Water Resources Research Report



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River stage and discharge records are essential for hydrological and hydraulic analyses. While stage is measured directly, discharge value is calculated from measurements of flow velocity, depth and channel cross-section dimensions. The measurements are affected by random and systematic measurement errors and other inaccuracies, such as approximation of velocity distribution and channel geometry with a finite number of measurements. Such errors lead to the uncertainty in both, the stage and the discharge values, which propagates into the rating curve established from the measurements. The relationship between stage and discharge is not strictly single valued, but takes a looped form due to unsteady flow in rivers.

In the first part of this research, we use a fuzzy set theory based methodology for consideration of different sources of uncertainty in the stage and discharge measurements and their aggregation into a combined uncertainty. The uncertainty in individual measurements of stage and discharge is represented using triangular fuzzy numbers and their spread is determined according to the ISO – 748 guidelines. The extension principle based fuzzy arithmetic is used for the aggregation of various uncertainties into overall stage discharge measurement uncertainty.

In the second part of the research we use fuzzy nonlinear regression for the analysis of the uncertainty in the single valued stage – discharge relationship. The methodology is based upon fuzzy extension principle. All input and output variables as well as the coefficients of the stage - discharge relationship are considered as fuzzy numbers. Two different criteria; the minimum spread and the least absolute deviation are used for the evaluation of output fuzziness. The results of the fuzzy regression analysis lead to a definition of lower and upper uncertainty bounds of the stage – discharge relationship and representation of discharge value as a fuzzy number.

The third part of this research considers uncertainties in a looped rating curve with an application of the Jones formula. The Jones formula is based on approximate form of unsteady flow equation, which leads to an additional uncertainty. In order to take into account of the uncertainties due to the use of approximate formula and measurement of discharge values, the parameters of the Jones formula are considered fuzzy numbers. This leads to a fuzzified form of Jones formula. Its spread is determined by a multi-objective genetic algorithm. We used a criterion to minimize the spread of the fuzzified Jones formula so that the measurements points are bounded by the lower and upper bound curves.

The study therefore considers individual sources of uncertainty from measurements to the single valued and looped rating curves. The study also shows that the fuzzy set theory provides an appropriate methodology for the analysis of the uncertainties in a nonprobabilistic framework.

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Department of Civil and Environmental Engineering, The University of Western Ontario


London, Ontario, Canada


Discharge calculation, Fuzzy arithmetic, Fuzzy number, Fuzzy nonlinear regression, Hysteresis, Jones formula, Measurement uncertainty, Stage-discharge relationship, Unsteady flow, Uncertainty aggregation


Civil and Environmental Engineering


Report no.: 061

A Fuzzy Set Theory Based Methodology for Analysis of Uncertainties in Stage-Discharge Measurements and Rating Curve