4.13. Aggregating activities of several aliquots¶
In certain cases, the measurement of a sample activity requires to determine the output quantity value from several compartments of the sample, or, regarding a surface contamination, from several measurements covering the entire surface.
This chapter describes how to proceed if several activity measurements, also of different types of measurements, need to be aggregated to obtain a single output quantity value. This value may be calculated as a sum or as an average of the single values. The simple aggregation method as described below also includes the calculation of the associated decision threshold and the detection limit.
If these measurements, however, represent repeated measurements in order to obtain a series of measurement values for a single input quantity, the recommended methods would be those described in section Using data sets for mean and variance and Gross quantity: Variance interpolation for a mean.
Activating the evaluation of several aliquot measurements
In the text field for equations, the following call is inserted for defining the activity as an aggregation of several values:
Asum = SumEval(mode, np, A1, A2, …)
The name of the symbol to the left of the = sign may be freely chosen. The name SumEval represents the internal procedure which does the calculations necessary for the aggregation of measured aliquot values. Its arguments are:
mode integer number, withe values:
1: calculate the mean value of the individual results,
2: calculate the sum of the individual results,
np integer, number of aliquot measurements
A1, A2, … the list np of symbol names (freely chosen) representing the activity or activity concentration values
Directly following the SumEval call, at first those main equations for calculating the activities Ai are inserted,
Ai = wi \* Rneti , for i=1 to np, one after another
These are followed by the lists of equations defining the calibration factors wi and the net count rates Rneti:
Rneti = Rbi – R0i
It is recommended to use further equations for explaining the count rates by their associated numbers of counts.
A complete example for two aliquot measurements may be defined as follows:
a = 1/F * Asum
Asum = SumEval(1, 2, A1, A2)
A1 = w1 * Rnet1
A2 = w2 * Rnet2
W1 = 1/eps1
W2 = 1/eps2
Rnet1 = Rb1 – R0
Rnet2 = Rb2 – R0
Rb1 = Nb1 / tb
Rb2 = Nb2 / tb
R0 = N0 / t0
Such factors found in all expressions of wi, may be extracted from the wi, i.e., not included in SumEval, as for instance the factor 1/F (1 / surface area) in the equation above that declaring Asum. This helps preventing covariances between the wi. An input quantity being part of several equations generates covariances between the quantities defined by these equations. This is true for the count rate R0 in the example given above, introducing a covariance between Rnet1 und Rnet2.
Such covariances, however, need not be identified explicitly by the user. They are considered by the uncertainty propagation applied within the SumEval procedure in the way, that covariance contributions of the form
induced by the independent input quantities \(x_{u,k}\) between dependent quantities \(x_{a,i}\), are taken into account; refer to Section 4.1.
Note: This procedure does not require further windows dialogs.
Notes about calculating the decision threshold and the detection limit
Calculations of the decision threshold and especially the detection limit require to vary the value \(a\) of the output quantity. Such an iteration step generates a modified value, denoted as \(a'\). This has to be transformed to new values \(A_{i}^{'}\) of the individual values \(A_{i}\) as part of SumEval. Two possible ways may be applied, which, based on the sample equations given above, are explained below.
If a mean value is to be calculated from the \(A_{i}\), a meaningful option would be to set all \(A_{i}^{'}\) to the same value \(a'\). The least-squares method is used as indicated in section 7.14 for calculating a weighed mean.
If instead a sum of aliquot values is to be derived, it may be meaningful, to modify the values \(A_{i}^{'}\) such that the original ratios between the \(A_{i}\) values are maintained. This may be achieved by applying relative “form“ factors \(h_{i}\)
\(h_{i} = \frac{A_{i}}{a = \sum_{j = 1}^{np}A_{j}}\)
such that
\({A_{i} = h}_{i}a\)
Then, the modified values \(A_{i}^{'}\) – and thereby the gross count rates \(R_{b,i}^{'}\) – are internally calculated from \(a'\):
\(Asum^{'} = a^{'}\ F\)
\(R_{net,i}^{'} = \frac{Asum^{'} \bullet h_{i}}{w_{i}}\)
\(R_{b,i}^{'} = \frac{Asum^{'} \bullet h_{i}}{w_{i}} + R_{0}\)
\(N_{b,i}^{'} = \left( \frac{Asum^{'} \bullet h_{i}}{w_{i}} + R_{0} \right)\ t_{m}\)
The associated uncertainties then are, again referring to the complete example given above:
\(u\left( N_{b,i}^{'} \right) = \sqrt{N_{b,i}^{'}}\)
\(u\left( R_{b,i}^{'} \right) = \sqrt{R_{b,i}^{'}/t_{m}}\)
\(u\left( R_{n,i}^{'} \right) = \sqrt{R_{b,i}^{'}/t_{m} + R_{0}/t_{0}}\)
From the uncertainties of the modified gross count rates, the uncertainty \(u(a^{'})\) associated with the activity value \(a'\) is calculated. Such pairs \((a^{'},\ u(a^{'}))\) are used for the iteration necessary for the detection limit calculation.
Example project:
sumEval_sum_EN.txp
sumEval_mean_EN.txp
