Derivation of instantaneous unit hydrographs using linear reservoir models

In this research, a new conceptual model for producing instantaneous unit hydrographs (IUHs) is introduced by a linear combination of the Nash model, which assumes that the discharge from a reservoir is a linear function of its storage, and a model called inter-connected linear reservoir model (ICLRM), which assumes that the discharge from a reservoir is a linear function of the difference of its storage and its adjacent downstream reservoir. By employing these assumptions, a system of firstorder linear differential equations with three degrees of freedom (storage coefficient, number of reservoirs, and weighting coefficient) is obtained as the governing equation for the proposed model. This model may be considered as the general form of the two models and is therefore capable of simulating IUHs laying between these two models. To show the capabilities of the model, linear and curvilinear soil conservation service (SCS) hydrographs are simulated using dimensionless hydrographs obtained by this model. Moreover, several real hydrographs were simulated by the proposed model and compared with hydrographs obtained by Nash, ICLRM, and SCS models. The results show that the model yields more accurate results compared to other studied models and may be considered as a new model for simulating IUHs.


INTRODUCTION
The number of floods causing damage and financial losses throughout the world is not low. For this reason, studies and investigations of flood prediction models for water emergency measures and management strategies are of great importance (Singh ). Among these models, hydrographs have been extensively used for surface runoff estimation.
The unit hydrograph, defined by Sherman () is one of the most widely used methods for generating a direct runoff hydrograph. Because of the lack of real flood records, different methods for developing synthetic unit hydrographs (SUHs) were then proposed. Traditional or empirical models are one of the methods that are used for developing SUHs. Snyder (), Bernard (), Taylor & Schwarz (), and Soil Conservation Service (Mockus ) have presented such models which are established based on empirical equations reflecting watershed characteristics (Bhunya et al. ). SUHs are still in use despite not being accurate (Singh et al. ).
Because of the similarity between probability distribution functions (PDFs) and unit hydrographs, and the fact that the area under these two is equal to unity, different Since the Nash model and other similar models such as ICLRM will produce a unique dimensionless IUH when plotted for a specific number of reservoirs, a general form of Nash and ICLRM models is studied in this research.
The new model, which is a linear combination of Nash and ICLRM models, is shown to produce a wider range of IUHs compared to the other two studied models. The validity of this new model is also investigated by the simulation of SCS linear and curvilinear, and several real hydrographs.

Model description
In this research, three different models for producing the IUH are studied and compared. The first one is the wellknown Nash model (Nash ), which assumes that a watershed could be simulated using a series of linear reservoirs connected to each other in a cascade form ( Figure 1(a)).
Using this assumption, the discharge from a reservoir can be written as: where L represents a linear function, Q is the discharge from the system, S is the storage of the system, and K is the storage coefficient.  have the same elevation (Figure 1(b)). This model can be formulated as: The third model, the modified linear model, being the focus of this study, is a linear combination of the first two models. This model can be mathematically expressed as: where ω is a weighting coefficient.

Nash model
The derivation of the Nash model is covered in most textbooks; however, since the governing equations for the ICLRM and the modified linear model are presented in a matrix form, the derivation of the Nash model in a matrix form is discussed. To obtain the governing equation for the Nash model, the continuity equation for the ith reservoir is written as: By differentiating Equation (1) and substituting it into Equation (4), the above equation is written as: For the first reservoir, the above equation can be in the form of: All of the above equations can be written in a matrix form as:
Since the first and last reservoirs are only connected to one reservoir, their governing equation does not follow Equation (11). The equation for the first reservoir is obtained by writing the continuity equation for the first and the second reservoirs as: Dividing Equations (12) and (13) by K 1 and K 2 and using the differentiated form of Equation (2) corresponding to the first reservoir results, one gets: Since the discharge from the last reservoir is as freefall, the discharge-storage relation for it is of the form: By writing the continuity equation for the last reservoir, and differentiating Equation (15), the formulation for the last reservoir is obtained as: By writing Equation (11) for interior reservoirs and Equations (14) and (16) for the first and last reservoirs, the general formulation of the model would be as follows: : 0 : : : : : : : : 0 0 :

Modified linear model
To derive the governing equation for the modified linear model, Equation (3b) is simplified to: By writing the continuity equation for all reservoirs and following the same procedures as in the derivation of the ICLRM equation, one yields: and the matrix form of the modified linear model would be as follows: The obtained matrix is a tri-diagonal matrix where the arrays of the main diagonal are obtained by the summation of inversed storage coefficients of the two reservoirs at the left and right sides of the considered pipe with a negative sign and weighting coefficients of 1 and 1 À ω, respectively; the diagonal below the main diagonal is the inversed storage coefficients of the left-side reservoirs, and the diagonal above the main diagonal is the inversed storage coefficients of the right reservoirs with a weighting coefficient of 1 À ω. For ω ¼ 1 and ω ¼ 0, the above equation would be identical to the Nash model and the ICLRM, respectively.

Solution and comparison of the models
As seen from Equations (7), (17), and (20), the governing equations of all these models are a first-order linear differential equation. The general equation of these models may be written as follows: where T is the transmissivity matrix presented in Equations (7), (17), and (20). The general solution of these equations is as follows: where λ i is the ith eigenvalue of the transmissivity matrices, and {v i } n×1 is their corresponding eigenvectors. c i s are constants that are determined using initial conditions as follows: : 0 : (1 À ω) K iþ1 0 : : : : : : : : : 0 0 : where {Q 0 } n×1 is the initial condition vector and is calculated by substituting initial elevations of each reservoir into Equation (18), and [V ] n×n is a matrix whose columns are eigenvectors of the matrix T.
In this research, the first reservoir is filled instantaneously with a unit volume of water, and thus {Q 0 } n×1 is a zero vector except for the first array which is equal to 1/K.
It should be noted that the solution of the Nash model using the matrix approach is the same as the traditional approach as one expects. These solutions for a value of  In this research, however, a linear relationship between storage and discharge is provided by assuming laminar flow through each pipe. It must be noted that linear reservoir models only simulate the IUH of a watershed, and therefore neither the hydraulics of these models nor the shape of reservoirs is representative of the watershed.
To have a laminar regime through connecting pipes, it is assumed that flow through each pipe follows the Hagen-Poiseuille equation. Hence, the velocity may be found as follows: In the above equation, γ is the specific weight of the fluid, μ is the dynamic viscosity of the fluid, D is the diameter of the pipe, L is the length of the pipe, and h is the water elevation at reservoirs.
Assuming that reservoirs are all cylindrical, storage may be replaced for h in Equations (25a)  as follows: where Q i is the flow rate in a pipe connecting reservoirs i and i þ 1, S is the storage of reservoirs, and A is the crosssectional area of reservoirs. The flow rate in each pipe is found to be: where K is the storage coefficient of reservoirs and is equal (27a) (1) and (2)).

RESULTS AND DISCUSSION
To investigate the applicability of the mentioned models, it is assumed that (1) all of the linear elements in these models are identical, in other words, K is uniform in these models; and (2) the number of reservoirs, n, must be an integer.
It should be noted that the dimensionless forms of Equations (22a) and (22b) are independent of K and are unique when plotted for a specified number of reservoirs, while the dimensionless IUH (DIUH) obtained by the  to 600, are plotted in Figure 4. It is seen from Figure 4(a) that by increasing the number of reservoirs, n, DIUHs obtained by the ICLRM lay between 150 and 300 SCS hydrographs, while those obtained by the Nash model lay within 200-600 SCS hydrographs. Therefore, it is expected that the modified linear model simulates SCS hydrographs with peak factors ranging from 150 to 600 as ω increases from 0 to 1 (Figure 4(b)).
An important aspect of Figure 4(a) is that for a specified value of n, the DIUH obtained by the ICLRM is always on the right side of that obtained by the Nash model. As a result, the hydrograph on the most right side of Figure 4(a) belongs to the ICLRM with n ¼ 2, and the one on the leftmost side of Figure 4(a) belongs to the Nash model with n approaching infinity. It is also observed, from Figure 4, that as the number of reservoirs, n, increases, the peaking factor for the DIUHs increases; t peak /t base decreases.
In general, by increasing one of the parameters n or ω, while the other one is kept fixed, t peak /t base will decrease. Therefore, However, as n increases, the resulting DIUH would be a better simulation for steeper catchments. The contour plot of a peak factor vs. n and ω is shown in Figure 5(b).
To investigate the applicability of the introduced model, at first, the IUHs obtained by the three studied models are compared with the widely used linear and curvilinear SCS dimensionless hydrographs reported in the NRCS National Engineering Handbook (). These hydrographs cover peaking factors ranging from 150 to 600. For this comparison, the number of reservoirs varying from 2 to 10 is studied. Using the least square method, the optimum value of ω is obtained.
These hydrographs are plotted in Figures 6 and 7, respectively.
The root-mean-square error (RMSE) and the Nash-Sutcliffe efficiency for the simulation of these hydrographs are also reported, respectively, in Tables 1 and 2.  Figures 8-11 and Table 3.
As can be seen in Figures 8-11, the modified linear model gives a better fit to the real hydrographs compared to other studied models. According to     Therefore, due to the simplicity of the proposed model and its capability of simulating a wider range of IUHs over the Nash model, the modified linear model may be considered as an appropriate model for the simulation of IUHs.

DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.