A Fuzzy Set Theory Based Methodology for Analysis of Uncertainties in Stage-Discharge Measurements and Rating Curve (original) (raw)
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Canadian Journal of Civil Engineering, 2010
The discharge and stage measurements in a river system are characterized by a number of sources of uncertainty, which affects the accuracy of a rating curve established from measurements. This paper presents 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 International Organization for Standardization (ISO) standard 748 guidelines. The extension principle based fuzzy arithmetic is used for the aggregation of various uncertainties into overall stage-discharge measurement uncertainty. In addition, a fuzzified form of ISO 748 formulation is used for the calculation of combined uncertainty and comparison with the fuzzy aggregation method. The methodology developed in this paper is illustrated with a case study of the Thompson River near Spences Bridge in British Columbia, Canada. The results of the case study show that the selection of number of velocity measurement points on a vertical is the largest source of uncertainty in discharge measurement. An increase in the number of velocity measurement points provides the most effective reduction in the overall uncertainty. The next most important source of uncertainty for the case study location is the number of verticals used for velocity measurements. The study also shows that fuzzy set theory provides a suitable methodology for the uncertainty analysis of stage-discharge measurements.
Fuzzy Nonlinear Regression Approach to Stage-Discharge Analyses: Case Study
Journal of Hydrologic Engineering, 2010
River discharge is typically derived from a single valued stage-discharge relationship. However, the relationship is affected by different sources of uncertainty, especially, in the measurement of discharge and stage values. The measurement uncertainty propagates into stage-discharge relationship curve and affects the discharge values derived from the relation. A fuzzy set theory based methodology is investigated in this paper for the analysis of uncertainty in the stage-discharge relationship. Individual components of stage and discharge measurement are considered as a fuzzy numbers and the overall stage and discharge uncertainty is obtained through the aggregation of all uncertainties using fuzzy arithmetic. Building on the previous work-fuzzy discharge and stage measurements, we use fuzzy nonlinear regression-in this case study for the analysis of uncertainty in the stage-discharge relationship. The methodology is based on fuzzy extension principle and considers input and output variables as well as the coefficients of the stage-discharge relationship as fuzzy numbers. Two different criteria are used for the evaluation of output fuzziness: ͑1͒ minimum spread and ͑2͒ least absolute deviation criteria. 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 methodology developed in this work is illustrated with a case study of Thompson River near Spences Bridge in British Columbia, Canada.
A New Approach of Fuzzy Methods for Evaluating of Hydrological Data
World Academy of Science, Engineering and Technology, International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering, 2011
The main criteria of designing in the most hydraulic constructions essentially are based on runoff or discharge of water. Two of those important criteria are runoff and return period. Mostly, these measures are calculated or estimated by stochastic data. Another feature in hydrological data is their impreciseness. Therefore, in order to deal with uncertainty and impreciseness, based on Buckley’s estimation method, a new fuzzy method of evaluating hydrological measures are developed. The method introduces triangular shape fuzzy numbers for different measures in which both of the uncertainty and impreciseness concepts are considered. Besides, since another important consideration in most of the hydrological studies is comparison of a measure during different months or years, a new fuzzy method which is consistent with special form of proposed fuzzy numbers, is also developed. Finally, to illustrate the methods more explicitly, the two algorithms are tested on one simple example and a ...
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Uncertainty associated with fuzzy membership functions for a water quality management problem is addressed through interval grey numbers. The lower and upper bounds of the membership functions are expressed as interval grey numbers, and the membership functions are modeled as imprecise membership functions. A grey fuzzy optimization model for water quality management of a river system is developed. Application of the optimization model with imprecise membership functions is illustrated with a hypothetical river system.
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Fuzzy Modelling for Uncertainty Propagation and Risk Quantification in Environmental Water Systems
Risk-based management of environmental systems, like rivers, lakes, groundwater aquifers and coastal areas, is a very useful approach for combating specific problems, like water pollution and loss of ecosystems biodiversity. Uncertainties that could be inherent to natural variabilities in space and time, such as those due to hydrological and climatic variations, together with uncertainties related to human activities or terrorist attacks, may produce various risks and failures affecting both human health and ecosystems integrity. The fuzzy set theory, in combination with mathematical modelling based on partial differential equations, is proposed in this paper, in order to propagate uncertainties in estimating output variables in water quality problems of water systems. Uncertainties in i nput variables and values of model parameters are first introduced as fuzzy numbers. Then, they are propagated using fuzzy arithmetic. Output variables, like water pollution and environmental risk, ...
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Uncertainty of model parameters in stream pollution using fuzzy arithmetic
Journal of Hydroinformatics, 2008
Fuzzy arithmetic is employed for the analysis of uncertainties in water-stream pollution, when the various model parameters involved are imprecise. The one-dimensional advection–dispersion equation, for both a conservative and a non-conservative substance, was solved analytically for point and Gaussian-hill input loads of pollution, considering the dispersion and decay coefficients involved as fuzzy numbers. The solution of the advection–dispersion equation was also conducted numerically for the same input loads with the finite-difference method, employing a Lagrangian–Eulerian scheme. The good agreement between analytical and numerical results presented in the form of fuzzy numbers confirms the reliability of the numerical scheme. The advection–dispersion equation of a non-conservative substance was then solved numerically for ten different water quality parameters, in order to study the water pollution in a water stream. The dispersion coefficient, the source terms and the input l...
Takagi–Sugeno fuzzy inference system for modeling stage–discharge relationship
Journal of Hydrology, 2006
Direct measurement of discharge in a stream is not only difficult and time consuming but also expensive. Therefore, the discharge in a stream is related to the stage through a number of carefully measured discharge values. A relationship between stages and corresponding measured discharges is usually derived using various graphical and analytical methods. As the relationship between stages and measured discharges is not linear, conventional methods based on least squares regression analysis for fitting a relationship are unable to model the non-linearity in the relationship and spatially in the cases when hysteresis is present in the data. The aim of the present study is to investigate the potential of Takagi-Sugeno (TS) fuzzy inference system for modeling stage-discharge relationships and the investigations are illustrated by application of the model to observed gauge and discharges of various gauging stations in Narmada river system, India. A step by step procedure for developing TS fuzzy model is also presented. The results show that the TS fuzzy modeling approach is superior than the conventional and artificial neural network (ANN) based approaches. Comparison of the models on hypothetical data set also reveals that the fuzzy logic based approach is also able to model the hysteresis effect (loop rating curve) more accurately than the ANN approach.