Complete formula of river flow routing based on Nash instantaneous unit hydrograph

Nash Instantaneous Unit Hydrograph (NIUH) is based on the assumption of zero initial conditions which is unable to represent river flood routing in real situation. The complete formula of NIUH for river flood routing which includes initial channel storage recession flow was deduced by the Laplace transform and mathematical induction. The physical meaning of each item of the complete formula was analyzed and discussed. It is demonstrated that outflow is composed of two portions: One is the recession of the initial channel storage, the other is the inflow routed through the channel. A real-time updating approach of the river flood routing was proposed based on the complete formula. The practicability and rationality of the formula were verified through the case study. The complete formula can be considered to be a more general case of the Nash river flood routing model. It is a development of the existed NIUH theory and can adequately improve the accuracy of the river flood routing.