Discussion of “Improved Nonlinear Muskingum Model with Variable Exponent Parameter” by Said M. Easa

Abstract

The discusser would like to thank the author for presenting an improved nonlinear Muskingum model with a variable exponent parameter and would like to draw attention to some points. The need to consider a variable exponent parameter for a nonlinear Muskingum model has been pointed out by the author. The variable exponent parameter accounts for the nonlinearity of flood wave and clearly produces better model performance than that with the constant exponent parameter. The author found that even a small number of inflow levels result in an improvement in model performance. As mentioned by the author, as the number of inflow levels increase, model performance improves, but the improvement was found to be small beyond five levels in different case studies. The variable exponent parameter proposed by the author accounts for the nonlinearity of flood wave in a disaggregate manner. When five inflow levels are used, the number of model parameters will be seven making it a highly overparameterized model. Additional parameters should not be added to a model unless the increased accuracy justifies the increase in model complexity. Also, model complexity can sometimes lead to an undesirable loss of predictive power. In a regression model, accuracy needs to be balanced against complexity, and slight improvements in accuracy do not compensate for the additional complexity. As it will be shown in this discussion, considering a suitable continuous functional form for exponent parameter of nonlinear Muskingum model results in a substantial improvement in model performance. The proposed general functional form has five parameters and thus will be comparable with the case of five inflow levels proposed by the author. The author and most of the previous researchers have used inaccurate explicit Euler’s method [Tung’s method (Tung 1985)] along with the manipulated discharge equation to get a better fit for observed data. This manipulation [using It−1 rather than It in Eq. (6) of the original paper] is not acceptable from a mathematical viewpoint. All explicit solution methods will produce similar results for small size of the time interval, Δt, but in practice for historical field data, the size of the time interval is fixed and may not be small enough, and thus a suitable solution method with sufficient accuracy should be used. As shown by Vatankhah (2010, 2014), fourth-order Runge-Kutta method is an accurate and suitable solution method among the explicit solution methods for solving the Muskingum model which will be used in this discussion.