CONTRIBUTION OF PRESENT RESEARCH TO UNCERTAINTY ASSESSMENT METHODS - Modelling Uncertainty in Flood Forecasting Systems - page 65

4.3 Qualitative scales for uncertainty interpretation
A qualitative method for uncertainty assessment based on expert judgement and fuzzy set
theory has been presented in Chapter 3 (Subsection 3.2.2). In this method the uncertainty
in each input for quality and importance is represented by linguistic variables and is
defined by a membership function. In contrast with the MF used in the fuzzy Extension
Principle-based method (Subsection 3.2.1), the base value of the MF in the expert
judgement-based qualitative method is defined in terms of an arbitrary unit.
Consequently, the estimated MF of the output also has its base value in an arbitrary unit.
It, therefore, does not directly show the uncertainty bounds (upper and lower) for the
output in the unit for the quantitative interpretation of the output variable. The result of
this method is therefore qualitative and hence suited for qualitative interpretation.
The present research proposed a Qualitative Uncertainty Scale (QUS) in which the
estimated qualitative result can be measured (Maskey et al., 2002a). The procedures of
deriving the QUS are explained in Subsection 4.3.1. An example is presented in
Subsection 4.3.2 to show how the estimated uncertainty is represented on the QUS.
4.3.1 Derivation of Qualitative Uncertainty Scales
The QUS scale is derived using the concept of the best-case and the worst-case
scenarios. The best-case scenario corresponds to the case where the Qualities of all the
sub-parameters are assigned to the highest level and the worst-case scenario corresponds
to the case where all the Qualities are assigned to the lowest level. For example, if five
variables very bad, bad, acceptable, good and very good are used, all qualities will be
very good for the best-case and very bad for the worst-case. In both cases the Importance of the
parameters are taken from the experts' evaluation. That is, from Equation (3.25), we obtain
(4.34)
(4.35)
The estimated uncertainty values corresponding to the best- and the worst-cases are then
used to represent the two extremes (the highest and the lowest levels, respectively) on the
qualitative scales. The scales between the two extremes are derived by linear
interpolations at different membership levels. The interpolations are performed
separately for the lower and upper bounds. Let each division of the scale be represented
by a fuzzy number U i ( i =1,…, n; n is the number of divisions (levels) on the scales). Then
the lower bound and upper bound of U i at each belief level ! is given by