Estimation of the Darcy–Weisbach friction factor for ungauged streams using Gene Expression Programming and Extreme Learning Machines

Abstract

The most uncertain parameter in determining the stage-discharge relationship of an ungauged stream is the friction factor, a parameter for which a gauged stream can be back-calculated using hydrometric survey data that has been compiled over a wide range of flow conditions. The existing friction factor models for ungauged streams that do not require hydrometric survey data as input can have very large prediction errors. In this study, 415 streamflow events collected from 52 rivers across three continents were assembled to allow for the use of a suitable global dataset size/range to develop a robust model for the prediction of the friction factor in ungauged streams using two machine learning algorithms (Gene Expression Programing (GEP) and Extreme Learning Machine (ELM)). The new GEP and ELM models outperformed existing models with significantly improved R-squared values of 0.76 and 0.78, respectively. The combination of the two input parameters (relative smoothness and the friction slope) allowed the new models to more accurately predict the friction factor for a wider range of streams and flow conditions. The forecasting uncertainty of the developed GEP and ELM models were compared with past models and found to be the least uncertain, with Width Uncertainty Band (WUB) values of ±0.0536 and ±0.0504, respectively. For the first time in this field of research, partial derivative sensitivity analysis was applied to the trends of the friction factor estimation by GEP and ELM. It was found that GEP and ELM models are extremely sensitive to friction slope in comparison with relative smoothness. The GEP and ELM model results for predicting the Darcy–Weisbach friction factor were compared with different existing relationships, and the superior performance of the proposed models was illustrated.