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.
The main difference lies in the way to estimate the initial storage state.
The GNM is the unique solution of the Nash cascade model.
The DLCM is an approximate solution of the Nash cascade model.
The DLCM can be transformed to the GNM.
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). Most of these models introduced the concept of the Nash model – cascade of linear reservoirs. In fact, the Nash model is also applicable to river flow routing (Yan et al. 2015, 2019), which has been done independently by Kalinin & Milyukov (1957), also known as a Characteristic Reach method. In the river flood forecasting, the initial state is usually thought to be insignificant as its effect will vanish after a sufficiently long simulation time. But for some short time prediction situations, just like the identification of the impulse-response function and the real-time forecasting, the initial state will produce a relatively great impact. Szollosi-Nagy (1982) formulated a state-space description of the Nash cascade model, i.e., the discrete linear cascade model (DLCM) in a matrix form whereby the initial state was included that can be thought of a generalization of the Nash cascade model. The determination of the initial state of the DLCM was then proposed by Szollosi-Nagy (1987) via observability analysis. The DLCM was discretized originally in a pulse-data system framework which seems more suitable for the irregularly changing precipitation but not necessarily for the gradually changing streamflow. Under a linear change assumption of the input, the DLCM was extended by Szilagyi (2003) to a sample-data system framework. Since then, Szilagyi and his team have made great efforts to develop this model (Szilagyi 2006; Szilagyi & Laurinyecz 2014). With so many advantages that have been summarized by Szilagyi (2006), the DLCM has been in operational use for over 30 years in Hungary. However, it has not yet been applied more broadly except in Hungary. One possible reason may be due to the complicated mathematical expression and calculation. The development of a simpler expression of the DLCM is necessary to make it more popular and applicable in practice.
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. What's more, the GNM has been physically interpreted, which makes it a conceptual model and not only a mathematical formulation. Compared with the DLCM, the GNM is also obtained from the Nash cascade model with the same initial condition. But whether the expressions or the simulation results of these two models are differently exhibited, there may be some confusion to the model users. It is necessary to distinguish these two models for the users. Hence, the relationship and difference between DLCM and GNM are studied in the following sections both mathematically and experimentally.
RELATIONSHIP BETWEEN THE DLCM AND THE GNM
That's just the GNM that has been proposed by Yan et al. (2015). Hence, the DLCM can be transformed to the GNM when the initial storage state is calculated by the linear storage-outflow relationship.
DIFFERENCE BETWEEN THE DLCM AND THE GNM
The main difference between the two models lies in the estimation of the initial storage state. In the DLCM, the initial storage state S(0) is expressed as a function of the first n input–output pairs, while, in the GNM, it is expressed as a function of the ith derivative of the initial outflow.
Comparison of Equations (22) and (23) suggests that the only difference between the recursive form of the two models in the case of n = 1 lies in whether the current outflow is estimated or observed. Similarly, for n > 1 in the GNM, the current storage state is estimated by current outflow and its derivatives, as shown in Equations (9) and (10), or equivalently current and antecedent observed outflows but not estimated ones used in the DLCM. Hence, the DLCM is an approximate solution but not the exact solution of the Nash cascade model. As an analytical solution, the GNM is applicable to the natural continuous streamflow system. However, in practice, the streamflow data are usually discretely measured. The derivative term in the GNM does not exist in the discrete streamflow data system. To make the GNM practically applicable, Yan et al. (2019) defined a variable Sn-curve to simplify the expression of the derivative term and further discretized the GNM. While the DLCM, as a discrete solution, can be directly applied to the discrete streamflow data system.
AN ILLUSTRATIVE EXAMPLE
The hydrograph was routed to distances of 20, 40, 60, 80, 100, and 120 km from the inflow section. To minimize the somewhat artificial nature of the upper and lower boundary conditions (Szilagyi 2006), the middle reach between 40 and 80 km was selected for flow routing, i.e., the flowrate values given by the Saint-Venant equations at 40 and 80 km served as the ‘observed’ upstream and downstream flow values, respectively. The SCE-UA global optimization algorithm (Duan et al. 1994) was used to optimize parameters in the two models by directly minimizing the root mean squared error, with same optimized values of n = 1 and K = 4.6 h. In the real-time forecasting, for example, take t = 16 h as the current time, then any time before t = 16 h can be taken as the initial time t0 in the DLCM. If t0 = 1 h, the current state can be estimated by combing Equations (14) and (21), and further the current outflow, i.e., O(t = 16 h) can be calculated by linear storage-outflow relationship, with a result of 126.13 m3/s. If t0 = 15 h, the current outflow O(t = 16 h) was estimated by the same procedure with a result of 118.58 m3/s. Then, this value was used to estimate the following outflow by using Equation (22). While for the GNM, the ‘observed’ value of O(t = 16 h) = 113.92 m3/s was directly used to estimate the outflow recursively by using Equation (23). The hydrographs obtained by the DLCM with different initial time as well as the GNM are illustrated in Figure 1. With different current outflow, the DLCM correspondingly provided different forecasted discharge values, especially the first few ones. The Nash–Sutcliffe efficiency coefficient (ENS) values for t0 = 1 h and t0 = 15 h were 0.9882 and 0.9919, respectively. The GNM provided the unique and also the best forecasted results, with a result of ENS = 0.9928. It is shown from this example that the DLCM has multiple solutions and the forecast precision depends upon the estimate accuracy of the current state.
Both the DLCM and GNM are derived from the Nash cascade model with a same non-zero initial condition. The DLCM formulated the Nash cascade model in a matrix form based on the principles of state-space analysis, while the GNM was written in a simpler algebraic expression after the complicated theoretical derivation. To clarify these two models, the relationship and difference have been investigated mathematically and experimentally. The main conclusions are summarized as follows:
The DLCM can be transformed to the GNM when the initial storage state is directly calculated by the linear storage-outflow relationship suggested in the Nash cascade model.
The essential difference between these two models lies in the identification of the initial state. In the DLCM, the initial state is estimated, while that in the GNM is observed.
The DLCM is an approximate solution of the Nash cascade model but not the exact solution due to its nonuniqueness of the initial estimated state. The GNM is the unique analytical solution of the Nash cascade model, whose initial state is implicitly written in a form of derivative and does not need to be estimated separately.
This study is financially supported by the National Natural Science Foundation of China (52079054) and the National Key R&D Program of China (2016YFC0402708).
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.