being developed at the National Physical Laboratory for the processing of TWSTFT measurements. The measurements are modelled in terms of phase and normalized frequency offsets together with random errors considered to be a linear combination of the well-known noise types of white phase modulation, white frequency modulation and random walk frequency modulation encountered in time and frequency measurements. A method is described for solving the model to provide estimates of the phase and normalized frequency offsets, together with their associated uncertainties, at any measurement epoch. The random errors associated with the measurements are first characterized through the use of second-difference statistics. An analysis of the second-difference statistics is undertaken to estimate a covariance matrix for the time series of measurements. Finally, with the covariance matrix so obtained, generalized least-squares regression is used to provide unbiased and efficient estimates of the required offset parameters.

A family of second-difference statistics has been designed for use on regular but unevenly spaced two-way satellite time and frequency transfer (TWSTFT) time series. The characteristics of these new statistics are compared with those of the Allan variance (AVAR) and modified Allan variance (MVAR). The new statistics are used to estimate values of AVAR, MVAR and TVAR. Direct comparisons are made with estimates of these quantities based on an interpolated time series. Confidence interval determination is discussed. The application of the new variances to TWSTFT and geodetic GPS time transfer data is presented.

A 3D lattice based on a high-power CO2 laser is considered in the context of laser cooling and trapping of atomic calcium. We expect to be able to realize a system with }10\,000$>>10 000 lattice sites each with more than 100 atoms, and the facility to laser cool all the atoms into the vibrational ground state using the intercombination line. The configuration allows the production of an array of small Bose?Einstein condensates, enabling an investigation of the build-up of phase coherence as a function of atom number.

The role of statistical error modelling as a tool in the evaluation of measurement uncertainty is discussed. The discussion is limited to the use of error modelling as a vehicle to evaluate measurement uncertainty in agreement with the Guide to the Expression of Uncertainty in Measurement. The discussion is based on practical experience gained in developing models for evaluating measurement uncertainty both in calibration and in testing. Until now, it has not been widely recognized that the development of uncertainty models may benefit from the use of classical error analysis techniques. The problems in modelling measurements are such that, based on current practice, this activity is a profession with its own merits and peculiarities. An application of error modelling is given by showing how the performance of laboratories can be assessed in linked interlaboratory comparisons.

The conditions under which results from different key comparisons in gas analysis can be combined are defined. The paper outlines the underlying model for computing degrees of equivalence. The key point in the development of the relevant expressions is the use of the error propagation law as given in the Guide to the Expression of Uncertainty in Measurement 1993 1st edn (Geneva: ISO), to convert the error model into an uncertainty model, including covariances. The covariance term in the generic expression derived is the key to developing tailored expressions for different cases. Some typical instances are explored, including the linking with a bilateral key comparison and the linking with an RMO key comparison. Additionally, a basis is established for linking two comparisons with different pilot laboratories.

We study polarization spectroscopy of Rb vapour. A weak probe beam analyses the birefringence induced in a room temperature vapour by a strong counterpropagating circularly polarized pump beam. In contrast to most other work on polarization spectroscopy, we use a polarization beam splitting cube and two detectors (rather than a polarizer and one detector) to analyse the probe beam. The signal is in the form of a derivative of a Lorentzian. For theoretical analysis we study the closed atomic transition 5 2S1/2 (F = 3) → 5 2P3/2 (F' = 4) in the D2 line of 85Rb. We study the time needed to redistribute population among the mF states, derive an expression for the expected lineshape and present experimental data in excellent agreement with theory. The polarization spectrum provides an ideal error signal for frequency stabilization of a laser. We describe the geometry and parameters for optimizing the error signal.

Guidelines containing two procedures are proposed for the evaluation of key comparison data. They apply to the simple circulation of a single travelling standard around all the participants. The application of the procedures to a specific set of key comparison data provides a key comparison reference value (KCRV) and the associated uncertainty, the degree of equivalence of the measurement made by each participating national institute and the degrees of equivalence between measurements made by all pairs of participating institutes. Procedure A is based on the use of the weighted mean, together with consistency checks based on classical statistics regarding its applicability. Should the checks fail, action to remedy the situation is suggested. If the remedy is inappropriate, Procedure B can be applied instead. It is based on the use of the median (or any another informed choice) as a more robust estimator in the circumstances.

We report the first measurement of effective electron g-factors in a doubly positively charged molecule (molecular dication); the measurements were made using a fast-ion-beam/laser-beam spectrometer. We measured Zeeman splittings of a rotational line within the v = 2←1 vibrational band of the X3Σ- electronic ground state of D35Cl2+. The Zeeman splittings allowed us to assign the line as a PQ23(6) fine structure transition and to measure the effective upper and lower state effective electron g-factors as g(v' = 2, N' = 5, J' = 5) = 1.0 ± 0.4 and g(v'' = 1, N'' = 6, J'' = 5) = 1.85 ± 0.05. The large differences in the g-factors arise from vibrationally dependent spin-orbit coupling with a repulsive A3Π state.

Collinear laser-beam/ion-beam spectroscopy provides a collision-free environment in which to study atomic and molecular ions. Ion-beam spectroscopy has particular advantages over other methods: spectra are relatively simple due to mass selection; kinematics allow sub-Doppler resolution of the whole velocity distribution; and the Doppler effect can be used to tune ions into resonance with fixed frequency light sources. We consider methods by which spectra have been obtained and review important results of ion-beam spectroscopy in fields from nuclear physics to astrophysical chemistry.

This special issue of Metrologia is the first that is not devoted to units, or constants, or measurement techniques in some specific field of metrology, but to the generic topic of statistical and probabilistic methods for metrology. The number of papers on this subject in measurement journals, and in Metrologia in particular, has continued to increase over the years, driven by the publication of the Guide to the Expression of Uncertainty in Measurement (GUM) [1] and the Mutual Recognition Arrangement (MRA) of the CIPM [2]. The former stimulated metrologists to think in greater depth about the appropriate modelling of their measurements, in order to provide uncertainty evaluations associated with measurement results. The latter obliged the metrological community to investigate reliable measures for assessing the calibration and measurement capabilities declared by the national metrology institutes (NMIs). Furthermore, statistical analysis of measurement data became even more important than hitherto, with the need, on the one hand, to treat the greater quantities of data provided by sophisticated measurement systems, and, on the other, to deal appropriately with relatively small sets of data that are difficult or expensive to obtain. The importance of supporting the GUM and extending its provisions was recognized by the formation in the year 2000 of Working Group 1, Measurement uncertainty, of the Joint Committee for Guides in Metrology. The need to provide guidance on key comparison data evaluation was recognized by the formation in the year 2001 of the BIPM Director's Advisory Group on Uncertainty. A further international initiative was the revision, in the year 2004, of the remit and title of a working group of ISO/TC 69, Application of Statistical Methods, to reflect the need to concentrate more on statistical methods to support measurement uncertainty evaluation. These international activities are supplemented by national programmes such as the Software Support for Metrology programme in the UK, which includes within its main themes generic items related to modelling, uncertainty evaluation and key comparisons. There are also teams concentrating on statistics within a metrology environment, the largest of which is the Statistical Engineering Division at NIST. There are, however, key pockets of mathematical and statistical expertise at all major and many of the smaller NMIs. Academia also makes considerable input to metrological thinking. The papers in this special issue reflect the above considerations—and more. There are several offerings relating to the GUM: (a) the manner in which the GUM is evolving, especially through Supplements to the GUM, (b) a comparison of the GUM, the GUM Supplement concerned with the propagation of distributions and Bayesian statistics, in the context of linear calibration, (c) theoretical and practical aspects of the use of a Monte Carlo method for propagating distributions, (d) the use of a generalization of the sensitivity coefficients in the GUM to correlated quantities, and (e) considerations on obtaining best estimates when the model is non-linear. At a more fundamental level, a systematic and versatile approach to developing the model of measurement, on which uncertainty evaluation is of course based, is presented, and a paper is included on principles of probability and statistics that promote sound decision-making. The evaluation of key comparison data is represented in terms of contributions relating to (a) models of key comparisons, with measures of operability and interoperability, (b) a Bayesian procedure for providing PDFs from which the measures required by the MRA can be extracted, (c) an extension of the En measure familiar to many metrologists, and (d) the use of the median and weighted median as the key comparison reference value in the presence of discrepant measurement results. The remaining contributions concern the analysis of measurement data, including spectral analysis. Covered are (a) a comparison of conventional and Bayesian approaches to the evaluation of data subject only to random effects, (b) element-wise weighted least squares and its comparison with conventionally weighted and total least squares, (c) the generalized weighted mean of correlated quantities (with an application to key comparisons), (d) the uncertainty associated with the average of autocorrelated quantities, (e) the fitting of three-dimensional geometric elements to coordinate data, (f) the fitting of calibration curves from data having general uncertainty structure, (g) the estimation of the power spectrum of clock noise, (h) Kalman filtering, also in the presence of clock noise, and (i) an improved Allan deviation-like statistic. The realization of such a rich issue involved considerable efforts from many contributors. Thanks are due to the former Editor, Peter Martin, who first conceived the idea of this issue, to Jeffrey Williams, the present Editor, for his support to its realization, to the authors for their contributions, and to the referees who gave their time to review and comment on the papers. References [1] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP and OIML 1995 Guide to the Expression of Uncertainty in Measurement (Geneva, Switzerland: International Organisation for Standardisation) ISBN 92-67-10188-9 [2] BIPM 1999 Mutual Recognition of National Measurement Standards and of Calibration and Measurement Certificates Issued by National Metrology Institutes (Sevres, France: Bureau International des Poids et Mesures)

We reply to the comment by showing how our formula for the evaluation of global consumer's risk on the one hand is mathematically equivalent to that proposed by the commenter. On the other hand it is safer from potential misuse, since it makes clear the role, in such an evaluation, of the distribution p(x), which governs the behaviour of the production process. Then we discuss some practical ways for obtaining such a distribution.

The relation between uncertainty evaluation according to the recent draft of GUM Supplement 1 and the application of Bayesian statistics including Bayes' theorem is considered. In the case of a Type A evaluation of uncertainty, repeated measurement indications are regarded as independently drawn from normal frequency distributions and, according to its suggestion, the numerical evaluation method of Supplement 1 is applied after having assigned scaled and shifted t-distributions to the corresponding input quantities. It will be shown that this approach is equivalent to a Bayesian analysis using Bayes' theorem with commonly used prior distributions.

Spectroscopy in the extreme ultraviolet region is important to magnetic fusion research as well as to astrophysics. We report on XEUS (X-ray and Extreme Ultraviolet Spectrometer) and LoWEUS (Long-Wavelength and Extreme Ultraviolet Spectrometer), which operate in the 5–400 A region on the NSTX (National Spherical Tokamak Experiment) tokamak. The instruments are being used to survey impurities, both for intrinsic elements present in the plasma and for metal impurities resulting from damage to various components. In addition, we have used XEUS and LoWEUS to investigate density-dependent and temperature-dependent emission lines for diagnostic use.

Metal on metal hip joint prostheses are now commonly implanted in patients with hip problems. Although hip replacements largely go ahead problem free, some complications can arise such as infection immediately after surgery and aseptic necrosis caused by vascular complications due to surgery. A recent observation that has been made at Manchester is that some Cobalt Chromium (CoCr) implants are causing chronic pain, with the source being as yet unidentified. This form of replacement failure is independent of surgeon or hospital and so some underlying body/implant interface process is thought to be the problem. When the synovial fluid from a failed joint is examined particles of metal (wear debris) can be found. Transmission Electron Microscopy (TEM) has been used to look at fixed and sectioned samples of the synovial fluid and this has identified fine (< 100 nm) metal and metal oxide particles within the fluid. TEM EDX and Electron Energy Loss Spectroscopy (EELS) have been employed to examine the composition of the particles, showing them to be chromium rich. This gives rise to concern that the failure mechanism may be associated with the debris.

Ammonia is an important compound in the chemical industry. It is widely used and is the basis for producing other compounds containing nitrogen. Ammonia is also very hazardous, and consequently emissions of ammonia need be controlled and monitored. In the past years, several national metrology institutes have developed facilities for the preparation of Primary Standard gas Mixtures (PSMs), dynamically generated ammonia mixtures and facilities for comparing and certifying gas mixtures containing ammonia. The amount-of-substance fraction level of ammonia chosen for this key comparison is 30–50 µmol/mol. The results of this key comparison revealed that there is at present no consensus among static and dynamic techniques for gas mixture preparation for this component in this range. As key comparison reference value (KCRV), the mean of the three methods is used. In its uncertainty, no allowance is made for the observed biases. With respect to the KCRV, only two laboratories report consistent results. When grouped in accordance with the employed methods, the results are consistent. Further experimental work is needed. Main text. 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 kcdb.bipm.org/. The final report has been peer-reviewed and approved for publication by the CCQM, according to the provisions of the CIPM Mutual Recognition Arrangement (MRA).

The CCPR K1-a key comparison of spectral irradiance (from 250 nm to 2500 nm) was carried out to meet the requirements of the Mutual Recognition Arrangement by 13 participating national metrology institutes (NMIs). Because of the fragile nature of the tungsten halogen lamps used as comparison artefacts, the comparison was arranged as a star comparison with three lamps per participant. NPL (United Kingdom) piloted the comparison and, by measuring all lamps, provided a link between participants' measurements. The other participants were BNM–INM (France), CENAM (Mexico), CSIRO (Australia), HUT (Finland), IFA–CSIC (Spain), MSL–IRL (New Zealand), NIM (China), NIST (United States of America), NMIJ (Japan), NRC (Canada), PTB (Germany) and VNIIOFI (Russian Federation). Before the analysis was completed and the results known, the pilot discussed with each participant which lamp measurements should be included as representative of their comparison. As a consequence of this check, at least one measurement was excluded from one third of the lamps because of changes due to transportations. The comparison thus highlighted the difficulty regarding the availability of suitable transfer standards for the dissemination of spectral irradiance. The use of multiple lamps and multiple measurements ensured sufficient redundancy that all participants were adequately represented. In addition, during this pre-draft A phase all participants had the opportunity to review the uncertainty budgets and methods of all other participants. This new process helped to ensure that all submitted results and their associated uncertainties were evaluated in a consistent manner. The comparison was analysed using a model-based method which regarded each lamp as having a stable spectral irradiance and the measurements made by an NMI as systematically influenced by a factor that applies to all that NMI's measurements. The aim of the analysis was to estimate the systematic factor for each NMI. Across the spectral region (250 nm to 2500 nm) there were 44 wavelengths at which a comparison was made. These were treated entirely independently and thus the report describes 44 comparisons. For wavelengths from 250 nm to 800 nm (apart from 300 nm) all participants had unilateral degrees of equivalence (DoEs) with values consistent with their uncertainties for a coverage level k = 2. At all other wavelengths (apart from 1400 nm) all participants achieved consistency at the k = 4 level for the unilateral DoEs and the vast majority within k = 3. The results are a significant improvement over those of the previous comparison in 1990, especially considering that the declared uncertainties of most participants have been substantially improved over the intervening decade. These results are evidence of the value of the effort devoted to the development of improved spectral scales (and of the evaluation of their uncertainty) by many NMIs in recent years. Main text. 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 kcdb.bipm.org/. The final report has been peer-reviewed and approved for publication by the CCPR, according to the provisions of the CIPM Mutual Recognition Arrangement (MRA).

A Fourier transform spectrometer designed to operate at high resolution at wavelengths down to 170 nm is described. The principal instrumental parameters are: mirror travel, 200 mm; resolving limit, 0.025 cm-1; collimator aperture ratio, f/25; overall dimensions of vacuum tank, 1.5 m*0.25 m*0.25 m. Test results show (i) a signal-to-noise ratio in the transformed spectrum at 200 nm better than 1000:1 for an iron-neon hollow cathode lamp at a resolving power of quarter of a million, (ii) fully resolved line profiles in the same source at a resolving limit of 0.03 cm-1 (resolving power 1.5*106), (iii) relative wavenumbers of Fe II emission lines reproducible to +or-0.0006 cm-1 (3.4 fm), and (iv) a significant luminosity gain over grating spectrometers operating in the same region.

A technique is described for the measurement of ion density in atmospheric pressure H2-O2-N2 flames containing hydrocarbon gases. A highly negative spherical electrostatic probe is used and is calibrated in unseeded flames of ion density 108-109 ions/cm3. The probe current-ion density characteristic is shown to be of the form n=κI145±008 where, in this case, κ=(79±32)×1018 (n is in ions/cm3 and I is in amperes). The expression is compared with a theoretical probe law deduced from the high pressure, continuum theory of Su and Lam. The experiment provides extensive data on the dependence of ion density and recombination rates on flame composition. This material is given in Part II of this paper.

Ion density measurements have been made, between 32 and 96 ms from the reaction zone, in constant velocity atmospheric pressure flames of H2-O2-N2, with additions of up to 20% of the following gases:methane, ethane, propane, ethylene, propylene and acetylene. Ion density is found to depend by a power law on the flow rate of reacting oxygen atoms, measured in fuel-rich flames by oxygen flow rate and in oxygen-rich flames by a parameter of the fuel. This parameter is found to differ significantly from chemical equivalence, suggesting that ion generation is associated with the formation of `over-oxidized' short-lived radicals. The paraffin and olefine additives examined show a dependence of ion density on the square of the flow rate of oxygen atoms, for wide ranges of stoichiometry, nitrogen dilution and hydrocarbon addition rate. In the case of acetylene additive this relation is found to be a cube law. The measurements also yield data on ion recombination coefficients. Maximum ion densities, in the reaction zone, of about 1011 ions/cm3 are implied, for the flames examined.

This instrument provides great flexibility for handling abnormal problems without loss of simplicity in normal operation. The oscillation angle can be adjusted to any value between 2° and 240°; the film records from 2θ = 4° to 170° with a camera of 2.75 cm radius; provision is made for fitting larger cameras and special accessories; the maximum inclination angle is 30°. The instrument may be used horizontally or vertically; its special features include a very compact demountable goniometer head of wide angular range and a simple device for setting the layerline screens accurately.

By Hans Neff Munich : R. Oldenbourg Verlag. Pp. 447. Price DM 73. In his foreword the author suggests that most workers in the field of X-ray structure analysis discover that there are gaps in their knowledge and experience, either in the fundamentals of crystallography, in the theory of diffraction, or in the use of equipment. The aim of his book is to deepen understanding of fundamentals, to help with practical investigations, and to supply some knowledge of the way in which the structures of crystals depend on the binding forces in them.

By Lynde Phelps Wheeler. (Yale University Press. London: Geoffrey Cumberlege.) Pp. xi + 264. Price $4.00 Many people think that Willard Gibbs was the greatest scientist the United States has ever produced and he has even been called "the greatest scientific philosopher since Newton," yet Gibbs the man has remained relatively unknown. The work under review, which incorporates a good deal of new material, was written by one of Gibbs' pupils at the request of members of Gibbs' family and others with the dual aim of presenting an authoritative personal biography and of illustrating the application of Gibbs' work to the subsequent development of the physical sciences.

By E. Cartmell and G. W. A. Fowles London: Butterworths Scientific Publications. Pp. 256 + xi. Price 32s. 6d. The quantum theory is fundamental for the development of modern theories of valency and molecular structure, and because of the overwhelming importance of the valency concept in chemistry many teachers nowadays are of the opinion that, in spite of the pedagogic difficulties, a student should be introduced to wavemechanics and some of its simpler chemical applications quite early in his university course. The average student of chemistry is not equipped, in his earlier years at least, to read the standard works on wavemechanics; the work under review is one of the latest of many attempts to present the subject to him in a (mathematically) less formidable guise.