Technical note: Comparison between two generalized Nash models with a non-zero initial condition

Abstract. Initial condition can impact the forecast precision especially in a real-time forecasting stage. The discrete linear cascade model (DLCM) and the generalized Nash model (GNM), expressed in different ways, are both the generalization of the Nash cascade model considering the initial condition. This paper investigates the relationship and difference between DLCM and GNM both mathematically and experimentally. Mathematically, the main difference lies in the way to estimate the initial storage state. In the case of n = 1, it was shown theoretically that the difference between the two models is whether the current outflow is estimated (DLCM) or observed (GNM). The GNM is the unique solution of the Nash cascade model with a non-zero initial condition, while the DLCM is an approximate solution and it can be transformed to the GNM when the initial storage state is calculated by the approach suggested in the GNM. At last, a test example obtained by the solution of the Saint-Venant equations is used to illustrate this difference. The results show that the GNM provides a unique solution while the DLCM has multiple solutions depending on the estimate accuracy of the current state.