EXISTING MATHEMATICAL METHODS FOR UNCERTAINTY ASSESSMENT - Modelling Uncertainty in Flood Forecasting Systems - page 33

Therefore, experts' judgements are used in one or other form to evaluate them. In gener-
al, experts are comfortable to make qualitative rather than quantitative judgement of such
parameters. Fuzzy set theory, which uses linguistic (qualitative) variables to represent
uncertainty, is very suitable to model uncertainty, particularly where experts' judgements
are used extensively. The present method is fully based on expert judgements (using
fuzzy variables) for the uncertainty in the input parameters, and it is qualitative also in the
sense that it propagates the uncertainty by evaluating the model rather qualitatively. This
method was originally reported by Sundararajan (1994 & 1998) to analyse the
uncertainties in computed natural frequencies of nuclear power plant piping systems.
Maskey (2001) and Maskey et al. (2002a) reported the method in the context of flood
forecasting. The procedures involved in the application of this method are as follows:
Identification of sources of uncertainty and decomposition
As in any other methods of uncertainty analysis, the first step consists of identifying the
sources of uncertainty in the output. These sources are then grouped into parameters and
sub-parameters (decomposition of parameters). This allows the expert to assess the
individual contributions of the decomposed parameters/sub-parameters rather than to assess the
uncertainty of a combination of many parameters. Also, the quality of a parameter may be
different for different events. This can be properly accounted for only if the parameter is
broken down into sub-parameters. As an example, suppose for a particular event the rainfall for
a basin was collected only from 4 gauge stations instead of usual 6 stations. The quality
of the rainfall measurement in the 4-station case may be different from the 6-station case.
Assessment of the Quality and Importance of parameters/sub-parameters
An important concept used in this method is that the uncertainty contribution of each
parameter is represented by two quantities: Quality and Importance . Quality refers to how
good the knowledge we have for the estimation of this parameter, and Importance refers
to how much this parameter contributes to the total uncertainty in the output. Evaluating
the Importance of a parameter can be viewed as giving a weight to the parameter for its
contribution to the total uncertainty in the output. If we compare Quality and Importance
with the probability-based FOSM method, Quality is analogous to the variance of the
parameter and Importance is analogous to the sensitivity of the output to the parameter.
Both Quality and Importance are evaluated qualitatively using a set of linguistic variables,
each of which is represented by a membership function. For example, if five variables are used,
the Quality may be evaluated as Very Good, Good, Acceptable, Bad and Very Bad. Equivalent
linguistic variables for Importance may be Very Large, Large, Moderate, Small and Very Small.
Membership functions are defined in arbitrary scale of 0 to 1. A typical set of linguistic variables
(both for Quality and Importance) with their membership functions is shown in Figure 3.3.