Modelling stage—discharge relationships affected by hysteresis using the Jones formula and nonlinear regression

Abstract Gauging stations where the stage—discharge relationship is affected by hysteresis due to unsteady flow represent a challenge in hydrometry. In such situations, the standard hydrometric practice of fitting a single-valued rating curve to the available stage—discharge measurements is inappropriate. As a solution to this problem, this study provides a method based on the Jones formula and nonlinear regression, which requires no further data beyond the available stage—discharge measurements, given that either the stages before and after each measurement are known along with the duration of each measurement, or a stage hydrograph is available. The regression model based on the Jones formula rating curve is developed by applying the monoclinal rising wave approximation and the generalized friction law for uniform flow, along with simplifying assumptions about the hydraulic and geometric properties of the river channel in conjunction with the gauging station. Methods for obtaining the nonlinear least-squares rating-curve estimates, while factoring in approximated uncertainty, are discussed. The broad practical applicability and appropriateness of the method are demonstrated by applying the model to: (a) an accurate, comprehensive and detailed database from a hydropower-generated highly dynamic flow in the Chattahoochee River, Georgia, USA; and (b) data from gauging stations in two large rivers in the USA affected by hysteresis. It is also shown that the model is especially suitable for post-modelling hydraulic and statistical validation and assessment.

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