4.1.1. Methods without linear unfolding

An equation, or a system of several equations describing the evaluation of the measurement quantity, is called evaluation model. It is stated in the form:

(4.1.1)\[Y = G\left( X_{1},\ldots,\ X_{m} \right)\]

Inserting estimates for the input quantities yields an estimate of the output quantity, the primary measurement result y:

(4.1.2)\[Y = G\left( x_{1},\ldots,\ x_{m} \right)\]

The standard uncertainty \(u(y)\) associated with the primary measurement result \(y\) is calculated according to the following equation, assuming that measured values of the input quantities are statistically independent:

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

This is known as uncertainty propagation. The partial derivatives are also known as sensitivity coefficients. It is often written such that G is replaced by Y:

(4.1.4)\[u^{2}(y) = \sum_{i = 1}^{m}\left( \frac{\partial Y}{\partial x_{i}} \right)^{2}u^{2}\left( x_{i} \right)\]

If the input quantities have been measured in a way that they are not statistically independent, associated covariances have to be taken into account. Then, the extended previous equation reads:

(4.1.5)\[u^{2}(y) = \sum_{i = 1}^{m}\left( \frac{\partial G}{\partial x_{i}} \right)^{2}u^{2}\left( x_{i} \right) + 2\sum_{i = 1}^{m - 1}{\sum_{j = i + 1}^{m}{\frac{\partial G}{\partial x_{i}}\frac{\partial G}{\partial x_{j}}}u\left( x_{i},x_{j} \right)}\]

Here, \(u\left( x_{i},x_{j} \right)\) represents a more general way of stating a covariance between two measured input quantities, which often are also given as \(cov\left( x_{i},x_{j} \right)\) or \(covar\left( x_{i},x_{j} \right)\), respectively. The following holds: \(u\left( x_{i},x_{i} \right) = u^{2}\left( x_{i} \right)\) and \(u\left( x_{i},x_{j} \right) = u\left( x_{j},x_{i} \right)\).

Equation (4.1.3) is the basic form of the uncertainty propagation used in UR for methods not requiring linear unfolding. This is done by using the values and standard uncertainties defined under the TAB “Values, Uncertainties”. In case that covariances have also been declared in this TAB, Equation (4.1.5) is applied.