I M Smith

Implementation of a generalized least-squares method for determining calibration curves from data with general uncertainty structures

The determination of a best-fit calibration curve that describes the response of a measuring system to the value of a standard is one of the most widely used procedures in metrology. The mathematical basis for a generalized least-squares solution to this problem is reviewed. Examples of the application of a software implementation of the method are presented to illustrate the treatment of calibration problems with different uncertainty structures for the calibration data, including correlated data. The examples concern the calibration of analysers to measure the composition of natural gas and the calibration of a gas flow dilutor.

A Monte Carlo method for uncertainty evaluation implemented on a distributed computing system

This paper is concerned with bringing together the topics of uncertainty evaluation using a Monte Carlo method, distributed computing for data parallel applications and pseudo-random number generation. A study of a measurement system to estimate the absolute thermodynamic temperatures of two high-temperature blackbodies by measuring the ratios of their spectral radiances is used to illustrate the application of these topics. The uncertainties associated with the estimates of the temperatures are evaluated and used to inform the experimental realization of the system. The difficulties associated with determining model sensitivity coefficients, and demonstrating whether a linearization of the model is adequate, are avoided by using a Monte Carlo method as an approach to uncertainty evaluation. A distributed computing system is used to undertake the Monte Carlo calculation because the computational effort required to evaluate the measurement model can be significant. In order to ensure that the results provided by a Monte Carlo method implemented on a distributed computing system are reliable, consideration is given to the approach to generating pseudo-random numbers, which constitutes a key component of the Monte Carlo procedure.

Mathematical modelling to support traceable dynamic calibration of pressure sensors

This paper focuses on the mathematical modelling required to support the development of new primary standard systems for traceable calibration of dynamic pressure sensors. We address two fundamentally different approaches to realizing primary standards, specifically the shock tube method and the drop-weight method. Focusing on the shock tube method, the paper presents first results of system identification and discusses future experimental work that is required to improve the mathematical and statistical models. We use simulations to identify differences between the shock tube and drop-weight methods, to investigate sources of uncertainty in the system identification process and to assist experimentalists in designing the required measuring systems. We demonstrate the identification method on experimental results and draw conclusions.

Software measurement standards for areal surface texture parameters: part 1—algorithms

Reference software for the evaluation of a set of areal surface texture parameters is described, focusing on the definitions of the parameters and giving details of the numerical algorithms employed in the software to implement those definitions. The main consideration in the design and development of reference software is its numerical correctness, and the algorithms chosen and the implementations of those algorithms reflect this consideration. The surface for which parameters are to be evaluated is a bicubic spline interpolant to the available discretely-sampled data, and parameters are evaluated for that continuous surface either to a high numerical precision or to a numerical precision that is under the control of the user.

Software simulation of a lock-in amplifier with application to the evaluation of uncertainties in real measuring systems

Lock-in amplifiers are widely used in metrology to extract a sinusoidal component of a known frequency from a signal that is noisy, returning estimates of the amplitude and phase of the component. A number of questions arise regarding the use by metrologists of the results returned by a lock-in amplifier. For example, what uncertainties should be associated with the estimates returned? How do these uncertainties vary as the signal-to-noise ratio changes? How do instrument errors affect the estimates and the associated uncertainties? In this paper a software simulation tool is described that may be used to process simulated or real (measured) signals, with the user allowed to associate uncertainties with the values of parameters that define the simulated signal and with those that define the operation of the instrument. The results of applying the software simulation tool to both simulated signals and real signals from infrared radiation and nanoindentation experiments are described.

Software measurement standards for areal surface texture parameters: part 2—comparison of software

A companion paper in this issue describes reference software for the evaluation of areal surface texture parameters, focusing on the definitions of the parameters and giving details of the numerical algorithms employed in the software to implement those definitions. The reference software is used as a benchmark against which software in a measuring instrument can be compared. A data set is used as input to both the software under test and the reference software, and the results delivered by the software under test are compared with those provided by the reference software. This paper presents a comparison of the results returned by the reference software with those reported by proprietary software for surface texture measurement. Differences between the results can be used to identify where algorithms and software for evaluating the parameters differ. They might also be helpful in identifying where parameters are not sufficiently well-defined in standards.

Bayesian analysis of a flow meter calibration problem

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.

A model for characterizing the frequency-dependent variation in sensitivity with temperature of underwater acoustic transducers from historical calibration data

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.

The detection of transient behaviour in environmental vibration for the Watt balance

The detection of transient behaviour in environmental vibration data is an important issue when considering the precision of the Watt balance, an electromechanical apparatus for the new definition of the kilogram in the International System of Units (SI). In this paper, the authors have developed a two-stage method for the analysis of large numbers of datasets containing measured vibration data. In the first stage, all the datasets are explored and statistical methods used to identify those datasets that potentially contain transient events. In the second stage, the transient events are separated from the background noise. This stage is implemented using a biorthogonal wavelet technique with Bayesian denoising.

Reference data sets for testing metrology software

Many fields of metrology rely on calculations that are implemented in software. When such software is used to provide a measurement result, which is required to be traceable, it is necessary to recognise explicitly the software and show it to be operating correctly. An approach to testing the performance of calculation software is based on using reference pairs each of which comprises reference input data applied as input to the software and corresponding reference output data against which the output data of the software is compared. However, to make the reference pair useful for verifying and validating calculation software, information must be provided about the numerical accuracy of the reference pair, the numerical sensitivity of the reference output data to perturbations in the reference input data, and the measurement uncertainty associated with the reference output data arising from simulated measurement uncertainty associated with the reference input data. Such information is important as a means to express quantitatively the quality of the reference pair as a numerical artefact to test calculation software, and as a basis for performance metrics to express quantitatively the numerical performance of software. In this paper these additional components of a reference data set are described, and various approaches to calculating them are discussed. An example, concerned with the calculation of the Gaussian (least-squares) best-fit plane to measured data, which is typical of calculations undertaken in coordinate metrology, is used to illustrate the ideas presented.