Determination of Activity and Measurement Uncertainty According to DIN ISO 11929
The following considerations are based on DIN ISO 11929 and its application to the evaluation of measurements in open geometry with experimental determination of intrinsic detector efficiency according to the Filß method.
Information about DIN ISO 11929 can be found in the section Standards – DIN ISO 11929 at a Glance.
Preliminary Remarks:
The starting point for the following considerations is a container (e.g., a 200 L waste package) that possibly contains gamma-emitting radioactive material of activity A . This is measured with a suitable detector in an energy-resolved measurement. The task is to quantify the activity A.
The activity A is the physical effect to be investigated in this case. A non-negative measurement quantity is assigned to this effect through the evaluation of the measurement, which quantifies the physical effect.
Note:
The assignment of a non-negative measurement quantity thus contains the a-priori information that there are no negative activities.
The application of DIN 11929 Part 1 for this measurement task follows the overview of the general procedure provided in section 5 (Summary of Procedures for Evaluating a Measurement and Calculating the Characteristic Limits) of the DIN standard.
Establishment of the Model
The starting point is the model for counted measurements of ionizing radiation according to DIN 11929 Part 1 (Section 6.2.2) for the primary estimate y of the measurement quantity Y after substituting the estimated values xi
\[ y = G(x_1, x_2, \cdots , x_m) = (x_1 - x_2 \cdot x_3 - x_4) \cdot w = \left( \frac{n_g}{t_g} - \frac{n_0}{t_0} \cdot x_3 - x_4 \right) \cdot w \]
with
\[ w = \frac{x_6 \cdot x_8 \cdots}{x_5 \cdot x_7 \cdots} \]
and the standard uncertainty u(y) associated with y
\[ u(y) = \sqrt{w^2 \cdot \left[ u^2(x_1) + x_3^2 \cdot u^2(x_2) + x_2^2 \cdot u^2(x_3) + u^2(x_4) \right] + y^2 \cdot u_{rel}^2(w)} \newline = \sqrt{w^2 \cdot \left( \frac{r_g}{t_g} + x_3^2 \cdot \frac{r_0}{t_0} + r_0^2 \cdot u^2(x_3) + u^2(x_4) \right) + y^2 \cdot u_{rel}^2(w)}\]
with
\[ u_{rel} ^2 (w) = \sum_{i=5}^m \frac{u^2(x_i)}{x_i^2} \]
The quantity x1 is the count rate of the gross effect and x2 is the count rate of the null effect. The remaining input quantities xi are calibration, correction, or influencing quantities or conversion factors.
The model for activity determination according to Filß for a fully homogeneously filled container with specific activity a
\[ A = M \cdot a = M \cdot \left[ \frac{1}{\epsilon} \cdot \frac{1}{\eta} \cdot \left( \frac{\mu }{\rho} \right) \cdot \frac{1}{F_0} \cdot \frac{K_2}{K_1} \right] \cdot Z \]
is transferred to this model description. Some adjustments to Filß's model must first be made to capture all dependencies:
- The count rate Z is replaced by the difference between the count rate of the gross effect (rg = ng/t0) and the count rate of the null effect (rr,l = nr,l/t0). ng is the gross peak area derived from the measured gamma spectrum, nr,l is the corresponding area of the background, and to is the measuring time (live time).
\[ Z = \frac{n_g}{t_0} - \frac{n_{r,l}}{t_0} \]Information: More information on the identification and quantification of the characteristic peaks of the measured gamma spectrum can be found in the sections on gamma spectrometry and in the video tutorials. - The area F0 is expressed in terms of its dependencies on distance S and radius r.
\[ F_0 = F_\infty \cdot \frac{(S + r)^2}{S^2} \] - The cross-sectional area F∞ of the active matrix of the container is expressed by its height h and width 2∙r.
Thus, for the Filß model, the expression is obtained
\[ A = M \cdot \left[ \frac{1}{\epsilon} \cdot \frac{1}{\eta} \cdot \left( \frac{\mu }{\rho} \right) \frac{1}{h \cdot 2 \cdot r} \cdot \frac{(S+r)^2}{S^2} \cdot \frac{K_2}{K_1} \right] \cdot \left( \frac{n_g}{t_0} - \frac{n_{r,l}}{t_0} \right) \]
and therefore
\[ G(r_g, r_{r,l},M,\cdots , K_1) = A = \left( \frac{n_g}{t_0} - \frac{n_{r,l}}{t_0} \right) \cdot w \]
with
\[ w = M \cdot \left[ \frac{1}{\epsilon} \cdot \frac{1}{\eta} \cdot \left( \frac{\mu }{\rho} \right) \frac{1}{h \cdot 2 \cdot r} \cdot \frac{(S+r)^2}{S^2} \cdot \frac{K_2}{K_1} \right] \]
The assignment of the individual factors of this equation to the estimated values for xi can be done by simple comparison (Note: any existing background (e.g., from nearby sources) is not considered in this analysis (i.e., x2 = x3 = 0); furthermore, the dependencies of K1 and K2 on the linear attenuation coefficients and material thicknesses are not explicitly considered).
Preparation of Input Data and Specifications
In the second step, for all input quantities Xi, the corresponding estimated values xi and the uncertainties u(xi) are determined, as well as the specified probabilities α, β, and γ.
| Estimated Value | Assigned Quantity | Partial Derivative | Determination of u(xi) (possibilities for determination) |
|---|---|---|---|
| x1 | \[ r_g =\frac{n_g}{t_0}\] | \[ \frac{\partial G}{\partial r_g} = \frac{\sqrt{n_g}}{t_0} \] | \[ \sqrt{n_g} \] from Poisson statistics |
| x2, x3 | 0 | - | not considered here |
| x4 | \[ r_{r,l} = \frac{n_{r,l}}{t_0} \] | \[ \frac{\partial G}{ \partial r_{r,l}}= \frac{\sqrt{n_{r,l}}}{t_0} \] | \[ \sqrt{n_{r,l}} \] from Poisson statistics |
| x5 | \[ \epsilon \] | \[ \frac{\partial g}{\partial \epsilon} = -\frac{A}{\epsilon} \] | from repeated calibration measurements (empirical value) |
| x6 | \[ M \] | \[ \frac{\partial g}{\partial M} = +\frac{A}{M} \] | from calibration certificate of the scale |
| x7 | \[ \eta \] | \[ \frac{\partial g}{\partial \eta} = -\frac{A}{\eta} \] | from information on the tabulated values |
| x8 | \[ \mu \] | \[ \frac{\partial g}{\partial \mu } = +\frac{A}{\mu} \] | tabulated values in connection with empirical values |
| x9 | \[ \rho \] | \[ \frac{\partial g}{\partial \rho} = -\frac{A}{\rho} \] | estimation from mass and volume information; empirical values |
| x10 | \[ K_2 \] | \[ \frac{\partial g}{\partial K_2} = +\frac{A}{K_2} \] | estimation using tools (K2 calculator); empirical values |
| x11 | \[ h \] | \[ \frac{\partial g}{\partial h} = -\frac{A}{h} \] | estimation from available information (e.g., transmission measurements); empirical values |
| x12 | \[ S \] | \[ \frac{\partial g}{\partial S} = -\frac{A \cdot 2 \cdot r}{S \cdot (S+r)} \] | repeated determination of the distance; data sheet of the measuring device; |
| x13 | \[ K_1 \] | \[ \frac{\partial g}{\partial K_1} = -\frac{A}{K_1} \] | estimation using tools (K1 calculator); empirical values |
| x14 | \[ r \] | \[ \frac{\partial g}{\partial r} = -\frac{A\ \cdot (S-r)}{r\cdot (S + r)} \] | estimation from available information (e.g., transmission measurements); empirical values |
For the partial derivatives with respect to x12 (i.e., with respect to S) and x14 (i.e., with respect to r), a deviation from the partial derivative formation specified in the standard was made due to the nonlinearity of these input quantities.
The probabilities for a type I and type II error are often set at 5%, i.e., α = β = 0.05, as well as for the probability of the confidence interval (1 - γ), i.e., γ = 0.05. With these specifications, the quantiles of the standardized normal distribution k1-α = k1-β = 1.65 and k1-γ/2 = 1.96 are obtained. The values for the quantiles are tabulated.
Calculation of the Primary Measurement Result A with Standard Uncertainty ũ(Ã)
The primary measurement result, i.e., the activity A, can be determined by inserting the quantities assigned to the estimated values xi into the above equation. Likewise, the standard uncertainty ũ(Ã) is determined by inserting the partial derivatives and the quantities assigned to the estimated values xi into the above equation.
Calculation of the Standard Uncertainty ũ(Ã)
For the determination of the standard uncertainty, the true value à is required. Since this value is not known, the non-negative primary measurement result (à ≈ A) is used as a close approximation. If this is negative, the true value is set to 0.
Since the input quantity rg (gross count rate) is Poisson-distributed in this case (counting measurement) and no null effect is considered, the estimator x1 is given by
\[ r_g = \frac{\tilde{A}}{w} + r_{r,l} \]
From this, the true standard uncertainty (based on the approximation (Ã ≈ A)) can be calculated.
\[ \tilde{u}(\tilde{A} ) = \sqrt{ w^2 \left[ \frac{\tilde{A}}{w} + r_{r,l} + u^2(r_{r,l}) \right] + \tilde{A}^2 \cdot u^2_{rel}(w) } \]
Calculation of the Detection Limit y*
To determine the detection limit, the approximations (Ã ≈ A) and ũ(Ã) ≈ u(A) and the specified probability α are used. The determination equation results in
\[ A^* = k_{1 - \alpha} \cdot \tilde{u}(0) = k_{1 - \alpha} \cdot w \cdot \sqrt{r_{r,l} + u^2(r_{r,l})} \]
If the detection limit A* is smaller than the determined primary measurement result A, it can be assumed that the physical effect is present. In the considered case, this would mean that the radionuclide for which the evaluation was conducted is present.
Calculation of the Indication Limit A#
The calculation of the indication limit A#, the smallest true value for the activity A, is performed with a specified probability β and takes into account the associated detection limit.
\[ A^{\#} = k_{1 - \alpha} \cdot \tilde{u}(0) - k_{1-\beta} \cdot \sqrt{w^2 \cdot \left[ \frac{A^{\#}}{w} + r_{r,l} + u^2(r_{r,l}) \right] + A^{\#} \cdot u^2_{rel}(w)} \]
This implicit equation for the indication limit can be analytically determined by solving for A#.
\[ A^{\#} = \pm \frac{1}{2} \left[ \sqrt{\left(-2 \cdot \sqrt{r_{r,l} + u^2(r_{r,l})} \cdot k_{1-\alpha} \cdot w - u_{rel}^2(w) \cdot (k_{1-\beta})^2 - (k_{1-\beta})^2 \cdot w\right)^2 +} \newline \overline{ + 4 \cdot (r_{r,l} + u^2(r_{r,l})) \cdot w^2 \cdot \left((k_{1-\beta})^2 - (k_{1-\alpha})^2\right)} + \; \; \; \; \; \; \; \; \; \; \; \; \;\; \; \newline + 2 \cdot \sqrt{r_{r,l} + u^2(r_{r,l})} \cdot k_{1-\alpha} \cdot w + u_{rel}^2(w) \cdot (k_{1-\beta})^2 + (k_{1-\beta})^2 \cdot w \right] \]
For the analytical solution of such (complex) equations, the online tool WolframAlpha (https://www.wolframalpha.com/) is extremely helpful.
Calculation of the Confidence Limits A< and A>
The calculation of the lower (A<) and upper limit (A>) of the confidence interval is performed using the primary measurement result A and the associated standard uncertainty ũ(Ã) for the specified probability 1-γ.
\[ A^{\triangleleft} =A - k_p \cdot u(y) \; \; \mathrm{with} \; p = w \cdot \left( 1-\frac{y}{2} \right)) \] \[ A^{\triangleright} =A + k_q \cdot u(y) \; \; \mathrm{with} \; q = 1- w \cdot \frac{y}{2} \]
with
\[ \omega = \Phi \left( \frac{A}{u(A)} \right) \]
The values of the standardized normal distribution Φ(t) are tabulated.
The true value of the measurement quantity is within the confidence interval with a probability of 1-γ.
Calculation of the Estimated Value ŷ of the Measurement Quantity with Standard Uncertainty u(ŷ)
The physical effect is considered recognized if the primary measurement result A is greater than the detection limit (A > A*).
The best estimate  of the measurement quantity is then given by
\[ \hat A = A + \frac{u(A) \cdot \exp \left\lbrace -A^2 /\left[ 2\cdot u^2(A)\right] \right\rbrace}{\omega \cdot \sqrt{2\pi}} \]
with the standard deviation
\[ u(\hat{A}) = \sqrt{u^2(A) - (\hat{A} - A) \cdot \hat{A}} \]
Creation of the Test Report – Documentation
The conclusion of a measurement with evaluation is the summary of all used data in a complete documentation. The goal is that the evaluation can be retraced at any later time.
In the following exemplary structure of a test report, the gamma spectrometric evaluations (i.e., the determination of the gross peak areas and the peak background) are not included. These should be recorded in a separate test report (see example). The corresponding data from the measurement should be entered into the gray shaded fields. From these, the entries for the empty fields are calculated (automatically). When describing the evaluation method used, the underlying references should be stated (possibly with revision and/or publication date).
Evaluation Procedure
Measurement in open geometry with evaluation model according to Filß [Ref] based on DIN ISO 11929 [Ref], Section 5.2.2, model in nuclear radiation measurements.
Specifications
| Preselected Parameters | Unit | Calculated Parameters | ||
|---|---|---|---|---|
| \[ \alpha \] | 1 | \[ k_{1-\alpha} \] | ||
| \[ \beta \] | 1 | \[ k_{1-\beta} \] | ||
| \[ \gamma \] | 1 | \[ k_{1-\gamma /2} \] | ||
Parameters
| Quantity | Symbol | \[ x_i \] | \[ u(x_i) \] | Unit | Type | \[ u_{rel}(x_i) \] |
|---|---|---|---|---|---|---|
| Measurement Time (live) | \[ t_p \] | \[ s \] | ||||
| Gross Peak Area | \[ n_g \] | \[ 1 \] | ||||
| Peak Background | \[ n_{r,l} \] | \[ 1 \] | ||||
| Distance | \[ S \] | \[ cm \] | ||||
| Intrinsic Detector Efficiency | \[ \epsilon \] | \[ 1 \] | ||||
| Transition Probability | \[ \eta \] | \[ 1 \] | ||||
| Net Mass | \[ M \] | \[ g \] | ||||
| Height of Active Matrix | \[ h \] | \[ cm \] | ||||
| Radius of Active Matrix | \[ r \] | \[ cm \] | ||||
| Linear Attenuation Coefficient of Active Matrix | \[ \mu_{matrix} \] | \[ cm^{-1} \] | ||||
| Density of Active Matrix | \[ \rho_{matrix} \] | \[ g \cdot cm^{-3} \] | ||||
| Thickness of the First Wall Layer | \[ d_1 \] | \[ cm \] | ||||
| Linear Attenuation Coefficient of the First Wall Layer | \[ \mu_1 \] | \[ cm^{-1} \] | ||||
| Thickness of the Second Wall Layer | \[ d_2 \] | \[ cm \] | ||||
| Linear Attenuation Coefficient of the Second Wall Layer | \[ \mu_2 \] | \[ cm^{-1} \] | ||||
| Correction Factor Active Matrix | \[ K_1 \] | \[ 1 \] | ||||
| Correction Factor Wall Layers | \[ K_2 \] | \[ 1 \] |
Results and Characteristic Values
| Quantity | Symbol | \[ x_i \] | \[ u(x_i) \] | Unit | \[ u_{rel}(x_i) \] |
|---|---|---|---|---|---|
| Activity | \[ A \] | \[ Bq \] | |||
| Detection Limit | \[ A^* \] | \[ Bq \] | |||
| Indication Limit | \[ A^{\#} \] | \[ Bq \] | |||
| Lower Confidence Limit | \[ A^{\triangleleft} \] | \[ Bq \] | |||
| Upper Confidence Limit | \[ A^{\triangleright} \] | \[ Bq \] | |||
| Best Estimate | \[ \hat A \] | \[ Bq \] |
Assessment
At this point, the results are summarized and assessed again. This should contain the following information:
- The primary measurement result is above/below the detection limit
- The indication limit is above/below the specified guideline value. The measurement procedure is suitable/not suitable for the measurement purpose. (Alternatively: Since no guideline value was provided, the evaluation of the measurement procedure does not apply)
- Lower and upper confidence limits
- Best estimate with assigned standard deviation
In gamma spectrometric measurements, often multiple nuclides need to be quantified. This type of text-based assessment is less suitable for this. In such cases, it is advisable to expand the table "Results and Characteristic Values" with additional rows.
Tools
A calculation tool is available in the section Measurement Uncertainty in Open Geometry for determining the characteristic values with the specifications and parameters.
