Iec tr 62461 2015

3 A.3 Calc lation of the complete res lt of the me s rement me s red value, pro a i ty den ity distribution, as ociated stan ard u certainty, an the coverage interval... 4 B.3 Calc latio

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Ce t re:c c@ie c

FOREWORD 5

INTRODUCTION 7

1 Sco e 8

2 Normative ref eren es 8

3 Terms an def i ition 9

4 List of s mb ls 12 5 The GUM an the GUM S1 con e t 14 5.1 General con e t of u certainty determination 14 5.1.1 Overview in four ste s 14 5.1.2 Summary of the analytical method f or ste s 3 an 4 15 5.1.3 Summary of the Monte Carlo method for ste s 3 an 4 15 5.1.4 Whic method to u e: Analytical or Monte Carlo 16 5.2 Example of a model fu ction 16 5.3 Col ection of data an existin k owled e for the example 18 5.3.1 General 1

8 5.3.2 Cal bration f actor for the example 19 5.3.3 Zero re din f or the example 2

5.3.4 Re din f or the example 21

5.3.5 Relative resp n e or cor ection factor f or the example 21

5.3.6 Comp rison of pro a i ty den ity distribution f or input q antities 2

5.4 Calc lation of the res lt of a me s rement an its stan ard u certainty (u certainty bu get 2

5.4.1 General 2

5.4.2 Analytical method 2

5.4.3 Monte Carlo method 2

5.4.4 Un ertainty bu gets 2

5.5 Statement of the me s rement res lt an its exp n ed un ertainty 2

5.5.1 General 2

5.5.2 Analytical method 2

5.5.3 Monte Carlo method 2

5.5.4 Re resentation of the output distribution fu ction in a simple form (Monte Carlo method) 31

6 Res lts b low the decision thres old of the me s rin device 31

7 Overview of the an exes 3

An ex A (informative) Example of an u certainty analy is f or a me s rement with an electronic ambient dose eq ivalent rate meter ac ordin to IEC 6 8 6-1:2 0 3

A.1 General 3

A.2 Model fu ction 3

A.3 Calc lation of the complete res lt of the me s rement (me s red value, pro a i ty den ity distribution, as ociated stan ard u certainty, an the coverage interval) 3

A.3.1 General 3

A.3.2 L w level of con ideration of me s rin con ition 3

A.3.3 Hig level of con ideration of me s rin con ition 3

An ex B (informative) Example of an u certainty analy is f or a me s rement with a p s ive integratin dosimetry s stem ac ordin to IEC 6 3 7:2 12 4

B.1 General 4

B.2 Model fu ction 4

B.3 Calc lation of the complete res lt of the me s rement (me s red value, pro a i ty den ity distribution, as ociated stan ard u certainty, an the coverage interval) 41

B.3.1 General 41

B.3.2 L w level of con ideration of workplace con ition 41

B.3.3 Hig level of con ideration of workplace con ition 4

An ex C (inf ormative) Example of an u certainty analy is f or a me s rement with an electronic direct re din neutron ambient dose eq ivalent meter ac ordin to IEC 610 5:2 0 4

C.1 General 4

C.2 Model fu ction 4

C.3 Calc lation of the complete res lt of the me s rement (me s red value, pro a i ty den ity distribution, as ociated stan ard u certainty, an the coverage interval) 4

C.3.1 General 4

C.3.2 Analytical method 4

C.3.3 Monte Carlo method 4

C.3.4 Comp rison of the res lt of the analytical an the Monte Carlo method 4

An ex D (inf ormative) Example of an u certainty analy is f or a cal bration of radon activity monitor ac ordin to the IEC 615 7 series 51

D.1 General 51

D.2 Model fu ction 51

D.3 Calc lation of the complete res lt of the me s rement (me s red value, pro a i ty den ity distribution, as ociated stan ard u certainty, an the coverage interval) 51

An ex E (informative) Example of an u certainty analy is f or a me s rement of s r ace emis ion rate with a contamination meter ac ordin to IEC 6 3 5:2 0 5

E.1 General 5

E.2 Model fu ction 5

E.3 Calc lation of the complete res lt of the me s rement (me s red value, pro a i ty den ity distribution, as ociated stan ard u certainty, an the coverage interval) 5

E.3.1 General 5

E.3.2 Ef fects of distan e 5

E.3.3 Contamination non-u if ormity 5

E.3.4 Sur ace a sorption 5

E.3.5 Other in uen e q antities 5

E.3.6 Un ertainty bu get 5

Biblogra h 5

Fig re 1 – Trian ular pro a i ty den ity distribution of p s ible values n for the cal bration factor N 2

Fig re 2 – Rectan ular pro a i ty den ity distribution of p s ible values g for the zero re din G 0 21

Fig re 3 – Gau sian pro a i ty den ity distribution of p s ible values g for the re din G 21

Fig re 4 – Comp rison of diff erent pro a i ty den ity distribution of p s ible values: rectan ular (broken l ne), trian ular (doted l ne) an Gau sian (sol d l ne) distribution 2

Fig re 5 – Distribution fu ction Q of the me s red value 2

Fig re 6 – Pro a i ty den ity distribution (PDF) of the me s red value 3

Fig re C.1 – Res lts of the analytical (red das ed l nes) an the Monte Carlo method (grey histogram an blue dot ed an sol d l nes) for H*(10)  5

Fig re D.1 – Res lt of the analytical (red das ed l nes) an the Monte Carlo method (grey histogram an blue dot ed l nes) for K T 5

Ta le 1 – Symb ls (an a breviated terms) u ed in the main text (ex lu in an exes) 1

2 Ta le 2 – Stan ard u certainty an method to compute the pro a i ty den ity distribution s own in Fig re 4 2

Ta le 3 – Example of an u certainty bu get f or a me s rement with an electronic dosemeter u in the model fu ction M = N K (G – G 0 ) an low level of con ideration of the workplace con ition , se 5.3.5.2 2

Ta le 4 – Example of an u certainty bu get f or a me s rement with an electronic dosemeter u in the model fu ction M = N K (G – G 0 ) an hig level of con ideration of the workplace con ition , se 5.3.5.3 2

Ta le A.1 – Example of an u certainty bu get for a dose rate me s rement ac ordin to IEC 6 8 6-1:2 0 with an in trument havin a logarithmic s ale an low level of con ideration of the me s rin con ition , se text f or detais 3

Ta le A.2 – Example of an u certainty bu get for a dose rate me s rement ac ordin to IEC 6 8 6-1:2 0 with an in trument havin a logarithmic s ale and hig level of con ideration of the me s rin con ition , se text f or detais 3

Ta le B.1 – Example of an u certainty bu get for a photon dose me s rement with a p s ive dosimetry s stem ac ordin to IEC 6 3 7-1:2 0 an low level of con ideration of the workplace con ition , se text f or detai s 4

Ta le B.2 – Example of an u certainty bu get for a photon dose me s rement with a p s ive dosimetry s stem ac ordin to IEC 6 3 7-1:2 0 an hig level of con ideration of the me s rin con ition , se text f or detais 4

Ta le C.1 – Example of an u certainty bu get for a neutron dose me s rement ac ordin to IEC 610 5:2 03 u in the analytical method 4

Ta le C.2 – Example of an u certainty bu get for a neutron dose rate me s rement ac ordin to IEC 610 5:2 03 u in the Monte Carlo method 4

Ta le C.3 – Res lts of the analytical an the Monte Carlo method 5

Ta le D.1 – List of q antities u ed in formula (D.1) 51

Ta le D.2 – List of data avai a le f or the input q antities of f ormula (D.1) 5

Ta le D.3 – Example of an u certainty bu get for the cal bration of a radon monitor ac ordin to IEC 615 7, se text f or detai s 5

Ta le E.1 – Example of an u certainty bu get for a s r ace emis ion rate me s rement ac ordin to IEC 6 3 5:2 0 , se text for detai s 5

Ta le E.2 – Example of an u certainty bu get for a s rf ace emis ion rate me s rement ac ordin to IEC 6 3 5:2 0 f or the determination of the u certainty at a me s red value of zero 5

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The main tas of IEC tec nical commit e s is to pre are International Stan ard However, a

tec nical commit e may pro ose the publcation of a tec nical re ort when it has col ected

data of a diff erent kin f rom that whic is normaly publs ed as an International Stan ard, f or

example "state of the art"

IEC 6 4 1, whic is a tec nical re ort, has b en pre ared by s bcommite 4 B: Radiation

protection in trumentation, of IEC tec nical commit e 4 : Nu le r in trumentation

This secon edition of IEC TR 6 4 1 can els an re laces the first edition, publ s ed in 2 0 ,

an con titutes a tec nical revision The main c an es with resp ct to the previou edition

are as f ol ows:

– add to the analytical method for the determination of u certainty the Monte Carlo method

f or the determination of u certainty ac ordin to s p lement 1 of the Guide to the

Expres ion of u certainty in me s rement (GUM S1), an

– ad a very simple method to ju ge whether a me s red res lt is sig ificantly diff erent f rom

zero or not b sed on ISO 1 9 9

The text of this tec nical re ort is b sed on the f ol owin doc ments:

En uiry draft Re ort o v tin

Ful information on the votin f or the a proval of this tec nical re ort can b f ou d in the

re ort on votin in icated in the a ove ta le

This publ cation has b en draf ted in ac ordan e with the ISO/IEC Directives, Part 2

The commit e has decided that the contents of this publ cation wi remain u c an ed u ti

the sta i ty date in icated on the IEC we site u der "htp:/we store.iec.c " in the data

related to the sp cific publ cation At this date, the publ cation wi b

• recon rmed,

• with rawn,

• re laced by a revised edition, or

A bi n ual version of this publcation may b is ued at a later date

IMPORTANT – The 'colour inside' logo on the cov r pa e of this publ c tion indic te

th t it contains colours whic are considere to be us f ul f or the cor e t

und rsta ding of its conte ts Us rs s ould theref ore print this doc me t using a

colour printer

The ISO/IEC Guide 9 -3:2 0 , U nc rtainty of me sureme t – Part 3: Guide to the e pres io

of u c rtainty in measureme t (GUM:19 5) as wel as its Sup lement 1:20 8, Pro a ation of

distributio s using a M ont e Carl o met hod (GUM S1), are general g ides to as es the

u certainty in me s rement This Tec nical Re ort lay emphasis on their a pl cation in the

area of radiation protection an serves as a practical introd ction to the GUM an its

s p lement 1 (GUM S1)

The proces of determinin the u certainty del vers not only a n merical value of the

u certainty; in ad ition it prod ces the b st estimate of the q antity to b me s red whic

may diff er fom the in ication of the in trument Th s, it can also improve the res lt of the

me s rement by u in inf ormation b yon the in icated value of the in trument, e.g the

energy de en en e of the in trument

RA DIA TION PROTECTION INSTRUMENTA TION –

This Tec nical Re ort gives g idel nes for the a pl cation of the u certainty analy is ac ord

-in to ISO/IEC Guide 9 -3:2 0 (GUM des ribin an analytical method f or the u certainty

determination) an its Sup lement 1:2 0 (GUM S1 des ribin a Monte Carlo method f or the

u certainty determination) f or me s rements covered by stan ard of IEC Subcommite 4 B

It do s not in lu e the u certainty as ociated with the con e t of the me s rin q antity,

e g the diff eren e b twe n H

p(10) on the ISO water sla phantom an on the p rson

This Tec nical Re ort explain the prin iples of the ISO/IEC Guide 9 -3:2 0 (GUM),its

Sup lement 1:2 0 (GUM S1) an the sp cial con ideration neces ary for radiation

protection at an example taken f rom in ivid al dosimetry of external radiation In the

informative an exes, several examples are given for the a pl cation on in truments, f or whic

SC 4 B has develo ed stan ard

This Tec nical Re ort is s p osed to as ist the u derstan in of the ISO/IEC Guide 9

-3:2 0 (GUM), its Sup lement 1: 2 0 (GUM S1), an other p p rs on u certainty analy is It

can ot re lace these p p rs nor can it provide the b c grou d an ju tif i ation of the

arg ments le din to the con e t of the ISO/IEC Guide 9 -3:2 0 (GUM) an its Sup lement

1:2 0 (GUM S1)

Final y, this Tec nical Re ort gives a very simple method to ju ge whether a me s red res lt

is sig ificantly dif ferent f rom zero or not b sed on ISO 1 9 9

For b t er re da i ty the cor ect terms are not alway u ed throu hout this tec nical re ort

For example, in te d of “ran om varia les of a q antity” only the “q antity” itself is stated

The f ol owin doc ments, in whole or in p rt, are normatively referen ed in this doc ment an

are in isp n a le f or its a pl cation For dated ref eren es, only the edition cited a ples For

u dated referen es, the latest edition of the referen ed doc ment (in lu in any

amen ments) a pl es

IEC 6 0 0 (al p rts): Intern t ion l Elect rotec nic l Vo a ulary (avaia le at

htp:/ www.electro edia.org)

ISO/IEC Guide 9 -3:2 0 , Un ert ainty of me sureme t – Part 3: Guid e t o t he ex pres ion of

u c rtainty in me sureme t (GUM:19 5)

ISO/IEC Guide 9 -3, Sup lement 1:2 0 , Un ertainty of me sureme t – Part 3: Guid e t o t he

expres ion of u c rtainty in me sureme t (GUM:19 5) – P ro a atio of d istribut ions usin a

Monte Carlo method

3 Terms and def initions

For the purp ses of this doc ment, the tec nical terms of IEC 6 0 0-151 [1] an

IEC 6 0 0-31 [2] as wel as the folowin definition taken fom the ISO/IEC Guide 9

q otient of the true value of a q antity an the in icated value f or a sp cified referen e

radiation u der sp cified ref eren e con ition

complete re ult of a me s reme t

set of values atributed to a me s ran , in lu in a value, the cor esp n in u certainty an

the u it of me s rement

Note 1 to e try: Th c ntral v lu of th wh le (s t of v lu s) c n b s le te a me sured val ue a d a

p rameter c ara terisin th dis ersio a un ert aint y

Note 2 to e try: Th re ult of a me s reme t is relate to th indicat i n given by t he inst rume t a d to th v lu s

of c re tio o tain d b c lbratio a d b th u e of a mo e

Note 3 to e try: In this Te h ic l Re ort, th “me s re v lu ”, s e Note 1 a o e, is a bre iate b M

Note 4 to e try: In this Te h ic l Re ort, th “in ic tio giv n b th in trume t , s e Note 2 a o e, is

a bre iate b G, a d c le “in ic te v lu ”

Note 5 to e try: In this Te h ic l Re ort, th “mo el”, s e Note 2 a o e, is c le “mo el fu ctio ”, s e 3.10 a d

3.6

de ision thre hold

m*

value of the estimator of the me s ran , whic when ex e ded by the res lt of an actual

me s rement u in a given me s rement proced re of a me s ran q antifyin a ph sical

ef fect, one decides that the ph sical ef fect is present

Note 1 to e try: Th d cisio thre h ld is d fin d s c th t in c s s wh re th me s reme t re ult, m, e c e s

th d cisio thre h ld, m*, th pro a i ty th t th tru v lu of th me s ra d is z ro is le s or e u l to a c o e

diff eren e b twe n the in icated values f or the same value of the me s ran of an in icatin

me s rin in trument, or the values of a material me s re, when an in uen e q antity

as umes, s c es ively, two diff erent values

Note 1 to e try: This d finitio is a plc ble to al me s rin in trume ts a d influ n e q a titie , b t it s o ld

mainly b u e in th s c s s, wh re this d viatio is in e e d nt of th in ic te v lu

q antity def i in an interval a out the res lt of a me s rement that may b exp cted to

en omp s a large f raction of the distribution of values that could re sona ly b at ributed to

q antity value provided by a me s rin in trument or a me s rin s stem

Note 1 to e try: An in ic tio is ofte giv n b th p sitio of a p inter o th dis la for a alo u o tp ts, a

dis la e or printe n mb r f or digital o tp ts, a c d p tern f or c d o tp ts, or a a sig e q a tity v lu f or

Th c lc latio s a c rdin to this mo el fu ctio are n t alwa s p rorme On main p rp s of this mo el fu

c-tio of th me s reme t is, th t it is n c s ary for a y d termin tio of th u c rtainty a c rdin to th GUM (s e

GUM, 3.1.6, 3.4.1 a d 4.1; s e als 5.2 of this Te h ic l Re ort)

Note 2 to e try: In th GUM th me sured val ue is c le val ue of t heme sura d

3.13

probabi ty de sity f unction <for a contin ou ran om varia le

f(x)

the derivative (when it exists) of the distribution fu ction: f(x)=dF(x) d

Note 1 to e try: f(x)·dx is th “pro a i ty eleme t: f(x)·dx=Prx<X<x+ x); in g n ral

set of sp cified values an /or ran es of values of influen e q antities u der whic the u cer

tainties, or l mits of er or, admis ible f or a me s rin in trument are the smalest

MG

q otient of the resp n e an the ref eren e resp n e u der sp cif ied con ition

Note 1 to e try: For th s e if i d refere c c n itio s, th re p n e is th re ipro al of th c lbratio f actor

3.17

re pons

R

ratio of the q antity me s red u der sp cif ied con ition by the eq ipment or as embly u der

test an the true value of this q antity

3.18

stan ard un ertainty

stan ard deviation as ociated with the me s rement res lt or an input q antity

Note 1 to e try: Se GUM:2 0 , 2.3.4

Note 2 to e try: Th sta d rd u c rtainty of th me s reme t re ult is s metime c le “c mbin d sta d rd

u c rtainty”

Note 3 to e try: Th q otie t of th sta d rd u c rtainty a d th me s reme t re ult is c le “ elativ sta d rd

u c rtainty” a d s metime giv n a p rc nta e

p rameter, as ociated with the res lt of a me s rement, that c aracterises the disp rsion of

the values that could re sona ly b atributed to the me s ran

Note 1 to e try: Th p rameter ma b , f or e ample, a sta d rd d viatio (or a giv n multiple of it), or th h lf

width of a interv l h vin a state le el of c nfid n e (c v ra e pro a i ty)

[SOURCE: GUM:2 0 , 2.2.3]

4 List of symbols

Ta le 1 gives a lst of the s mbols (an a breviated terms) u ed in the main text of this

Tec nical Re ort (ex lu in an exes)

Table 1 – Symbols (a d abbre iate terms) us d

L wer lmit of a interv l for p s ible v lu s of a q a tity As q a tity

a Up er lmit of a interv l for p s ible v lu s of a q a tity As q a tity

α

Pro a i ty to d te t a eff ect (state a re ult a o e z ro) alth u h in re lty n

ef fe t is pre e t (th tru v lu is z ro) als c le “pro a i ty of als p sitiv

z Ra d m n mb r o t of th interv l 0 1 (e ta g lar distrib tio ) –

5 The GUM a d the GUM S1 conce t

5.1 Ge eral conc pt of unc rtainty determination

5.1.1 Ov rview in f our steps

The GUM:2 0 an its s p lement 1, GUM S1:2 0 :

– con ider avai a le q antities influen in the me s rement, e.g the exp rien e of the

p rson p rf ormin the me s rement,

– are p rtly b sed on the Bayes statistic (esp cial y the GUM S1),

– are international y ac e ted

NOT Th meth d of th GUM a d th GUM S1 are d s rib d a d e plain d in ma y p p rs [3] to [1 ]

The a plcation of the GUM (analytical method) an GUM S1 (Monte Carlo method), not the

ju tification or the mathematic b hin it, wi b des rib d in a simp fied e ample in the

f ol owin s bclau es Further detai s can b fou d in the l terature

The f ol owin f our ste s are neces ary for the pro agation (determination) of u certainty

Esp cial y, for the first two ste s, the exp rtise of the evaluator is es ential

– Ste 1: A mathematical model f un tion (or an algorithm) has to b stated des ribin the

The model fu ction s ould contain every q antity, in lu in al cor ection an cor ection

f actors that can contribute a sig ificant comp nent of u certainty to the res lt of the

me s rement; detai s are given in 5.2

– Ste 2: The avai a le inf ormation f or the input q antities X

) of the output q antity has to b calc lated u in

either the analytical method (explained in 5.1.2) or the Monte Carlo method (explained in

5.1.3) F r this ste , only the a pl cation of mathematic is req ired This tas can,

therefore, b p rf ormed completely by a computer program, f or example, the sof tware

“GUM Workb n h” [12] or “Un ertRadio” [13] detais are given in 5.4

– Ste 4: The exp n ed u certainty U(m

̂

) an or the cor esp n in coverage interval have

to b stated; detai s are given in 5.5

5.1.2 Summary of the a alytic l method for steps 3 a d 4

In this s bclau e, a s ort s mmary is given in the f ol owin to i u trate the analytical method:

b) Secon ly, the sen itivity co ff i ient, i.e the p rtial derivative of the output q antity with

resp ct to e c input q antity, has to b calc lated: c

q antity to the output q antity, th s, it is the “lever arm” or “imp ct of the cor esp n in

input q antity

c) Thirdly, the u certainty contribution to the output q antity d e to e c input q antity has

to b calc lated by multiplyin the sen itivity co f ficient an the stan ard u certainty:

u

i(m

d) F urthly, the combined stan ard u certainty for the output q antity is computed as the

s uare ro t of the s uared u certainty contribution : { }

∑

=

=n

iic

mum

u

12

)ˆ

)ˆ

; in case some

(ran om varia les expres in the state of k owled e a out the ac ordin ) input q antities

are cor elated with one another (i.e they de en on e c other), f urther terms ne d to b

ad ed to the s m u der the s uare ro t sig , as detai ed in 5.2 of the GUM:2 0

e) Final y, the exp n ed u certainty for the output q antity has to b calc lated by

multiplyin the stan ard u certainty with the a pro riated coverage f actor (u ual y k = 2):

U

c

m̂) = 2 · u

c

m̂); if the pro a i ty distribution of the output q antity is not a proximately

Gau sian (or normal), the coverage f actor may have another value, se 6.3 of the

5.1.3 Summary of the Monte Carlo method f or steps 3 a d 4

In this s bclau e, a s ort s mmary, taken fom the introd ction an f rom 5.9.6 of the

GUM S1:2 0 , is given in the f ol owin to i u trate the Monte Carlo method:

This Su pleme t to the GUM is c n ern d with th pro a atio of pro a ility d istrib t ions

thro g t he math matic l model of me sureme t [GU M:19 5, 3.1.6] as a b sis for t he

e aluation ofu c rtainty of me sureme t, a d it s impl eme tatio b a M onte Carlo meth d

Th tre t me t a p es to a mod el h ving a y n mb r ofinp t q a tities, a d a singl e o t put

q a tity Th described Monte Carl o met hod is a pra t ic l al tern t ive t o t he GUM u c rtainty

famework [GUM:19 5, 3.4.8] I h s valu whe

a) l i e rizatio ofthe model pro id es a inad eq ate re rese tation or

b) th pro a il ity de sity fu ct ion (PDF) for the o t put q a t ity de arts a pre ia ly fom a

Ga s ian d istribut ion or a sc led a d shift ed t -d istribution, e.g d ue to mark d asymmet ry

ofd ominating infl ue c q a t ities (i.e those wit h larg u c rtainties) or du to a mod el

fu ct ion with o ly v ry few infl ue c q a t ities whic are, in ad d it ion, n n-Ga s ian

d istributed

The Monte Carlo method can b stated as a ste -by-ste proced re, se 5.9.6 of the

GUM S1:2 0 :

a) sele t th n mb r L of M ont e Carlo trials t o b made;

b) g n rat e L v ctors, b samp n fom the as ig ed PDFs, as re lizat ions ofthe (set of

jwith j = 1 L;

d) sort these L model v lu s into in re sin ord er, using the sort ed model v lues to provid e

th d istribution fu ct ion for th o tput q a tity Q;

e) c lc l at e t he avera e of M

1

Lwhic is a estimate m

f) use Q t o form a a pro riat e c v ra e int erv l for M, for a st ipul at ed c v ra e pro a il ity

p, se 5.5.3

5.1.4 Whic method to us : Analytic l or Monte Carlo?

The Monte Carlo method u ual y del vers b t er estimates of the res lt an the u certainty if

the me s rement con ition are modeled pro erly as no a proximation is a pl ed; this is

confirmed by exp rimental f i din s [1 ] However, the analytical method is e sier to a ply f or

a large n mb r of me s rements as they, for example, oc ur in services p rormin dai y a

large n mb r of simi ar me s rements, an may therefore prefera ly b a pl ed

If the model fu ction is l ne r an the input q antities are l mited s mmetrical y arou d their

centre value, then the analytical method can b u ed

Otherwise, the res lts of b th method s ould b given in order to display their dif feren e

When the 9 % coverage intervals of the Monte Carlo method an of the analytical method do

not deviate by more than 10 %, then the analytical one may be u ed f or the u certainty

determination in simi ar cases, i.e a simi ar model f un tion an simi ar or smal er values of

the u certainty of the input q antities

5.2 Ex mple of a model fun tion

The b sis of an me s rement an the first (an pro a ly most imp rtant ste of the u cer

tainty evaluation is the definition of the me s rement model This is a mathematical

relation hip b twe n al the influen e q antities However, dif ferent evaluators may wel have

diff erent k owled e of the proces , an dif ferent u derstan in s of how the q antities in play

interact an by that state diff erent model fu ction This is an image of the s ientific re lty:

one evaluator is aware of a sp cif i in uen e q antity an th s in lu es it in the model

f un tion, whi e the other is not As a res lt, diff erent u certainties (an mayb even dif ferent

me s rin res lts) can b calc lated by dif ferent evaluators It is, therefore, imp rtant to

explain in detai whic input q antities have b en taken into ac ou t, even when they are

regarded as negl gible

Sin e dif ferent me s rement models typical y wi le d to dif ferent u certainty evaluation ,

this is a source of u certainty, to , often cal ed “model u certainty" [14] [15] If diff erent

models a p ar comp ra ly re sona le to the evaluator, then alternative u certainty

evaluation s ould b p rf ormed to as es the sen itivity of the res lts to the model n

as umption , an p s ibly also to q antify the comp nent u certainty that derives f om the

multiplcity of s c models

The model fu ction is in most cases an analytical f un tion, but the GUM S1 method do s not

req ire this: it can also b an algorithm It is imp rtant that the model gives an u ambig ou

value of the me s ran To explain the model, an example of a direct re din in ivid al

dosemeter wi b con idered The dosemeter’s display in icates the dose directly in u its of

the q antity to b me s red, for example, in µSv or mSv for the q antity H

p(10)

A proven method to set up the model f un tion is to start f rom the prin iple of cau e an ef fect

The cau e – an the aim of the me s rement – is the dose M whic prod ces, d e to the

a solute resp n e R

a s, an in ication of M × R

a s, whic is in re sed by the zero in ication

G

0 Theref ore, the in ication of the dosemeter is given by

G = M R

a s+ G

M is the cau e, f or example, the p rsonal dose eq ivalent H

p(10), whic s al b

me s red;

R

a s

is the a solute resp n e;

RN

is the resp n e relative to the resp n e at cal bration con ition an , th s, ac ou ts for

the diff erent in uen e q antities, f or example, for energ an an le of radiation

in iden e;

K is the cor esp n in cor ection factor for deviation fom cal bration con ition an , th s,

ac ou ts f or the dif ferent in uen e q antities, for example, f or energ an an le of

radiation in iden e

In order to have s mmetrical intervals a out the b st estimate of the influen e q antity, either

R

rel

or K is u ed de en in whic one is l mited s mmetrical y to u ity in the resp ctive

in trument sp cific stan ard, e.g 1,0 ± 0,4 If none is l mited s mmetrical y, the one with the

interval closer to u ity s ould b u ed Ex e tion: If the analytical method is a pl ed K s ould

b u ed in case the stan ard u certainty ex e d 10 % The re son is that a l ne r

a proximation of the model f un tion is implcitly u ed for the analytical method an the

a proximation is not go d enou h f or stan ard u certainties ex e din 10 %, se 5.1.2 of the

GUM:2 08, 7.9 of GUM S1:2 0 , an [10]

Note 1 Wh n th distrib tio of R is lmite s mmetric ly a d it is relativ ly wid , e.g 1,0 ± 0,4, th relatio

K = 1 / R

rel

is n t trivial ie it d e n t le d to a s mmetric l distrib tio of K a d it le d to a oth r (u u ly n t

trivial) pro a i ty d n ity f un tio (P F) For e ample, a re ta g lar distrib tio le d to a h p rb lc o e

Howe er, this is ig ore in this re ort f or two re s n : Firstly, f or th s k of simplcity Se o dly, in trume t

s e if i sta d rd o ly la d wn lmits for th re p n e or c re tio fa tor Th tra sf ormatio of th s lmits via

K = 1 / R

rel

o ly le d to n w lmits Th s, in b th c s s th prin iple of ma imum e tro y (PME) imple a

re ta g lar distrib tio

NOT 2 For a d vic a c mulatin ra iatio o er a lo g p rio of time (or e ample, a p rs n l d s meter b in

worn for s v ral h urs u to mo th ), th v lu of R u u ly is th me n of al v lu s th in ut q a tity to k d rin

th time of me s reme t

Final y, the model f un tion is given by

00

rel

GGKNGG

RN

The model fu ction (5) gives the relation b twe n the me s ran (me s rin q antity) M,

cal ed output q antity of the evaluation (whic is the me s red value), an the input q anti

-ties N, R

rel, (or K,) G an G

0

If one or more input q antity is in the nominator of the model fu ction, the res lts of the

analytical method ne d to b verified u in Monte Carlo method This can b done in the

fol owin way: Determine the 9 % coverage intervals res ltin f rom the Monte Carlo method

an f rom the analytical method: they s ould not deviate by more than 10 %, se 5.1.4 A

p s ible f al ac when p rormin the u certainty analy is is to p rorm the analy is with

f ormula (2) for the in icated value, but this ig ores that the aim of the me s rement is the

cau e M an its as ociated u certainty an not the in icated value G

An alternative method to define a me s rement model is of interest in case some of the input

q antities de en on the me s ran (i.e an implcit relation) In s c cases the so cal ed

o servation f ormula is a s ita le alternative [16]

For routine me s rements, often N = R

rel

= K = 1 a d G

0

= 0 is as umed res ltin in M = G,

whic me n that no cor ection at al is con idered However, when the u certainty

as ociated with the me s rement is dis u sed, the model f un tion in lu in al cor ection

mu t b con idered Th s, in an me s rement, the model f un tion is implcitly in lu ed in

the me s rement proces

The imp rf ect k owled e of the true value wi b taken into ac ou t in s c a way that for the

evaluation b th the input q antities N, R

rel, K, G, G

0

an the output q antity M are b in

re laced by ran om varia les Their p s ible values are denoted by smal leters, for

example, n an r, where s al q antities are writ en in ca ital let ers as in f ormula (5) For

e c q antity the p s ible values are c aracterized by a distribution, whic has an exp

cta-tion value (me n value) denoted by the cor esp n in smal let er with a circ mf lex ac ent,

f or example, n

̂

̂

an a cor esp n in stan ard deviation (stan ard u certainty) of the

exp ctation value, denoted by the let er s an the in ex given by the me n value, f or example,

q antities N, R

rel, (or K,) G an G

0

via the model fu ction Therefore, the distribution of the

p s ible values of the input q antities le d to a distribution of the p s ible values of the

output q antity M This is des rib d by the cor espon in exp ctation value m̂ an its

stan ard deviation In analogy to the s mb ls u ed f or the input q antities this could b given

the s mb l s

m̂

but in al the l terature the s mb l u is u ed, so this is f ol owed here The aim

of the u certainty analy is ac ordin to the GUM an the GUM S1 is the determination of

u(m̂), this s ould b re d as ”u as ociated with m̂” The prin iple method to determine it is to

vary al the input q antities within their ran es of p s ible values This res lts in a variation of

the p s ible values m of the output q antity, whic is determined by the model fu ction This

variation determines a distribution of the output values m whose me n value is m̂ an whose

stan ard deviation is u(m

̂

)

00

rel

ggkngg

rn

is a p s ible value of the resp n e relative to the resp n e at calbration con ition an ,

th s, ac ou ts f or the diff erent influen e q antities, for example, for the energ an the

an le of radiation in iden e;

k is a p s ible value of the cor ection f actor f or deviation f rom cal bration con ition an ,

th s, ac ou ts for the diff erent influen e q antities, f or example, for the energy an the

an le of radiation in iden e;

g is a p s ible value of the in icated value, for example, the re din of the dosemeter in

u its of H

p(10);

g

0

is a p s ible value of the zero re din ;

m is a p s ible value of the me s rement res lt, f or example, of the p rsonal dose

eq ivalent H

p(10), an is calc lated f rom f ormula (6) with the p s ible values for n, …,

The secon ste of the u certainty analy is is the col ection of data an existin k owled e

This in lu es b th mathematical method l ke statistical analy is an other method l ke col

-lectin data fom data s e ts, f or example, cal bration certificates, or u in s ientific an ex

-p rimental exp rien e These other method are the most imp rtant new item introd ced by

the GUM method an they are most imp rtant f or re lstic u certainty calc lation This

sec-on ste of the u certainty analy is de en s as wel as the f irst ste to a gre t extent on the

exp rien e an the k owled e of the evaluator Diff erent evaluators may wel as ig (or

estimate) dif ferent values f or the u certainties of the input q antities an by that calc late

diff erent u certainties for the output q antity This is again an image of the s ientif i re l ty

But this s ould not b interpreted as an u certainty of the u certainty; this is d e to the

diff eren e in information col ected by dif ferent evaluators If the evaluators started with the

same inf ormation (an calc lated cor ectly) the u certainty determined by the evaluators

would b the same

In p rtic lar, the other method mentioned a ove can only b reviewed if the u certainty

analy is is cle rly doc mented An adeq ate doc mentation method, the u certainty bu get,

wi b given in 5.4 In the f ol owin , these method wi b demon trated f or the mentioned

example of an in ivid al electronic dosemeter with the model fu ction of formula (5)

Therefore, the input q antities N, R

rel, K, G an G

0are dis u sed one af ter the other in the

f ol owin s bclau es

5.3.2 Cal bration fa tor f or the e ample

The in ivid al dosemeter is cal brated at the f actory u der ref eren e con ition , for example,

Cs-13 radiation, 0° radiation in iden e an a dose of 0,3 mSv an a dose rate of 5 mSv h

1

Durin the cal bration proces , the dosemeter is adju ted so that the cal bration factor is close

to u ity Theref ore, the calbration f actor N in formula (5) s ould only cor ect f or remainin

imp rection in the adju tment proces Su h imp rection could b d e to the u certainty

of the f ield p rameters of the cal bration faci ty at the f actory – given in the cal bration certifi

-cate of that f aci ty – an l mits for the adju tment given by the factory proced re, for example,

adju tment u ti the deviation of the re din f rom the referen e value is les than 10 %

For simplcity the u certainty of the f ield p rameters of the cal bration faci ty is as umed to

b mu h les than 10 % an can, theref ore, b neglected The tec nician are ad ised to

adju t u ti the deviation of the re din f om the ref eren e value is les than 10 % an ,

f urthermore, p rf orm the adju tment as thorou hly as p s ible Theref ore, no p s ible value

of n is b low 0,9 or a ove 1,1 an most values are very close to u ity The existin k owled e

a out the cal bration f actor N is given by

an by the as umption of a trian ular pro a i ty den ity distribution of n, se Fig re 1 In this

example, the c oice of the trian ular pro a i ty den ity distribution is the decision of the

evaluator, other condition or other evaluators may le d to other distribution

= 0,0 1 for the analytical method A ran om n mb r f om this distribution is given by

n = 0,9 + 0,1 × (z

1+ z ) with the two in e en ent ran om n mb rs z

1

an z f rom the u iform

distribution in the interval 0 1 (ne ded for the Monte Carlo method)

Figure 1 – Tria gular probabi ity de sity distribution

of pos ible v lue n f or the c l bration f actor N

5.3.3 Zero re din for the ex mple

As mentioned a ove, the dosemeter in icates the dose directly in u its of the q antity to b

me s red, th s, a digital display with a resolution of 1 µSv is as umed Durin the adju tment

proced re at the factory, the tec nician are ad ised to adju t the zero re din u ti the dos

e-meter in icates 0 µSv So the zero re din G

0

in formula (5) s ould only cor ect f or remainin

imp rf ection in the adju tment proces Due to a resolution of 1 µSv, the adju tment can

only b done within ± ,5 µSv, otherwise the in ication would b +1 µSv or −1 µSv Dosem

e-ters wi normal y not display negative values, but this is as umed to b p s ible f or

i u tration purp ses The b st estimate (me n value) of G

has the same pro a i ty, as the in ication is alway

0 µSv Con eq ently, the existin k owled e a out the zero re din G

ample, the c oice of the rectan ular pro a i ty den ity distribution is the decision of the

evaluator, other con ition or other evaluators may le d to other distribution

= 0,2 µSv for the analytical method A ran om n mb r f rom this distribution is

Fig re 2 – Re ta gular pro abi ity d nsity distribution

of pos ible value g

0 for the zero re ding G

relative stan ard deviation of the re din s of 4 % is as umed This is not mu h smal er than

the req irement given in IEC 615 6:2 10 [17] A b st estimate of ĝ = 5 0 µSv (arbitrari y

c osen) le d to a distribution given by g = (5 0 + 2 × y) µSv with y a draw f rom the stan ard

Gau sian distribution, se 5.3.6, an to a stan ard deviation of s

several se arate relative resp n es f or diff erent influen e q antities, R

rel

b in the prod ct of

al these In case of in ivid al monitorin , these in uen e q antities are determined by the

workplace con ition , for example radiation energ an direction of radiation in iden e,

cl matic con ition given by temp rature an h midity, dose rate, prevai n d rin dose

me s rement Diff erent levels of con ideration of these workplace con ition are p s ible

The lowest level is the as umption that the dosemeter is adeq ate f or the workplace This

me n that the values of influen e q antities prevai n at the workplace are within the rated

ran es sp cified in the data s e t of the dosemeter This level may b adeq ate for low dose

values far b low the dose l mit

An even worse level could b that the workplace con ition are not covered by the rated

ran es, but this wi not b con idered here

The hig est level of con ideration is given when the workplace con ition of a given dose

me s rement are con idered in detai The values of the influen e q antities are determined

by on site in estigation an the cor ection val d f or these sp cial con ition are a pl ed to

the dose value The cor ection can, f or example, b taken fom the resp n e values

determined in the course of a typ test This level of con ideration may b adeq ate in case

of an ac ident or when the dose value is ne r or a ove the dose lmit

In the fol owin , two examples (low an hig level of con ideration of workplace con ition )

are given

5.3.5.2 Ex mple of low le el of consid ration of workpla e con itions

The workplace con ition are covered by the rated ran es of the influen e q antities given in

the relevant stan ard, f or example, IEC 615 6:2 10, for the dosemeter u ed In other word ,

the dosemeter was adeq ately selected f or the me s rement tas , but the actual values of

the in uen e q antities are not k own or not con idered d rin dose evaluation Becau e the

combined influen e q antity “radiation energ an direction of radiation in iden e” is most

imp rtant, this example wi foc s on this influen e q antity an neglect al the others If n

ec-es ary other in uen e q antities can b in lu ed in analog to the method given here by

introd cin further relative resp n es or cor ection factors If the dosemeter f ulfi s the

requirements of IEC 615 6:2 10 the relative resp n e to photon radiation (relative to the

resp n e to referen e radiation, f or example, Cs-13 ) is b twe n 0,71 an 1,6 within the

whole rated ran e As this ran e is non-s mmetric to u ity, a tran f ormation of varia les fom

the relative resp n e to the cor ection factor, K = 1/R

rel, is done This res lts in a cor ection

f actor b twe n 1,4 an 0,6 Therefore, al p s ible values k of the cor ection factor K are

within this ran e: 0,6 ≤ k ≤ 1,4 This tran f ormation of varia les is done in this case as the

centre value of the res ltin varia le is closer to the exp cted value (u ity) than the centre

value of the original varia le, se 5.2

The c oice of the distribution of k within the ran e given a ove is g ided by the f ol owin f acts

for a me s rin p riod of one day:

a) The p rson we rin the dosemeter is c an in his orientation in the radiation f ield d rin

work b cau e of the p rson ’ movement Therefore, the me n value of the an le of radi

a-tion in iden e is estimated to b close to the centre of the interval vald for the an le In

general, the cor ection factor has its extreme values for extreme values of the an le of

radiation in iden e Theref ore, the me n value for the cor ection factor is exp cted to b

close to the centre of the interval vald f or k

b) The workplace f ield , given for example, by the sp ctral distribution of the photon , are

bro der than the radiation f ield u ed d rin the typ test This also cau es the cor ection

factor to b close to the centre of the interval val d for k

c) The movement of the p rson also c an es the radiation f ield he is in This makes the

ran e of photon energies impin in on the dosemeter even bro der, en an in the

pro a i ty of a cor ection f actor close to the centre of the interval val d for k even more

Al these statements give rise to a distribution that is even more p aked than the trian ular

distribution given in 5.3.2 One p s ible distribution is a normal (Gau sian) distribution where

9 ,7 % of al p s ible k values are within the given interval ( he interval half width is 3 × s

k

̂)

Con ernin the normal distribution, there are 0,15 % of the p s ible k values b low the l mit

of 0,6 an 0,15 % a ove the lmit of 1,4 This is smal enou h to b neglected

The existin k owled e a out the cor ection f actor K is given by

an by a Gau sian pro a i ty distribution of k p aked at the centre of the interval The

Gau sian pro a i ty distribution was c osen as resp n es in workplace con ition are often

q ite close to 1,0 [1 ] As alway , the c oice of the (Gau sian) pro a i ty den ity distribution

is the decision of the evaluator, other con ition may le d to other distribution

= 0,13 A ran om

n mb r f om the cor esp n in distribution is given by k = (1 + 0,13 × y) with y a draw f rom

the stan ard Gau sian distribution, se 5.3.6

5.3.5.3 Ex mple of high le el of consideration of workpla e conditions

The workplace u der con ideration is an X- ay testin eq ipment f or aluminium whe l rims for

cars In the resp ctive energ ran e, the relative resp n e of the dosemeter (relative to the

resp n e to ref eren e radiation, f or example, Cs-13 ) is low, alway b low u ity Therefore, it

is as umed that the cor ection factor is b twe n 1,0 an 1,4 Again, the in icated dose value,

the re din , was 5 0 µSv af ter one workin day As this is an u exp cted hig value, the

me s red dose value s ould b determined con iderin al k owled e of the workplace

Al p s ible values k of the cor ection f actor K are within the ran e: 1,0 ≤ k ≤ 1,4 The

arguments f or the pro a i ty distribution of the k values given a ove are sti val d f or a

workin p riod of one day an wi , therefore, b a pl ed as wel

Therefore, the existin k owled e a out the cor ection f actor K for this example is given by

an by a Gau sian pro a i ty distribution of k p aked at the centre of that interval Again, the

Gau sian pro a i ty distribution was c osen as resp n es in workplace con ition are of ten

= 0,0 7 A ran om

n mb r f om the cor esp n in distribution is given by k = (1,2 + 0,0 7 × y) with y a draw fom

the stan ard Gau sian distribution, se 5.3.6

The cor ected me s red value is m̂ = 1,2 × 5 0 µSv = 6 0 µSv with an as ociated u certainty

smal er than in case of low level con ideration of workplace con ition , this is s own in 5.4

5.3.6 Comparison of probabi ty de sity distribution f or input qua titie

For input q antities that were determined as me n value of several me s rements the

stan ard u certainty is given by the stan ard deviation of a sin le me s rement divided by

the s uare ro t of the n mb r of the me s rements – in the GUM cal ed typ A evaluation of

u certainty To these input q antities u ual y a t distribution can b as ig ed (se 6.4.9.2 an

Ta le 1 of the GUM S1:2 0 )

For al the other input q antities the stan ard u certainty has to b o tained by other than

statistical method , i.e f om an as umed pro a i ty den ity fu ction based on the degre of

b l ef a out the value for the input q antity [often caled s bjective pro a i ty] – in the GUM

cal ed typ B evaluation of u certainty In most cases one of the f ol owin pro a i ty den ity

f un tion can b as umed: a rectan ular, trian ular or Gau sian distribution with its half

width, denoted here by the s mb l a Further distribution are given in ta le 1 of the GUM S1

For al these pro a i ty distribution , the most pro a le value, the b st estimate, is the centre

of the distribution, denoted here by the s mb l x̂ In practice, either the b st estimate, x̂ is

rection factor dis u sed in 5.3.5, an the b st estimate is the me n value

2+

−+

=aa

xˆ

(1 )

For comp rison purp ses, the pro a i ty distribution mentioned in this Tec nical Re ort are

s mmarized in Fig re 4 an the values f or the stan ard u certainty an the cor esp n in

method of computation are given in Ta le 2 Other distribution may also b u ed, if

a pro riate Further examples are given in 6.4 of the GUM S1:2 0

Figure 4 – Comparison of dif f ere t probabi ity de sity distributions of pos ible v lu s:

re ta gular (brok n l ne), tria gular (dot e l ne) a d Ga s ia (sol d l ne) distribution

Table 2 – Sta d rd unc rtainty a d method to compute

the pro abi ty d nsity distributions s own in Figure 4

T pe of distrib tio Stan ard

u c rtainty

Comp tatio

meth d1

Remark

Re ta g lar

3a

x = a

–+ 2 a z

10 % of al p s ible v lu s are within th

interv l fom a

–

toa

+with th c ntre at x

x = a

–+ a (z

1+ z )

10 % of al p s ible v lu s are within th

interv l f rom a

–

toa

+with th c ntre at x̂ a d

a h lf width of a

Ga s ia

3a

x = x

̂

+

3a

a h lf width of a

1

z, z

1, a d z d n te ra d m n mb rs o t of th interv l 0 1 (e ta g lar distrib tio );

y d n te a ra d m n mb r f rom th sta d rd Ga s ia distrib tio

NOT Two v lu s of th sta d rd Ga s ia distrib tio c n b o tain d u in two in e e d nt draws z

1

a d z

21

1

2

c os)ln(

21

2

2sin)ln(

5.4 Calc lation of th re ult of a me s reme t a d its sta d rd unc rtainty

(unc rtainty budget)

The third ste of the u certainty analy is is the calc lation of the res lt of a me s rement an

the as ociated stan ard u certainty ac ordin to the model fu ction This is done u in

esta ls ed mathematical method an may, therefore, also b p rf ormed by sof tware, se

-dard u certainties, s, of the input q antities For every input q antity, the “amou t of this d

e-p n en e is denoted by the s mb l u(m̂) with a s bs ript in icatin the input q antity, for

ex-ample, u

n(m̂) , u

k(m̂) , u

g(m̂) or u

g(m̂) for the input q antities given in formula (5) This

“amou t is given by the “extent to whic the output quantity is in uen ed by variation of the

input q antity multipl ed by the stan ard u certainty of the input q antity The “extent is

cal ed “sen itivity co f ficient , denoted by the s mb l c with a s bs ript in icatin the input

q antity, for example, c , c , c or c

0

f or the input q antities given in f ormula (5) In

mathematical lan uage, the “extent is the c an e of the output q antity, ∆m, d e to a c an e

of a p rtic lar input q antity, for example, ∆n Their q otient ∆m/∆n is the sen itivity

co ff i ient Usin dif ferential calc lu , this is the p rtial derivative of the model f un tion of the

me s rement with resp ct to the p rtic lar input q antity Th s, the sen itivity co ff i ients

ac ordin to formulas (5) an (6) are:

)ˆˆˆ

ˆ

,ˆ

,ˆ

,ˆ

0

00

ggk

NM

c

gGgGkKnNn

ˆ

,ˆ

,ˆ

,ˆ

0

00

ggn

KM

c

gGgGkKnNk

GM

c

gGgGkKnNg

ˆ

ˆ

ˆ,ˆ,ˆ

,ˆ

kn

GM

c

gGgGkKnNg

ˆ

ˆ

ˆ

,ˆ

,ˆ

,ˆ

0

The contribution of the stan ard u certainties of the input q antities to the stan ard u cer

tainty as ociated with the output q antity are then given by:

u

n(m̂) = |c | s

k

̂

u

g(m

g0

m̂) are p sitiv , s th a s lute v lu s of

th s n itivity c eff i ie ts are u e in formula ( 3)

The total stan ard u certainty u(m̂), as ociated with the output q antity m̂ is given by the

ge metrical s m of al these contribution

)ˆ()ˆ)ˆ)ˆ)

mu

gg

kn

22

22

0++

+

pro er c oic of th mo el f un tio For c relate in ut q a titie , s e 5.2 of th GUM:2 0

The cor esp n in u certainty bu get is given in 5.4.4

5.4.3 Monte Carlo method

The pro a i ty den ity fu ction (PDF) for the output q antity M an its stan ard u certainty

has to b o tained f rom the PDFs of the input q antities via the f ol owin ste s:

a) Select the n mb r L of Monte Carlo trials to b made, at le st 1 0 0 0 0 ( his fig re

serves as an example f or the f ol owin ), se also 7.9 in GUM S1:2 0 an [10] The

cor esp n in f i ures for this example of 1 0 0 0 0 trials are given in the f ol owin in

c rly brac ets { ;

b) Generate L vectors, by sampl n fom the as ig ed PDFs, as re l zation of the (set of

i = 1 4) input q antities X

i(N, K, G, an G

0)t

1 L

;

c) For e c s c vector, form the cor esp n in model value of M = h(X

i):

0ggn

jjm

Lm

11

Lmu

1

2

11

)

ˆ

(ˆ

= 7 µSv as ociated with m̂

̂

wi in g n ral n t a re with th mo el e alu te at th b st e timate of th in ut q a titie ,

sin e, f or a n n-ln ar mo el h(X), th e p ctatio v lu of h(X), E[h(X)], is u u ly n t e u l to th mo el

v lu of th e p ctatio v lu s of th in ut q a titie , h[E(X)] (s e 4.1.4 in th GUM:2 0 ) Ir e p ctiv of

wh th r h is ln ar or n n-ln ar,in th lmit a Lte d to infinity, m̂ a pro c e E[h(X)] wh n it e ists

5.4.4 Unc rtainty bu gets

The complete u certainty analy is f or a me s rement – sometimes cal ed the u certainty

bu get of the me s rement – s ould in lu e a l st of al sources of the u certainty together

with the as ociated pro a i ty den ity distribution , stan ard u certainties an the method

of evaluatin them F r re e ted me s rements, the n mb r of o servation also has to b

stated For the sake of clarity, it is recommen ed to present the data relevant f or this analy is

in the f orm of a ta le An example of s c a ta le f or the a ove example of a dose

me s rement with an electronic dosemeter u in the model fu ction of f ormula (5) is given in

Ta le 3 for low level of con ideration of the workplace con ition an in Ta le 4 for hig level

of con ideration of the workplace con ition Column 1, 2, 3, an 4 are relevant f or the

Monte Carlo method whi e column 1, 2, 3, 5, an 6 are relevant f or the analytical method

It can b se n that in case of hig level of con ideration of the workplace con ition , the b st

estimate of the dose is en an ed fom 5 0 µSv to 6 0 µSv This is ac omp nied by a red

c-tion of the stan ard u certainty f rom 7 µSv to 4 µSv, whic is eq ivalent to a relative stan

-dard u certainty of 15 % an 8 %, resp ctively

It can also b se n, that the res lts fom the analytical an the Monte Carlo method are

eq ivalent The re son is that a l ne r a proximation of the model f un tion is val d in the

ran e of the u certainties of the input q antities In this case, it would b s f ficient to a ply

the analytical method f or simi ar cases

Table 3 – Ex mple of a un ertainty bud et f or a me s reme t with a

Table 4 – Ex mple of a un ertainty bud et f or a me s reme t with a

ele tronic dos meter usin the mod l fu ction M = N K (G – G

0) a d

high le el of con ideration of th workpla e conditions, s e 5.3.5.3

) whic covers 6 % of the p s ible values of the output q antity that could

re sona ly b at ributed to the me s rement In general, a larger certainty (coverage pro

-a i ty or level of con den e) is as ed for, therefor, typical y the 9 % coverage interval is

stated to re resent the exp n ed u certainty

For other distribution the p rcentages mentioned a ove dif fer, however, the pro a i ty

distribution of output q antities is of ten q ite simi ar to a Gau sian, se G.2.1 of the

GUM:2 0

5.5.2 Analytic l method

In order to o tain the exp n ed u certainty, the stan ard u certainty is multipl ed by a f actor

larger than one The f actor is cal ed 'coverage factor', u ual y given the s mb l k but to

distin uis it f rom the cor ection factor the s mb l k

c v

is u ed The exp n ed u certainty is

u ual y given the s mb l U (ca ital leter)

For the case of low level of con ideration of the workplace con ition the res lt is

NOT In th e ample, th in re s d k owle g le d to a smaler u c rtainty This is n t alwa s th c s , it is

als p s ible th t a in re s of k owle g le d to a e h n e u c rtainty, for e ample, b c u e n w influ n e

q a titie were id ntif i d whic wereig ore pre io sly

To this statement an explanation s ould b ad ed whic in the general case wi have the

f ol owin content:

The u certainty stated is the exp nded me s rement u certainty o tained by multiplyin

the stan ard u certainty by a coverage f actor k

c v

= 2 It has b en determined in

ac ordan e with the Guide to the Ex res ion of Unc rtainty in Measureme t The value of

the me s ran then normal y l es, with a pro a i ty of a proximately 9 %, within the

atributed coverage interval

As mentioned in 5.5.1 the 9 % (an ac ordin ly k

c v

= 2) are only val d for Gau sian output

distribution whic can, however, mostly b as umed In case other output distribution have

to b as umed, G.6.4 of the GUM:2 0 s ould b con idered

5.5.3 Monte Carlo method

In order to o tain the exp n ed u certainty, the folowin ste s have to b a pl ed:

a) Sort the L model values m

j(at le st L = 1 0 0 0 0 values o tained ac ordin to 5.4.3) into

in re sin order; u e these sorted model values to provide the distribution fu ction for the

output q antity Q, se Fig re 5 for the distribution fu ction of the example;

NOT 1 As me tio e in 5.4.3, 1 0 0 0 0 v lu s is th minimum n mb r of Mo te Carlo trials to b u e In

a ditio , this fig re s rv s a a e ample for th folowin Th c re p n in f i ure for this e ample of

1 0 0 0 0 are giv n in th folowin inc rly bra k ts {

b) As emble the values m

j

into a histogram (with s ita le cel width ) to f orm a f req en y

distribution normal zed to u it are This distribution provides an a proximation to the PDF

f or M, se Fig re 6 for the distribution of the example Calc lation are not general y

car ied out in terms of this histogram, the resolution of whic de en s on the c oice of cel

width , but in terms of Q (se Fig re 5) The histogram can, however, b u eful as an aid

to u derstan in the nature of the PDF, e.g the extent of its as mmetry

c) Use Q to f orm an a pro riate coverage interval [m

low, m

hig] f or M, for a c osen coverage

pro a i ty p, f or example p = 0,9 = 9 % by the fol owin : L t q = pL {= 0,9 × 1 0 0 0 0

= 9 0 0 0} If q is no integer it s ould b rou ded to an integer Then (L – q) {= 5 0 0}

9 % coverage intervals [m

low, m

hig] exist f or M, where m

j = 1 (L – q) {= 1 5 0 0} That me n (L – q) {= 5 0 0} diff erent coverage intervals

exist Two of them are of sp cial interest:

1) The pro a i stical y s mmetric p = 9 % coverage interval is given by takin j = (L –

of the distribution are located

2) The s ortest p = 9 % coverage interval is given by determinin j* s c that, for

j = 1 (L – q) = {1 5 0 0}, the ineq alty m

j*+q– m

j*

≤ m

j+q– m

j

is val d, i.e the

dif feren e m

j*+q– m

In this case the s ortest interval is only 0,3 % s orter than the pro abi stical y s mmetric

one as the PDF is ne rly s mmetric to its me n value an u imodal, i.e it has only one

maximum In case the PDF is non-s mmetric, the len th of the two coverage intervals can

b sig ificantly diff erent; a cor esp n in example is given in C.3.4

For more detai ed inf ormation, Clau e 7 of the GUM S1:2 0 may b u ed as a g ide

Figure 5 – Distribution fun tion Q of the me s re v lue

IEC

Figure 6 – Probabi ity de sity distribution (PDF) of th me s re v lue

For the a ove example, in the case of low level of con ideration of the workplace con ition ,

the complete res lt of the me s rement is given by

M = m̂

lowhig

UU

−+

= (5 0

141

14

−+

M = m

̂

lowhig

UU

−+

= (5 0

13

14

−+

) µSv at p = 9 % (pro a i stical y s mmetric interval) (16 2)

an in the case of hig level of con ideration of the workplace con ition , the complete res lt

of the me s rement is given by

M = m̂

lowhig

UU

−+

= (6 0

99

−+

M = m

̂

lowhig

UU

−+

= (6 0

99

−+

) µSv at p = 9 % (pro a i stical y s mmetric interval) (16 2)

In b th cases, the two intervals overla , th s, these res lts are con istent

To this an explanation s ould b ad ed whic in the general case wi have the f ol owin

content:

The u certainty stated is the exp n ed me s rement u certainty with a coverage

pro a i ty of p = 9 % o tained f om the distribution f un tion of the output q antity It has

b en determined in ac ordan e with Sup lement 1 of the Guid e t o the Ex res ion of

Unc rtainty in Measureme t The value of the me s ran then normal y l es, with a

pro a i ty of a proximately 9 %, within the atributed coverage interval (s ortest or

pro a i stical y s mmetric interval)

NOT 2 In th la t ln in bra k ts eith r th word “pro a i stic ly s mmetric interv l” or “s orte t interv l”

d p n in o whic is th c s s o ld b giv n

Us al y, the s ortest coverage interval s ould b stated b cau e the cor esp n in ran e of

p s ible values is smal est

IE C