Parameter estimation for the nonlinear Muskingum model using the BFGS technique

In the past, various methods have been used to estimate the parameters in the nonlinear three-parameter Muskingum model to allow the model to more closely approximate a nonlinear relation compared to the original two-parameter Muskingum model. In this study, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique, which searches the solution area based on mathematical gradients, is introduced. The technique found the best parameter values compared to previous results in terms of the sum of the square deviation between the observed and routed outflows, using the smallest number of computational iterations. A sensitivity analysis showed that the initial values of certain parameters were critical when finding the optimal solution. Although this gradient-based technique makes use of initial value assumptions and involves complicated calculus, different initial values reach the same optimal or near-optimal solution within less time. Moreover, this mathematical technique does not require the algorithm parameters that are essential factors in meta-heuristics such as genetic algorithm or harmony search. The technique also considers the hydrologic parameters to be continuous rather than discrete variables for pure structures.