Fuzzy Nonlinear Regression Approach to Stage-Discharge Analyses: Case Study

Abstract

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: (1) minimum spread and (2) 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.