Measurement Uncertainties
An important and often neglected topic in the execution and evaluation of segmented gamma scan measurements is the determination of measurement uncertainties, which we now want to specifically address from a practical application perspective. Details on this very, very complex topic can be found in the sections Standards – DIN ISO 11929 in Brief or Application Examples: Measurement Uncertainty in Open Geometry.
In practice, historically, only the uncertainties of the net peak areas ΔA for an activity A were often considered to determine the uncertainty. This usually led to uncertainties that were in the low single-digit percentage range or less. This approach completely disregarded the uncertainty about the content of the examined container.
Questions such as
- Is the matrix homogeneous?
- Is the container filled to the top?
- Can the matrix be accurately characterized by an average density?
- Are inner containers and/or shielding containers included?
- Is the activity of the various radionuclides homogeneously distributed or are there local “hot spots”?
- Were all areas of the container captured in the segmented gamma scan measurement, or were there “gaps”?
- etc.,
The number of such questions can be almost infinitely expanded, illustrating that there are many factors that influence the calculation of activities based on segmented gamma scan measurements and are associated with sometimes considerable uncertainties.
Another aspect is the evaluation model used to calculate the activities. These models are always based on certain assumptions. If these are not met or only partially fulfilled, this also has a (sometimes substantial) influence on the determined activities and their uncertainties.
What does this mean for practice and the determination of uncertainties?
The central task is to gather and evaluate all available information about the container and its content to be characterized. Based on this information, a suitable scan mode can then be selected, and (thereby) the appropriate evaluation model can be determined.
Assuming that the assumptions for the chosen evaluation model are (mostly) fulfilled, it is best to create a table with all parameters used in the model description (Note: these may depend on energy, i.e., they must be specified for all energy values of the determined characteristic lines!). If individual parameters depend on further parameters, these must also be captured. For each parameter, the uncertainties must then be determined.
Various sources can be used for the quantitative determination of the uncertainties of the various parameters:
- Repeated measurements (which is normally not practical in practice due to time constraints since the available time is preferably invested in maximizing the measurement time of one measurement);
- Statistical considerations (e.g., for counting measurements, the Poisson statistics can usually be assumed, i.e., the uncertainty for the net peak area \( n_{r,l} \) can for example be set to \(\sqrt{ n_{r,l}} \));
- Information from calibration certificates, data sheets, certificates, etc. (e.g., regarding the measurement uncertainty of a weighing unit, tolerance values for container dimensions, etc.);
- Additional information on tabulated values (e.g., uncertainties are usually provided for emission probabilities and half-lives in the tables);
- Own (well-founded!) estimates and/or experiential values, possibly including further measurement data (and their uncertainties);
- Use of tools;
- etc.
The respective uncertainties belong to the evaluation and must be documented for later tracking of the evaluation steps.
Note:
For the determined uncertainties, realistic values must be provided and not values that are too small in terms of minimal overall uncertainty.
Example of a parameter table
The following table was created for the evaluation model according to Filß for a measurement in open geometry (see Application Examples: Measurement Uncertainty in Open Geometry).
In the column xi, the parameter values are to be entered, and in the column u(xi), the associated uncertainties are to be specified, as well as in the column Type, the information about the type of measurement uncertainty. The relative measurement uncertainty urel(xi) can be (automatically) calculated from the respective parameter value and the associated uncertainty (\( u_{rel}(x_i) = \frac{u_i}{x_i} \)).
Type of measurement uncertainty:
Type A: Determination from the statistical analysis of several statistically independent measurement values from a measurement repetition.
Type B: Determination without statistical methods, for example, by taking values from a calibration certificate, from the accuracy class of a measuring device, or based on personal experience and previous measurements.
| Quantity | Symbol | \[ x_i \] | \[ u(x_i) \] | Unit | Type | \[ u_{rel}(x_i) \] |
|---|---|---|---|---|---|---|
| Measurement time (live) | \[ t_p \] | \[ s \] | ||||
| Gross peak area | \[ n_g \] | \[ 1 \] | A | |||
| Peak background | \[ n_{r,l} \] | \[ 1 \] | A | |||
| Distance | \[ S \] | \[ cm \] | A | |||
| Intrinsic detector efficiency | \[ \epsilon \] | \[ 1 \] | A / B | |||
| Emission probability | \[ \eta \] | \[ 1 \] | B | |||
| Net mass | \[ M \] | \[ g \] | A / B | |||
| Height of active matrix | \[ h \] | \[ cm \] | A | |||
| Radius of active matrix | \[ r \] | \[ cm \] | A / B | |||
| Linear attenuation coefficient of the active matrix | \[ \mu_{matrix} \] | \[ cm^{-1} \] | B | |||
| Density of active matrix | \[ \rho_{matrix} \] | \[ g \cdot cm^{-3} \] | A | |||
| Thickness of first wall layer | \[ d_1 \] | \[ cm \] | A / B | |||
| Linear attenuation coefficient of the first wall layer | \[ \mu_1 \] | \[ cm^{-1} \] | B | |||
| Thickness of second wall layer | \[ d_2 \] | \[ cm \] | A / B | |||
| Linear attenuation coefficient of the second wall layer | \[ \mu_2 \] | \[ cm^{-1} \] | B | |||
| Correction factor active matrix | \[ K_1 \] | \[ 1 \] | Tool | |||
| Correction factor wall layers | \[ K_2 \] | \[ 1 \] | Tool |
Once you have determined all values, you can (for example according to DIN 11929) calculate the characteristic quantities for evaluating your measurement: best estimate, detection limit, limit of quantification, lower and upper confidence interval.
Since the creation of the corresponding calculation formulas is very complex, we have already created these for various evaluation models:
- Application Examples: Measurement Uncertainty in Open Geometry
- Application examples for further models will follow
We also provide various tools for estimating the uncertainties of different parameters:
- Influence of the K1 factor in evaluations in open geometry according to the evaluation model of Filß. By varying the parameters included in the K1-factor, you can estimate their influence on the value of K1 and the uncertainty range.
- Influence of the K2 factor in evaluations in open geometry according to the evaluation model of Filß. By varying the parameters included in the K2-factor, you can estimate their influence on the value of K2 and the uncertainty range.
After you have determined all data, you can use the tool for the corresponding evaluation model to calculate the characteristic quantities (best estimate, detection limit, limit of quantification, lower and upper confidence interval) for a specific energy. You can also use the tools to check the suitability of a measurement method for the application case for given parameters and uncertainties.
- Evaluation for Measurement in Open Geometry According to the Evaluation Model of Filß
- Evaluations for further models will follow
