Reverse stream flood routing determines the upstream hydrograph in a stream reach given the downstream hydrograph. The Muskingum model of flood routing involves parameters that govern the routed hydrograph. These parameters are herein estimated using simulation methods coupled with optimization tools to achieve optimized parameters. Different simulation methods are shown to perform unequally in the estimation of nonlinear Muskingum parameters. This paper presents two simulation methods for nonlinear Muskingum reverse flood routing: (1) Euler equations and (2) Runge-Kutta 4th order equations. Moreover, the generalized reduced gradient (GRG) is used as the optimization tool that minimized the sum of the squared deviations (SSQ) between observed and routed inflows in a benchmark flood routing problem. Results show the Runge-Kutta 4th order equations yield better routed hydrographs with smaller SSQ than obtained in previous research and with the first simulation method (Euler equations).

This paper introduces a hybrid non-linear Muskingum model for flood routing. The proposed hybrid model has more degrees of freedom for fitting observed data than other non-linear Muskingum models. The main goal of this work is to develop a comprehensive model for outflow routing. The proposed hybrid model's predictive skill is evaluated with experimental, real and multimodal hydrograph-routing problems. The results confirm the predictive skill of the hybrid model based on the minimisation of the sum of the square deviation (SSD) between observed and routed outflows, the sum of the absolute value of the deviations (SAD) between the observed outflow and the computed outflow, and the deviations between the peak of the routed and actual outflows (DPO). Results from this study show the hybrid model improved the SSD by 79, 15 and 5%, SAD by 50, 2 and 5%, and the DPO by 77, 4 and 34% compared with the best alternative Muskingum model in solving the experimental, real and multimodal example problems, respectively.

The nonlinear form of the Muskingum model has been widely applied to river flood routing. There are four variants of the nonlinear Muskingum model based on alternative formulations of the nonlinear storage equation. This paper proposes a new Muskingum model with an improved, seven-parameter, nonlinear storage equation. The proposed model provides more degrees of freedom in fitting observed hydraulic data than other nonlinear Muskingum models. The proper estimation of the proposed Muskingum nonlinear model’s parameters is essential to achieve accurate flood-routing predictions. This paper introduces a hybrid method for the estimation of Muskingum parameters. The parameter-estimation method combines the shuffled frog leaping algorithm (SFLA) and the Nelder-Mead simplex (NMS). The proposed Muskingum model and parameter estimation method were applied to the routing of several hydrographs. Our results indicate improved performance of the methodology described in this work when compared with those of other Muskingum models.