Принципы нанометрологии, страница 10

Also errors can appear as random orsystematic dependent on how they are treated.The VIM definition of resolution is:smallest change in a quantity being measured that causesa perceptible change in the corresponding indicationand the definition of error is:measured quantity value minus reference quantity value.2.8.3 Uncertainty in measurementAs discussed in the introductory text for section 2.8 all measurements aresubject to some degree of imperfection. It follows that a measured value canbe expected to differ from the true quantity value, and measured valuesobtained from repeated measurement to be dispersed about the true quantityvalue or some value offset from the true quantity value.

A statement1718C H A P T ER 2 : Some basics of measurementof uncertainty describes quantitatively the degree of imperfection ofa measurement.A basic introduction to uncertainty of measurement is given elsewhere[26] although some of the more important terms and definitions aredescribed briefly here.

The Guide to the Expression of Uncertainty inMeasurement (GUM) [27] is the definitive text on most aspects of uncertainty evaluation and should be read before the reader attempts an uncertainty evaluation for a particular measurement problem. A working group ofthe Joint Committee for Guides in Metrology (JCGM), the body responsiblefor maintaining the GUM, is in the process of preparing a number of documents to support and extend the application of the GUM [28].

The first ofthese documents, Supplement 1 to the GUM on the propagation of distributions using a Monte Carlo method [29], has been published.The VIM definition of measurement uncertainty is:non-negative parameter characterizing the dispersion of the quantityvalues being attributed to a measurand, based on the information usedWhen measurement uncertainty is evaluated and reported as a coverageinterval corresponding to a specified coverage probability p, it indicates aninterval that is expected to contain 100p % of the values that could beattributed to the measured quantity.2.8.3.1 The propagation of probability distributionsThe basis for the evaluation of measurement uncertainty is the propagationof probability distributions. In order to apply the propagation of probabilitydistributions, a measurement model of the generic formY ¼ fðX1 ; .; XN Þ(2.2)relating input quantities X1, ., XN, about which information is available,and the measurand or output quantity Y, about which information isrequired, is formulated.The input quantities include all quantities that affect or influence themeasurement, including effects associated with the measuring instrument (suchas bias, wear, drift, etc.), those associated with the artefact being measured (suchas its stability), those associated with the measurement process, and ‘imported’effects (such as the calibration of the instrument, material properties, etc.).Information concerning the input quantities is encoded as probabilitydistributions for those quantities, such as rectangular (uniform), Gaussian(normal), etc.

The information can take a variety of forms, including a series ofindications, data on a calibration certificate, and the expert knowledge of themetrologist. An implementation of the propagation of probability distributionsAccuracy, precision, resolution, error and uncertaintyprovides a probability distribution for Y, from which can be obtained an estimateof Y, the standard uncertainty associated with the estimate, and a coverageinterval for Y corresponding to a stipulated (coverage) probability.

Particularimplementations of the approach are the GUM uncertainty framework (section2.8.3.2) and a Monte Carlo method (section 2.8.3.3).In a Type A evaluation of uncertainty, the information about an inputquantity Xi takes the form of a series of indications xir, r ¼ 1, ., n,obtained independently. An estimate xi of Xi is given by the average of theindications, i.e.,xi ¼ x ¼n1Xxir ;n r¼1(2.3)with associated standard uncertainty u(xi) given by the standard deviationassociated with the average, i.e.,vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiunXu1(2.4)uðxi Þ ¼ sðxÞ ¼ tðxir xi Þ2 ;nðn 1Þ r ¼ 1and degrees of freedom ni ¼ n 1.In a Type B evaluation of uncertainty, the information about Xi takessome other form, and is used as the basis of establishing a probabilitydistribution for Xi in terms of which an estimate xi and the associatedstandard uncertainty u(xi) are determined.

An example is the case that theinformation about Xi takes values between the limits a and b (a b). Then,Xi could be characterized by a rectangular distribution on the interval [a, b]from which it follows that xi and u(xi) are the expectation and standarddeviation of Xi evaluated in terms of this distribution, i.e.,xi ¼bþa;2uðxi Þ ¼ðb aÞpffiffiffi :2 3(2.5)Note that there are other types of distribution, for example triangular andU-shaped.2.8.3.2 The GUM uncertainty frameworkThe primary guide in metrology on uncertainty evaluation is the GUM [27].

Itpresents a framework for uncertainty evaluation based on the use of the law ofpropagation of uncertainty and the central limit theorem. The law of propagation of uncertainty provides a means for propagating uncertainties throughthe measurement model, i.e., for evaluating the standard uncertainty u(y)associated with an estimate y of Y given the standard uncertainties u(xi)1920C H A P T ER 2 : Some basics of measurementassociated with the estimates xi of Xi (and, when they are non-zero,the covariances u(xi, xj) associated with pairs of estimates xi and xj).

Thecentral limit theorem is applied to characterize Y by a Gaussian distribution(or, in the case of finite effective degrees of freedom, by a scaled and shiftedt-distribution), which is used as the basis of providing a coverage interval for Y.In the GUM uncertainty framework, the information about an inputquantity Xi takes the form of an estimate xi, a standard uncertainty u(xi)associated with the estimate, and the degrees of freedom ni associated withthe standard uncertainty.The estimate y of the output quantity is determined by evaluating themodel for the estimates of the input quantity, i.e.y ¼ fðx1 ; .; xN Þ:(2.6)The standard uncertainty u(y) associated with y is determined by propagating the standard uncertainties u(xi) associated with the xi througha linear approximation to the model.

Writing the first-order Taylor seriesapproximation to the model asNXY y ¼ci ðXi xi Þ(2.7)i¼1where ci is the derivative of first order of f with respect to Xi evaluated at theestimates of the input quantities, and assuming the Xi are uncorrelated, u(y)is determined fromu2 ðyÞ ¼NXc2i u2 ðxi Þ:(2.8)i¼1In the equation (2.8), which constitutes the law of propagation ofuncertainty for uncorrelated quantities, the ci are called (first-order) sensitivity coefficients.

A generalization of the formula applies when the modelinput quantities are correlated.An effective degrees of freedom neff associated with the standard uncertainty u(y) is determined using the Welch-Satterthwaite formula, i.e.NXc4i u4 ðxi Þu4 ðyÞ¼:nineffi¼1(2.9)The basis for evaluating a coverage interval for Y is to use the central limittheorem to characterize the random variableY yuðyÞ(2.10)Accuracy, precision, resolution, error and uncertaintyby the standard Gaussian distribution in the case that neff is infinite ora t-distribution otherwise. A coverage interval for Y corresponding to thecoverage probability p takes the formy U:(2.11)U is called the expanded uncertainty given byU ¼ kuðyÞ(2.12)where k is called a coverage factor, and is such thatProbðjZj kÞ ¼ p(2.13)where Z is characterized by the standard Gaussian distribution in the casethat neff is infinite or a t-distribution otherwise.There are some practical issues that arise in the application of the GUMuncertainty framework.

Firstly, although the GUM uncertainty frameworkcan be expected to work well in many circumstances, it is generally difficultto quantify the effects of the approximations involved, which include linearization of the model in the application of the law of propagation ofuncertainty, the evaluation of effective degrees of freedom using the WelchSatterthwaite formula, and the assumption that the output quantity ischaracterized by a Gaussian or (scaled and shifted) t-distribution. Secondly,the procedure relies on the calculation of the model sensitivity coefficients cias the basis of the linearization of the model. Calculation of the ci can bedifficult when (a) the model is (algebraically) complicated, or (b) the model isspecified as a numerical procedure for calculating a value of Y, for example, asthe solution to a differential equation.2.8.3.3 A Monte Carlo methodA Monte Carlo method for uncertainty evaluation is based on the followingconsideration.

The estimate y of Y is conventionally obtained, as in theprevious section, by evaluating the model for the estimates xi of Xi. However,since each Xi is described by a probability distribution, a value as legitimate asxi can be obtained by drawing a value at random from the distribution. Themethod operates, therefore, in the following manner. A random draw is madefrom the probability distribution for each Xi and the corresponding value of Yis formed by evaluating the model for these values.

Many Monte Carlo trialsare performed, i.e., the process is repeated many times, to obtain M, say,values yr, r ¼ 1, ., M, of Y. Finally, the values yr are used to provide anapproximation to the probability distribution for Y.2122C H A P T ER 2 : Some basics of measurementAn estimate y of Y is determined as the average of the values yr of Y, i.e.,y ¼M1Xyr :M r¼1(2.14)The standard uncertainty u(y) associated with y is determined as thestandard deviation of the values yr of Y, i.e.,u2 ðyÞ ¼M1 Xðyr yÞ2 :M 1 r¼1(2.15)A coverage interval corresponding to coverage probability p is an interval[ylow, yhigh] that contains 100p % of the values yr of Y. Such an interval is notuniquely defined.