Processing factors uncertainty

Processing effects are modelled either by a fixed processing factor, or by a lognormal or logistic-normal distribution (depending on the distribution type of the processing type). In case of a fixed factor, the uncertainty distribution is lognormal or logistic-normal with the same mean $\mu$ as the fixed value, and with a standard deviation ${\sigma }_{unc}$ which is calculated from the specified central value $\mu$ (or nominal) and an estimate of the p95 of the uncertainty distribution (set NominalUncertaintyUpper in the table for Processing factors).

The calculation is:

$${\sigma }_{unc}=\frac{\mathit{f}\mathit{(}\mathit{NominalUncertaintyUpper}\mathit{)}-f(\mu )}{1.645}$$

with $f()=logit$ for the logistic-normal distribution (distribution type 1) and $f()=ln$ for the lognormal distribution (distribution type 2). Values lower than 0.01 or higher than 0.99 (distribution type 1 only) are replaced by default values (0.01 and 0.99); this is useful computationally to avoid problems. In each iteration of the uncertainty analysis a new value is drawn from this distribution to be used as a fixed factor in the Monte Carlo calculation. In case of distribution based processing factors (describing the variability of processing factors) two uncertainties can be specified. For ${\sigma }_{unc}$, specification and calculation is as before (set NominalUncertaintyUpper in the table for Processing factors).

The uncertainty about the variability standard deviation

$${\sigma }_{var}=\frac{f(Upper)-f(\mu )}{1.645}$$

can be specified by the UpperUncertaintyUpper value. This value is specified as the p95 upper limit on Upper. The specified value is used to derive in a iterative search the number of degrees of freedom $\mathit{df}$ (van der Voet et al. (2009)). In the uncertainty analysis, a modified chi-square distribution with $\mathit{df}$ degrees of freedom is used to generate new values of ${\sigma }_{var}$. A very high value of $\mathit{df}$ means little uncertainty and ${\sigma }_{var}$ will be almost equal in all iterations of the uncertainty analysis. A $\mathit{df}$ close to 0 means a large uncertainty and very different values of ${\sigma }_{var}$ will be obtained in the iterations of the uncertainty analysis. The p95 upper limit on Upper is set through parameter UpperUncertaintyUpper.