The Heterogeneous Discrete Generalized Nash Model for Flood Routing

The topographic heterogeneity of the rivers has great effects on the river flood routing. The discrete generalized Nash model (DGNM), developed on the basis of the Nash’s instantaneous unit hydrograph (IUH), is a lumped model that can’t reflect the spatial heterogeneity of the river topography. The heterogeneous DGNM (HDGNM) with a consideration of such heterogeneity has been developed 10 by the conceptual interpretation of the DGNM. Two compositions of the downstream outflow generated by the recession of the old water stored in the river channel and the discharge of the new water from upstream inflow were deduced respectively with the help of the heterogeneous IUH and the corresponding heterogeneous S curve. The HDGNM is finally expressed as a linear combination of the inflows and outflows, whose weight coefficients are calculated by the heterogeneous S curve. 15 The HDGNM expands the application scope, and becomes more applicable, especially in river reaches where the river slopes and cross-sections change greatly. The middle Hanjiang River was selected as a case study to test the model performance. It is suggested that the HDGNM performs better than the DGNM, with higher model efficiency and smaller relative error in the simulated flood hydrographs. 20 https://doi.org/10.5194/hess-2020-17 Preprint. Discussion started: 18 February 2020 c © Author(s) 2020. CC BY 4.0 License.


Introduction
In rainfall-runoff modelling, the instantaneous unit hydrograph (IUH) proposed by Nash (Nash, 1957) is one of the most widely used methods for overland flow routing. Under the assumption that watershed was represented as a cascade of equal linear reservoirs, IUH was obtained in a form of 25 gamma distribution with two parametersn, the number of linear reservoirs, and K, reservoir storage coefficient. The linear cascade concept has greatly promoted the development of the flow routing theory. Many linear cascade -based models have been developed since then. However, as a lumped model, IUH cannot reflect the spatial heterogeneity of rainfall and landforms. Great efforts have been made to make IUH be semi-distributed or distributed mainly by two approaches. The first approach 30 has been mostly performed by replacing the equal reservoirs in the IUH with unequal ones. Dooge (1959) conceptualized the watershed as a combination of unequal linear reservoirs and linear channels, and developed a general theory for unit hydrograph. Singh (1964) derived the IUH using a nonlinear model considering the overland and channel flow components separately, in which two unequal linear reservoirs with different storage coefficients were used. To solve the flow routing in urban areas, 35 Diskin et al. (1978) proposed an urban parallel cascade IUH model by representing the basin system as the combination of two parallel branches having a series of equal linear reservoirs. Bhunya et al. (2005) developed a hybrid model by splitting the single linear reservoir into two serially connected reservoirs of unequal storage coefficients (one hybrid unit), and obtained the analytical expression of the model for two hybrid units in series. Later, to consider the translation time, Singh et al. (2007) 40 extended this hybrid model by inserting a linear channel between each hybrid unit. Bhunya et al. (2008) formulated a rainfall-runoff model incorporating a variable storage coefficient instead in the two-https://doi.org/10.5194/hess-2020-17 Preprint. Discussion started: 18 February 2020 c Author(s) 2020. CC BY 4.0 License. reservoir Nash cascade model. Li et al. (2008) derived the IUH with different K values for each reservoir using the Laplace transform and developed a general rule for the equation of the IUH of any order. The other approach is to divide the watershed into a number of subwatersheds to consider the 45 nonuniformly distributed rainfall. For example, based on the structure of a stream network, Wang and Chen (1996) divided the watershed into a number of subwatersheds and obtained the outflow hydrograph of each subwatershed based on the concept of linear cascade reservoirs. This linear, spatially distributed model can be capable of predicting runoff from non-uniformly distributed rainfall and geographical conditions over an entire watershed. Similarly, Wan et al. (2016) divided the 50 watershed into subareas by isochrones, and established an independent linear reservoir-channel cascade model in each subarea. Finally, the generalized concentration curve that can be applied to large heterogeneous watersheds was derived. All of these modifications of IUH have made a certain improvement in the rainfall-runoff modeling.
As a general method of flow routing, Nash's IUH is also applicable to river flow routing, which 55 has been done independently by Kalinin and Milyukov (1958), also known as Characteristic Reach method. Nash's IUH can also be obtained by solving the nth order differential equation of a linear system with the zero initial conditions (Chow, 1988). Zero initial conditions represent that the linear reservoirs in the Nash cascade model are empty at initial time, or equivalently the initial river storages are empty when IUH is applied in river flow routing, which does not match the fact. To consider the 60 influence of the initial state, Szollosi-Nagy (1982) formulated a state-space description of the Nash cascade model in a matrix form, whereby the initial storage of the river system should be estimated separately via observability analysis (Szollosi-Nagy, 1987). Szilagyi (2003) then extended this model https://doi.org/10.5194/hess-2020-17 Preprint. Discussion started: 18 February 2020 c Author(s) 2020. CC BY 4.0 License.
to a sample-data system framework and made some modifications to make it more applicable (Szilagyi, 2006;Szilagyi and Laurinyecz, 2014). Recently, Yan et al. (2015) exactly solved the nth order 65 differential equation of the Nash cascade model with the same non-zero initial condition, and obtained the generalized Nash model (GNM) with a simpler expression, in which the initial state was directly included and should not be estimated separately anymore. To make the GNM be applied easily to the sample-data system, Yan et al. (2019) further discretized its analytical expression by introducing a variable Sn-curve, and obtained the discrete generalized Nash model (DGNM). The DGNM expresses 70 the outflow as a linear combination of the old water stored in the river reach and new water from the upstream inflow. The DGNM is based on the lumped IUH that it cannot reflect the spatial heterogeneity of the river topography too. So the DGNM is less applicable for those the topography changes large along the river. In this case, the unequal reservoir concept should be used instead.
However, under the non-zero initial conditions, the solving of high order differential equation of the 75 Nash cascade model with unequal storage parameters will become very difficult, which makes the generalization of the Nash model impossible by directly solving the differential equation. A new way is proposed in this paper to obtain the heterogeneous DGNM (HDGNM) through the conceptual interpretation and mathematical derivation of the DGNM.
Where is the combination formula; ± represents the downstream outflow at time t ± i∆t; ∆t is the time interval; represents the upstream inflow at time t; ∆ +1 represents the inflow increment during the time interval [ , + ∆ ]; n and K are model parameters; can be computed by 85 According to the definition of the S curve, S represents the outflow generated by the unit continuous inflow after the routing of a series of i reservoirs at the end of the time. The calculation formula of the DGNM shows that the downstream outflow is composed of three items. The first item is the recession flow of the current water storage capacity in the channel, which is the superposition of 90 the flow generated by the current storage of each reservoir routed by the subsequent reservoirs. The second term is the recession flow generated by the current inflow I t routed by river channel, or equivalently by a series of n cascade linear reservoirs. According to the definition of S as well as the storage-discharge relation of the linear reservoir, KS i represents the water stored in each reservoir for a unit continuous inflow, and ∑ ∆ ⁄ =1 represents the ratio of water stored in the channel during 95 represents the ratio of water discharges from the channel. So the third term is the outflow generated by the inflow increment during the time interval [ , + ∆ ] after the channel routing. In summary, the downstream outflow is generated by the old water stored in the river channel and the new water from upstream inflow. Part of the new water flows out of the downstream section and becomes one part of the outflow, the other part remains in the river channel 100 to supplement the old water. The old water recedes and becomes the other component of the outflow.
In such circulation, the outflow process of the downstream section is formed. Through the conceptual interpretation of the DGNM, the downstream outflow is physically generated by the old water stored https://doi.org/10.5194/hess-2020-17 Preprint. Discussion started: 18 February 2020 c Author(s) 2020. CC BY 4.0 License.
in the river reach and new water from the upstream inflow, and formally expressed as a linear combination of the inflows and outflows, whose weight coefficients are calculated by the S curve that 105 gives another way to deduce the HDGNM by introducing the heterogeneous S curve.

Heterogeneous S curve
The routing storage capacity of the basin is affected by geographical features and has spatial heterogeneity. The storage routing effect of the basin is equated to a cascade reservoir routing in Nash's IUH, and each reservoir has the same storage coefficient. This generalization has certain rationality 110 for basins with homogeneous topography variation. But for the basins with large topographic changes, the spatial difference in storage routing effect will be more significantly affected by topography and geomorphology. To consider such spatial heterogeneity in storage routing effect, Dooge (1959) conceptualized the watershed as a combination of unequal linear reservoirs and linear channels, and developed a general theory for unit hydrograph. Li et al. (2008) further deduced the IUH with different 115 storage parameters, here we call it heterogeneous IUH (HIUH) to distinguish with Nash' IUH: where K ( = 1, ⋯ , ) is the storage parameter of the i-th reservoir (the numbers here are sorted in the forward direction, that is, the most upstream reservoir is the number 1, and the most downstream reservoir is the number n). Correspondingly, the heterogeneous S curve formed by HIUH is (Li et al., 120 2008) https://doi.org/10.5194/hess-2020-17 Preprint. Discussion started: 18 February 2020 c Author(s) 2020. CC BY 4.0 License.
where ( ) represents the outflow of the nth reservoir yielded by a continuous unit upstream inflow.
If further define the storage curve, we obtain represents the detention storage of the nth reservoir yielded by a continuous unit upstream inflow.
The HIUH is a more accurate generalization of the watershed storage routing, and is a theoretical expansion of the Nash's IUH. With consideration of the spatial heterogeneity in the storage routing, HIUH is especially applicable to the basin with large topographic changes. The DGNM is developed 130 on the basis of the Nash's IUH, which leads to its theoretical limitations when applied to the river reach with large changes of cross-sections and slopes. The introduction of HIUH can reflect the difference of flood routing in each sub-river, thus can improve the flood simulation precision in the whole river reach theoretically.

Derivation of the heterogeneous DGNM 135
The conceptual interpretation of the DGNM shows that the downstream outflow is generated by the old water stored in the channel and the new water from upstream inflow, denoted by O old and O new respectively, we have ( ) ( ) represents the ratio of water stored in the channel during the period ∆t, based on the former conceptual interpretation of the O new . Hence, for the linear reservoir system with unequal storage parameters, the outflow generated by "new water" 145 can be obtained by replacing the storage parameter K and S curve in Eq. (7) with variable Ki and heterogeneous S curve, respectively, i. e.
Therefore, on the basis of the conceptual interpretation of the DGNM, the outflow O new formed by the new water can be deduced directly. But it seems impossible to obtain the outflow O old by 150 directly using the heterogeneous S curve instead of in Eq. (7) due to the coefficient of − is also varying with it. For the sake of simplicity, we assume that the most downstream reservoir is numbered 1, and the most upstream reservoir is numbered n, that is to say, the n reservoirs are reversely numbered.
Then the storage routing equation of the j-th reservoir can be obtained from the water balance equation: It can be known from Eq.(9) that the outflow of each reservoir at the current time is as follows: Based on the physical interpretation of the GNM (Yan et al., 2015), the recession flow of the current water storage in river channel is the superposition of the recession flow generated by the current water storage in each reservoir. According to the conception of linear reservoir, the current water storage of the j-th reservoir is K ( ), which can be treated as an instantaneous inflow into each 165 reservoir, then the outflow at the end of the period generated by that is K ( ) (∆ ). Based on the principle of superposition, the outflow at the end of the period formed by the current water storage of all reservoirs is ( ) ( ) The formula shows that the recession process can finally be expressed as a linear combination of 0~(n-1) derivatives of the current time O( ), which is is derived in the case of reverse numbering, and its calculation formula shows that has symmetry. Therefore, in the case of forward numbering, or equivalently, let 1 , , n K K  replace https://doi.org/10.5194/hess-2020-17 Preprint.
Substituting equation (13) and (14) into equation (12), we obtain ( ) According to the conceptual interpretation of the DGNM, the downstream outflow is jointly produced by the old water stored in the river channel and the new water from upstream inflow, then we have Eq. (16) is the calculation formula of HDGNM, and is also the discrete solution of linear cascade 190 model with unequal reservoirs under non-zero initial conditions. Like the HIUH, with a consideration of the spatial heterogeneity in the storage routing, HDGNM should be more applicable to the river reach with large changes of cross-sections and slopes.

Case study
To test the applicability of the HDGNM, the river reach between gauging stations Huangjiagang  Fig. 1. The studied river reach is located in the hilly and plain areas, where hills, terraces, artificial narrows and wide valleys distribute alternatively, and showing obvious lotus root node shape on the plane. The main channels in wide sections have large swings and many beaches, but become single in narrow sections, which makes a large change of the shape in the sections along the river reach, as shown in Fig.1. The mean slopes of sub-reaches Huangjiagang -Guanghua, Guanghua -Taipingdian, 210 Taipingdian -Niushou and Niushou -Xiangyang are 0.000176, 0.000276, 0.000221 and 0.000214 (Gong, 1982), respectively. It is indicated that the channel slope of the studied river reach changes largely, especially from sub-reach 1 to sub-reach 2. In short, the topography of the studied river reach varies greatly. For such significant spatial heterogeneity, the proposed HDGNM should be more applicable as interpreted above. 215 Fig.1 The sketch map of the middle Hanjiang River and the studied river reach According to the flood data of the Huangjiagang and Xiangyang hydrological stations from the year 1974 to 2011, 10 floods with a low proportion of the lateral inflows (time interval ∆t = 3h) were selected, of which 8 floods were used for model calibration, and the other 2 floods were used for 220 validation. In order to demonstrate the simulation effect and test the forecast capability of the HDGNM, the DGNM was selected for comparison. For application, the model parameters in these two models should be calibrated by using the observed flood data. The SCE-UA algorithm, with the advantages of robust and reliable performance, global search capability, has become a commonly used optimization method for hydrological model parameters (Duan et al., 1994), and hence was used to obtain the 225 optimized parameters of these two models. Take the root mean square error of the observed 8 floods from the year 1974 to 2005 as the objective function, and run the SCE-UA algorithm to mimimize this objection function, we obtain the optimized parameters of n = 3, K = 3.51h for the DGNM and https://doi.org/10.5194/hess-2020-17 Preprint. Discussion started: 18 February 2020 c Author(s) 2020. CC BY 4.0 License. n = 3 , K 1 = 1.58h , K 2 = 8.80h , K 3 = 1.59h for the HDGNM. The parameter K reflects the difference in the storage capacity of linear reservoirs. The three linear reservoirs of the DGNM are 230 equivalent, in essence, it is a homogenization of the topographical differences of the sub-reaches.
While two of the three linear reservoirs of the HDGNM are approximately the same, and the other is quite different from the two. Therefore, it can more objectively reflect the influence of topographical differences on the storage routing of the river channel. Theoretically, it can improve the accuracy of the flood routing in river channel. The performances of these two models were assessed by the 235 following two commonly used statistics (Wu et al. 2012): (1) Relative error (RE) of peak discharge , .
. 100% p est p obs in which Op,est and Op,obs are the estimated and observed peak discharge, respectively.
(2) Nash-Sutcliffe efficiency coefficient (ENS). 240 in which Ot,est and Ot,obs are estimated and observed discharge at time t, respectively. t O represents the mean of observed discharge. To further test the validity of the parameter estimations from calibration, the verification 245 experiment was also conducted. The other 2 observed floods in the year 2007 and 2011 were adopted to verify the calibration results. The accuracy evaluation results of these two models in calibration and validation periods were both shown in Table 1. In calibration period, the average values of RE and ENS were 3.94% and 0.9690 for DGNM, respectively. Compared with the DGNM, the HDGNM has made some improvements in the simulation. The average value of RE has reduced to 2.42% and that of ENS 250 has increased to 0.9773. The similar improvements can be found in the validation period with values from 1.55% to 0.49% for RE , and from 0.9798 to 0.9852 for ENS, respectively. Comparison of the observed and simulated hydrographs for the selected 10 floods was shown in Fig. 2. The HDGNM with a consideration of the topographical heterogeneity of the river reach, makes the simulated hydrographs much closer to the measured flood hydrographs, especially near the flood peaks. Except 255 for the October 1974 flood event, the simulations of other floods have different improvements. The simulation results show that, compared with the DGNM, the spatial difference of the storage parameter is considered, thus the HDGNM is more applicable, especially in the river reaches where the river slopes and cross -sections change greatly.

Conclusions
The heterogeneous DGNM for flood routing was deduced indirectly by conceptual interpretation of the DGNM. It is suggested that the downstream outflow is generated by the recession of the old water stored in river channel and the discharge of the new water from upstream inflow, and can formally expressed as a linear combination of the inflows and outflows, whose weight coefficients are 270 calculated by the S curve. When such old water and new water are routed by a series of unequal reservoirs, the DGNM becomes to HDGNM. Hence, these two compositions of the outflow were deduced respectively. The discharging part produced by the new water can be easily deduced with the help of the heterogeneous S curve. The recession part produced by the old water is obtained by the superposition of the recession process for each linear reservoir, which can be calculated by the impulse 275 response of the current stored water with the help of HIUH. At last, the HDGNM is expressed as a linear combination of the inflows and outflows, whose weight coefficients are calculated by the heterogeneous S curve.
The proposed HDGNM was applied to a reach of the middle Hanjiang River with large changes of river slopes and cross-sections. A different linear reservoir with a much larger storage coefficient 280 was detected from the conceptualized reservoirs in the studied river reach. Considered of the topographical heterogeneity of the river reach, the HDGNM performs better than the DGNM, with higher model efficiency and smaller relative error in the simulated flood hydrographs. The HDGNM enriches the existing generalized Nash flow routing theory and becomes more applicable, especially in river reaches where the river slopes and cross-sections change greatly. The river storage is 285 conceptually as a series of unequal linear reservoirs, thus the HDGNM may have the potential for https://doi.org/10.5194/hess-2020-17 Preprint. Discussion started: 18 February 2020 c Author(s) 2020. CC BY 4.0 License.
semi-distributed modelling, e.g. river flow routing with multiple tributaries inflows, which will be further studied in the future.

Acknowledgments
This study is financially supported by the National Key

Data Availability
The data that support the findings of this study are available from the corresponding author upon