The rainfall–runoff conceptual model as a cascade of submerged linear reservoirs with particular outflows depending on storages of adjoining reservoirs is developed. The model output contains different exponential functions with roots of Chebyshev polynomials of the first kind as exponents. The model is applied to instantaneous unit hydrograph (IUH) and recession curve problems and compared with the analogous results of the Nash cascade. A case study is performed on a basis of 46 recession periods. Obtained results show the usefulness of the model as an alternative concept to the Nash cascade.
The significance of the rainfall–runoff relation conceptual model introduced by Nash as a linear cascade of reservoirs (Nash, 1957) and developed later as parallel cascades (Wittenberg, 1975; Oben-Nyarko, 1976) known nowadays as the Diskin model (Diskin et al., 1978; Diskin, 1980) cannot be overestimated. These models have been widely applied in the mathematical modeling of catchments for many years and are still in use. Undoubtedly, one of the advantages of these models is the simplicity related to the linearity, what corresponds inter alia to the real baseflow features (Fenicia et al., 2006). However, the Nash and Diskin models do not represent many real hydrographs correctly enough, including peak flows (Singh, 1976). Bárdossy (2007) noticed the great uncertainty of the identified cascade parameters and related difficulties in the determination of the optimum parameters set for a particular catchment. These problems, together considered with the high diversity of real hydrographs shapes including recession curves (Stoelzle et al., 2003), force a search for new solutions. One of the modern tendencies are nonlinear models (e.g., Liu and Todini, 2002; Ding, 2011; Kim and Georgakakos, 2014). This direction of research may be perceived as an expression of disappointment due to unsatisfactory results of linear model applications. On the other hand, it seems that the possibilities of linear models have not been exploited enough. The linear model of cascaded reservoirs generating outputs different from the classical Nash hydrographs, which may be an alternative solution to standard ones, is presented below.
The peculiarity of the model is replacing classical reservoirs of the Nash
cascade by submerged ones (Fig. 1), where outflows depend on storages of
adjoining reservoirs (except the last reservoir in a chain). Assuming the
linearity of the system, it is described by the set of constitutive
equations:
Conceptual model of submerged reservoirs.
Determination of the eigenvalues' vector requires the solution to the
following equation:
The matrix of equations set constituting the SC2 model has the form
Graphs
Roots of the Chebyshev polynomials of any degree satisfy the relation
Graphs
The conditions of the filling/emptying rates for cascades of reservoirs is the basic feature differentiating (in a physical sense) the SC2 and Nash models. In the SC2 model, this rate depends on storages of both adjoining reservoirs (except the last reservoir in a chain), while in the Nash one it depends on the upper reservoir storage only. In other words, the present state of the reservoir in the Nash model does not affect the upper part of the cascade. This difference is analogous to the distinction between supercritical and subcritical flows in open channels, where any action can affect the upper part of a stream in the subcritical flow only. It is worth noting that the difference between storages of two neighboring reservoirs may be perceived analogously to the hydraulic slope in the groundwater flow; therefore, the SC2 model is a conceptual performance of the Darcy law. This analogy allows anticipation of the usefulness of the SC2 application first of all with regard to baseflow modeling.
Doubling of the storage coefficient for the last reservoir is a measure to obtain a simple, transparent algorithm for analytical solutions at any number of reservoirs; however, in real catchments, the last phase of outflow transformation takes place in watercourses and is characterized by distinctly different features in relation to the previous phases, i.e., surface, subsurface and baseflow. Similarly to the real conditions, the last reservoir in an SC2 cascade shows higher ability to empty itself in comparison with the upper ones.
Numerical values of constants of integration to the instantaneous unit hydrograph (IUH) in the SC2
model (
IUH at different numbers of reservoirs in the SC2 model (
Considering the instantaneous unit hydrograph (IUH) problem, the
following initial conditions are introduced:
Figure 4 shows the IUHs for consecutive reservoirs of the SC2 cascade for
number of reservoirs varying from
Recession curves at different numbers of reservoirs in SC2 model
(
Numerical values of constants of integration to recession curves in
the SC2 model (
Initial conditions for recession curves in the Nash model may be determined
by considering the equal storage for each reservoir with no rainfall supply.
Such assumption is rational and justified in particular for long-lasting
rainfall before the recession period. However, in the SC2 model, such
rainfall does not lead to the situation of equal storage of reservoirs since
in that case no flows between adjoining reservoirs exist. Therefore, the
initial conditions for SC2 may be formulated as
Comparison of IUH in SC2 and Nash models (
IUHs and recession curves yielded by SC2 were compared with analogous Nash
model results. In order to ensure the similarity of both cascades, the
storage coefficient
IUH peak values in SC2 and Nash models versus storage coefficient
IUH lag time in SC2 and Nash models versus storage coefficient
Comparison of recession curves in SC2 and Nash models
(
Comparison of the reaction of SC2 and Nash models to precipitation.
Figures 7 and 8 show peak flows (Fig. 7) and lag time (Fig. 8) versus
storage coefficient
Figure 9 shows recession curves for both models (in order to obtain better comparativeness of all graph pairs, values of storage coefficients for a particular number of reservoirs are differentiated). Differences of both hydrographs' shapes are apparent; in particular, curves generated by SC2 in their upper parts tend to decrease faster than the Nash ones. This leads to the conclusion that SC2 can be a good alternative to the Nash cascade at rapid transitions of hydrographs' curvature from a concave to convex one.
Figure 10 shows the reaction of both cascades to the precipitation occurring during the recession period. Rainfall with constant intensity lasting one time unit was introduced to the recessive scenario. Independently of the number of reservoirs, the peak flow generated by the time-distributed rainfall appears earlier and is more distinct in the SC2 cascade than in the Nash one. This testifies the rationality of further attempts of SC2 application not only to the baseflow but to the surface flow as well.
To examine the usefulness of the SC2 model for practical purposes, 12 catchments of the Vistula and Oder river basins with areas of
500–1000 km
Since each of the selected periods was preceded by rainfall of different
height and intensity, application of initial conditions neither relating to
the equal storage of all reservoirs in the Nash cascade nor to the condition
of Eq. (22) in the SC2 model was possible. Therefore, the initial conditions
defined by vector in the SC2 model: Eq. (18); in the Nash model:
Exemplary values of storage coefficient
Mean values of storage coefficient
Figures 9 and 10 show the optimization results. Despite the fact that the
SC2 model does not allow application of a non-integer number of reservoirs and
the Nash model was not analyzed from this point of view, graphs are
presented as continuous lines, which facilitates the analysis of the
variability of the optimized values. Figure 11 shows exemplary results of
the optimization for one of the catchments (Ścinawka River,
Gorzuchów gauge station) and Fig. 12 shows the averaged values of
storage coefficients
Comparison of graphs for both models leads to the following regularities:
In most cases, At low Optimized values of storage coefficient
In this study, the rainfall–runoff conceptual model as a cascade of submerged
linear reservoirs is proposed. The supply of each reservoir (except the
first one in a chain) depends on the storage of the upper reservoir and the
considered one as well. Additionally, to obtain the recurrence solution to
the set of equations describing water flow throughout the cascade, the value
of the storage coefficient
Comparison of features of IUHs and theoretical recession curves generated by
SC2 and Nash models suggests a possibility and even advisability of further
attempts to replace the Nash model by the SC2 one, in particular with
regards to baseflow modeling. This is confirmed by the analysis of measured
recession curves. Results of the analysis show that the optimized values of
storage coefficients
Data can be accessed by contacting the corresponding author.
Determination of the matrix
The author declares that he has no conflict of interest.
The author is grateful to Ezio Todini and Paolo Reggiani as well as to Michael Stoelzle for their very valuable comments and suggestions on the previous draft of the paper. Edited by: Albrecht Weerts Reviewed by: Paolo Reggiani and Ezio Todini