Blair D Hall

Callaghan Innovation · Measurement Standards Laboratory of New Zealand

It is now more than 40 years since the BIPM outlined the current harmonised approach to the expression of uncertainty in measurement, and 30 years since the publication of the Guide to the Expression of Uncertainty in Measurement. Despite several decades of use and application to a wide variety of measurements, there remains a lack of clarity about...

Data processing in metrology is often laborious, because calculations must include uncertainty analysis, and must be independently checked to meet the requirements of quality systems. This tutorial introduces an approach to data processing that automatically performs the steps prescribed in the Guide to the Expression of Uncertainty in Measurement...

The International System of Units (SI) has two units for temperature: the kelvin and the degree Celsius. The kelvin is the base unit for thermodynamic temperature and the degree Celsius is, by definition, the same size as the kelvin. However, when expressing absolute temperature, zero on the Celsius temperature scale is equivalent to 273.15 kelvin,...

It is now more than 40 years since the BIPM outlined the current harmonized approach to the expression of uncertainty in measurement, and 30 years since the publication of the Guide to the Expression of Uncertainty in Measurement. Despite several decades of use and application to a wide variety of measurements, there remains a lack of clarity about...

This article elaborates on a proposal for expressing measurement data in digital systems. Systems will deal with compact three-component expressions of data, comprising: an aspect, value and scale. The aspect and scale will be compactly encoded as unique digital identifiers, which can also serve as keys to access information held in central registe...

A simple model for data obtained by measurement is described. The data model satisfies requirements for evaluation and reporting of measurement uncertainty given in the Guide to the expression of uncertainty in measurement (GUM). The model supports what is called internally consistent and transferable calculations, which are qualities favoured by t...

When metrological traceability is considered in the context of modern digital systems, a model of the staged collaborative nature of traceable measurements emerges. The key elements of this model are instances of external factors that influence the measurement, and intermediate measurement stage results. Digital implementation of the uncertainty ca...

The Asia Pacific Metrology Programme created a special focus group to address the emerging trends of digitalisation and digital transformation in metrology at the end of 2021. The group will serve as a regional forum for coordination of activities and collaborations. It is tasked with: acquiring the latest information about international trends; id...

This article elaborates on a proposal for expressing measurement data in digital systems. Systems will deal with compact three-component expressions of data, comprising: an aspect, value and scale. The aspect and scale will be compactly encoded as unique digital identifiers, which can also serve as keys to access information held in central registe...

Systems of quantities and units were introduced by Maxwell in the late 19th century. However, adapting these ideas for use in modern digital systems has proven difficult. We present an interpretation of units and dimensions that clarifies the main reasons for this difficulty. We then suggest how a central register of metrological information would...

In the age of big data, it is time to ensure that units are routinely documented for easy, unambiguous exchange of information. In the age of big data, it is time to ensure that units are routinely documented for easy, unambiguous exchange of information.

When the principles of metrological traceability are considered in the context of modern digital systems, a model emerges that captures the staged collaborative nature of traceable measurement. Key elements of the model require unique digital identification of instances of external factors that influence a measurement and intermediate measurement s...

The Asia Pacific Metrology Programme created a special focus group to address the emerging trends of digitalisation and digital transformation in metrology at the end of 2021. The group will serve as a regional forum for coordination of activities and collaborations. It is tasked with: acquiring the latest information about international trends; id...

A simple data model for data obtained by measurement is described. The model satisfies requirements for evaluation and reporting of measurement uncertainty given in the Guide to the expression of uncertainty in measurement (GUM). The model is inherently capable of producing internally consistent and transferable data, which are qualities favoured b...

There is currently interest in the digitalisation of metrology because technologies that can measure, analyse, and make critical decisions autonomously are beginning to emerge. The notions of metrological traceability and measurement uncertainty should be supported, following the recommendations in the Guide to the Expression of Uncertainty in Meas...

Many different quantities can be described as 'dimensionless' and have the same SI unit "one'', so their different natures cannot be determined from the unit name alone. In scientific unit systems like the SI, a set of base quantities and associated units can lead to such many-to-one relationships between quantities, dimensions, and units. So, it...

Systems of quantities and units were introduced by Maxwell in the late 19th century. However, adapting these ideas for use in modern digital systems has proven a challenge. We provide an interpretation of units and dimensions that clarifies the main reasons for difficulties. We then suggest how a digital system can provide adequate support for quan...

Many different quantities can be described as 'dimensionless' and have the same SI unit "one", so their different natures cannot be determined from the unit name alone. In scientific unit systems like the SI, a set of base quantities and associated units can lead to such many-to-one relationships between quantities, dimensions, and units. So, it is...

This paper considers a future scenario in which digital reporting of measurement results is ubiquitous and digital calibration certificates (DCCs) contain information about the components of uncertainty in a measurement result. The task of linking international measurement comparisons is used as a case study to look at the benefits of digitalizatio...

The International Committee for Weights and Measures has recognised the need to provide support for digital representations of the International System of Units (SI) and related metrological concepts, such as traceability and measurement uncertainty. A specialised Task Group was established to address this in 2019. This summary reports on the activ...

Widespread use of digital systems provides an opportunity to improve the quality of information reported for traceable measurements. By using digital technology to pass information along a traceability chain, the effects of common influences can be rigorously accounted for, so the provenance of any influence becomes traceable. This can be achieved...

Unit systems and dimensions emerged in the late 19th century. However, efforts to adapt these ideas for use in modern digital systems are proving a challenge. We provide an interpretation of units and dimensions that clarifies the main reasons for difficulties. We then suggest how a digital system would provide adequate support for quantities and u...

Successful digital transformation of measurement quality infrastructures must capture and share information about the metrological traceability of measurement results. So appropriate representations of measurement uncertainty, which is the cornerstone of traceability, are critical. Conventional uncertainty reporting is not ideal, which raises quest...

Relationships between physical quantities are often expressed as mathematical equations, but evaluation of these equations is subject to restrictions related to the quantities involved and the scales of measurement. Calculations will be prone to error if quantity values are assigned to standard numeric data types in digital systems. To address this...

The validity of calibration and measurement capability (CMC) claims by national metrology institutes is supported by the results of international measurement comparisons. Many methods of comparison analysis are described in the literature and some have been recommended by CIPM Consultative Committees. However, the power of various methods to correc...

A Python package for data processing with automatic evaluation of uncertainty is described. The software follows recommendations in the Guide to the Expression of Uncertainty in Measurement and implements some extensions that support complex-valued quantities and situations involving a small number of repeated measurements. The methodology implemen...

There are no generally agreed principles on how digital systems should handle data about physical quantities. From the late 19th century, scientific equations have been expressed in a unit-independent form that hides the existence of measurement scales (units), and are governed by the rules of quantity calculus. To clarify the meaning of such unit-...

Widespread use of digital systems provides an opportunity to improve the quality of information reported for traceable measurements. By using digital technology to pass information along a traceability chain, the effects of common influences can be rigorously accounted for and the provenance of any influence becomes traceable. This can be achieved...

Relationships between physical quantities are often expressed as mathematical equations, but evaluation of these equations is subject to restrictions related to the quantities involved and the scales of measurement. Calculations will be prone to error if quantity values are assigned to standard numeric data types in digital systems. To address this...

The Guide to the Expression of Uncertainty in Measurement advocates the use of an 'effective number of degrees of freedom' for the calculation of an interval of measurement uncertainty. However it does not describe how this number is to be calculated when (i) the measurand is a vector quantity or (ii) when the errors in the estimates of the quantit...

This report sets out the mathematics for solving the `linking comparison' problem. The degrees of equivalence (DoE) for participants in a regional measurement comparison should be linked to a primary comparison. This can be achieved when some laboratories participate in both comparisons, because the DoEs of linking participants will be known from t...

This introduction to measurement uncertainty is intended for metrology professionals working in calibration laboratories and metrology institutes, as well as students in tertiary-level science and engineering programmes. The subject matter is presented with an emphasis on developing models of the physical measurement process. The level of mathemati...

Software support for calculations involving physical quantities is discussed. A Python package (called Quantity-Value) is presented that allows a measured value to be paired with a measurement unit (scale), in an entity that can be used in mathematical operations. The software applies the rules of quantity calculus so that the nature of quantities,...

In a digital agriculture system, Internet of Things (IoT) enabled sensors generate farm data including measurements related to environment, soil, plant, and animal status. Farm models (statistical, verbal, visual, deterministic, etc) transform these data to actionable farm knowledge and information. Sensor data is always associated with measurement...

What is the relationship between 'data interoperability' and 'metrological traceability'? How do existing national measurement infrastructures relate to emerging digital technologies? What needs to change? What opportunities are there for innovation? How can metrological principles be applied in digital systems? Presented at the Digital Metrology...

When reporting expanded uncertainty, rounding the number of significant figures can affect the associated level of confidence. A simple numerical method is used to measure the level of confidence obtained after different rounding methods have been applied. This also offers some insight into the meaning of level of confidence. The findings are consi...

A metrologically traceable measurement is related to its expressed units by a chain of calibrations, each contributing to the measurement uncertainty. At present, however, traceable measurements report the combined measurement uncertainty but give no details about the different contributions to that uncertainty. The contributions from various measu...

Some important vector network analyzer (VNA) calibration procedures, such as thru-reflect-line (TRL), are not amenable to uncertainty analysis by conventional methods, because the calibration equations do not refer to the standards used. Nevertheless, by relating small changes in standard S-parameter values to corresponding changes in VNA response,...

This report proposes that two formulae be changed in the latest (third) edition of the EURAMET calibration guide cg-12, "Guidelines on the Evaluation of Vector Network Analysers". The formulae in question evaluate the uncertainty of residual directivity and residual source match measurements made using the so-called 'Ripple Method'. The report deve...

This report describes a method of evaluating the uncertainty associated with measurements performed using a calibrated system when the calibration equations for that system assume ideal calibration standards. The report presents a solution to a particular problem that arises in radio and microwave frequency metrology. In this field, the so-called T...

This introduction to measurement uncertainty is intended for metrology professionals working in calibration laboratories and metrology institutes, as well as students in tertiary-level science and engineering programmes. The subject matter is presented with an emphasis on developing models of the physical measurement process. The level of mathemati...

This paper reviews a selection of topics related to the evaluation and expression of measurement uncertainty in complex quantities, which are prevalent in electromagnetic measurements at radio and microwave frequencies. Methods appropriate for complex quantities are presented that extend those described, for real-valued quantities, in the Guide to...

Some important vector network analyser (VNA) calibration procedures, such as thru-reflect-line (TRL), are not amenable to propagation of measurement uncertainty by conventional methods, because the equations involved do not explicitly refer to estimates of calibration standards used. However, the sensitivity of calibration constants to those estima...

The expanded measurement uncertainty of a complex quantity is a region in the complex plane surrounding the measured value. When an estimate is expressed in polar coordinates, simultaneous intervals expressing the uncertainty in the radial and angular coordinates define an annular sector. The performance of annular sectors as uncertainty regions is...

The realization of a direct method for measuring signal generator source match is described. Full metrological traceability is achieved by propagating the uncertainty of influence quantities through the data processing algorithms used.

The Guide to the Expression of Uncertainty in Measurement advocates the use of an 'effective number of degrees of freedom' for the calculation of an interval of measurement uncertainty. However, it does not describe how this number is to be calculated when (i) the measurand is a vector quantity or (ii) when the errors in the estimates of the quanti...

The expanded measurement uncertainty of a complex quantity is a region in the complex plane surrounding the measured value. This paper considers different shaped uncertainty regions in the form of ellipses, circles, rectangles and parallelograms. The different types of region are compared, under a variety of measurement error conditions, with regar...

An earlier publication (Hall 2006 Metrologia 43 L56–61) introduced the notion of an uncertain number that can be used in data processing to represent quantity estimates with associated uncertainty. The approach can be automated, allowing data processing algorithms to be decomposed into convenient steps, so that complicated measurement procedures ca...

This report describes methods of evaluating measurement uncertainty in complex quantities with a particular emphasis on RF and microwave measurements. The intent is to provide a concise reference. The report has been prepared in two parts. Part 2 (this document) discusses a general method of treating complex measurement uncertainty that can be used...

Simple expressions are obtained for the measurement uncertainty of complex quantities when no information about phase is available. Three different cases are considered: a known magnitude, an upper bound on the magnitude and an estimate of the magnitude with an associated uncertainty. An expression is also given for the uncertainty of a product whe...

A simulation approach is described for testing the performance of uncertainty calculations based on the Guide to the Expression of Uncertainty in Measurement (GUM) 1st edn (Sèvres, Paris: BIPM Joint Committee for Guides in Metrology). Performance is measured in terms of the long-run success rate of an uncertainty calculation when applied to many si...

This document summarizes a number of topics relating to the measurement uncertainty of complex quantities with particular emphasis on RF and microwave measurements. The intent is to provide a concise reference. The full report is in two parts. Part 1 (this document) applies to measurements where certain simplifying assumptions can be made about the...

This report examines the error models referred to in the the EURAMET Calibration Guide cg-12/v.01 for evaluating VNA performance. An analysis is carried out using a formulation of the {\it Law of Propagation of Uncertainty\/} developed recently for complex quantities. The standard equations for VNA raw-data correction are used to determine compone...

The uncertain-numbers method (Hall B D 2006 Metrologia 43 L56–61) is an alternative computational procedure to the Law of Propagation of Uncertainty (LPU) described in the Guide to the Expression of Uncertainty in Measurement. One advantage of the method is that data processing can be carried out in an arbitrary series of steps and much of the math...

An uncertain complex number is an entity that encapsulates information about an estimate of a complex quantity. Simple algorithms can propagate uncertain complex numbers through an arbitrary sequence of data-processing steps, providing a flexible tool for uncertainty calculations supported by software. The technique has important applications in mo...

Some RF and microwave measurements pose difficulties for uncertainty calculations. A case in point is when the magnitude of the complex reflection coefficient of a weakly reflecting device is measured, such as a load or power sensor. To investigate this, simulations are used to generate a large number of independent sets of data for testing differe...

The uncertain complex number is a simple construct that can be used to achieve full uncertainty propagation in multi-step processing of measurement data. The entity encapsulates information about a measured complex quantity and propagates it through equations by using a few simple algorithms. Software can easily automate the operations involved. Th...

This communication demonstrates the need for independent validation when an uncertainty calculation procedure is applied to a particular type of measurement problem. A simple measurement scenario is used to highlight differences in the performance of two general methods of uncertainty calculation, one from the Guide to the Expression of Uncertainty...

Several issues are discussed that relate to the evaluation of measurement uncertainty for complex-valued quantities in radio frequency measurements. In situations where there is information about the magnitude of a complex quantity, but not phase, uniform distributions in the form of a disc or a ring may be appropriate representations for the uncer...

This report addresses the question of assessing the measurement uncertainty in a set of correction terms obtained for a vector network analyser (VNA) by a standard calibration procedure (the OSLT -- open-short-load-thru -- procedure). The report collects together material from diverse sources to provide a useful resource for developing measurement...

A computational method for processing measurement data is discussed that applies the law of propagation of uncertainty, which is described in the Guide to the Expression of Uncertainty in Measurement. The method introduces the notion of an 'uncertain number', which is an entity that encapsulates information about the value and uncertainty of a quan...

Three statistical distributions have been tested as candidates to represent the uncertainty of complex-valued quantities in Monte Carlo measurement uncertainty calculations. Two candidates are Bayesian multivariate t-distributions with 'non-informative' priors. The other is the distribution of a 'generalized pivotal quantity' (GPQ) for a multivaria...

A method for designing modular instrumentation systems that can propagate measurement uncertainty is described. The approach adheres to current international best-practice in uncertainty calculation, as presented in the “Guide to the Expression of Uncertainty in Measurement” [International Organization for Standardization, Geneva, 1995]. System com...

An algorithm for propagating measurement uncertainty in a system of interconnected modules is presented. The method adheres strictly to current best-practice in the evaluation and reporting of measurement uncertainty. It allows modular instrumentation systems to be designed that will propagate uncertainty automatically. The algorithm is simple, gen...

A method is described for designing modular instrumentation systems and software frameworks that inherently propagate measurement uncertainty according to current international best-practice. The method uses a mathematical procedure that is equivalent to the 'Law of Propagation of Uncertainty' presented in the "Guide to the Expression of Uncertaint...

In some measurement uncertainty calculations there is uncertainty in the scale factor of a range of values that may be attributed to a quantity. This poses special problems. For real-valued calculations, international guidelines indicate two possible approaches, one loosely based on classical statistics, the other on Bayesian statistics. Complex-va...

This is the final report on the APMP bilateral supplementary comparison in RF power measurement P1-APMP.EM.RF-S4. The purpose of the comparison is to determine the level of consistency of calibration results given by two national standards laboratories. This is a comparison of one of the high-frequency key quantities. The comparison protocol was ba...

The paper describes a simple approach to software design in which the 'Law of propagation of uncertainty' is used to obtain measurement results that include a statement of uncertainty, as described in the Guide to the Expression of Uncertainty in Measurement (ISO, Geneva, 1995). The technique can be used directly for measurement uncertainty calcula...

This paper explores a recent suggestion to use a bivariate form of the Gaussian 'error propagation law' to propagate uncertainty in the measurement of complex-valued quantities (Ridler N M and Salter M J 2002 Metrologia 39 295–302). Several alterative formulations of the law are discussed in which the contributions from individual input terms are m...

Electron diffraction is a direct method which can be used to probe insitu the structure of unsupported nanoparticles in molecular beams. The technique is important because it permits a determination of structure, free of the perturbing effects of a substrate, matrix or chemical contamination. This chapter reviews diffraction experiments with partic...

A software technique is described that provides support for uncertainty calculations for complex-valued measurements. The technique is based on classical methods of multivariate statistics and propagation of variance as well as automatic differentiation, which is an established computational method. It facilitates propagation of covariance data by...

A method of uncertainty analysis based on classical statistical principles is presented for a measurand that is a linear combination of multidimensional input quantities. The method assigns the measurand a combined standard uncertainty matrix and an effective degrees of freedom, which allows the measurand to be estimated by an ellipsoidal confidenc...

This paper builds on recent contributions to ARFTG on the calculation and propagation of measurement uncertainty for complex-valued quantities according to N.M. Ridler and M.J. Salter (2000) and N.M. Ridler and M.J. Salter (2001). It describes the design of a simple software framework that substantially reduces the complexity of uncertainty calcula...

A computational technique is described that can perform the simultaneous evaluation of a measurement function and the associated combined standard uncertainty. The uncertainty equation is automatically derived from the measurement function and need not be provided separately. The technique is based on automatic differentiation. It is simple, effici...

A method of encapsulating and propagating measurement uncertainty information in modular instrumentation systems is described. The technique is simple to implement and fully compliant with international best practice. Measurement uncertainty is a fundamental aspect of any measurement system and is currently difficult to manage when systems are subj...

Early in the 20th century, expressions involving abstract physical quantities gained acceptance. For example, Newton's third law, $F=ma$, holds regardless of units, provided we understand and apply the rules of quantity calculus. Scientists, engineers and technicians are trained to use quantity equations. Perhaps unsurprisingly, this skill is diffi...

A beam of Pb clusters is produced with the inert gas aggregation method and probed by electron diffraction. Analysis of the diffraction patterns indicates that average cluster size can vary between 3 and 7 nm, according to nucleation conditions. The diffraction patterns from beams with larger average cluster size are very similar to patterns calcul...

: We present new results of electron diffraction experiments on unsupported nanometer-sized bismuth clusters. The high intensity cluster beam, necessary for electron diffraction, is provided by an inert-gas aggregation source. The cluster beam contains particles with average cluster sizes between 4.5 and 10 nm. When using Helium as a carrier gas we...

The Welch-Satterthwaite (W-S) formula is a classical, approximate method for combining several sample estimates of normal population variances. However, the W-S formula is routinely applied to the results of physical measurements even though uncertainties may include those of Type B. This paper considers the validity of the W-S formula as a means o...

The design of software for automatic calculation of uncertainty is described. Using a mathematical model of a measurement, expressions for uncertainty can be derived, establishing the sensitivity of a result to any variation in the uncertain input terms. We show how software suitable for measurement systems can be designed to automate the evaluatio...

Molecular dynamics simulations of nanometer-sized lead clusters have been performed using the Lim, Ong and Ercolessi glue potential (Surf. Sci. {\bf 269/270}, 1109 (1992)). The binding energies of clusters forming crystalline (fcc), decahedron and icosahedron structures are compared, showing that fcc cuboctahedra are the most energetically favoured...

Nanometre-sized particles are of considerable current interest because of their special size-dependent physical properties. Debye±Scherrer diffraction patterns are often used to characterize samples, as well as to probe the structure of nanoparticles. Unfortunately, the well knowScherrer formula' is unreliable at estimating particle size, because t...

Chemically synthesized nanoparticles can be regarded as building blocks for more complex systems. Hence, the characterization of their attributes is a key step toward controlling and tuning the properties of new nanocomposite materials. We present a structural characterization of thiol-capped gold nanoparticles as a function of size (2−4 nm) by X-r...

Two Monte Carlo algorithms that calculate measurement uncertainties are presented. The algorithms are discussed in the context of the International Organisation for Standardization's (ISO's) published guidelines for the expression of uncertainty in measurement. They are intended for use in problems where more than one estimated input quantity has f...

The special features of diffraction from particles in the nanometer size range are discussed and the problem of structure determination is considered. A direct method of structure analysis, known as Debye Function Analysis, is presented and evaluated. The method is capable of obtaining information about both the size and structure of domains in a s...

A simple software framework to manage the concurrent execution of small, programmable, tasks is described. The system is intended for use with a digital signal processor-based scanning probe microscope and is written in ANSI C. The design provides independent memory management, message passing, and structured programming of tasks. © 2000 American I...