ISO 11929-3:2019

INTERNATIONAL ISO
STANDARD 11929-3
Second edition
2019-02
Determination of the characteristic
limits (decision threshold, detection
limit and limits of the coverage
interval) for measurements of ionizing
radiation — Fundamentals and
application —
Part 3:
Applications to unfolding methods
Détermination des limites caractéristiques (seuil de décision, limite
de détection et extrémités de l'intervalle élargi) pour mesurages de
rayonnements ionisants — Principes fondamentaux et applications —
Partie 3: Applications aux méthodes de déploiement
Reference number
©
ISO 2019
© ISO 2019
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ii © ISO 2019 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 2
3 Terms and definitions . 2
4 Quantities and symbols . 6
5 Evaluation of a measurement using unfolding methods . 8
5.1 General aspects . 8
5.2 Models of unfolding and general uncertainty evaluation . 8
5.3 Unfolding as a sub-model .10
5.4 Input quantities and their uncertainties .10
5.5 Parameters of unfolding.11
5.6 Procedure for unfolding .12
5.7 Modification for Poisson distributed count numbers for unfolding .14
5.8 Evaluation of the primary results and their associated standard uncertainties .15
5.9 Standard uncertainty as a function of an assumed true value of the measurand .16
5.10 Decision threshold, detection limit and assessments .17
5.10.1 Specifications .17
5.10.2 Decision threshold.18
5.10.3 Detection limit .18
5.10.4 Assessments . . .18
5.11 Coverage interval and the best estimate and its associated standard uncertainty.18
5.11.1 General aspects .18
5.11.2 The probabilistically symmetric coverage interval .19
5.11.3 The shortest coverage interval .20
5.12 Documentation .20
Annex A (informative) Correlations and covariances .22
Annex B (informative) Spectrum unfolding in nuclear spectrometric measurement .25
Bibliography .37
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
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ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
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on the ISO list of patent declarations received (see www .iso .org/patents).
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.org/iso/foreword .html.
This document was prepared by This document was prepared by ISO/TC 85, Nuclear energy, nuclear
technologies, and radiological protection, Subcommittee SC 2, Radiological protection.
This second edition of ISO 11929-3 together with ISO 11929-1, ISO 11929-2, cancels and replaces
ISO 11929:2010 which have been technically revised, specifically with reference to the type of statistical
treatment of the data and extended with respect to the methodology of uncertainty assessment from
the ISO/IEC Guide 98-3:2009, to the ISO/IEC Guide 98-3-1:2008.
A list of all the parts in the ISO 11929 series can be found on the ISO website.
iv © ISO 2019 – All rights reserved

Introduction
Measurement uncertainties and characteristic values, such as the decision threshold, the detection limit
and limits of the coverage interval for measurements as well as the best estimate and its associated
standard measurement uncertainty, are of importance in metrology in general and for radiological
protection in particular. The quantification of the uncertainty associated with a measurement result
provides a basis for the trust an individual can have in a measurement result. Conformity with
regulatory limits, constraints or reference values can only be demonstrated by taking into account and
quantifying all sources of uncertainty. Characteristic limits provide, at the end, the basis for deciding
under uncertainty.
This standard provides characteristic values of a non-negative measurand of ionizing radiation. It is
also applicable for a wide range of measuring methods extending beyond measurements of ionizing
radiation.
The limits to be provided according to the ISO 11929 series for specified probabilities of wrong decisions
allow detection possibilities to be assessed for a measurand and for the physical effect quantified by
this measurand as follows:
— the “decision threshold” allows a decision to be made on whether or not the physical effect quantified
by the measurand is present;
— the “detection limit” indicates the smallest true quantity value of the measurand that can still be
detected with the applied measurement procedure; this gives a decision on whether or not the
measurement procedure satisfies the requirements and is therefore suitable for the intended
measurement purpose;
— the “limits of the coverage interval” enclose, in the case of the physical effect recognized as present,
a coverage interval containing the true quantity value of the measurand with a specified probability.
Hereinafter, the limits mentioned are jointly called the “characteristic limits”.
NOTE According to ISO/IEC Guide 99:2007 updated by JCGM 200:2012 the term “coverage interval” is used
here instead of “confidence interval” in order to distinguish the wording of Bayesian terminology from that of
conventional statistics.
All the characteristic values are based on Bayesian statistics and on the ISO/IEC 98-3 Guide to the
Expression of Uncertainty in Measurement as well as on the ISO/IEC Guide 98-3-1 and ISO/IEC 98-3-2.
As explained in detail in ISO 11929-2, the characteristic values are mathematically defined by means of
moments and quantiles of probability distributions of the possible measurand values.
Since measurement uncertainty plays an important part in ISO 11929, the evaluation of measurements
and the treatment of measurement uncertainties are carried out by means of the general procedures
according to the ISO/IEC Guide 98-3 and to the ISO/IEC Guide 98-3-1; see also References [9] to [13].
This enables the strict separation of the evaluation of the measurements, on the one hand, and the
provision and calculation of the characteristic values, on the other hand. The ISO 11929 series makes
use of a theory of uncertainty in measurement [14] to [16] based on Bayesian statistics (e.g. [17] to [22])
in order to allow to take into account also those uncertainties that cannot be derived from repeated or
counting measurements. The latter uncertainties cannot be handled by frequentist statistics.
Because of developments in metrology concerning measurement uncertainty laid down in the
ISO/IEC Guide 98-3, ISO 11929:2010 was drawn up on the basis of the ISO/IEC Guide 98-3, but using
Bayesian statistics and the Bayesian theory of measurement uncertainty. This theory provides a
Bayesian foundation for the ISO/IEC Guide 98-3. Moreover, ISO 11929:2010 was based on the definitions
[9] [10] [11]
of the characteristic values , the standard proposal , and the introducing article . It unified and
replaced all earlier parts of ISO 11929 and was applicable not only to a large variety of particular
measurements of ionizing radiation but also, in analogy, to other measurement procedures.
Since the ISO/IEC Guide 98-3-1 has been published, dealing comprehensively with a more general
treatment of measurement uncertainty using the Monte Carlo method in complex measurement
[12]
evaluations. This provided an incentive for writing a corresponding Monte Carlo supplement to
ISO 11929:2010 and to revise ISO 11929:2010. The revised ISO 11929 is also essentially founded on
Bayesian statistics and can serve as a bridge between ISO 11929:2010 and the ISO/IEC Guide 98-3-1.
Moreover, more general definitions of the characteristic values (ISO 11929-2) and the Monte Carlo
computation of the characteristic values make it possible to go a step beyond the present state of
standardization laid down in ISO 11929:2010 since probability distributions rather than uncertainties
can be propagated. It is thus more comprehensive and extending the range of applications.
The revised ISO 11929, moreover, is more explicit on the calculation of the characteristic values. It
corrects also a problem in ISO 11929:2010 regarding uncertain quantities and influences, which do
not behave randomly in measurements repeated several times. Reference [13] gives a survey on the
basis of the revision. Furthermore, this document gives detailed advice how to calculate characteristic
values in the case of multivariate measurements using unfolding methods. For such measurements, the
ISO/IEC Guide 98-3-2 provides the basis of the uncertainty evaluation.
Formulas are provided for the calculation of the characteristic values of an ionizing radiation
measurand via the “standard measurement uncertainty” of the measurand (hereinafter the “standard
uncertainty”) derived according to the ISO/IEC Guide 98-3 as well as via probability density functions
(PDFs) of the measurand derived in accordance with ISO/IEC Guide 98-3-1. The standard uncertainties
or probability density functions take into account the uncertainties of the actual measurement as well
as those of sample treatment, calibration of the measuring system and other influences. The latter
uncertainties are assumed to be known from previous investigations.
vi © ISO 2019 – All rights reserved

INTERNATIONAL STANDARD ISO 11929-3:2019(E)
Determination of the characteristic limits (decision
threshold, detection limit and limits of the coverage
interval) for measurements of ionizing radiation —
Fundamentals and application —
Part 3:
Applications to unfolding methods
1 Scope
The ISO 11929 series specifies a procedure, in the field of ionizing radiation metrology, for the
calculation of the “decision threshold”, the “detection limit” and the “limits of the coverage interval” for
a non-negative ionizing radiation measurand when counting measurements with preselection of time
or counts are carried out. The measurand results from a gross count rate and a background count rate
as well as from further quantities on the basis of a model of the evaluation. In particular, the measurand
can be the net count rate as the difference of the gross count rate and the background count rate, or
the net activity of a sample. It can also be influenced by calibration of the measuring system, by sample
treatment and by other factors.
ISO 11929 has been divided into four parts covering elementary applications in ISO 11929-1, advanced
applications on the basis of the ISO/IEC Guide 98-3-1 in ISO 11929-2, applications to unfolding methods
in this document, and guidance to the application in ISO 11929-4.
ISO 11929-1 covers basic applications of counting measurements frequently used in the field of ionizing
radiation metrology. It is restricted to applications for which the uncertainties can be evaluated on
the basis of the ISO/IEC Guide 98-3 (JCGM 2008). In Annex A of ISO 11929-1:2019, the special case of
repeated counting measurements with random influences is covered, while measurements with linear
analogous ratemeters, are covered in Annex B of ISO 11929-1:2019.
ISO 11929-2 extends the former ISO 11929:2010 to the evaluation of measurement uncertainties
according to the ISO/IEC Guide 98-3-1. ISO 11929-2 also presents some explanatory notes regarding
general aspects of counting measurements and on Bayesian statistics in measurements.
This document deals with the evaluation of measurements using unfolding methods and counting
spectrometric multi-channel measurements if evaluated by unfolding methods, in particular, for
alpha- and gamma-spectrometric measurements. Further, it provides some advice on how to deal with
correlations and covariances.
ISO 11929-4 gives guidance to the application of the ISO 11929 series, summarizes shortly the general
procedure and then presents a wide range of numerical examples.
ISO 11929 Standard also applies analogously to other measurements of any kind especially if a
similar model of the evaluation is involved. Further practical examples can be found, for example,
[7] [2] [3] [4] [5] [1] [8]
in ISO 18589 , ISO 9696 , ISO 9697 , ISO 9698 , ISO 10703 , ISO 7503 , ISO 28218 , and
[6]
ISO 11665 .
NOTE A code system, named UncertRadio, is available for calculations according to ISO 11929- 1 to
[35][36]
ISO 11929-3. UncertRadio can be downloaded for free from https: //www .thuenen .de/en/fi/fields -of
-activity/marine -environment/coordination -centre -of -radioactivity/uncertradio/. The download contains a
setup installation file which copies all files and folders into a folder specified by the user. After installation one
has to add information to the PATH of Windows as indicated by a pop-up window during installation. English
language can be chosen and extensive “help” information is available.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in
probability
ISO 80000-1, Quantities and units — Part 1: General
ISO 80000-10, Quantities and units — Part 10: Atomic and nuclear physics
ISO/IEC Guide 98-3, Uncertainty of measurement — Part 1: Guide to the expression of uncertainty in
measurement, JCGM 100:2008
ISO/IEC Guide 98-3-1, Evaluation of measurement data — Supplement 1 to the “Guide to the expression of
uncertainty in measurement” — a Propagation of distributions using a Monte Carlo method, JCGM 101:2008
ISO/IEC Guide 98-3-2, Evaluation of measurement data — Supplement 2 to the “Guide to the expression of
uncertainty in measurement” — Models with any number of output quantities, JCGM 102:2011
ISO/IEC Guide 99, International vocabulary of metrology — Basic and general concepts and associated
terms (VIM)
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 80000-1, ISO 80000-10,
ISO/IEC Guide 98-3, ISO/IEC Guide 98-3-1, ISO/IEC 98-3-2, ISO/IEC Guide 99 and ISO 3534-1 and the
following apply.
— ISO Online browsing platform: available at https: //www .iso .org/obp
— IEC Electropedia: available at http: //www .electropedia .org/
3.1
quantity value
value of a quantity
value
number and reference together expressing magnitude of a quantity
[SOURCE: JCGM 200:2012, 1.19]
3.2
measurement
process of experimentally obtaining one or more quantity values that can reasonably be attributed to a
quantity
[SOURCE: JCGM 200:2012, 2.1]
3.3
measurand
quantity intended to be measured
[SOURCE: JCGM 200:2012, 2.3]
2 © ISO 2019 – All rights reserved

3.4
coverage interval
interval containing the set of true quantity values of a measurand with a stated probability, based on
the information available
[SOURCE: JCGM 200:2012, 2.36]
Note 1 to entry: A coverage interval does not need to be centred on the chosen measured quantity value (see
JCGM 101:2008).
Note 2 to entry: A coverage interval should not be termed “confidence interval” to avoid confusion with the
statistical concept.
3.5
measurement method
method of measurement
generic description of a logical organization of operations used in a measurement
[SOURCE: JCGM 200:2012, 2.4]
3.6
measurement procedure
detailed description of a measurement according to one or more measurement principles and to a
given measurement method, based on a measurement model and including any calculation to obtain a
measurement result
[SOURCE: JCGM 200:2012, 2.6]
3.7
measurement result
result of measurement
set of quantity values being attributed to a measurand together with any other available relevant
information
[SOURCE: JCGM 200:2012, 2.9]
3.8
measured quantity value
value of a measured quantity
measured value
quantity value representing a measurement result
[SOURCE: JCGM 200:2012, 2.10]
3.9
true quantity value
true value of a quantity
true value
quantity value consistent with the definition of a quantity
[SOURCE: JCGM 200:2012, 2.11]
Note 1 to entry: In the Error Approach to describing measurement, a true quantity value is considered unique
and, in practice, unknowable. The Uncertainty Approach is to recognize that, owing to the inherently incomplete
amount of detail in the definition of a quantity, there is not a single true quantity value but rather a set of
true quantity values consistent with the definition. However, this set of values is, in principle and in practice,
unknowable. Other approaches dispense altogether with the concept of true quantity value and rely on the
concept of metrological compatibility of measurement results for assessing their validity.
Note 2 to entry: When the definitional uncertainty associated with the measurand is considered to be negligible
compared to the other components of the measurement uncertainty, the measurand may be considered to have
an “essentially unique” true quantity value. This is the approach taken by the ISO/IEC Guide 98-3 and associated
documents, where the word “true” is considered to be redundant.
3.10
measurement uncertainty
uncertainty of measurement
uncertainty
non-negative parameter characterizing the dispersion of the quantity values being attributed to a
measurand, based on the information used
[SOURCE: JCGM 200:2012, 2.26]
Note 1 to entry: Measurement uncertainty includes components arising from systematic effects, such as
components associated with corrections and the assigned quantity values of measurement standards, as well
as the definitional uncertainty. Sometimes estimated systematic effects are not corrected for but, instead,
associated measurement uncertainty components are incorporated.
Note 2 to entry: The parameter may be, for example, a standard deviation called standard measurement
uncertainty (or a specified multiple of it), or the half-width of an interval, having a stated coverage probability.
Note 3 to entry: Measurement uncertainty comprises, in general, many components. Some of these may be
evaluated by Type A evaluation of measurement uncertainty from the statistical distribution of the quantity
values from series of measurements and can be characterized by standard deviations. The other components,
which may be evaluated by Type B evaluation of measurement uncertainty, can also be characterized by standard
deviations, evaluated from probability density functions based on experience or other information.
Note 4 to entry: In general, for a given set of information, it is understood that the measurement uncertainty is
associated with a stated quantity value attributed to the measurand. A modification of this value results in a
modification of the associated uncertainty.
3.11
model of evaluation
set of mathematical relationships between all measured and other quantities involved in the evaluation
of measurements
Note 1 to entry: The model of evaluation does not need to be an explicit function; it can also be an algorithm
realized by a computer code.
3.12
decision threshold
value of the estimator of the measurand, which when exceeded by the result of an actual measurement
using a given measurement procedure of a measurand quantifying a physical effect, is used to decide
that the physical effect is present
Note 1 to entry: The decision threshold is defined such that in cases where the measurement result, y, exceeds
*
the decision threshold, y , the probability that the true value of the measurand is zero is less or equal to a chosen
probability for a wrong decision, α.
*
Note 2 to entry: If the result, y, is below the decision threshold, y , it is decided to conclude that the result cannot
be attributed to the physical effect; nevertheless it cannot be concluded that it is absent.
3.13
detection limit
smallest true value of the measurand which ensures a specified probability of being detectable by the
measurement procedure
Note 1 to entry: With the decision threshold (3.12), the detection limit is the smallest true value of the measurand
for which the probability of wrongly deciding that the true value of the measurand is zero is equal to a specified
value, β, when, in fact, the true value of the measurand is not zero. The probability of being detectable is
consequently (1 − β).
Note 2 to entry: The terms detection limit and decision threshold are used in an ambiguous way in different
standards (e.g. standards related to chemical analysis or quality assurance). If these terms are referred to one
has to state according to which standard they are used.
4 © ISO 2019 – All rights reserved

3.14
limits of the coverage interval
values which define a coverage interval
Note 1 to entry: A coverage interval is sometimes known as a credible interval or a Bayesian interval. Its limits
are calculated in the ISO 11929 series to contain the true value of the measurand with a specified probability
(1 − γ).
Note 2 to entry: The definition of a coverage interval is ambiguous without further stipulations. In this standard
two alternatives, namely the probabilistically symmetric and the shortest coverage interval are used.
3.15
best estimate of the true quantity value of the measurand
expectation value of the probability distribution of the true quantity value of the measurand, given the
experimental result and all prior information on the measurand
Note 1 to entry: The best estimate is the one, among all possible estimates of the measurand on the basis of given
information, which is associated with the minimum uncertainty.
3.16
guideline value
value which corresponds to scientific, legal or other requirements with regard to the detection
capability and which is intended to be assessed by the measurement procedure by comparison with the
detection limit
Note 1 to entry: The guideline value can be given, for example, as an activity, a specific activity or an activity
concentration, a surface activity or a dose rate.
Note 2 to entry: The comparison of the detection limit with a guideline value allows a decision on whether or not
the measurement procedure satisfies the requirements set forth by the guideline value and is therefore suitable
for the intended measurement purpose. The measurement procedure satisfies the requirement if the detection
limit is smaller than the guideline value.
Note 3 to entry: The guideline value shall not be confused with other values stipulated as conformity requests or
as regulatory limits.
3.17
background effect
measurement effect caused by radiation other than that caused by the object of the measurement itself
Note 1 to entry: The background effect can be due to natural radiation sources or radioactive materials in or
around the measuring instrumentation and also to the sample itself (for instance, from other lines in a spectrum).
3.18
background effect in spectrometric measurement
number of events of no interest in the region of a specific line in the spectrum
3.19
net effect
contribution of the possible radiation of a measurement object (for instance, of a radiation source or
radiation field) to the measurement effect
3.20
gross effect
measurement effect caused by the background effect and the net effect
3.21
shielding factor
factor describing the reduction of the background count rate by the effect of shielding caused by the
measurement object
3.22
relaxation time constant
duration in which the output signal of a linear-scale ratemeter decreases to 1/e times the starting value
after stopping the sequence of the input pulses
4 Quantities and symbols
The symbols for auxiliary quantities and the symbols only used in the annexes are not listed. Physical
quantities are denoted by upper-case letters but shall be carefully distinguished from their values,
denoted by the corresponding lower-case letters.
A response matrix of the spectrometer
A elements of the response matrix A
ik
a , a parameters in an algebraic expression of the standard uncertainty of a net counting rate
0 1
b width of a gamma peak, in channels
c position parameter of a peak j, in gamma-ray or alpha-ray spectrometry
j
diag indicator for a diagonal matrix
D matrix converting measured activities to decay corrected activity concentrations
d set of statistically independent quantities
f function representing the analogue of the total peak area method design factor [1 + b/(2L)]
B
for the peak fitting case (gamma-ray spectrometry)
f self-attenuation correction factor for gamma-line i
att,i
f true-coincidence-summing correction factor for gamma-line i
TCS,i
f decay correction factor including the decay during the measurement
d
G function of the input quantities X (i = 1, …, m)
k i
G column matrix of the G
k
h full width at half-maximum of a peak, in channels
h(.) function as part of an implicit model
H()ϑ functional relationship representing the spectral density at ϑ of a multi-channel spectrum
i i
i number of a channel in a multi-channel spectrum obtained by a spectrometric nuclear
radiation measurement (i = 1,.1 m)
+
J matrix of partial derivatives of y with respect to parameters y
L width of a background region (in channels) adjacent to a gamma peak
L k-th element of a system of functions describing spectral densities, which constitute by
k
superposition the total fitting function
m number of input quantities; or number of channels in the spectrum; number of lines per
nuclide used for activity calculation; or a parameter index
6 © ISO 2019 – All rights reserved

N Poisson-distributed random variable of events counted in channel, i, during the measuring
i
time, t (i =, …, m)
n number of events counted in a channel, i, during the measuring time, t (i =, …, m), esti-
i
mate of N
i
n number of output quantities in unfolding
n gross counts in a peak region
g
n average background counts per channel (spectrum)
p estimate of an input quantity which is not subject to fitting (parameter); contained in the
i
response matrix A
p column matrix of the p
i
p values of non-linear parameters held fixed at their calibrated estimates
c
p alpha emission probability of gamma-line i
α,i
p gamma emission probability of gamma-line i
γ,i
q column matrix of input quantities considered as parameters; mainly contained in the
matrix D
Q matrix of partial derivatives of y with respect to parameters p
+
Q′ matrix of partial derivatives of y with respect to parameters q
R net counting rate of the peak i of interest
ni
R net counting rate of a background spectrum peak at the position of the peak i of interest
ni,0
R gross counting rate of the peak i of interest
gi
R counting rate of the trapezoidal background continuum of the peak i of interest
Ti
t duration of measurement
X random variable of the rate of events counted in channel i during the measuring time, t,
i
input quantity of the evaluation, X = N /t (i = 1, …, m)
i i
X column matrix of the X
i
x rate of events counted in channel, i, during the measuring time, t, x = n /t (i = 1, …, m), esti-
i i i
mate of X
i
x column matrix of the x
i
x column matrix of net counting rates
net
u(x , x ) covariance associated with x and x
i j i j
u(y ) standard uncertainty associated with y
k k
U uncertainty matrix of X
x
U uncertainty matrix of Y
y
T
w column matrix of input estimates; w = (x , …, x , p , p , …) (transposed row matrix)
1 m 1 2
Y output quantity (parameter) derived from the multi-channel spectrum by unfolding meth-
k
ods (k = 1, …, n)
Y column matrix of the Y
k
y estimate of the output quantity Y (k = 1, …, n) resulting from (primary) unfolding
k k


y column matrix y after replacement of y with y
+
Y column matrix of final output quantity values after conversion to decay corrected activity
concentrations
Y column matrix of background counting rates
z column matrix of values z fitted to the values x
i i
Δ fractional size of a parameter j, used for the parameter increment in partial derivatives
j
with respect to this parameter
ϑ continuous parameter, e.g. energy or time) related to the different channel numbers in a
gamma-ray spectrum
value of ϑ connected with channel (i = 1, …, m)
ϑ
i
ε detection efficiency of a nuclide i or of a gamma-line i
i
η area fraction of tailing component l of an alpha peak, shape parameter in alpha spectrometry
l
τ tailing parameter of tailing component l of an alpha peak, shape parameter in alpha spec-
l
trometry
σ width of a Gaussian, parameter in alpha spectrometry
function describing the shape of an individual spectral line or of a background contribu-
ψ ()ϑ
k
tion (k = 1, …, n)
5 Evaluation of a measurement using unfolding methods
5.1 General aspects
This clause is based on the ISO/IEC Guide 98-3 and the ISO/IEC Guide 98-3-2. The latter extends the
ISO/IEC Guide 98-3 framework to any number of output quantities. Stipulations are made regarding the
evaluation of nuclear radiation counting and spectrometric measurements by unfolding methods and
the calculation of the characteristic values.
5.2 Models of unfolding and general uncertainty evaluation
When simultaneously measuring more than one output quantity, their individual probability
distributions are superimposed with respect to an independent quantity such as radiation energy
or time, which may yield (e.g. an energy spectrum or a time-dependent decay-curve) as the primary
output of the measurement. Most often, the superposition is linear. A problem occurs if their individual
probability distributions suffer from smearing or broadening (e.g. by a non-ideal detector response
distribution function). The process of reconstructing the original probability density functions from
the measured one, an energy spectrum or a decay-curve, and from the (known) detection response
density function is termed as “unfolding”.
Thus, measuring values y of physical quantities Y (rank n), like radionuclide-specific activities or
counting rates, starts from measuring values x of X (rank m) (e.g. which represent the channel contents
of a multichannel spectrum (energy spectrum) or measured counting rates forming a time-dependent
8 © ISO 2019 – All rights reserved

decay-curve). In the context of this standard, such a measurement is treated as a linear superposition of
the source activity and background related distribution functions (or contributions) A of the
k,i
radionuclide k to each of the components i of the measured x: xA= y .
ik ,ik
∑
k
Although functional representations of detector response functions A (e.g. gamma line-shape) may
k,i
depend non-linearly on parameters like the width parameter, their associated net areas are always
linearly superimposed.
A measurement of more than one output quantity requires a multivariate measurement model. Such
quantities are generally mutually correlated because they depend on common input quantities.
Depending on how Formulas for evaluating the values of each Y can be formulated, two forms of such a
k
model exist. The case of an explicit model is given, if it is possible to formulate separate functions G (X),
k
depending only on X, for calculating any of the values of Y ; G is the multivariate measurement function
k
(see ISO/IEC Guide 98-3-2:2008, Clause 6). An implicit model is encountered, if components of Y are
involved in such functions also, thereby requiring an iterative process for solving. Such a model for Y is
specified by a set of n Formulas
T
hh= ,.,h or hY ,X =0 (1)
()
()
1 n
The explicit multivariate model is given by a set of n functional relationships
YG==(,XX., ); (,kn1 ., ) (2)
kk 1 m
Estimates y of the n measurands Y are obtained from Formula (2) by inserting estimates x for the m
k k i
input quantities X (i = 1,., m)
i
yG==(,xx., ); (,kn1 ., ) (3)
kk 1 m
The standard uncertainties, u(x ), and covariances, u(x , x ), associated with the x are the elements of
i i j i
the symmetric uncertainty matrix U and meet the relations u(x , x ) = u (x ) and u(x , x ) = u(x , x ). If
x i i i i j j i
they are given, the analogous standard uncertainties u(y ) and covariances u(y ,y ) associated with the
k k l
y follow from
k
m
∂G ∂G
k l
uy(,y ))= ⋅ ⋅=ux(,xk;( ,1ln,., ) (4)
kl ij
∑
∂x ∂x
i j
i,j=1
One obtains u()y = u(,yy ) and u(y , y ) = u(y , y ) (k ≠ l). For convenience, the partial derivatives
k l l k
kk k
∂∂GX with all the input quantities X substituted by their estimates x are briefly denoted by
i i
ki
∂∂Gx in Formula (4) and in the following.
ki
The model functions G need not be explicitly available as arithmetical expressions. They can also be
k
given as an algorithm, for instance, in form of a computer code. In such cases, or when more complicated
model functions are involved, the partial derivatives possibly cannot be explicitly derived but can
numerically be approximated sufficiently exactly using half of the standard uncertainty u(x ) as an
i
increment of x
i
∂G
k
=+G xx,., ux()/,2 ., xG−−xx,., u(()xx/,2 .,  (5)
{}
ki 11im ki im
∂xu()x
ii
NOTE 1 Formulas (2) to (4) apply for model functions G which can be taken as sufficiently linear in the
k
uncertainty ranges between x − u(x ) and x + u(x ). Otherwise, more refined procedures can be applied as
i i i i
described in the ISO/IEC Guide 98-3:2009, 5.1.2.
−6
NOTE 2 In practise, u(x ) in Formula (5) is replaced by a much smaller value Δ (e.g. Δ = 2 · 10 x ) for improving
i j j j
the precision of the differential quotient.
It has to be emphasized that in multivariate measurements, it is more convenient to use matrix notation.
Therefore, those quantities, values and functions being denoted by the same symbol are in the following
combined to form a column matrix, written as a transposed row matrix and denoted by the same
T T T
symbol, but in bold face. Examples are x = (x ,., x ) and y = (y ,., y ) and G(x) = (G ,., G ) . In addition,
1 m 1 n 1 n
the uncertainty matrices U = [u(x , x )] and U = [u(y , y )] and also the sensitivity matrix GG=∂ /∂x
x i j y k l ()
xk i
are introduced. Formulas (3) and (4) then read
T
yG==()xU; GU G (6)
yx xx
An often-encountered situation in multivariate measurements with an explicit model is described by
linear equations which can be combined into a matrix equation of the form X = A · Y. It is solved by
the method of weighted linear least-squares, also called generalized least-squares, if the system of
equations is over-determined.
5.3 Unfolding as a sub-model
The primary output quantities Y obtained from unfolding are activities or counting rates. Most often
Y is not the desired measurand, but activity concentrations. The latter may need to be corrected for
radionuclide dependent radioactive decay, chemical yield or other influences. Therefore, another
+
measurand of interest, Y , has often to be calculated
++ ++
YD=⋅Y  or  YD=⋅()YY− (7)
with a diagonal matrix D. Its diagonal elements generally are functions D (q,y) of input quantities q;
j,j
they may also depend on Y, if any of the elements y is used in them. The latter occurs if for instance the
k
design of the simultaneous measurement of activities of Strontium isotopes is extended such that the
measurement of Sr, added with a known activity to the sample as a radiochemical tracer, is included
in unfolding in order to calculate from it the chemical Strontium yield. The second case of Formula (7)
may occur for instance in fitting peak areas in gamma-ray spectrometry, where Y are counting rates
of fitted peak areas from which possible peak contributions Y still are to be subtracted, which are
determined from a separately measured background spectrum.
The extension to a two-step uncertainty propagation implied by utilizing the transformation D is
outlined in 5.6.
5.4 Input quantities and their uncertainties
The input quantities encompass all quantities from which measured or other values are used in the
unfolding and which have uncertainties associated with them. The count number or counting rate input
quantities, denoted as X are separated from the other input quantities, considered as parameters, the
i,
values of which are denoted as p (see 5.5).
i
Depending on the type of measurement, the count rate related input quantities, X , where n events are
i i
counted during a measuring time, t , may be linked to a common single channel analysis or to individual
c
channel analyzers in the case of multiple counting channels or of even a multi-channel analysis, as in
multi-window liquid-scintillation counting or in an alpha- or gamma-ray spectrum.
For a count rate X = R with the given counting result, n recorded during the measuring time, t , and if
i i i, c
independent Poisson statistics can be assumed for the individual channels, the specifications x = r n /
i i = i
2 2
t and u (x ) = n /t = x /t apply. In addition, the covariances often can be set at zero, i.e. u(x ,x ) = 0
c i i c i c i j
(i ≠ j). The counting times may vary between measurements associated with different X The
i.
components of uncertainty of measurement comprise uncertainty matrices U = [u(x ,x)] and
x i j
U = [u(y ,y )]. U is often diagonal with the diagonal elements ux =nt/ .
()
y k l x
ii c
10 © ISO 2019 – All rights reserved

It is useful to have Formulas by which the uncertainty matrix U of the input quantities X can be
x
quantified. Assuming that X represents a vector of net counting rates, it is desirable to put the variances
of U into the following general form
x
x 
x
0,k
2 i 2
ux =+ +ux (8)
() ()
 
i ∑ 0,k
t t
ci ci
 
k
where the symbols x and t denote p
...