Abstract
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), though 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 DLCM, the initial state is estimated and not unique, while that in the GNM is observed and unique. Hence, the GNM is the exact solution of the Nash cascade model, 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. As a discrete solution, the DLCM can be directly applied to the practical discrete streamflow data system. However, the numerical calculation approach such as the finite difference method is often used to make the GNM practically applicable. Finally, 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, whose forecast precision depends upon the estimate accuracy of the current state.
Highlights
In hydrology, the concept of linear reservoir cascade suggested by Nash (1957) is widely used in connection with the mathematical modeling of surface runoff
Many Nash cascade based models have been developed to model the rainfall-runoff process, e.g., the urban parallel cascade model proposed by Diskin et al (1978), the hybrid and extended hybrid model, respectively, represented by Bhunya et al (2005) and Singh et al (2007), the two-reservoir variable storage coefficient model formulated by Bhunya et al (2008), the cascade of submerged reservoirs model developed by Kurnatowski (2017), the inter-connected linear reservoir model (ICLRM) introduced by Khaleghi et al (2018), and the linear combination model of Nash model and ICLRM recently developed by Monajemi et al (2021)
Both the discrete linear cascade model (DLCM) and generalized Nash model (GNM) are derived from the Nash cascade model with a same non-zero initial condition
Summary
The concept of linear reservoir cascade suggested by Nash (1957) is widely used in connection with the mathematical modeling of surface runoff. Many Nash cascade based models have been developed to model the rainfall-runoff process, e.g., the urban parallel cascade model proposed by Diskin et al (1978), the hybrid and extended hybrid model, respectively, represented by Bhunya et al (2005) and Singh et al (2007), the two-reservoir variable storage coefficient model formulated by Bhunya et al (2008), the cascade of submerged reservoirs model developed by Kurnatowski (2017), the inter-connected linear reservoir model (ICLRM) introduced by Khaleghi et al (2018), and the linear combination model of Nash model and ICLRM recently developed by Monajemi et al (2021). Yan et al (2015) applied the Laplace transform and the principle of mathematical induction to solve the nth order nonhomogeneous linear ordinary differential equation (NLODE) of the Nash cascade model with a non-zero initial condition, and obtained the generalized Nash model (GNM) with a simpler expression. The relationship and difference between DLCM and GNM are studied both mathematically and experimentally
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