Metrology and theory of measurement

De Gruyter Studies in Mathematical Physics 20

Editors

Michael Efroimsky, Bethesda, USA

Leonard Gamberg, Reading, USA

Dmitry Gitman, São Paulo, Brasil

Alexander Lazarian, Madison, USA

Boris Smirnov, Moscow, Russia

Valery A. Slaev

Anna G. Chunovkina

Leonid A. Mironovsky

Metrology and Theory of

Measurement

De Gruyter

Physics and Astronomy Classification Scheme 2010: 06.20.-f, 06.20.F-, 06.20.fb, 03.65.Ta,

06.30.-k, 07.05.Rm, 07.05.Kf, 85.70.Kh, 85.70.Li

ISBN 978-3-11-028473-7

e-ISBN 978-3-11-028482-9

ISSN 2194-3532

Library of Congress Cataloging-in-Publication Data

A CIP catalog record for this book has been applied for at the Library of Congress.

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;

detailed bibliographic data are available in the Internet at http://dnb.dnb.de.

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Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen

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www.degruyter.com

“To measure is to know.”

Lord Kelvin

“We should measure what is measurable

and make measurable what is not as such.”

Galileo Galilei

“Science begins when they begin to measure ...”

“Exact science is inconceivable without a measure.”

D.I. Mendeleyev

“. . . for some subjects adds value only an exact match

to a certain pattern. These include weights and mea￾sures, and if the country has several well-tested stan￾dards of weights and measures it indicates the pres￾ence of laws regulating the business relationship in

accordance with national standards.”

J.K. Maxwell

Preface

During the 125th anniversary of the Metre Convention in 2000, Steve Chu, now Pres￾ident Obama’s Energy Secretary and a one-time metrologist, spoke and included the

now famous quotation: “Accurate measurement is at the heart of physics, and in my

experience new physics begins at the next decimal place”.

Metrology, as the science of measures or measurements traceable to measurement

standards [323], operates with one of the most productive concepts, i.e., with the con￾cept of measurement accuracy, which is used without exception in all natural and tech￾nical sciences, as well as in some fields of social sciences and liberal arts.

Metrology, by its structure, can be considered a “vertically” designed knowledge

system, since at the top level of research it directly adjoins the philosophy of natural

sciences, at the average level it acts as an independent section of the natural (exact)

sciences, and at the bottom level it provides the use of natural science achievements

for finding solutions to particular measurement tasks, i.e. it performs functions of the

technical sciences. In such a combination, it covers a range from the level of a knowl￾edge validity criterion to the criterion of correctness in the process of the exchange of

material assets.

For metrology the key problem is to obtain knowledge of physical reality, which

is considered through a prism of an assemblage of quantity properties describing the

objectively real world. In this connection, one of the fundamental tasks of metrology is

the development of theoretical and methodological aspects of the procedure of getting

accurate knowledge relating to objects and processes of the surrounding world which

are connected with an increase in the measurement accuracy as a whole. Metrology,

as the most universal and concentrated form of an organizing purposeful experience,

allows us to make reliability checks of the most general and abstract models of the

real world (owing to the fact that a measurement is the sole procedure for realizing the

principle of observability).

Metrology solves a number of problems, in common in a definite sense with those

of the natural sciences, when they are connected with a procedure of measurement:

the problem of language, i.e. the formalization and interpretation of measurement

results at a level of uniformity;

the problem of structuring, which defines the kind of data which should be used

depending on the type of measurement problem being solved, and relates to a system

approach to the process of measurements;

viii Preface

the problem of standardization, i.e. determination of the conditions under which the

accuracy and correctness of a measurement result will be assured;

the problem of evaluating the accuracy and reliability of measurement information

in various situations.

At present, due to the development of information technologies and intelligent mea￾surement systems and instruments, as well as the growing use of mathematical meth￾ods in social and biological sciences and in the liberal arts, there were a number of

attempts to expand the interpretation of metrology [451] not only as the science about

measurements of physical quantities, but also as a constituent of “gnoseo-techniques”,

information science, “informology” and so on, the main task of which is "to construct

and transmit generally recognized scales for quantities of any nature", including those

which are not physical. Therefore, it remains still important to determine the place of

metrology in the system of sciences and its application domain more promptly, i.e., its

main directions and divisions [9, 228, 429, 451, 454, 503, 506, and others].

It is possible to analyze the interconnection of metrology and other sciences from

the point of view of their interaction and their mutual usefulness and complemen￾tariness, using as a basis only its theoretical fundamentals and taking into account a

generally accepted classification of sciences in the form of a “triangle” with vertexes

corresponding to the philosophical, natural, and social sciences [257], but paying no

attention to its legislative, applied, and organizational branches.

Among such sciences one can mark out philosophy, mathematics, physics, and tech￾nical sciences, as well as those divisions of the above sciences, the results of which are

actively used in theoretical metrology, and the latter, in its turn, provides them with

materials to be interpreted and given a meaning to.

It is known that in an application domain of the theoretical metrology two main

subdivisions can be singled out: the general theory of measurements and the theory of

measurement assurance and traceability.

The general theory of measurements includes the following directions:

original regulations, concepts, principles, postulates, axiomatic, methodology,

terms, and their definitions;

simulation and investigation of objects, conditions, means, and methods of perform￾ing measurements;

theory of scales, measures, metrics, references, and norms;

theory of measurement transformation and transducers;

theory of recognition, identification, estimation of observations, and data process￾ing;

theory of measurement result uncertainties and errors;

theory of dynamic measurements and signal restoration;

Preface ix

theory of enhancement of the measurement accuracy, sensitivity, and ultimate capa￾bilities taking into account quantum and other limitations;

automation and intellectualization of measurement information technologies, inter￾pretation and use of measurement information in the process of preparing to make

decisions;

theory of the optimal planning for a measurement experiment;

theory of metrology systems;

theory of measurement quality estimation, as well as of technical and social-eco￾nomic efficiency of metrology and measurement activities.

The theory of measurement assurance and traceability consists of the following direc￾tions:

theory of physical quantities units, systems of units, and dimensionality analysis;

theory of measurement standards;

theory of reproducing, maintaining, and transferring a size of quantity units;

theory of estimating normalized metrological characteristics of measuring instru￾ments;

methodology of performing metrology procedures;

theory of metrological reliability and estimation of intercalibration (interverifica￾tion) intervals;

theory of estimating the quality of metrology systems, and the methodology of op￾timizing and forecasting their development.

Let the particular features of measurement procedures be considered in the following

order: interaction of an object with a measuring instrument ! recognition and selec￾tion of a measurement information signal ! transformation ! comparison with a

measure ! representation of measurement results.

Interaction of an object under investigation with a measuring instrument assumes

searching, detecting, and receiving (reception) a quantity under measurement, as well

as, if necessary, some preparatory procedures of the probe-selection or probe prepara￾tion type, or exposure of the object to some outside influence for getting a response

(stimulation), determining an orientation and localization in space and time.

Discrimination or selection assume marking out just that property of an object to

which the quantity under measurement corresponds, including marking out a useful

signal against a noise background and applying methods and means for noise con￾trol.

Transformation includes changes of the physical nature of an information carrier or

of its form (amplification, attenuation, modulation, manipulation, discretization and

quantization, analog-digital and digital-analog conversion, coding and decoding, etc.),

as well as the transmission of measurement information signals over communication

x Preface

channels and, if necessary, recording, storing, and reproducing them in memory de￾vices.

Comparison with a measure can be realized both directly and indirectly with the

help of a comparator or via some physical or technical mechanism. A generalization

of this procedure is the information comparison with an image.

Representation of measurement results assumes data processing according to a cho￾sen algorithm, evaluation of uncertainties (errors) of a measurement result, represen￾tation of measurement results with a digital display, pointer indicator device, analog

recorder, hard copy (printing), or graphical data representation, their use in automatic

control systems, semantic interpretation (evaluation) of the results obtained, identifi￾cation, structuring, and transmission of the results into data and knowledge bases of

artificial intelligence systems.

In performing a measurement procedure it is very important to use a priori and a pos￾teriori information. Under a priori information one understands the models selected

for an object, conditions, methods and measuring instruments, type of measurement

scale, expected ranges of amplitudes and frequencies of a quantity to be measured,

etc. Examples of using a posteriori information include improvement of the models

being used, recognition of patterns (images) or their identification, determination of

the equivalence classes to which they refer, as well as structurization for data base

augmentation and preparation for making decisions.

The key procedures to which the measuring instruments are subjected are checks,

control operations, tests, graduation, calibration, verification (metrological validation),

certification, diagnostics, and correction. From the above, in the first turn, the calibra￾tion, verification, certification, and validation can be considered to be metrological

procedures.

It should be noted that many repeated attempts to create a general theory of measure￾ment have been made, using various approaches. Among them there are approaches of

the following types [54, 84, 143, 164, 209, 216, 218, 256, 268, 280, 297, 320, 362, 368,

382, 384, 404, 412, 437, 438, 481, 487, 489, 493, 496, 510, 551, and others]: energet￾ical, informational, thermodynamical, algorithmical, theoretical-set, theoretical–bulk,

statistical, representational, analytical, quantum ones, as well as approaches using al￾gebraic and geometric invariants, and others. Moreover, there are some particular the￾ories of measurement which have been quite well developed and which presuppose an

experimental determination of particular physical quantities in various fields of mea￾surements.

Taking into account the technology of performing measurements, which includes

the above indicated procedures of interaction between an object and measuring instru￾ment, detection and selection of a signal, transformation of measurement information,

comparison with a measure and indication of measurement results, it is possible to

consider various scales used in performing measurements (and a corresponding axiom￾atic), i.e. from the nominal to absolute ones, and to discuss a relationship between a

set of scales and a sequence of measurement procedure stages.

Preface xi

On the basis of an analysis of links of the general theory of measurements, as one

of the main components of the theoretical metrology, with philosophy, mathematics,

physics, and technical sciences, it is possible to determine divisions of mathematics

[242, 450], the results of which are used in the theoretical metrology. They are as

follows:

set theory, including measures and metrics;

theory of numbers, including the additive and metric ones;

mathematical analysis, including the differential, integral, calculus of variations, op￾erational, vector, and tensor calculus;

higher algebra, including the algebra of sets, measures, functions; the theory of

groups, rings, bodies, fields, grids, and other algebraic systems; the theory of math￾ematical models and algebraic invariants, etc.;

higher geometry, including the Euclidean, affine, projective, Riemannian, Loba￾chevskian geometries, etc.;

theory of functions and functional analysis, including the theory of metric spaces,

norms, and representations;

spectral and harmonic analysis, including the theory of orthogonal series and gen￾eralized functions;

mathematical physics with models as well as with direct, inverse, and boundary

problems;

probability theory and mathematical statistics, including a statistical analysis: con￾fluent, covariance, correlation, regression, dispersion, discriminant, factor, and clus￾ter types, as well as a multivariate analysis and the theory of errors, observation

treatment, and statistical evaluation; the theory of optimal experiment planning and

others;

game theory, including the utility theory and many-criterion problems;

theory of systems, including dynamic ones; ergodic theory, perturbation and stability

theory, optimal control theory, graph theory, theory of reliability and renewal theory,

and others;

mathematical logic, including theory of algorithms and programming, calculation of

predicates, propositional calculus, mathematical linguistics, evidence theory, train￾able system theory, and so on;

computational mathematics.

An object of measurement, as the well-known Polish metrologist Ya. Piotrovsky [384]

has remarked, is “the formation of some objective image of reality” in the form of a

sign symbol, i.e. a number. Since we wish to get the results of measurement in the

form of a number, particularly a named one, then the scale of a physical quantity has

xii Preface

to correspond to axioms, postulates, and foundations [313] of a number system being

in use.

At the same time, I. Newton [365] gave the following definition to the number:

“Under a number we understand not so much a set of units, as an abstract ratio of

some quantity to that of the same kind which is accepted as a unit”. This definition, in

an explicit form, corresponds to the proportional measurement scale or scale of ratios

[268, 382].

The present volume sums up the work of many years done by the authors in the fields

indicated above and includes the results of activities on investigation of some actual

aspects and topics of measurement theory (printed in italics) over a period of more

than 30 years [36, 37, 95-115, 347, 349-351, 353, 354, 451-457, 462-464, and oth￾ers]. The priority of research is confirmed by the number of the authorship inventions

certificates [42–44, 265, 272, 383, 448, 449, 465, 469–476, 480]. The basic part of

the investigations was conducted at the D. I. Mendeleyev Institute for Metrology and

the St. Petersburg State University of Aerocosmic Instrumentation, where the authors

have been working during the course of nearly 40 years.

It should be noted that in the text of this book data accumulated for a long time pe￾riod in Russian metrological institutes is presented.This data does not always coincide

with the presemt-day and commonly accepted ideas and points of view of the global

metrological community. However, they firstly do not conflict with the basic (funda￾mental) metrological statements and concepts, and secondly, they accentuate special

features of metrology application in Russia and the member-states of the Common￾wealth of Independent States (CIS).

The authors are grateful to T. N. Korzhakova for her help in translating the text into

English.

Contents

Preface vii

Abbreviations xix

1 International measurement system 1

1.1 Principles underlying the international measurement system . ....... 1

1.2 Classification of key comparisons of national measurement standards . 5

1.3 Basic approaches to evaluating key comparison data . . . ........... 9

1.4 Expression of the degree of equivalence of measurement standards on

the basis of a mixture of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Evaluation of regional key comparison data . . . . . . . . . . . . . . . . . . . . . 15

1.5.1 Two approaches to interpreting and evaluating data of regional

key comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5.2 Equation of linking RMO and CIPM KC. Optimization of the

algorithm of evaluating degrees of equivalence . . . . . . . . . . . . 18

1.5.3 Different principles for transforming the results of regional

comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6 Bayesian approach to the evaluation of systematic biases

of measurement results in laboratories . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.7 Evaluation of measurement results in calibrating material measures

and measuring instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.7.1 Formulating a measurement model . . . . . . . . . . . . . . . . . . . . . . 32

1.7.2 Evaluation of measurement uncertainty . . . . . . . . . . . . . . . . . . 39

1.7.3 Calculation of measurement uncertainty associated with a

value of a material measure using Bayesian analysis . . . . . . . . 42

1.7.4 Determination of the linear calibration functions of measuring

instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2 Systems of reproducing physical quantities units and transferring their

sizes 53

2.1 Classification of reproducing physical quantities units and systems for

transferring their sizes (RUTS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.1.1 General ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

xiv Contents

2.1.2 Analysis of the RUTS systems . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.1.3 Varieties of the elements of a particular RUTS system . . . . . . 74

2.1.4 Interspecific classification of particular RUTS systems . . . . . . 77

2.1.5 Some problems of the specific classification of particular

RUTS systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.1.6 The technical-economic efficiency of various RUTS systems . 92

2.2 Physical-metrological fundamentals of constructing the RUTS systems 96

2.2.1 General ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

2.2.2 Analysis of the state of the issue and the choice of the

direction for basic research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

2.2.3 Foundations of the description of RUTS systems . . . . . . . . . . . 108

2.2.4 Fundamentals of constructing a RUTS system . . . . . . . . . . . . . 129

2.2.5 System of reproduction of physical quantity units . . . . . . . . . . 145

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

3 Potential measurement accuracy 164

3.1 System approach to describing a measurement . . . . . . . . . . . . . . . . . . . 164

3.1.1 Concept of a system approach to the formalized description

of a measurement task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.1.2 Formalized description of a measurement task . . . . . . . . . . . . . 166

3.1.3 Measurement as a process of solving a measurement task . . . . 167

3.1.4 Formalization of a measurement as a system . . . . . . . . . . . . . . 169

3.1.5 Target function of a system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

3.2 Potential and limit accuracies of measurements . . . . . . . . . . . . . . . . . . 171

3.3 Accuracy limitations due to the components of a measurement task . . 173

3.3.1 Measurand and object models . . . . . . . . . . . . . . . . . . . . . . . . . . 173

3.3.2 Physical limitations due to the discontinuity of a substance

structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3.3.3 Quantum-mechanical limitations . . . . . . . . . . . . . . . . . . . . . . . 183

3.4 Influence of external measurement conditions . . . . . . . . . . . . . . . . . . . 189

3.5 Space–time limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

4 Algorithms for evaluating the result of two or three measurements 197

4.1 General ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

4.2 Evaluation problem and classical means . . . . . . . . . . . . . . . . . . . . . . . . 201

4.2.1 Classification of measurement errors . . . . . . . . . . . . . . . . . . . . 201

4.2.2 Problem definition and classification of evaluation methods . . 204

4.2.3 Classical means and their properties . . . . . . . . . . . . . . . . . . . . . 207

4.2.4 Geometrical interpretation of the means . . . . . . . . . . . . . . . . . . 220