1.1.  General

The SIR (International Reference System for activity measurements of $\mathrm{\gamma }$-ray emitting radionuclides) was established in 1976 at the Bureau International des Poids et Mesures (BIPM) to complement the international comparisons of activity measurements organized by Section II of the CCRI (Comité Consultatif des Rayonnements Ionizants). Participating laboratories submit SIR glass ampoules containing their standardized solutions of radionuclides to the BIPM, where the current produced by these samples in a $4\mathrm{\pi }$-ionization well chamber (ionization chamber) is compared with the current obtained with a ²²⁶Ra reference source [40]. The simplicity and rapidity of the measurement as well as the long-term stability of the ionization chamber has allowed the comparison over 30 years of hundreds of radioactive solutions for a total of about 60 radionuclides.

Some of the radionuclides considered are also beta emitters. The consequent bremsstrahlung (radiation emitted by charged particles under deceleration) produced mainly in the wall of the ionization chamber contributes to the measured ionization current and must be taken into account.

The detection efficiency of the SIR ionization chamber for a given photon of energy $E$ or electron of energy $W$ is the ionization current produced in the ionization chamber by a source emitting 10³ such radiations per second, expressed relative to the ionization current produced in the ionization chamber by the BIPM ²²⁶Ra reference source $I_{\text{Ra},5}$.

Similarly, the detection efficiency $\epsilon$ of the SIR ionization chamber for a given radionuclide is the ratio of (a) the ionization current $I$ produced in the ionization chamber per activity $A$ of that radionuclide measured by a laboratory in $\mathrm{kBq}$ and (b) the ionization current $I_{\text{Ra},5}$.

Efficiency curves (detection efficiency versus photon energy and detection efficiency versus electron energy) are required for the calculation of the response of the ionization chamber for radionuclides not previously measured in the SIR. They are needed to evaluate the correction for a photon-emitting impurity present in an ampoule when the efficiency for this impurity is not known experimentally [30], or to give a point of comparison when a radionuclide is measured in the SIR for the first time. Each SIR measurement can be considered as a determination of the efficiency of the ionization chamber for a given radionuclide [36], [38]. In consequence the whole set of SIR measurement data can be used, in principle, to establish the required efficiency curve.

However, in order to obtain reliable efficiency curves, a selection of the data is made: only the results based on NMI primary standardizations are used, which are those retained for the calculation of the key comparison reference values (KCRVs) of the CIPM MRA [7]. In addition, for physical reasons, gaseous radionuclides and strong ${\mathrm{\beta }}^{+}$ emitters are excluded. Further, for low-energy photon emitters and strong $\mathrm{\beta }$ emitters, radioactive solutions of low acid and carrier concentrations are selected in order to minimize the influence of self-attenuation.

Since physical models for the efficiency curves are not available, previous methods have used empirical representations of these curves in the form of polynomials, exponentials, and products of polynomials and exponentials. The adjustable parameters (e.g., polynomial coefficients) in the representations are determined by iteration. This is to account for the influence of gamma-ray-emitting impurities and also to deal with the multi-photon emitters, the efficiency for one photon being deduced from the measured equivalent activity by subtracting the contribution of all other photons as estimated from the previous iteration [31], [41], [42]. The iterative schemes have been based on a physical knowledge of the relative magnitudes of the contributory effects. Least-squares minimizations were also applied by several authors [23], [43], [44]. However, corrections for impurities were not included in the model equations and the associated uncertainty evaluation was not fully treated.

The approach considered here is different. A least-squares formulation is used that accounts for available physical information and measurement uncertainties (Chapter 3). The formulation, as in the above cited references, allows for the presence of radioactive impurities [30]. Families of empirical functions are used within this formulation to represent the efficiency curves (Section 3.2). Although as in previous approaches these functions are empirical in nature, they are chosen to take a form that ensures they are capable of possessing physically possible behaviour.

In the context of data modelling, the nature of the problem is unconventional. The measurement data would not be expected to lie on the desired model curve, even if it were possible to make each measurement perfectly and correct it precisely for the respective radioactive decays and any impurity content. Such a ‘perfectly adjusted’ measured value of a radionuclide in solution would only lie on the curve if it decayed by a single energy photon emission. The model curve is a continuous function of energy, whereas each radionuclide has a number of discrete energy emissions and so cannot be represented by a single energy.

The use of appropriate least-squares modelling principles permits a statistically valid solution to be obtained. Such a solution would not be obtainable were some of the data inconsistent with the model, for example as a consequence of an incorrectly recorded measured value or the stipulation of too small an uncertainty associated with a measured value. The approach simultaneously provides the required efficiency curves and corrects the measurements for impurities (Chapter 5).

The parameters of the empirical functions occur non-linearly within the overall model. Accordingly, recognized, high-quality algorithms for solving generalized non-linear least-squares problems are applied [2]. In particular, the uncertainties associated with the various physical quantities involved are properly taken into account [18].

The resulting efficiency functions enable efficiency values to be provided for any radionuclide with known nuclear reference data, whether previously measured in the SIR or not, together with the associated uncertainties.

Validation procedures (Chapter 7) (a) permit the selection of appropriate members of the families of empirical functions considered, (b) check the adequacy of the numerical quadrature rule used when evaluating the model, © provide an assessment of the consistency of the model and the data and the uncertainties associated with the data, and (d) consider the physical feasibility of the computed model.

The formulation caters for the uncertainties associated with the measured values of radionuclide activity provided by the laboratories. The uncertainties associated with the estimates of the model parameters can be formed from the information provided by the algorithm used to solve the generalized non-linear least-squares problem. In turn, these uncertainties are propagated through the models for a given energy to obtain the uncertainties associated with the corresponding efficiency values. There are further uncertainty sources that are taken into account and that influence the uncertainties associated with these efficiency values. These sources relate to the tabular values of the nuclear reference data used in the model for photon and beta-transition energies and the corresponding probabilities. They also relate to the relative impurity activities. The uncertainties associated with these sources are propagated through the model, and combined with the above-mentioned uncertainties.

In this work, a distinction is made between (a) quantities themselves and (b) estimates of quantities or data values of quantities. Quantities are regarded as random variables in that they are not known exactly. Knowledge of a quantity is characterized by a probability distribution. A best estimate of that quantity is then the expectation of that quantity so characterized, and the standard uncertainty associated with that estimate is the standard deviation of the quantity. An example of a quantity is a parameter in a model, such as the gradient of a straight-line calibration function. An estimate of that quantity would generally be obtained by fitting that function to calibration data. Another example of a quantity is the emission probability of a photon. A data value of the quantity would be provided in nuclear reference data tables. These considerations are consistent with those of the GUM [9].

Appendix 1 contains a glossary of symbols used.

A distillation of the detailed exposition here is available [32], [36].

1.2.  Overview of approach

The starting point of the approach is the consideration of the data that is available relating to the problem of concern and the physical quantities of which this data constitutes particular values (section 2). There are four categories of data:

1. data vector $\widehat{R}$, regarded as a best estimate of $R$, the vector representing the relative impurity activities (Section 2.2);

2. data vector $\widehat{D}$, regarded as a best estimate of $D$, the vector of equivalent activity measurement quantities for the radionuclides and laboratories of concern (Section 2.1);

3. data vector $\widehat{N}$, regarded as a best estimate of $N$, the vector representing nuclear reference data quantities (Section 2.3);

4. data vector $\widehat{K}$, regarded as a best estimate of $K$, the vector representing those KCRVs available for the radionuclides of concern (Section 2.4).

Associated with each of these data vectors is an uncertainty matrix (covariance matrix). The uncertainty matrix associated with $\widehat{R}$ is denoted by $V_{\widehat{R}}$, with a similar notation for the other uncertainty matrices. Mostly, these uncertainty matrices are diagonal, with the diagonal elements containing the squares of the standard uncertainties associated with the individual data items or estimates. When a covariance associated with a pair of data items, the $i$th and $j$th, say, is available, it is included as elements ($i$, $j$) and ($j$, $i$) of the corresponding uncertainty matrix.

Account is not taken here of the uncertainties associated with the elements of $\widehat{K}$, since their influence compared with those of $\widehat{D}$, $\widehat{N}$ and $\widehat{R}$ would be expected to be small. See Chapter 11.

It is necessary to incorporate correction factors that compensate for the fact that a solution containing a particular radionuclide measured by a laboratory may contain impurities (Section 2.1). Such correction is not necessarily purely numerical because in general it depends on the vector $B$ of parameters of the efficiency curves that is to be determined and also on $R$ and $N$. The manner in which this aspect is handled is covered in the approach used for the modelling (Section 3.3).

An established generic form for the model [31] for the efficiency (the reciprocal of equivalent activity) ${\epsilon }_i$ of the ionization chamber for radionuclide $i$ is used (Section 3.1). Letting $E$ denote photon energy and $W$ electron energy, this model is expressed in terms of the photon efficiency function $\epsilon \left(E\right)$ and the electron efficiency function ${\epsilon }_{\beta }\left(W\right)$. Since functional forms for the photon and electron efficiency functions are not available from physical considerations, they are modelled empirically in terms of appropriate functions involving unknown parameter vectors $B^{\left(\begin{array}{c}1\end{array}\right)}$ and $B^{\left(\begin{array}{c}2\end{array}\right)}$, respectively, constituting the (combined) parameter vector $B$ (Section 3.2.1).

The empirical modelling functions used are transformed polynomial functions having the properties that they exhibit physically possible behaviour, namely they are positive for all energy values and they tend to zero at low energy. To ensure numerical stability of the consequent computation, these polynomials are expressed in Chebyshev form (Section 3.2.2).

To account for impurities, a functional form for impurity correction based on the considerations of Section 2.1 is used that depends on $B$, $N$ and $R$ (Section 3.3). This form is generalized through the use of a mixing ratio to reflect the use of previously evaluated but not totally reliable equivalent activity values. This use also results in a simplification of the formulae subsequently required in the uncertainty evaluation.

In order to estimate the parameter vector $B$, an appropriate measure of match of modelled and measured values of equivalent activity is minimized. A measure based directly on the considerations so far would give rise to an unconventional formulation because of the dependence of the ‘measured’ equivalent activities on $B$ through the modelled correction for impurities. It is shown, however (Section 4.1), that a simple transformation of the problem results in a formulation in conventional form, in which a model consisting of the quotient of modelled equivalent activity and impurity correction factor is matched to $D$. The resulting measure is discussed (Section 4.2.1) in the context of a formulation that incorporates the given uncertainty information. Minimization of this measure constitutes solving a generalized non-linear least-squares problem.

In the full formulation of this problem, $D$, $B$, $N$ and $R$ are estimated to provide posterior estimates given prior estimates $\widehat{D}$ (laboratories’ measured equivalent activity values not corrected for impurities), $\widehat{B}$ (e.g. from previous work on determining efficiency curves), $\widehat{N}$ (published nuclear reference data), and $\widehat{R}$ (relative impurity activity data). Any evidence of a mis-match of posterior and prior estimates could be used to propose a consequent adjustment to the nuclear data but in practice would be used to focus the radionuclide community on making appropriate investigations. Moreover, this full formulation, although appropriate for the problem, gives rise to computational difficulties because of the large number of quantities and hence the large matrices involved. Instead, therefore, a partial formulation is given that avoids these difficulties, which has good rather than near-optimal statistical properties (Section 4.2.2). All quantities are fixed at their prior estimates apart from the curve parameters $B$. Solution of this formulation also constitutes solving a generalized non-linear least-squares problem, but of much smaller size. Although they are not needed in solving this least-squares problem, equations characterizing the resulting solution (Section 4.3) are given. They are required subsequently in order to evaluate the uncertainty matrix associated with the estimate $B^{*}$ of $B$ (Chapter 6).

The algorithm recommended for solving generalized non-linear least-squares problems is iterative in nature, at each iteration producing what is generally an improved approximation to the solution. For each approximation to $B^{*}$, the algorithm requires the values of the partial derivatives of first order of the impurity-corrected equivalent activity model. Expressions are derived for these derivatives (Section 5.2 and Appendix 2). The use of a generalized non-linear least-squares solver (Section 5.3) and the determination of initial approximations to the model parameters (Section 5.4), required for iterative solution, are considered.

Were the solution of the full formulation feasible, the uncertainty matrix $V_{B^{*}}$ associated with the estimate $B^{*}$ of the model parameter vector $B$ would be available as a by-product of the least-squares procedure. This uncertainty matrix would reflect the uncertainties associated with $\widehat{D}$, $\widehat{N}$ and $\widehat{R}$. For the partial formulation, the $V_{B^{*}}$ provided would account only for the uncertainties associated with $\widehat{D}$, and take the form [29]

where $J_s\left(B\right)$ is the (Jacobian) matrix containing the partial derivatives of first order of the model residual deviations $s$ with respect to the model parameters.

The further sources of uncertainty to be taken into account, associated with the estimates $\widehat{N}$ of $N$ and $\widehat{R}$ of $R$, are propagated through the least-squares solution, to be combined with the above-mentioned uncertainties. These further uncertainties are taken into account using a generalization of the GUM uncertainty framework.

The solution obtained as described corresponds to the use of estimates regarded as the expectations of distributions for the corresponding quantities. On the basis of the principles of GUM Supplement 1 [8], such distributions would be assigned to be Gaussian with expectations equal to the estimates and standard deviations equal to the standard uncertainties associated with these estimates.

At the solution to the generalized non-linear least-squares problem, the partial derivatives with respect to the adjustable quantities $B$ are zero. The resulting expressions provide a measurement model relating input quantities $D$, $R$ and $N$ to output quantities $B$ that can be used as the basis for applying the GUM uncertainty framework. In terms of a classification of measurement models [19], this model is categorized as (a) multivariate (having a number of output quantities), (b) implicit (being defined in such a way that $B$ cannot be expressed directly in terms of $D$, $R$ and $N$), and © real (not involving complex quantities).

2.1.  Equivalent activity measurement data

Equivalent activity measurement data are derived from more fundamental measurement data. Consider an SIR measurement of a radioactive solution, in a glass ampoule, containing a radionuclide, generally with impurities. Let $A$ denote the activity of the solution at a reference date $t_{\text{r}}$, that is when the measurement was made by a participating laboratory.

The equivalent activity $A_{\text{e}}$ is the activity of the solution that would produce the same ionization current as the reference ²²⁶Ra source number 5 at the fixed SIR reference date $t_0$.

Let $I_{\text{Ra,s}}$ and $I$ denote the ionization currents produced by the ²²⁶Ra source number s and the ampoule, respectively. Let $I_f$ denote the background current. There will be a value for each of these three currents corresponding to each laboratory’s measurement.

Let $t_{\text{m}}$ be the date of the SIR measurement of the solution, and ${\lambda }_{\text{Ra}}$ and ${\lambda }_i$ the decay constants of ²²⁶Ra and a measured radionuclide $i$, respectively.

Three multiplicative factors are used to transform $A$ to $A_{\text{e}}$ [30]:

1. The product of the quotient

   $\frac{I_{\text{Ra,s}}-I_f}{I-I_f}$

   of the respective ionization currents, allowing for the background current, and a factor $F_s$ that is the quotient of currents of radium source numbers 5 and $s$.

2. The decay-correction factor

   $\frac{\text{exp}\left(-{\lambda }_i\left(t_m-t_r\right)\right)}{\text{exp}\left(-{\lambda }_{\text{Ra}}\left(t_m-t_0\right)\right)}$

   due to the respective decay constants and the times elapsed to the date of the SIR measurement.¹

3. An impurity correction factor $C$ that accounts for the impurities in the solution.

Thus [30],

where

and $C$ is considered below. The $A$ and $M$ are combined to form the current- and decay-corrected measurement quantity

Equation (1) corresponds to the equivalent activity per se that is current-, decay- and impurity-corrected.

The above notation is qualified as follows when a measurement is made of radionuclide $i$ by a laboratory (indexed by) $𝓁$:

where

The superscript in ${\left(A_{\text{e}}\right)}_{i,𝓁}^{\text{meas}}$ is used to emphasize that equivalent activity is based on measurement. (Subsequently, another superscript will indicate when equivalent activity is based on a model.) The $A_{i,𝓁}$ denotes the measured value for the activity of radionuclide $i$ provided by laboratory $𝓁$. The $M_{i,𝓁}$ denotes the value of $M$ in Equation (1) relevant to radionuclide $i$ and laboratory $𝓁$, namely,

The $C_{i,𝓁}$ is the correction factor that compensates for the fact that the solution containing radionuclide $i$ that is measured by laboratory $𝓁$ may contain impurities, and is given [30] by

(see Section 2.2).

Let $D$ denote the vector of values of the $D_{i,𝓁}$. It constitutes the set of current- and decay-corrected measurement quantities, or in brief the data quantity vector.

The modelling approach described here utilizes the $D_{i,𝓁}$ and therefore requires the standard uncertainties $u\left({\widehat{D}}_{i,𝓁}\right)$ associated with estimates ${\widehat{D}}_{i,𝓁}$ of the $D_{i,𝓁}$ given by

using Equation (3), where ${\widehat{A}}_{i,𝓁}$ is the measured value provided by laboratory $𝓁$ of the activity of radionuclide $i$ and ${\widehat{M}}_{i,𝓁}$ is the value of $M_{i,𝓁}$ given by substituting measured values for the various quantities in the right-hand side of Equation (4).

These uncertainties are given by applying the law of propagation of uncertainty to Equation (3), under the assumption that the quantities involved are mutually independent, to give

where $u\left({\widehat{A}}_{i,𝓁}\right)$ is the standard uncertainty associated with ${\widehat{A}}_{i,𝓁}$ as declared by laboratory $𝓁$, and $u\left({\widehat{M}}_{i,𝓁}\right)$ that associated with ${\widehat{M}}_{i,𝓁}$. The $u\left({\widehat{M}}_{i,𝓁}\right)$ is obtained by applying the law of propagation of uncertainty to Equation (4), given the standard uncertainties associated with the measured values of the quantities involved.

Generally the $C_{i,𝓁}$ in Equation (5) will depend on the model parameters $B$, say, which are to be determined. They also depend on $N$ and $R$, estimates of which, with associated standard uncertainties, are available (see Section 3.3).

2.2.  Data relating to relative impurity activities

Consider the activity ratio quantities ${\stackrel{\sim }{R}}_{i,𝓁,k}$, $k\in K_{i,𝓁}$ corresponding to the activities at the reference date of impurity $k$ in a solution of radionuclide $i$ for laboratory $𝓁$, for all relevant $i$, $𝓁$ and $k$. The $K_{i,𝓁}$ denotes the index set relating to impurities associated with that solution. Let $R$ denote the vector of decay-corrected quantities $R_{i,𝓁,k}$ for relevant $i$, $𝓁$ and $k$. Data constituting a best estimate $\widehat{R}$ of $R$ is available, as are the associated standard uncertainties.

The $R_{i,𝓁,k}$ is given by

where

is the appropriate decay correction.

Consider the correction factors $C_{i,𝓁}$ in Equation (5). The value provided by laboratory $𝓁$ in its measurement of a solution containing radionuclide $i$ will fall into one of three categories, corresponding to the measurement of a radionuclide

1. that is pure, in which case $C_{i,𝓁}=1$,

2. together with impurities that have all previously been well characterized in terms of reliable measurement, meaning that KCRVs for the equivalent activities ${\left(A_e\right)}_i$ and ${\left(A_e\right)}_k$ in Equation (5) would be available, and hence a numerical value for $C_{i,𝓁}$ could be determined, and

3. together with impurities not all of which have previously been well characterized. For the well-characterized impurities, KCRVs for the ${\left(A_e\right)}_k$ in Equation (5) would again be available, but the efficiency model is used to estimate the remaining impurities.

2.3.  Nuclear reference data

For a radionuclide (indexed by) $i$ let

$J_i$

denote the set of indices of the photons associated with this radionuclide,

${}_{\beta }{J}_i$

the set of indices of the beta transitions associated with this radionuclide,

$E_{i,j}$

the energy associated with the $j$th photon,

$P_{i,j}$

the emission probability for the $j$th photon²,

$W_{i,j}$

the maximum energy associated with the $j$th beta-transition,

${}_{\beta }{P}_{i,j}$

the emission probability for the $j$th beta-transition,

$S_{i,j}\left(W\right)$

the energy distribution function for the spectrum corresponding to the $j$th beta-transition, normalized such that

where $W=E/\left(m_e{c}^2\right)+1$ is the reduced total electron energy, $m_e$ being the rest mass of the electron and $c$ the speed of light.

Values of the quantities $\left\{E_{i,j}:j\in J_i\right\}$, $\left\{P_{i,j}:j\in J_i\right\}$, $\left\{W_{i,j}:j\in {}_{\beta }{J}_i\right\}$ and $\left\{{}_{\beta }{P}_{i,j}:j\in {}_{\beta }{J}_i\right\}$ are available as published nuclear reference data, together with associated standard uncertainties (see e.g. [4]). These published energies and probabilities are regarded as best-available estimates $\widehat{N}$ of quantities collectively labelled $N$. In general, only a subset of the nuclear reference data is used and $N$ and $\widehat{N}$ are interpreted accordingly.

For the problem here most of the quantities concerned are independent and hence $V_{\widehat{N}}$ is predominantly diagonal. However, there are some correlation effects associated with (a) the nuclear reference data relating to the normalization of the relative emission probabilities³, and (b) an emission probability in the case where two laboratories provide activity estimates having small associated uncertainties. The $V_{\widehat{N}}$ will consequently contain some off-diagonal non-zero elements.

The beta energy distribution function $S_{i,j}$, for the $j$th beta transition, is calculated as in the publications of Wilkinson [45], [46], [47], [48], [49]. The first- and second-forbidden non-unique decays are approximated by allowed and first-forbidden unique decays, respectively. The required values of the Coulomb functions ${\lambda }_n$ are approximated as independent of energy and estimated from the tables of Behrens and Jänecke [5]. The ‘finite nuclear radius’ is estimated by the relation given in Grau Malonda [28]. The electron shielding correction is calculated by the method of Rose [39], where the total positron/electron energy in the Fermi function is shifted by a screening potential $\pm V_s$.

The emission of conversion electrons of energy $W_c$ is taken into account by defining a special case of energy distribution: $S_{i,j}=\mathrm{\delta }\left(W-W_c\right)$.

2.4.  Key comparison reference values

Let ${\left(A_e\right)}_i^{\text{KCRV}}$ denote a key comparison reference value (KCRV) for radionuclide $i$. For many of the radionuclides, such a KCRV is published in the BIPM Key Comparison Database (KCDB) [6] and is considered to be a best estimate, together with an associated standard uncertainty, of the equivalent activity. The model may be used ultimately to derive KCRVs for other radionuclides [37]. The set of such KCRVs is denoted by $\widehat{K}$ and the vector of quantities of which $\widehat{K}$ is a realization is denoted by $K$.

Account is not taken here of the uncertainties associated with the elements of $\widehat{K}$, since their influence compared with those of $\widehat{D}$, $\widehat{N}$ and $\widehat{R}$ would be expected to be small. See Chapter 11.

3.1.  Generic form for the efficiency function of the ionization chamber

Denote by $\epsilon \left(E\right)$ the efficiency of the ionization chamber for photons of energy $E$, and by ${}_{\epsilon \beta }\left(W\right)$ its efficiency for electrons of energy $W$. The function $\epsilon \left(E\right)$ is known as the photon efficiency function or photon efficiency curve and ${}_{\epsilon \beta }\left(W\right)$ as the electron efficiency function or curve. Since analytical forms for these functions derived from physical principles are not available, $\epsilon \left(E\right)$ is modelled by an appropriate empirical function $F\left(B^{\left(\begin{array}{c}1\end{array}\right)},E\right)$ and ${}_{\epsilon \beta }\left(W\right)$ by $G\left(B^{\left(\begin{array}{c}2\end{array}\right)},W\right)$. The $B^{\left(\begin{array}{c}1\end{array}\right)}$ and $B^{\left(\begin{array}{c}2\end{array}\right)}$ denote sets of adjustable model parameters, with $B=\left({\left(B^{\left(\begin{array}{c}1\end{array}\right)}\right)}^{\top }\right.$, ${\left(B^{\left(\begin{array}{c}2\end{array}\right)}\right)}^{\top }{)}^{\top }$, that is

representing the aggregated set of adjustable model parameters.

The model for the efficiency of the chamber for radionuclide $i$ is given [31] by

The corresponding model for the equivalent activity [31], [38], [41] of radionuclide $i$, indicating explicitly that it depends on $B$ and $N$, is

This quantity is the model value for the equivalent activity of radionuclide $i$.

3.2.  Representation of the photon and electron efficiency functions

3.3.  Accounting for impurity corrections

Any particular correction $C_{i,𝓁}$ (Section 2.1) is either known or depends on $B$, $N$ and $R$, and can thus generally be represented by $C_{i,𝓁}\left(B,N,R\right)$.

To account for the categories of correction factor considered in Section 2.1, and in order to generalize the treatment, a mixing ratio ${\theta }_i$ relating to each radionuclide of concern is introduced. This ratio is taken as unity or zero according to whether or not radionuclide i had previously been well-characterized, that is whether or not a reliable value ${\left(A_{\text{e}}\right)}_i^{\text{KCRV}}$ for the equivalent activity ${\left(A_{\text{e}}\right)}_i$ is available. Then, Equation (5) can be interpreted as

where

and, with the superscript cmptd denoting ‘computed’,

This approach has the property that a value of ${\theta }_i$ between zero and one can be chosen to reflect the use of a previously evaluated but not totally reliable value of ${\left(A_{\text{e}}\right)}_i$. The use of ${\theta }_i$ also results in a simplification of the formulae used for derivatives when solving the generalized non-linear least-squares problem and in the uncertainty evaluation (Appendix 2).

4.1.  Matching modelled and measured values

Let $I$ denote the set of indices of all radionuclides of concern and $L_i$ the set of indices representing the laboratories that have measured radionuclide $i$.

The current-, decay- and impurity-corrected quantity (Equation (2)) is expressed as

The requirement is to match in some sense the Equation (8) and Equation (21). Both the ‘measured’ and modelled equivalent activities over all relevant radionuclides $i$ and laboratories $𝓁$ generally depend on $B$, $N$ and $R$. The latter activities (Equation (21)) also depend on $D$. Matching is achieved by estimating $B$ in such a way that a suitable measure of closeness of the two expressions (Section 4.2) is as small as possible. However, the form of the problem is unconventional as a consequence of the ‘measured’ equivalent activities depending on the unknown parameters $B$ rather than being fixed. The problem can, however, be transformed as follows.

Divide the right-hand sides of Equation (8) and Equation (21) by $C_{i,𝓁}\left(B,N,R\right)$. The advantage of this simple problem transformation is that the $f_{i,𝓁}\left(B,N,R\right)$ given by

which depend on $B$, $N$ and $R$, can then be matched to the $D_{i,𝓁}$, which depend only on $D$.

The function $f_{i,𝓁}\left(B,N,R\right)$ will be known as the equivalent activity model for radionuclide $i$ and laboratory $𝓁$. The vector of the $f_{i,𝓁}\left(B,N,R\right)$ will be denoted by $f\left(B,N,R\right)$ and known as the equivalent activity model vector.

To reiterate, the information available for use in the determination of a match includes

1. the vector estimate $\widehat{D}$ of $D$ provided by the laboratories’ measured values of the activities of the radionuclides and by the ionization current measurements in the SIR,

2. a vector estimate $\widehat{B}$ of the efficiency curve parameters $B$, e.g. from previous work on determining efficiency curves (see Section 5.4),

3. a vector estimate $\widehat{N}$ of $N$ from published nuclear reference data,

4. a vector estimate $\widehat{R}$, the elements of which are provided by the laboratories, of the relative impurity activities $R$, and

5. the uncertainty matrices associated with the vector estimates in 1)-4).

Denote by $Z$ the composite vector of quantities $B$, $N$ and $R$:

It is taken that $D$ and $Z$ are mutually independent.

4.2.  Measure of deviation of model from data

4.3.  Equation characterizing the solution

Specifically, $B^{*}$ is the $B$ that minimizes the Equation (24). At the solution, the partial derivatives of the measure with respect to $B$ are zero. The solution hence satisfies [26]

where

Vector equation Equation (25), constituting a system of non-linear algebraic equations, is not generally solved as such for $B=B^{*}$, but $S_{\text{part}}\left(B\right)$ in Equation (24) is minimized using a generalized nonlinear least-squares algorithm to provide $B^{*}$. Equation (25) is, however, important regarding uncertainty evaluation for the solution of the problem (Chapter 6).

5.1.  General

The generalized non-linear least-squares algorithm mentioned in Section 4.2 is iterative. Starting with an initial approximation to $B$, at each iteration it constructs what is generally an improved approximation. The sequence of approximations ultimately converge to $B^{*}$. At each iteration, the model deviation

and the partial derivatives of that deviation with respect to $B$ are evaluated for the current value of $B$, with

5.2.  First-order derivative evaluation

These derivatives constitute the Jacobian matrix

The model vector $f\left(B,N,R\right)$ has components $f_{i,𝓁}\left(B,N,R\right)$ and the model deviation vector $s\left(B,D,N,R\right)$ has component deviations $s_{i,𝓁}\left(B,D,N,R\right)$, with respect to the elements of the efficiency curve parameter vector $B$, the decay-corrected measurement vector $D$, the nuclear reference data vector $N$, and the relative impurity activities vector $R$. The components are related by

where, to recapitulate, $D_{i,𝓁}$ is an element of $D$, and $f_{i,𝓁}\left(B,N,R\right)$ is equal to the quotient of the model function ${\left(A_{\text{e}}\right)}_i^{\text{model}}\left(B,N\right)$ and the parametrized correction term $C_{i,𝓁}\left(B,N,R\right)$. The first set of such derivatives is needed when using an algorithm to solve the generalized non-linear least-squares problem to determine a vector estimate $B^{*}$ of $B$ when the quantities $D$, $N$ and $R$ are set to their best prior estimates (Section 4.2.2). All these derivatives are needed as part of the evaluation of the uncertainties associated with $B^{*}$.

The various first-order derivatives are given in Appendix 2.

5.3.  Use of a generalized non-linear least-squares solver

The solution of the generalized non-linear least-squares formulation of Section 4.2.2 is carried out using a high-quality library software implementation [24] of a recognized algorithm for such problems.

Such an algorithm requires for its operation:

1. A procedure for calculating the residual deviations $s\left(B,D,N,R\right)$ given $B$, $D$, $N$ and $R$. This procedure is based on Equation (27), Equation (22), Equation (12)–Equation (17), Equation (18)–Equation (20);

2. A procedure for calculating the first-order partial derivatives of these deviations with respect to the elements of $B$. Appendix 2 gives details.

3. An approximation to the required estimate of $B$, to initialize the iterative solution procedure used by the algorithm. Section 5.4 gives details.

Let ${\left(B^{\left(\begin{array}{c}1\end{array}\right)}\right)}^{*}$ denote the estimate of $B^{\left(\begin{array}{c}1\end{array}\right)}$, and ${\left(B^{\left(\begin{array}{c}2\end{array}\right)}\right)}^{*}$ that of $B^{\left(\begin{array}{c}2\end{array}\right)}$, given by minimizing $S_{\text{part}}\left(B\right)$ in Equation (24). Let $B^{*}={\left\{{\left[{\left(B^{\left(\begin{array}{c}1\end{array}\right)}\right)}^{*}\right]}^{\top },{\left[{\left(B^{\left(\begin{array}{c}2\end{array}\right)}\right)}^{*}\right]}^{\top }\right\}}^{\top }$.

The generalized non-linear least-squares software also provides the uncertainty matrix $V_{B^{*}}$ associated with $B^{*}$. This uncertainty matrix accounts only for $V_{\widehat{D}}$, and not $V_{\widehat{N}}$ and $V_{\widehat{R}}$.

5.4.  Initial approximations to the model parameters

The following approach is used to provide initial approximations to the model parameters $B$:

1. Provide a set of points $\left(E_s,{\epsilon }_s\right),s=1,...,q$, adequately covering the energy spectrum of concern. Previously published curves or preliminary results of Monte Carlo simulations or points derived from single photon emitters may be used for this purpose.

2. Solve the (unweighted) linear least-squares problems of fitting the function

   $\sum _{h=1}^n{B}_h^{\left(\begin{array}{c}1\end{array}\right)}{\varphi }_h\left(\text{ln}E\right)$

   to the data $\left(E_s,\text{ln}\left({\epsilon }_s/E_s\right)\right),s=1,...,q$, to obtain approximations to the values of the $B_h^{\left(\begin{array}{c}1\end{array}\right)}$. In carrying out this step, use the approach described in Section 7.1 to provide an appropriate value for $n$.

3. Carry out the counterpart of Steps 1 and 2 for the electron efficiency curve to obtain approximations to the values of the $B_h^{\left(\begin{array}{c}2\end{array}\right)}$.

6.1.  General

This section is concerned with the evaluation of the uncertainty matrix $V_{B^{*}}$ associated with the vector estimate $B^{*}$ of the model parameter vector $B$. The evaluation constitutes the propagation of the uncertainties associated with all the data used in the least-squares modelling, namely, the vector $\widehat{D}$ containing the set of the laboratories and SIR decay-corrected measurement data used, the vector $\widehat{N}$ of relevant published values of the nuclear reference data $N$ used, and the estimate $\widehat{R}$ of the relative impurity activities vector $R$ used. For this purpose, the vector estimates $\widehat{D}$, $\widehat{N}$ and $\widehat{R}$ are available, as are the associated uncertainty matrices $V_{\widehat{D}}$, $V_{\widehat{N}}$ and $V_{\widehat{R}}$.⁷

6.2.  Implicit model

The contribution to $V_{B^{*}}$ from $V_{\widehat{D}}$ is already provided as part of the least-squares solution process (Section 5.3). That contribution is identical to that obtained by uncertainty propagation as considered here applied to $V_{\widehat{D}}$ alone.