Open Access e-Journal Cardiometry - No.14 May 2019

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REVIEW Submitted: 21.7.2013; Accepted: 12.8.2013; Published online: 30.11.2019Justification for measurementequation: a fundamental issue intheoretical metrologyAleksandr V. Prokopov 1*1National Scientific Center "Institute of Metrology"Ukraine, 61002, Kharkov, 42 Mironositskaya str.*Corresponding author:e-mail: info@metrology.kharkov.ua,phone: +38 (057) 700-34-09AbstractA review and a critical analysis of the specialized literature onjustification for the measurement equation and an estimation ofa methodical error (uncertainty) of the measurement result arepresented in the paper, and some prospects for solving of theissue are discussed herein.KeywordsError, Uncertainty, Measurement equation (model equation),Physical and mathematical model, AlgorithmImprintAleksandr V. Prokopov. Justification for measurement equation: afundamental issue in theoretical metrology; Cardiometry; No.3;November 2013; p.58-68 ; doi: 10.12710/cardiometry.2013.3.5868Available from: www.cardiometry.net/no3-november-2013/justification-for-measurement-equationIntroductionStatement of the problem andanalysis of publicationsIn recent years the measurement equation (modelequation) has become an object enjoying close attentionby researchers in metrology in many countries throughoutthe world. The measurement equation in metrologyis interpreted [1, 2] as a functional relationship linkingthe quantity intended to be measured (measurand)with the quantity values immediately measured, as wellas other quantity values which are essential for the measurementprocedure under consideration. The measurementequation is required for solving many metrologicalissues, particularly for development of newmeasurement methods and measuring instruments,including an analysis of measurement result accuracy.The problem of measurement modeling is widelyrepresented in scientific programs of the UK’s NationalPhysical Laboratory (NPL) devoted to the developmentof mathematical models and software for metrologyapplications [3], and issues on modeling measurementequations have a significant place in papers by German[4 - 6], Russian [2, 7 - 10] and Ukrainian [11 - 18] metrologists.It should be noted that most papers give ananalysis of the already known models to select the bestsuited measurement equations. At the same time, thegeneral principles of justification for the equation ofmeasurement and an estimation of a methodical error(uncertainty) of measurement result are still insufficientlytreated.In the specialized literature on the subject, an extremecomplexity of the issue on derivation (justification)of the measurement equation is emphasized;among other things, it is suggested that some specificapproaches relating basically to arts rather than scienceshould be required to derive the measurementequation [2]. It is also argued [8] that the measurementequation is one of the components of a priori knowledgeobtained not by the procedures known from thepoint of view of metrology, but produced by methodsused in some allied sciences. Perhaps that is the reasonwhy there are no instructions in the Guide [1] on howto justify the measurement equation for metrologicalanalysis. Instead, such unusual for scientific and technicalliterature recommendations as to rely on "criticalthinking", "intellectual honesty" and "professionalskill" are used therein.It should be noted that the authors of the Guide tothe Expression of Uncertainty in Measurement [1] areaware of the necessity to improve this document. A fewyears ago, a special working group WG1 was establishedunder the Aegis of the International Bureau of Weightsand Measures (BIPM). One of the main tasks of thisGroup is to prepare specific Supplements to the Guide,including a separate Supplement (JCGM 103) dedicatedto the measurement procedure modeling and methodologyof the measurement equation justification [19].Thereby it confirms the fact that the problem of arigorous justification for an algorithm of obtaining themeasurement equation remains highly topical, and thegiven problem is a subject of the theory of measuringwhich should be referred to the fundamental issues inmetrology.44 | Cardiometry | Issue 14. May 2019
The aim of this paper is to review and criticize theexisting publications on the measurement equationjustification and estimation of a methodical error (uncertainty)of the measurement result as well as discusssome promising ways for solving this problem.Summary and outlinesThe developments [4-6, 11-18], where the main aspectsof the above problem are analyzed and generaloutlines for obtaining the measurement equation areprovided, can be considered as a remarkable step onthe way of finding the proper solution to the aboveissue. In the papers [4-6], for example, an outline ofthis sort consists of five successive stages which are asfollows:I. Description of the measurement, identification of significantquantities (including the measurand and influencequantities) and applied method of measurement.II. Analysis of the measurement, its decompositioninto separate elements , graphical representation ofthe cause-and-effect relations between the elementsfor a certain ideal (not exposed to external influences)measurement within the scope of the standard modelingcomponents.III. Consideration of all distortions, effects of an incompleteknowledge of quantities and external factorscapable of affecting the ideal measurement. Graphicaland mathematical interpretation of the cause-and-effectrelations for the real (exposed to external influences)measurement. Use of corrections for takinginto account distortions (external influences).IV. Identification of reciprocal influence of quantities,introduction of correlations.V. Transformation of mathematical relations whichdescribe the cause-and-effect relationship for obtainingthe model equation.It should be pointed out that the operations necessaryfor the mathematical formulation of the problemare not specifically treated in the above mentionedpublications.In articles [11–18] an algorithm of obtaining themeasurement equation is represented by seven stagesas given below (you can find below the description ofthe algorithm for the case with the only one quantityto be measured to simplify the matter, and for the casewith measuring more than one quantities the procedureshould be similar):I. Selection of a measurement object and preliminaryidentification of its properties and attributes whichare essential for formation of a data signal carryinginformation relevant to the quantity value to be measured(unknown quantity value). Concretization ofthe said quantity.II. Preliminary identification of physical processeswhich may influence the data signal and lead to achange in the latter during signal traveling from theobject of measurement to the measuring device includingtraveling of the signal within the measuring device.Selection of quantities to be immediately measured.III. Formulation of a set of equations (with appropriateinitial and boundary conditions) to describe thephysical processes responsible for forming the datasignal and influencing the latter during traveling ofthe signal from the measurement object to the measuringdevice including traveling of the signal withinthe measuring device (the mathematical formulationof the problem).IV. Analysis of the initial equations. A quantitativeassessment of influence of individual physical effectsand processes on characteristics of the data signalduring transmission of the signal from the measurementobject to the measuring device, includingtransformation of the signal in the measuring device.Simplification of the mathematical formulation of theproblem based on elimination of some physical effectsand processes, which are insignificant for a given levelof accuracy, from consideration. Evaluation of an error(uncertainty) caused by the said negligible effectsand processes.V. Selection and justification of the methods for solvingthe simplified equations for solving the problemwhich allow establishing the relationship between theunknown quantity (the measurand) and the immediatelymeasured quantity values. Generally, this relationshipmay contain known constants, parameterswith fixed values as well as some additional unknownparameters which consider, as an example, an influenceof some external factors and which are to be determinedseparately.VI. Solution of the equations solving the problem,analysis of the established relationship and transformingit to get the most convenient form for derivationof the measurement equation. In this case, two variantsshould be considered as follows: a) additionalunknown parameters are represented as correctionsprovided they are empirically calculated in an independentway; b) the established relationship is usedto formulate a set of equations that allows determin-Issue 14. May 2019 | Cardiometry | 45
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ANALYTICAL NOTECardio-oculometric(c
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