Standards – DIN ISO 11929 in Brief

In the DIN ISO 11929, procedures for determining the characteristic quantities are described

• detection limit y^*,
• quantification limit y^# and
• confidence limits y^⊲ and y^⊳

for a non-negative measured quantity.

This section summarizes the essential points necessary for understanding the application of measurements from various fields of non-destructive characterization of radioactive waste. Specific applications can be found in other sections (Application to measurements in open geometry).

Measured Quantity

By means of a measurement, a non-negative measured quantity Y is assigned to a physical effect. The quantity Y is referred to as an estimator and is a random variable.

The value y obtained from the measurement for the estimator Y is then an estimate for the measured quantity. Together with the standard uncertainty u(y), the value y forms the primary complete measurement result for the measured quantity. Its determination is based on a mathematical relationship of the measurement data and additional information through a suitable (evaluation) model. This must be capable of considering all involved quantities.

General Model Description

The measured quantity Y generally depends on multiple input quantities X_i and can be described in its most general form by the equation

\[ Y = G(X_1, X_2, \cdots, X_m) \]

The function G is the evaluation model. By substituting given estimates x_i for the input quantities X_i into this model function, the primary measurement result y of the measured quantity results

\[y = G(x_1, x_2, \cdots , x_m) \]

Note:
An evaluation is only as good as the underlying model. If the model does not adequately describe the physical effect, the results are usually unusable.

The standard uncertainty u(y) associated with the primary measurement result y is determined by the relationship

\[ u^2(y) = \sum_{i=1}^m \left( \frac{\partial G}{\partial X_i} \right) ^2 \cdot u^2(x_i) \]

A prerequisite for this description is the mutual independence of the input quantities X_i from one another. In the partial derivatives of the model function G, the input quantities X_i are to be replaced by their estimates x_i.

Model for Nuclear Radiation Measurements

In measurements of radioactive material, the measured quantity Y is determined with the true value ỹ from the count rates of the gross effect and the net effect. A (possible) model for this is

\[ Y = G(X_1, X_2, \cdots , X_m) = (X_1 - X_2 \cdot X_3 - X_4) \cdot \frac{X_6 \cdot X_8 \cdots}{X_5 \cdot X_7 \cdots} = (X_1 - X_2 \cdot X_3 - X_4) \cdot W \]

with

\[ W = \frac{X_6 \cdot X_8 \cdots}{X_5 \cdot X_7 \cdots} \]

In this model, the count rate of the gross effect (r_g = n_g/t_g) is described by the input quantity X₁ and the count rate of the net effect (r₀ = n₀/t₀) is described by X₂. X₃ can be an additional shielding factor, X₄ an additional correction factor for the background. All other input quantities X_i (i ≥ 5) are calibration, correction, or influencing quantities or conversion factors. The number of these depends on the respective model description.

The primary estimate y of the measured quantity Y results from substituting the estimates x_i into

\[ y = G(x_1, x_2, \cdots , x_m) = (x_1 - x_2 \cdot x_3 - x_4) \cdot w = \left( \frac{n_g}{t_g} - \frac{n_0}{t_0} \cdot x_3 - x_4 \right) \cdot w \]

with

\[ w = \frac{x_6 \cdot x_8 \cdots}{x_5 \cdot x_7 \cdots} \]

By substituting the estimates x_i, w, and y into the partial derivatives

\[\frac{\partial G}{\partial X_1} = W, \quad \frac{\partial G}{\partial X_2} = -X_3 \cdot W, \quad \frac{\partial G}{\partial X_3} = -X_2 \cdot W, \newline \frac{\partial G}{\partial X_4} = -W, \quad \frac{\partial G}{\partial X_i} = (-1)^i \frac{Y}{X_i} \quad (i \geq 5) \]

results in 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 estimates x_i and the standard uncertainties u(x_i) for i >3 are determined experimentally in preliminary trials or taken as empirical values based on other information.

Calculation of Standard Uncertainty as a Function of the Measured Quantity

For determining the detection limit and the quantification limit, the true standard uncertainty ũ(ỹ) is needed for the true measured quantity ỹ (with ỹ ≥ 0).

The specification of the true value ỹ results in the estimate x₁

\[ x_1 = \frac{\tilde{y}}{w} + x_2 \cdot x_3 + x_4 \]

and thus for the true standard uncertainty

\[ \tilde{u}(\tilde{y}) = \sqrt{w^2 \cdot \left[ \frac{\tilde{y}}{w} + x_2 \cdot x_3 + x_4 + x_3^2 \cdot u^2(x_2) + x_2^2 \cdot u^2(x_3) + u^2(x_4) \right] + y^2 \cdot u_{rel}^2(w)} \]

It has been assumed that in the case considered, it is a counting measurement for the pulses n₀ of the gross effect. The Poisson statistics can be assumed, i.e., u²(x₁) = x₁.

If this assumption is not met, either explicit calculations or approximate solutions should be applied (see DIN ISO 11929).

Detection Limit

The determined primary measurement result y only significantly indicates that the true value is different from zero (i.e., ỹ > 0) if it is greater than the detection limit y^*, meaning it must hold: y > y^*.

The detection limit y^* is determined by the relationship

\[ y^* = k_{1-\alpha} \cdot \tilde{u}(0) \]

The specified value α in the factor k_1-α (quantile of the standardized normal distribution to probability) indicates the probability that despite satisfying the condition y > y^*, the physical effect does not exist, i.e., ỹ = 0.

Quantification Limit

The quantification limit y^# is the smallest true value ỹ of the measured quantity, for which it is assumed with the specified probability β that the physical effect is not present. Thus, the quantification limit is the smallest true value that can still be detected using the measuring method. It is the smallest solution of the equation

\[ y^{\#}= y^* + k_{1-\beta} \cdot \tilde{u}(y^{\#}) \; \; \mathrm{ with } \; y^{\#} \geq y^{*} \]

The implicit equation for the quantification limit can be determined analytically by solving for y^# or by iteration (using the approximation value ỹ_i for y^#)

\[ \tilde{y}_{i+1} = y^* + k_{1-\beta} \cdot \tilde{u}(\tilde{y}_i) \]

with the initial condition ỹ₀ = 2∙y^*.

Confidence Limits

The confidence interval contains the true value of the measured quantity with the specified probability 1-γ.

Given the primary measurement result y and the associated standard uncertainty u(y) of the measured quantity, the lower limit of the confidence interval y^< and the upper limit of the confidence interval y^> are determined by

\[ y^{\triangleleft} =y - k_p \cdot u(y) \; \; \mathrm{with} \; p = \omega \cdot \left( 1-\frac{\gamma}{2} \right) \newline y^{\triangleright} =y + k_q \cdot u(y) \; \; \mathrm{with} \; q = 1- \omega \cdot \frac{\gamma}{2} \]

with

\[ \omega = \Phi \left[ \frac{y}{u(y)} \right] \]

The values of the standardized normal distribution Φ(t) are tabulated.

Note: The confidence limits are usually neither symmetric to y nor to ŷ. The relationship 0 < y^⊲ < y^⊳ holds.

Best Estimate and Standard Uncertainty

The physical effect is considered detected when the primary measurement result y is greater than the detection limit (y > y^*).

The best estimate ŷ of the measured quantity is then given by

\[ \hat{y} = y + \frac{u(y) \cdot \exp(\frac{-y^2}{2\cdot u^2(y)})}{\omega \cdot \sqrt{2\ \pi}} \]

with the associated standard deviation

\[ u(\hat{y}) = \sqrt {u^2 (y) - (\hat{y} - y)\cdot \hat{y}} \]

Assessment of the Measurement Method

By comparing the quantification limit y^# with a specified reference value y_r, a decision can be made regarding the suitability of the measurement method.

If the quantification limit cannot be determined or y^# > y_r, then the measurement method is not suitable for the intended measurement purpose.