A generalized concentration curve (GCC) method for storm flow hydrograph prediction in a conceptual linear reservoir-channel cascade

Predicting storm runoff magnitude, timing and recurrence interval is one of the most common practical tasks undertaken by hydrologists. The two main approaches to modelling the translation of effective rainfall inputs to storm flow outputs are data-driven, both empirical black-box and conceptual model types, and knowledge-driven distributed physically-meaningful model types (Todini 2007). A basic foundation of the data-driven approach is the unit hydrograph (UH), a lumped, linear conceptual model that describes storm runoff over time at a point resulting from one unit of effective rainfall distributed uniformly over a watershed at a uniform rate during a unit period of time (Dooge 1973). Since it was first introduced by Sherman (1932), the UH concept has been widely used in rainfall-runoff models to predict total event runoff and flood peak magnitude. A range of non-parametric and parametric methods has been used to evaluate the UH (Yang & Han 2006). This paper is concerned only with the parametric approach, in particular, models of hydrographs having a limited number of parameters (usually two or three) that are widely applied to the problem of predicting storm flow hydrographs in ungauged or poorly gauged watersheds (Razavi & Coulibaly 2013).

The above review illustrates that historical development of the UH conceptual model has progressed from TA curves to linear channel systems, single reservoir to multiple reservoirs, equal to unequal storages, and simple to more complex formulations. Despite these improvements, some limitations remain. The isochronal, or TAC curve, method represents the physical process of watershed subareas contributing to the outlet hydrograph to some extent, but it does not consider flow attenuation, and the common practice of plotting isochrones equidistantly unrealistically suggests homogeneous flow transfer times throughout the watershed. In contrast with the TAC method, the UH method includes the attenuation, while this is the result of routing the flow through a system of reservoirs and channels rather than by representing the physical process of attenuation of flow. Dooge (1959) established a general theory of UH with channel-reservoir system but he did not provide an analytical solution for his original general equation of the UH, which he recognized would have involved impracticably complex and tedious calculation. Rather, Dooge (1959) simplified the problem by lumping the channels (Figure 2), and assuming that the order of the channel-reservoir system was immaterial and that the unrestricted reservoirs were all equal. In retrospect, Dooge's (1959) simplifying assumptions might seem unrealistic, but should be viewed in light of the limited understanding of watershed hydrology and relatively crude information technology available at that time. Another limitation is that, by adopting the assumption of uniformly distributed rainfall, the established UH methods are essentially constrained to relatively small watersheds (<500 km² in area) where the distribution of rainfall can be regarded as homogeneous (Shaw 1994).

It is apparent from the literature that the theory of UH generated considerable research interest before the 1970s, but little progress has been made since. It is timely then to advance UH theory by overcoming the above mentioned limitations and simplifications, potentially allowing this approach to take advantage of modern technologies such as remote sensing, geographic information systems (GIS) and weather radar (Moon et al. 2004; Finsen et al. 2014; Kang & Merwade 2014; Wu et al. 2015), improved understanding of catchment hydrology, and advancements in hydrological methods (Li et al. 2008; Zhang et al. 2010; Li et al. 2014).

The objective of this paper is to make a contribution to the development of UH theory by presenting a new physically distributed conceptualization of the linear reservoir-channel cascade that overcomes the limitations identified above. The new UH method proposed here includes a riverbed slope-based isochronal method that more realistically divides the watershed into runoff generation and transfer subareas. In each subarea, the runoff is attenuated by a reservoir and then translated downstream by channels. By considering both attenuation and translation processes, effective rainfall is progressively concentrated along the river so that the order of reservoirs and channels is of significance. The theoretical development of this new method, named general concentration curve (GCC), is outlined in the following section where it is also demonstrated that some well-known UH models are special cases of the GCC, differentiated only by their simplifying assumptions. Rainfall and runoff data from 18 historical storm events from Dagutai, Bailianhe and Kaifengyu watersheds in China are then used to demonstrate application of the GCC method and compare its performance with that of the Nash IUH.

Derivation of the GCC method is presented here in four sections, with the first covering conceptual and theoretical development, the second describing the method for deriving the S curve and UH of duration T, the third showing a suggested method based on reach-averaged riverbed slope for determining the storage parameter, K_i, and translation time parameter, τ, and the fourth briefly demonstrating the relationship between the new GCC method and existing concentration curves.

The IUH is derived under the assumption that the input is one instantaneous pulse, which is unrealistic, so the UH of duration T (TUH) must be transferred in order to convert an effective rainfall into surface flow (Chow et al. 1988; Li et al. 2008). Fortunately, the S curve can be applied to obtain the TUH from the IUH. The S curve, or S hydrograph, is defined as the runoff response to unit intensity of effective rainfall from the beginning to time t and continues indefinitely, close to unity. The S curve and TUH are derived below.

Therefore, there are only three parameters in the GCC method, i.e. the number of reservoirs n, watershed average storage coefficient K and channel translation time τ.

The GCC is ‘generalized’ in the sense that some common UHs can be shown to be special cases of this new formulation, but with specific parameters n, K and ⁠. Five examples are illustrated below.

• 1. When ⁠, ⁠, and ⁠, Equation (18) becomes:

  

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• 2. When and ⁠, Equation (18) becomes:

  

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⁠

• 3. When ⁠, ⁠, and if all effective rainfall begins in the nth subarea, Equation (18) can be written as:

  

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• 4. When ⁠, ⁠, and if all effective rainfall begins in nth subarea and taking the channel translation effort into consideration, Equation (18) can be written as:

  

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• 5. When and all effective rainfall is routed from the river headwater, Equation (18) can be written as:

  

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The Dagutai watershed covers an area of 727 km² located between longitude 111°25′E and 111°48′E and latitude 31°21′N and 31°42′N. The Dagutai River is 65.6 km long, with the elevation ranging from 1,158 m above mean sea level (a.m.s.l.) in the headwaters to 268 m a.m.s.l. at the outlet. Average annual precipitation is 1,003.6 mm, of which 85% occurs from April to October.

The Bailianhe watershed covers an area of 1,800 km² located between longitude 115°31′E and 116°4′E and latitude 30°39′N and 31°9′N. The river originates in the Dabie Mountain and has two main branches with lengths of 72 km and 63 km, respectively. The elevation varies from 464 m a.m.s.l. in the headwaters to 96 m a.m.s.l. at the outlet, which also corresponds with the inlet to Bailianhe Reservoir. Average annual precipitation is 1,366 mm, of which 85% occurs from June to August. The Bailianhe watershed is prone to flood hazard, and accurate flood forecasting is important for management of the Bailianhe Reservoir.

The Kaifengyu watershed covers an area of 5,253 km² located between longitude 110°16′E and 111°25′E and latitude 31°26′N and 32°24′N. The Kaifengyu River rises on the southern side of Shennongjia Mountain and falls 470 m over its 140 km long course. Average annual precipitation ranges from 855 mm in the north to 1,140 mm in the south of the watershed, with most rain falling from June to September.

The main purpose of this paper was the introduction and verification of the GCC method. Investigation of the runoff generation process was not our focus, so a traditional and reliable runoff generation method was sufficient for our purpose. The direct runoff hydrograph (DRH) was separated from base flow using the objective and repeatable digital filter method (Chapman 1999; Lin et al. 2007). The effective rainfall hyetograph (ERH) was derived according to the traditional precipitation-runoff correlation diagram derived from the API (Antecedent Precipitation Index) Model (Sittner et al. 1969), which has previously been successfully applied in humid and semi-humid regions of China (Zhang 2010). Appropriate values were assigned for the model parameters, including maximum areal mean tension water storage W_m (mm) for each watershed, and initial soil moisture W₀ (mm) for each storm (Table 2). The isochrones of the watershed were determined from the DEM by the method of Zhang et al. (2010) using GIS software. GIS was also used to measure the reach-average riverbed slopes for each subarea from the DEM (Table 3).

The isochrones drawn by the method of Zhang et al. (2010) were completely different from equidistant lines. For example, Dagutai watershed was divided according to the method of Zhang et al. (2010) into an upper subarea of 29 km length and a lower subarea of 16 km length, with corresponding subarea weights of 0.69 and 0.31, respectively (Table 3). Had the isochrones been drawn from equidistant lines, the watershed would have been divided into equal upper and lower lengths with corresponding subarea weights of about 0.54 and 0.46, respectively. The method of Zhang et al. (2010) produced upper stream subareas much larger than the lower subareas, while the equidistant lines method results in relatively small differences in area. The reason for this is that the method of Zhang et al. (2010) plots isochronal lines based on riverbed slope, rather than distance from the outlet (Appendix, available with the online version of this paper). The upper headwater area of Dagutai watershed is mountainous with steep slopes (mean 0.025) while the downstream area is plain with low slopes (mean 0.006) (Table 3), implying higher flow velocities in the upper stream than in the lower stream. Therefore, for the same translation time, the upper stream translation distance was longer than that of the lower one. For the three watersheds investigated, the larger watershed area was associated with a greater number of subareas, which reflects the greater heterogeneity of larger watersheds.

The calculated reservoir storage coefficient (K) of GCC was of the same magnitude as that of Nash, which indicated that the SCE-UA algorithm was appropriate for derivation of K and translation time with a pre-determined number of reservoirs (n). Due to a slightly larger value of n, most calculated K values of GCC were a little smaller than those of Nash, while the time required for water to move from the river headwaters to its outlet which is often presented by NK (Singh 1990), was almost the same between the two methods. Therefore, the derived GCC UHs compared reasonably with the Nash IUHs.

After applying the derived UHs to hydrograph prediction, visual comparison of the observed and predicted hydrographs of GCC and Nash method (Figure 5) suggested that the GCC method generally provided better predictions than the Nash method with respect to the magnitude of the peak, and the timing of the rise, peak and recession. This was supported by the goodness-of-fit statistics (Table 4). In the calibration period, the average values of E_NS, E_WB, and PDE of the three watersheds were 87.90%, 1.01 and 8.71%, respectively, for the GCC method and 82.49%, 0.99 and 12.58%, respectively, for the Nash method. The best values of E_NS, E_WB, and PDE were 95.30%, 1.00 and 1.99%, respectively, for the GCC method and 92.39%, 1.00 and 0.58%, respectively, for the Nash method. The worst values of E_NS, E_WB, PDE were 70.10%, 1.06 and 18.12%, respectively, for the GCC method and 68.00%, 0.95 and 27.09%, respectively, for the Nash method. Similar results were obtained in the verification period.

In the verification period, the average values of E_NS, E_WB, and PDE of the three watersheds were 89.06%, 1.00 and 9.01%, respectively, for the GCC method, with 10.79% higher E_NS, 0.02 better E_WB, and 6.42% lower PDE than those for Nash method. Overall, the GCC outperformed the Nash IUH model with an average Nash–Sutcliffe efficiency coefficient increased by 7.66%, and an average PDE decreased by 4.14%. The differences in performance between GCC and Nash IUH varied with watershed scale. The averages of improved performance for E_NS were 2.28, 5.34 and 8.62%, respectively, for Dagutai, Bailianhe, and Kaifengyu watersheds in the calibration period, which suggests that the advantage of the GCC method over the Nash IUH increases with the area of the watershed.

The superior performance of the GCC in this case is possibly explained by the three watersheds, all greater than 500 km² in area, being too large for reliable application of the Nash IUH. The linear reservoir-channel cascade of the GCC method can represent the spatial heterogeneity of these large watersheds, and thus generates a more realistic hydrograph (Xu et al. 2013). The 18 storm events examined here varied in terms of rainfall magnitude (indicated by average, maximum and minimum total rainfall volume), and spatial variability of rainfall (indicated by ratio of maximum to minimum rainfall and coefficient of spatial variation) (Table 1). A positive relationship between storm event rainfall spatial variability indicators and improved model goodness-of-fit implied that the higher the spatial variability of rainfall the greater was the performance advantage of the GCC method compared with the Nash method. This explains the observed superior performance of the GCC compared with the Nash IUH as watershed area increased, because for our case studies, the larger watershed area was associated with generally higher storm event rainfall variability. For storm events with spatially homogeneous rainfall across the watershed under consideration, the advantage of the GCC over the Nash IUH method would be diminished.

The above point can be illustrated by the examples of storm events ID 14 and ID 18 in Kaifengyu watershed. Storm ID 14 was caused by local convective weather which brought high intensity precipitation only to the headwater area. In this case, the calculated coefficient of spatial variation was 1.76 and the maximum rainfall volume was greater than the minimum by a factor of 5.75 (Table 1). The assumption of uniform distribution of precipitation over the entire watershed by Nash IUH led to an earlier rise in the hydrograph. On the other hand, the GCC method divided Kaifengyu watershed into five subareas and routed the uneven effective rainfall separately through the reservoir-channel cascade, which retained the heterogeneous characteristics of the storm. By considering rainfall as a distributed phenomenon, a more realistic rising limb and peak of the flood hydrograph was estimated. In contrast to storm ID 14, storm ID 18 was characterized by relatively uniform precipitation over the watershed, with a low coefficient of spatial variation of 0.33 (Table 1). In this case, the hydrographs generated by the GCC and Nash methods were similar, and both were a realistic representation of the observed hydrograph (Figure 5).

The theory of the GCC was developed to overcome the problem of heterogeneous rainfall distribution that is characteristic of larger watersheds (i.e. area >500 km²). Application of the GCC method to multiple storm events in three watersheds in the subtropical monsoon climate zone of China with heterogeneous rainfall distribution demonstrated its superior performance compared to the traditional Nash method. The robustness and applicability of the GCC method across other physiographic areas requires further investigation. In addition, the work required to divide a large watershed into subareas to consider uneven rainfall distribution can be relatively tedious, although less so for experienced GIS operators. This might be a potential drawback for application of the GCC compared with the simpler traditional UH methods. Thus, the traditional Nash IUH will remain a satisfactory approach for watersheds with rivers of even riverbed slope and homogeneous rainfall distribution.

A new representation of the UH named the GCC method was proposed by combining a simple isochrone method based on riverbed slope with a conceptual cascade of reservoirs and channels. The GCC, which simulates watershed runoff by considering flood attenuation and translation processes, and by progressively concentrating rainfall excess along the river, was devised as a conceptual model with physical meaning. The GCC was tested using data from 18 storm events at a 6-hourly time step from three watersheds located within the Yangtze River Basin, China. The main conclusions drawn from this paper are as follows:

1. The four well-known existing concentration curves of Clark, Kalinin–Milyukov, Nash and Dooge were found to be special cases of the GCC, with different simplifying assumptions. Thus, the GCC is a generalized method that can be applied across a wide range of conditions.

2. The way that rainfall excess is progressively concentrated along the river in the GCC eliminated the assumption of uniform rainfall distribution across the watershed, thus opening the potential to expand application of the UH concept to large watersheds.

3. Application of the GCC method to data from three watersheds in China indicated that both the GCC and the Nash IUH methods simulated storm hydrographs with reasonable accuracy. However, goodness-of-fit statistics suggested that the GCC method outperformed the Nash method, with average Nash–Sutcliffe efficiency coefficient higher by 7.66%, and average PDE lower by 4.14%. This result appears to be due to the Nash IUH method being less suited to application in watersheds larger than around 500 km² with heterogeneous rainfall.

4. The water storage coefficient (K) influenced the peak of the UH in subareas close to the watershed outlet, and influenced both peak and time to peak in subareas distant from the outlet. The water translation time had no impact on hydrograph characteristics in the subarea closest to the watershed outlet, but it influenced time to peak in subareas distant from the outlet.

5. In modelling storm events for a watershed with even riverbed slope and homogeneous rainfall distribution, compared with a simpler traditional UH, the GCC could mean a larger calculation effort that realizes an insignificant benefit in improved results. For this situation, the traditional Nash IUH will continue to find satisfactory application.

It is noted that, in this paper, the GCC method was verified using data from three watersheds in the subtropical monsoon climate zone of China as a preliminary demonstration of its utility. Further work is required to test the robustness and applicability of the GCC method in other watersheds covering a wider range of morphometries and climates.

This study was supported by the National Natural Science Foundation of China (51279139 and 51279140) and National Grand Science and Technology Special Project of Water Pollution Control and Improvement (No. 2014ZX07204006).