EXISTING MATHEMATICAL METHODS FOR UNCERTAINTY ASSESSMENT - Modelling Uncertainty in Flood Forecasting Systems - page 36

value obtained from defuzzification (Appendix I) give the idea of the amount of
uncertainty represented by the fuzzy membership function, which is like using the first
and second moments to express uncertainty in the probabilistic approach. The amount of
uncertainty represented by the fuzzy membership function can also be looked at in terms
of fuzziness and nonspecificity (Klir and Yuan, 1995). Maskey (2001) presented a
comparison of uncertainty expressed in terms of the support, defuzzification by the
centre-of-area and max-membership methods, nonspecificity and fuzziness.
As discussed earlier, the quality of a parameter is inversely proportional to the
uncertainty contribution of the parameter. Similar is the case with the standard deviation
(in probabilistic representation). Taking this as a basis Sundararajan (1994 & 1998)
presented a comparison of the fuzzy presentation of uncertainty to standard deviation.
This comparison however requires a known value of fuzzy uncertainty corresponding to a
known value of a standard deviation, called a bench-marking value.
An alternative way is to express the uncertainty using linguistic variables, such as
highly likely, likely, less likely etc., or expressing the uncertainty levels, such as small
uncertainty, large uncertainty, etc. The present research proposed a Qualitative
Uncertainty Scale to express the uncertainty levels, which is explained in Section 4.3.
3.3 Bayesian theory-based uncertainty assessment methods used in flood
forecasting
Although various theories of uncertainty representation have shown good potential in
other fields of engineering, only the methods based on probability theory have been used
widely in flood forecasting problems. While the sampling method, MC simulation, and
the approximate method, FOSM, have been widely used, the Bayesian forecasting system
(Krzysztofowicz, 1999) and the generalised likelihood uncertainty estimation (Beven and
Binley, 1992) can be considered as emergent methods in assessing uncertainty in flood
forecasting. The latter two methods, primarily based on Bayesian theory, are briefly
described here with recommendations to relevant literature for detail coverage. It should
be noticed that these two methods also require sampling techniques like MC simulation.
3.3.1 Bayesian forecasting system (BFS)
The Bayesian forecasting system (BFS) by Krzysztofowicz (1999) is a recent
development in the quantification of predictive uncertainty through a deterministic
hydrological model. The BFS decomposes the total uncertainty about the variable to be
forecasted (the “predictand”) into input uncertainty and hydrological uncertainty. In BFS
classification, the input uncertainty is associated with those inputs into the hydrological
model which constitute the dominant sources of uncertainty and which therefore are
treated as random and are forecasted probabilistically. The hydrological uncertainty is
associated with all other sources of uncertainty such as model, parameter, estimation and
measurement errors. The input uncertainty and the hydrological uncertainty are processed
separately through the so-called input uncertainty processor and the hydrological
uncertainty processor, respectively. These two uncertainties obtained from the two
independent processors are then integrated by a so-called integrator. The BFS structure