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MetrologyThe science of measurement

Year 2013-2014

Jean Virieux

The scientific investigation requires oftenquantitative measurements of physical and chemicalquantities.

Other analysis could require other evaluations: more qualitative related to subjective analysis(folded, unfolded, unclassified) which could occur as well in our Earth disciplines.

What is the most difficult one ?

The technical measurement

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MetrologyThe science of measurement

Measurements could be comparative (Jean is taller than Pierre) or could be absolute (Jean is 1.82 m high).

The second case is also a comparison with a global reference (a standard as the standard meter for example)

Relative versus absolute measurements

Examples ?

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Measurements are crucial in Earth sciencesand more specifically in M2P - GERboth in Geology and Geophysics

This presentation comes partiallyfrom teaching lectures ofJean-Paul Masson (engineer at ISTerre laboratory)

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Measurement Outline1. Measurand and output signal « o »2. External parameters3. Calibration4. Measurement errors5. Metrological characteristics of sensors6. Uncertainties estimation7. Practical estimation of uncertainties8. Standard mistakes9. Promoting results through graphics10. Theory11. Conclusion

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Documentation

See Amazon website for textbooks and Wikipedia for other writtensupport

See various websites for measurementwww.omega.com or fr.mt.com (Mettler in French!)

Other suggestions ?

Build up a bibliography ?

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experiment

osensorm

"Ce monde est pénétré des applications de la mesure ;toute connaissance,non mesurable, est frappée d'un jugement de dépréciation.Le nom de "science" se refuse de plus en plus à tout savoir intraduisible

en chiffre."Paul Valéry (1871-1945) : Poète, critique littéraire, essayiste , académicien

Measurand m: the quantity to be measured – ideal and never reached

Output signal o from the sensor (potential difference or current)

The sensor is characterized by the relation o=F(m)

We shall discuss here only a specific measurement and, whenmeasurements are performed through time, we leave the analysis the time series investigation (static case) to other trainings …

1) Measurand and output signalAny measure has an error which could not be determined

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m1 m2 m3

o3Non-linearsensor

S is the sensitivity of the sensor and has physical units related to the output signal and the measurand m

Example: flowmeterMeasurand m has units en m3/so is in VoltS is in Volt /(m3/s)

When linear sensor, the sensitivity is independent of the measurand m:S = a . m + b

Calibration curve F andsensitivity function S

Example: seismometerMeasurand m has units en m/so is in VoltS is in Volt /(m/s)

o2

o1

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Pressuresensor

StabilizedalimentationMeasurand =

pressure amperemeter

4-20 mA

Screen output (analogic)

A/D converter: conversion fromaanlog to digits

Abits

Computer Discrete values Compression Backup Visualisation

interrface

Example of a measurement workflow

Key elements in the chain for reaching the measurand: the final element gives discrete digital quantities

Measuring chain

Discussion about the music

Sigma-Delta ADC Basics

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Illustrative chain

ADC DSP DAC AnalogSignal

AnalogSignal

Digital Signal Processing

Digitizer Converter

A DAC converts an abstract finite-precision number (usually a fixed-point binary number) into a physical quantity (e.g., a voltage or a pressure). In particular, DACs are often used to convert finite-precision time series data to a continually varying physical signal.

AmplificationGainOffsetResponseNoiseSNRGlitch

coming from the sensor

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Physical, chemical and environmentalquantities that may influence the output

signal o or the calibration curve

o

For a specific measurand m, the output signal o varies with the temperature

timemorning evening

Frequent experimental perturbation parametersTemperature : often a strong impact on electronic circuits Magnetic field (terrestrial one CMP)Champ magnétique Vibrations: for many sensors related to pressure and particle velocities Hydrometryo = F (m,g1, g2, ... g i)g i are external parametersg1, g2,... g i are often taken as constants: not entirely true

Remedy : periodic calibration

2) External parametersDrift related to the temperature

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BIPMMass standard

Standard BNM3. 10-9

primer

StandardBNM1. 10-8

transfert

Metrology serviceReference:10-7

Working :10-6

Workingstandards 10-5

to10-6 for industry

Example of the cascade of standards for mass

measurements

Estimation of the relation o = F(m) often with a a graphical description and an algebricexpression (zeroes and poles of this function through a rational representation): we often call it a transfert function.

Often simple calibationOne should include all components of the acquisition chainPeriodicity : often before and after the experiment.« closure« of the calibration

3) Calibration

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Calibration curve for a pressure sensor with up and down curves for pressure

o Relation linéaireo = 0.5 m + 2R2 = 0.95

mb

o = a . m + b

The closure is performed through thre relation o = F(m) betweenthe standard of rank n+1 and the standard of rank n.Calibration certificat: o = F(m) following a strict experimentalprotocol.The standard of rank n+1 provides a known measurand.

3) Calibration

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Etalon de rang nIndication m tient lieu de mesurande

Bouteilleair comprimé

Capteur à étalonner (ou étalon de rang n+1)indication s

Manomètre indicateur

Vanne étanche

Circuit parfaitement étanche

Calibration of a pressure instrument

Case 1 : Fixed pressure generator: watertight for pressure Pressure measurement: standard rank n « connected"

3) Calibration

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Masse M1

Plateau tournant de masse M2

Section cylindre = AM1, M2, A parfaitement connus

Pression P connue

Pressure system to be calibrated

Case 2 : generator of a reference pressure:

3) Calibration

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4) Measurement error

By repeating the measure of an object (the length of the table), we can see variations on the estimation of the measurand.We define the error as the misfit between the measureand the measurand (or the true value), even if this value could not be estimated.

External informations may help braketing this true value

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The need of a modelNot error definition without the introduction of

a model of the measurand …

Example: the length of the table does not vary with time … therefore manymeasurements may improve our estimation of the « true » value …

Without a model of an expected constant length …. Probability framework

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nm...mmm n21 Probability

densityfunction(PDF)

Output signal ooa ob

o- o+

a) systematic errors- Improper instrument: degraded response- Improper use of the instrument: wrong calibration; paraxial reading

(please follow the protocol)b) Accidental errors- Various origins- Unexpected and, therefore difficult to identifiy- Often randomized signatures

If we have a model (constant value for example of the measurand), we mayIncrease the number of measurements=> probability density function (implicit model formulation). Dispersion of measurements through standard deviation Standard deviation

1n...σ

2m)n(m

2m)2(m

2m)(m

1

Error typesExamples ?

,

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Random errorsHow to reduce systematic errors (accuracy)

How to reduce accidental (random?) errors(precision)

An accurate measurement requires a precisemeasurement

A precise measurement may be not accurate

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nm...mmm n21 Probability

densityfunction(PDF)

Output signal ooa ob

o- o+

If we have a model (constant value for example of the measurand), we mayIncrease the number of measurements=> probability density function (implicit model formulation). Dispersion of measurements through standard deviation standard deviation

1n...σ

2m)n(m

2m)2(m

2m)(m

1

The probability of having a measurand between oaand ob is given by the hashed surface.

Gaussian distribution

,

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Accuracy

s = F(m)

True value

Low uncertainty

We may have an upper limit of on the error for the probabilistic point of view

If Gaussian distribution (n independent variables), we may find o+ et o- such that

P(o ∈ [o-,o+])=95% to 99%

normemP %.5.95)2(_

m

3m

3m

%7.99)3(_

mP

Precision

Gaussian distribution

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Accuracy versus precisionThese two entities are confusing because we use the

same word in French for these two concepts.The accuracy of the measurement tells us how near/far

we are from the measurement to the true value expected from the model we have

The precision of the measurement tells us about the variability of the measurement itself without the need of any model representation

Absolute precision and relative precision

Again the table length example

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Relation between repeatability and precision

The repeatability of an instrument ischaracterized by the flatness of the probabilitydensity function

More the curve is narrow, better is the repeatability and, therefore the precision of the measurement.

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m

Capteur : pas fidèle, pas juste * Capteur : juste, pas fidèle **

dp(m)

m

dp(m)

m

Truevalue

m

Capteur : fidèle, juste ****

m

Capteur : fidèle, pas juste ***

dp(m) dp(m)

mm

mm

Classification of sensors

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Significant numbers and rulesfor rounding approximations

Explain 1,234 123,400 123,4 1.000 1.001 100,0

Rounding rules 1,24 1,27 1,2451 (2nd digit after comma) 1.2452 (2nd digit after comma)

(avoid a systematic error)

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The error is of the order of the last selected figure of the number: Cancel out the figures which are not significative.

Do not write: pression = 4500.125843 Pa P = 1. 10-6 Pa

T = 5.025 ±0.053 N

Do write:pression = 45.0 Hecto Pa P = 10 Pa

T = 5.1 ± 0.3

When using a spreadsheet, please use the number of figures for yournumber driven by the physics.

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UncertaintiesBy repeating the measurement, we may see

variations: this disagreement is bounded by a limit which is the uncertainty and which cantake a positive or negative value.

Two kinds of uncertaintiesone comes from the operatus of measurementthe other comes from the definition of the measurand itself: this is a theoreticaluncertainty(example of the length of the table if not a rectangle!)

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The expected or formal errrorIf we are able to determine the pdf, we may

have an estimation of the inaccuracy of the measurand.

If one measurement is done, it is probable that it could not be less accurate that the formal error.

Consequently, disagreement between twomeasurements could not go beyond the formal error if we have a good description of the model.

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Estimation of errors and reductionof uncertainties

Our goal is the estimation of errors, in order to better constrain parameters of our model (which could not be the correct

Therefore, we may want to reduce measurementuncertainties for a better precision but we wantto improve our model description for a betteraccuracy

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Another way to say is that it is not possible to define it unambigously.

We may a tentative estimation

Absolute error= true value – measured value

Estimation, evalutionof upper bounds of errors

Uncertainty rectangles in x & y Practical work

y

x

x

y

Y = ax +y0a min < a <a max y min< y < ymax

Probability Density Function

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Absolute error = measured value – true valueAbsolute uncertainty: uncertainty of a quantity with physical units

l - l < true value « L » < l + l

ll

example : measurement of a water level in a tank with limnimeter.Absolute uncertainty: 0.2 mmMeasured value: 45 .3 mm45.1 mm < True value < 45.5 mmOne must write : L = 45.3mm 0.2 mm

%44.03.452.0

ll

Rule of length l

length l-L

length L+L

6) Uncertainties estimation

Relative uncertainty: uncertainty without any units dimensionless value

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Errors propagationThe error of the sum of two measurements is

equal to the sum of errors of eachindependent measurement

The error of the difference of twomeasurements is equal to the difference of errors of each independent measurement

Give the rules for the product and the division (hinttake the logarithm)

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Uncertainties propagationThe uncertainty of a sum of two independent

measurements is equal to the sum of individual uncertainties

The uncertainty of a difference of twoindependent measurements is equal to the sum of individual uncertainties

Please provide rules for the product and the division operations

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Do it using the logarithmic transformation

QVT

LogQ LogV LogT dQQ

dVV

dTT

QQ

VV

TT

Variational calculus

dVVx

dxVy

dyVz

dz

x dxy dxz dz

VVx

xVy

yVz

z

sumV = x + y - z

zz

yy

xx

VV ΔΔΔΔ

dV = dx + dy - dzV = x + y + z

VΔzΔyΔx

VΔV

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productV = x * y * zdV = y * z * dx + x * z * dy + x * y * dz

ratio

ydy

xdx

VdV

yxV

..

Specific case when we have the same parameter in different parts of the global expression: please locate them and group for a better reduction of the error estimation

)y)-(x

1(dy)y-x

1x1(dx

vdv

y-xdy

y-xdx

xdx

y)-(xy)-d(x

xdx

vdv

yxxv

y-yx1x -y)(

1x1

vv

x

→ΔV

Δ Δy

Δz

ratio→

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- Measurements in hydrodynamics: flow, velocityprecision < 5 % : oftenprecision < 2% : cautiousprecision < 0.5% : expertprecision better than 0.1% : specific standard center

- Measurements of lengthsAn operator could appreciate half the division of standardrule and 1/20 for a caliper for the initial division

- Measurements of time (or frequencies)Time of reaction of the operator and of the detection methodstandard : t = 0.05 à 1 soften t = 0.2 à 0.4 s

7) Practical estimation ofuncertainties

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V = x + y +z

8) Standard mistakes

+ + ΔVV

Δ Δ ΔV

Other mistakes?

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3) Promoting resultsthrough graphics

Scale selectionCurve should filled up the main partReadability

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?

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Careful presentation

Caption : SHOULD BE SELF-CONSISTENT

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Readability: stamp style

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Order of magnitude: absurd or anomalousvaluesMake sure that values make sense

What is the speed of a seismic wave? What isthe age of the Earth?

Meaning of the graph

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Meaning of the plot

CAPTION: what is p and q, what is D, what to see in thispicture, what are units?

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In summary: a good plot

A plot has a title and a caption. Drawing zone should be adapted Scale selection: curve should occupied roughly 2/3 of

the spreadsheet Characters should be adapted for reading. Values in coordinates should have their units

specified (SI). If a quantitative reading is required, use a grid Curves should have uncertainty rectangles: control

EXCEL. Trend curve and correlation coefficient are welcom Values should be of the expected order Caption should help the reader seeing the picture

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A useful graph

Banc soufflerie : trainée du cylindre = f(hauteur d'eau)

y = 43,891x - 0,317R2 = 0,9994

0,00

0,50

1,00

1,50

2,00

2,50

3,00

3,50

4,00

4,50

5,00

0,000 0,020 0,040 0,060 0,080 0,100 0,120

hauteur d'eau : m

forc

e de

trai

née

: N

force mesuréepar le capteur

Linéaire (force mesuréepar le capteur)

Figure a: Measured net force along vertical line and the water levelalong the horizontal line as a pink point of measurement. A lineartrend has been found with a high correlation coefficient.

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Anthology

Le TP marche bien! Nous obtenons des erreurs tout à fait négligeables par rapport aux expériences réalisées dans d’autres TP.

résultat de traînée du cylindre : T = 6 ± 9 N C’est pourquoi le calcul manuel avec le chronomètre n’est pas si superflu

que semble l’indiquer les résultats l’air passe ensuite dans la chambre d’essai, qui, comme son nom l’indique

est le lieu ou se déroulent les essais .

En sortie de soufflerie, un système permet de modifier la vitesse en entrée « On ne peut guère que faire des suppositions dessus, tout comme le sexe

des anges… Cette méthode est à la science ce que le tire bouchon est à la boite de conserve, dans la mesure ou elle est hors de propos. En effet, si on ne connaît pas le résultat, elle ne sert à rien et si on le connaît déjà, elle ne sert pas davantage. »

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More mathematicalpresentation

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Probability distribution and reference sampling

If we have n measurements xi of a measurandx, we may expect to guess the structure of these measurements

In case of an infinite number of measurements, a distribution may explain each value: we call it the probability distribution and we considerthat the set of measurements are the reference sampling

(reference parameter = lim (discretized parameter)

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Mean, median, expected value

The determined parameter should converge towards the parameter of the referencedistribution

Mean :

Median :

Expected value :

1i

ix

1/ 2 1/ 2( ) ( ) 0.5i iP x P x

max max( ) ( )P P x

Symmetrical distribution: values will be identical

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Probability distribution1/ 2

max

The surface under the distribution is normalizedto one.

The probability of having the value x is ( )p x dx

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DeviationsThe deviation of the measurement with respect

to the mean of the reference distribution isdefined as the difference

The mean of the deviation should be equal to zero (show it)

The mean deviation is the mean of the amplitudes of deviations

i id x

1lim ( )iNx

N

lim 0iNd

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Variance and standard deviationThe variance is defined as the limit of the mean

of the square of the deviations

And the standard deviation is

2 2 2 21 1lim ( ) limi iN Nx x

N N

21lim ( )iNx

N

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SignificationThe mean with the median and with the expected

value are parameters which characterize the information we are looking for during the experiment.

In case of any distribution, we may consider thatthe mean value gives the best representation of the reality and, therefore, of the true value.

The variance and the standard deviationcharacterize the uncertainties with ourmeasurements and sample the best the fluctuations of our measurements.

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Discrete distributionWe may define a mean and a standard

deviation of the reference distribution P(x) through following expressions

22

lim ( )

lim ( )

i iN

i jN

x P x

x P x

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Continuous distributionGoing to the limit, we have following

expressions

But in practice we have only a finite numberof measurements

2 2 2 2

( )

( ) ( ) ( )

xP x dx

x P x dx x P x dx

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Sampling Mean

Variance

Unbias variance

1ix x

N

2 2 2 2

2 2 2

1 1( ) 2

1

i i

i

x x xN N

xN

2 2 21 ( )1 is x x

N

N-1 is the number of degrees of freedom, i.e. the number of degrees of freedom minus the number of parameters

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Normal distributionRandom errors are expected to be distributed

using the following theoretical distribution with two parameters (mean and standard deviation)

which represents the probability of measuring the value x

21 1( , , ) exp22G

xP x

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Properties of the normal distribution

Half-length is defined such that, betweenthese values, the probability is equal to 0.5, which gives

( 1/ 2 , , ) 1/ 2 ( , , )G GP P

The tangente curve where is the highest slopegives un interval of values x=±2

1/ 2( 2 , , ) ( , , )G GP e P

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Poison distributionWe may have physical phenomena as the

radioactivity which do not follow a normal/gaussian distribution but anotherlaw which is for this specific case the Poisson distribution

( , ) exp( )!

x

PP xx

The standard deviation is equal to the square root of the mean of this distribution: only one parameter.

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Gaussian Law

Poisson Law Poisson Law

Graphical presentation

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Quality testHow to estimation the correctness of the

distribution law we selectWe may consider the test of X2 which provides

an estimation of the measurementdispersion from those expected from a theoretical Gaussian distribution

If N measurements are performed and N values are obtained, we may look at the expected frequency distribution of thesevalues given by ( ) ( )j jf x NP x

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Quality testFor each value xj, we have a standard deviation

j(f) related to the uncertainty of the frequencydetermination f(xj).

We may estimate the following quantity

Value which could be equat to the number of observations and which indicate that the dispersion of frequencies should be equal for all values.

In fact, we consider the reduced value related to the number of degrees of freedom =(n-2) for the Gaussian distribution

2

22

1

( ) ( )( )

nj j

j j

f x NP xf

22 1

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Conclusion

Measurements should be givent with the uncertainties

Measurements allow to bracket the truevalue by repeating the procedure if errorsare randomly distributed.

It is crucial that other sources of errorsshould be corrected by the operatoir in the experimental protocol.

END OF THE STORY

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-distinction entre la précision intrinsèque de l'instrument de mesure et LA PRÉCISION EFFECTIVE de la mesure.

Exemple :mesure de niveau d'une surface librelimnimètre à pointe l = 0.2 mmfluctuations de l'ordre de 1 mm = vrai l

- TOLÉRANCE DIMENSIONNELLE de réalisationplan du bureau d'étude + 0.4

orifice de diaphragme D = 134 + 0

D = 0.2 mm

ordres de grandeur : bâtiment 1cm

chaudronnerie : 5mmusinage classique : 0.1 mmusinage de précision : 0.01 mm ou mieux

7) Practical estimation ofuncertainties

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5) CARACTÉRISTIQUES MÉTROLOGIQUES D'UN CAPTEURPrécision exactitudeINDICATIONS PROCHES DE LA VALEUR VRAIE DE LA GRANDEUR À MESURER.fidèle et justePRÉCISION EN POURCENTAGE DE L'ÉTENDUE DE MESURE Etendue de mesure : EMPlage de mesure située entre les valeurs extrêmes (mini- maxi)pour lesquelles les caractéristiques métrologiques sont garanties.Capteur de pression

Non destruction 3 EMNon détérioration 1.5 EM

Nominal 1 EMDynamique : rapport entre les valeurs extrêmes de l'étendue de mesureExpression en dB : 20 log10 (Vmax / Vmin)Sensibilité : rapport entre la variation du signal de sortie et la variation dumesurande.Résolution mobilité, quantification : plus petite variation du mesurandemodifiant le signal de sortie s.Seuil : résolution autour du zéroValeur minimale de la mesure.

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étendue de mesure

Ecart de linéarité max

s

mesurande

Linéarité adéquation avec une formulation du type :s = a . m + bSensibilité indépendante de la valeur du mesurandelinéarité : caractérisée par le coefficient de corrélation

Décalage de zéro : offsetDécalage à l'origine du signal de sortie (ex : dérive thermique)

Hystérésis réversibilitéDifférence de valeurs de s suivant l'approche d'une valeur donnée variant de façon croissante ou décroissante.

Résultat quantitatif : Coefficient de corrélation1 = parfait0.9 : médiocre

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Dérive : variation monotone de la sortie pour une entrée constante

Répétabilité reproductibilité, répétitivité

Caractérisé par les différentes valeurs successives obtenues avec un mesurande constant durant un temps court devant une variationsignificative du mesurande. Finesse discrétionInfluence du capteur sur le mesurandeEx : mesure de force avec un dynamomètre : force "prélevée"Ex : mesure de déplacement optique : pas de modification du mesurande

Étendue de mesure

Ecart d'hystérésis max = h/2

s

mesurande

h

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CRITÈRES DE CHOIX D'UN CAPTEUR

-Type de grandeur à mesurer- Etendue de mesure- Précision requise : prix exp (1/précision )- Possibilité d'implantation : poids, encombrement, possibilité de câblage- Grandeurs d'influence spécifiques : haute température, pression.Une précision de mesure élevée peut demander l'utilisation de plusieurs capteurs dont les EM se chevauchent.*Prix*Impossibilité technique : contrainte de poids (spatial)

Ex : On veut mesurer une pression entre 2 et 20 Bars avec une précision de 1/100 sur toute cette gamme soit 0.02 Bar absolu à 2 Bar.Définir 3 capteurs de précision courante : gamme et précision permettant d’obtenir une précision de 1% sur toute cette gamme de mesure

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capteur 0 => 20 Bars à 0.5 %10

5 102

capteur 0 => 5 Bars à 0.4 %5

20Bars

capteur 0 =>10 Bars à 0.5 %

2 Bars

solution 1 : Un seul capteur EM 20 Bars à 0.1%Prix élevésolution 2 : capteur 0 à 5 Bars à 0.4 % + capteur 0 à 10 Bars à 0.5 % + capteur 0 à 20 Bars à 0.5%

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Exemple 1Mesure de débit par mesure de volume transité pendant un temps donné t.

Q = V / t = (S.h)/t = (L .l. h) / tRemarque : 1 t au déclenchement du chronomètre + 1 t à l'arrêt du chronomètre

tΔt2h

ΔhlΔl

LΔL

QΔQ

Exemple 2On cherche à calculer le débit du ventilateur à la vitesse de rotation N0 sachant que la mesure a été effectuée à la vitesse N.Soit No la vitesse de référence visée correspondant à Qo et Pco.Soit N la vitesse correspondant à Q et PcCp est un coefficient de correction.S est la section du diaphragme (diamètre D).Pc est la différence de pression aux bornes du diaphragme mesurée par une hauteur d’alcool(masse volumique al) dans un tube en U.g est l’accélération de la pesanteur

air

PcSCpQ *2**

On sait que :

NNoQQo *

Calculez ?.QQ

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NN

NoNo

hh

gg

DD

CpCp

QoQo

NNohgDCpQo

PcSCpQ

air

air

al

al

air

al

air

21

21

21

212

****2*4

**

.*2**

2

Les grandeurs d'entrée variables (grandeurs mesurées) sontN h alcoolLes grandeurs calculées sont Q et Hs

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Cp correspond aux erreurs d'interpolation pour la détermination de Cp(à évaluer sur la courbe) et l’imprécision de cette valeur.

D correspond à la tolérance de l'usinage des diaphragmes(~ 0.2mm).

h correspond à l'erreur sur la mesure de la colonne d'alcool et sa fluctuation(~ 0.5mm).

N correspond à l'erreur sur la mesure de la vitesse de rotation et sa fluctuation. (~ 1 à 2

NN

hh

DD

CpCp

QoQo

212

Certaines erreurs sont négligeables.La masse volumique de l'alcool al, la masse volumique de l'air air(calculée dans les conditions de l’expérience), la valeur de g sont connues avec une précision très correcte (< 0.2%).No étant une valeur fixée , l'erreur est nulle par principe.