Introduction

As part of ongoing efforts for continuous software improvements, work is currently under way to upgrade the internal uncertainty propagation of Mirion Technologies (Canberra)’s Genie 2000 gamma analysis software [1] to include statistically rigorous treatment of the most significant correlation effects. Correlations arise in gamma ray spectroscopic analysis whenever fitting methods are employed, introducing covariant parameter sets, or when input data are correlated. These effects are often small, but their impact on uncertainty estimates and activity quantification can be important for high precision applications where other sources of uncertainty are well controlled.

The primary source of correlation effects is the measured efficiency calibration, in which a parameterized function is fit to a convenient set of measured efficiency data points. This function can then be used to calculate efficiencies at energies of interest differing from the calibration measurements. Correlations between the regression parameters must be taken into account in evaluating the uncertainty of calculated efficiency values. These also introduce correlations between the efficiencies calculated at different energies, with consequences for downstream analysis steps. Activities for multi-line nuclides, calculated as the weighted mean of measured individual line activities, are also affected by correlations between efficiency values. A more complex instance of this effect applies to the unfolding of interferences between nuclides which share gamma emissions at common energies.

Further complexity arises if the measured efficiency data points with which the calibration is performed are themselves correlated, altering the covariance structure of the fit parameters. This occurs when multi-line nuclides are used in the calibration reference source and when multiple data points share a dependence on the activity of a common nuclide.

To properly account for these effects, new Genie 2000 Efficiency Calibration and Nuclide Identification (NID) algorithms were prototyped using the Python programming language [2]. The primary modifications to both algorithms consisted of (a) generalizing the least squares fitting methods to correctly apply the full covariance matrix between the input data points and (b) saving the full output covariance matrix between the fit parameters to the spectral data file for later use. The effects of these changes on analysis results were demonstrated on simulated spectra created using MCNP-CP [3]. Spectra for several multi-line nuclides were generated singly and in combination, for a point source positioned 30 cm from an n-type HPGe detector with 45% relative efficiency. The spectra were analyzed using standard Genie 2000 algorithms except for the efficiency calibration and NID where the prototype algorithms were used.

Correlations in efficiency calibrations

In a measured efficiency calibration, efficiency data triplets \( \{ \varepsilon_{i} ,E_{i} ,\sigma (\varepsilon_{i} )\} \) are fit with a parameterized regression function of linear form

$$ y(x) = \sum\limits_{\alpha = 0}^{n} {a_{\alpha } f_{\alpha } (x)} , $$
(1)

with basis functions \( f_{\alpha } (x) \) and parameters \( a_{\alpha } \). Here \( x \) and \( y \) may be the efficiency and energy, or their logarithms, depending on the choice of regression function.

The fit is performed by minimizing with respect to the parameters \( \alpha \) the Chi squared function, given in vector notation by

$$ \chi^{2} = ( {\vec{\varvec{y}}( {\vec{\varvec{x}}} ) - {\mathbf{F}} \cdot \vec{\varvec{a}}} )^{\varvec{T}} \cdot {\mathbf{V}}^{ - 1} \cdot ( {\vec{\varvec{y}}( {\vec{\varvec{x}}} ) - {\mathbf{F}} \cdot \vec{\varvec{a}}} ) $$
(2)

where \( \vec{\varvec{y}} \) is the vector of measured data, \( \vec{\varvec{a}} \) is the parameter vector, and \( {\mathbf{F}} \) is a matrix of basis functions with \( F_{\alpha i} = f_{\alpha } \left( {x_{i} } \right). \) The weight function V−1 is the inverse of the covariance matrix of the data; for independent data it is diagonal and (2) becomes the usual Chi squared for uncorrelated fitting.

The outputs of the fit are the parameter values \( \alpha \) as well as the parameter covariances, \( \text{cov} (a_{\alpha } ,a_{\beta } ) \), given by the inverse of the Hessian matrix of second derivatives of the Chi squared function. Parameter covariances appear even when the measured data are uncorrelated. They are necessary to correctly calculate both the uncertainties of values calculated from the calibration function and also the correlations between values at different energy values \( x_{i} \) and \( x_{j} \). In general,

$$ {\text{cov}}(y(x_{1} ), y(x_{2} )) = \sum\limits_{\alpha } {\left\{ {f_{\alpha } (x_{1} )f_{\alpha } (x_{2} )\sigma^{2} (a_{\alpha } ) + \sum\limits_{\beta \ne \alpha } {f_{\alpha } } (x_{1} )f_{\beta } (x_{2} ) {\text{cov}}(a_{\alpha } a_{\beta } )} \right\}} . $$
(3)

When multi-line nuclides are used in the calibration measurement, additional covariance effects occur. The efficiency \( \varepsilon_{i} \) measured at energy \( E_{i} \) from nuclide A is normally calculated as

$$ \varepsilon_{i} = \frac{{N_{i} }}{{I_{{i{\text{A}}}} A_{\text{A}} T}} $$
(4)

where \( N_{i} \) is the peak area at energy \( E_{i} \), \( I_{{i{\text{A}}}} \) is the gamma intensity for nuclide A at energy \( E_{i} \), \( A_{\text{A}} \) is the activity of nuclide A, and \( T \) is the live time of the measurement. The variance of the measured efficiency is

$$ \sigma^{2} (\varepsilon_{i} ) = \varepsilon_{i}^{2} \left\{ {\frac{{\sigma^{2} (N_{i} )}}{{N_{i}^{2} }} + \frac{{\sigma^{2} (I_{{i{\text{A}}}} )}}{{I_{{i{\text{A}}}}^{2} }} + \frac{{\sigma^{2} (A_{\text{A}} )}}{{A_{\text{A}}^{2} }} + \frac{{\sigma^{2} (T)}}{{T^{2} }}} \right\}. $$
(5)

Neglecting the usually trivial uncertainty in the count time T, the covariance between efficiencies measured from two lines of the same nuclide A depends on the variance of the common activity:

$$ {\text{cov}}(\varepsilon_{i} , \varepsilon_{j} ) = \frac{{\sigma^{2} (A_{\text{A}} )}}{{A_{\text{A}}^{2} }}\varepsilon_{i} \varepsilon_{j} . $$
(6)

The data covariance matrix V is constructed with diagonal elements (5), and with off-diagonal elements given by (6) for lines of a common nuclide or zero for lines of different nuclides. This methodology for handling correlated input data in the efficiency calibrations was described in greater detail in [4].

To demonstrate the effects of correlations in the measured data, simulated spectra were created to perform efficiency calibrations for three different calibration sources, with varying degrees of correlations, using standard Genie 2000 efficiency functions:

  • A “Mixed Gamma” source containing seven single-line nuclides and two two-line nuclides shows the effect of a low degree of correlation in the fit;

  • An “AmBaCsCo” source (241Am, 133Ba, 137Cs, 60Co) shows the fit with stronger correlations, as 133Ba has six lines and 60Co has two lines, while the other two nuclides have one line each;

  • A 241Am–152Eu source was simulated to show a high degree of correlation, as 152Eu contributes 9 correlated lines, while 241Am adds one uncorrelated point at low energy.

The nuclides in each calibration standard and the emission energies used for the calibrations are listed in Table 1. In all simulations, the uncertainty on each nuclide activity was taken to be ± 3%.

Table 1 Nuclide content and gamma energies of example calibration sources
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Figure 1 shows the effect of correlations on the fit for the three cases. On the top row, the ratio of the efficiency values with correlated data to the values without correlated data is shown. This variation is very small; on the order of 1% over most of the range of the calibrations, although larger values are observed at the extrema of the functions.

Fig. 1
figure 1

Top: ratios of efficiency functions with and without data correlations. Bottom: calculated efficiency uncertainties with (solid line) and without (dashed line) data correlations. Energies and uncertainties of measured efficiencies and nuclides of origin indicated by points

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The uncertainties, calculated with and without correlated data, are plotted in the bottom row of Fig. 1. The energies and uncertainties of the efficiency data points used in the fit, as well as their nuclides of origin, are indicated by the points. When data correlation are taken into account the uncertainties are higher in regions where nearby data points are correlated, but tend to be slightly lower in regions between clusters of correlated data. The overall effect is greatest of 152Eu, as expected. Neglecting correlations in this case would underestimate the uncertainty at most energies, by a maximum of around 1% (absolute) out of 3% near 1100 keV.

Correlation coefficients between different points on the curve are shown in Fig. 2 for the “mixed gamma” simulation, with correlations (right) and without (left). Strong correlation is seen in the region along the diagonal for both cases, as nearby energies are correlated even when the input data are not. Adding the effects of correlated data increases the region of positive correlation between the outputs, in this case primarily at higher energies, as seen in the right hand plot.

Fig. 2
figure 2

Correlation coefficients between calculated efficiency values with data correlation effects (right) and without (left)

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NID and interference correction

Weighted mean activities

Covariances in the efficiency calibration have consequences in the Nuclide Identification (NID) analysis step. The weighted mean activities for multi-line nuclides are calculated from individual line activities, evaluated at different energies, which are correlated with one another through their common dependence on the efficiency calibration parameters.

The formula for calculating the weighted mean activity \( A \) with covariances is a special case of the least squares fitting problem:

$$ A = \sum\limits_{i,j = 1}^{n} {V_{ij}^{ - 1} } A_{j} \left/\sum\limits_{i,j = 1}^{n} {V_{ij}^{ - 1} } \right.. $$
(7)

If the individual line activities for nuclide A at energies \( E_{i} \) are calculated as

$$ A_{{i{\text{A}}}} = \frac{{N_{i} }}{{I_{{i{\text{A}}}} \varepsilon_{i} T}} $$
(8)

then the covariances between different lines are

$$ {\text{cov}}(A_{{i{\text{A}}}} ,A_{{j{\text{A}}}} ) = A_{i} A_{j} \frac{{{\text{cov}}(\varepsilon_{i} , \varepsilon_{j} )}}{{\varepsilon_{i} \varepsilon_{j} }},\quad i \ne j. $$
(9)

The matrix V is constructed with off-diagonal elements given by (9) and diagonal elements given by the line activity variances. For uncorrelated data, the inverse covariance matrix V−1 is diagonal.

To demonstrate these effects, spectra were simulated for several multi-line nuclides: 57Co, 60Co, 133Ba, 134Cs, and 152Eu. The statistical (counting) uncertainties on the most significant lines were < 1%. The spectra were analyzed in three different ways for each of the three calibrations created previously: with no correlation effects, with correlations considered between the sample nuclide line activities but treating the efficiency calibration data as uncorrelated, and including correlation effects between the sample line activities and between the efficiency data points from like nuclides. Weighted mean activities were calculated for each case.

The recovery fractions for each nuclide are plotted in Fig. 3 for all three analyses using the Mixed Gamma calibration—the results are similar for the other calibrations except as noted below. The effect on the activity result is modest, with the greatest impact seen for 133Ba with a ~ 1% change (absolute) from the completely uncorrelated analysis to the most correlated. The more appreciable difference between the analyses is in the estimated uncertainties, which increase as additional levels of correlation are accounted for.

Fig. 3
figure 3

Weighted Mean Activity recovery fractions for multi-line nuclides, using the Mixed Gamma calibration, for analyses including increasing levels of correlation

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The uncertainties obtained from each analysis are listed in Tables 2–4. For each calibration, the weighted mean activity uncertainties increase as correlations between lines of the sample nuclide are accounted for in each example. Adding in the effects of the correlations in the calibration standard has varied effect, depending on the degree of correlation for that standard. In Table 2, the least correlated calibration standard has little additional effect on the activity uncertainties when the correlations are included. The effect is somewhat more pronounced for the more correlated AmBaCsCo calibration results in Table 3, and even more so for the AmEu source in Table 4. In the AmEu calibration results in Table 4, the uncertainty for the weighted mean 152Eu activity for example goes from 1.0% in the uncorrelated analysis, to 3.1% when all correlations are accounted for. This is in accordance with expectation; since the calibration was based on a 152Eu source with 3% uncertainty, the activity uncertainties should not generally be smaller than that. This is true for all the activities except 133Ba. This nuclide has strong lines in the region where the influence of the uncorrelated 241Am calibration point reduces the degree of covariance in the efficiency curve.

Table 2 Uncertainties in weighted mean activities for multi-line nuclides with different analyses for the Mixed Gamma calibration source
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Table 3 Uncertainties in weighted mean activities for multi-line nuclides with different analyses for the AmBaCsCo calibration source
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Table 4 Uncertainties in weighted mean activities for multi-line nuclides with different analyses for the AmEu calibration source
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Interference corrections

Interference correction is performed when one or more nuclides identified in the spectrum contribute to one or more common peaks. A least-squares fit optimizes the nuclide activities from the line activity data. The Chi squared function has the form given in (2), with data vector elements \( y_{i} = N_{i} /(\varepsilon_{i}T) \), where \( N_{i} \) is the net peak area at energy \( E_{i} \) and \( \varepsilon_{i} \) is the efficiency. The basis functions are \( F_{\alpha i} = I_{\alpha i} \), the gamma intensity of nuclide \( \alpha \) at energy \( E_{i} \). The parameters \( a_{\alpha } \) are the nuclide activities. The data covariance matrix elements are given by

$$ V_{jk} = \frac{{N_{j} N_{k} }}{{\varepsilon_{j} \varepsilon_{k}T^2 }}\frac{{\text{cov} (\varepsilon_{j} ,\varepsilon_{k} ) }}{{\varepsilon_{j} \varepsilon_{k} }}. $$
(10)

The prototype version of these new algorithms assumes all peak areas are uncorrelated.Footnote 1

To demonstrate the effects of correlations on the interference correction, MCNP-CP was used to simulate the spectrum of a sample containing three multi-line nuclides with interfering lines—75Se, 57Co, and 152Eu—which was then analyzed using the previously described calibrations. The emission energies of the three nuclides are listed in Table 5. Both lines of 57Co are also present in the other nuclides, so that it is only possible to identify and quantify 57Co through the interference correction.

Table 5 Emission energies of interfering nuclide demonstration set
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The simulated spectrum was analyzed in three different ways, as described for the multiline nuclides. Recovery fraction results for the Mixed Gamma and AmEu calibrations are plotted in Fig. 4 for each analysis approach.

Fig. 4
figure 4

Changes in recovery fraction and uncertainty when different degrees of correlations are accounted for. Shown are results of the Mixed Gamma (left) and AmEu (right) calibrations

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As observed for the Weighted Mean results without interference corrections, there is little change in the reported activities in most cases and the primary difference is found in the uncertainties reported on the activities. For the directly observed nuclides 75Se and 152Eu, the error bars increase with the inclusion of additional degrees of correlation in the analysis. Somewhat counterintuitively, the uncertainty on the 57Co activity, quantified by inference in the interference correction, decreases with added correlation. The correlations between line activities and efficiency values actually provide additional constraints on the fit, giving better precision for this activity that is not directly observable.

The relative uncertainties for each nuclide as obtained with all three calibration standards under each of the analysis assumptions described above, are listed in Table 6.

Table 6 Uncertainties in weighted mean activities for multi-line nuclides, with different anlyses, for the three calibration sources
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Conclusions

Correlations introduced through the efficiency calibration process can have important effects on the quantification of radionuclide activities and uncertainties. These can be particularly important in high-precision applications, where measurement and sample geometry are well controlled and counting statistics do not dominate the uncertainties. Efforts are currently underway to incorporate more rigorous and complete uncertainty quantification in the Genie 2000 software suite by including correct handling of correlations. New spectroscopic analysis engines incorporating these changes have been prototyped to demonstrate the concepts and methodology. In the examples shown here, the prototype engines accurately reproduced the expected nuclide activities. The additional correlation effects from multi line calibration nuclides may affect line activities by as much as a percent or two, in normal applications, and generally will tend to increase the reported uncertainties for line and weighted mean activities. However these effects are somewhat complex and counterintuitive behavior was observed; for instance, the additional constraints provided by correlation information can reduce the uncertainties in some regions of the calibration curve and for some interference corrected activities.