Use of basin outlet velocity to determine the basin concentration time and storage coefficient

A synthetic unit hydrograph (UH) is generally applied to estimate the runoff characteristics at an ungauged location. The synthetic UH is generally derived by considering basin characteristics rather than rainfall–runoff data, which may be classified as traditional, conceptual, probabilistic, and geomorphological models (Singh et al. 2014). Recently, due to the availability of high-resolution digital elevation model (DEM) and the development of geographic information system (GIS) analysis tools, more studies have been focused on the geomorphological models such as the geomorphological UH by Rodríguez-Iturbe & Valdés (1979) and width-function UH by Mesa & Mifflin (1986) (Noto & La Loggia 2007; Grimaldi et al. 2012a; Tarahi et al. 2022).

In Korea, while various UHs have been considered in the past, recently the Clark UH (Clark 1945) has mostly been used. The Clark UH is represented by two parameters, such as the concentration time (or the time of concentration) and the storage coefficient, which are based on the linear channel and the linear reservoir theory, respectively. These two parameters play a crucial role in estimating peak time and peak discharge. In an ungauged basin, these parameters are estimated using some empirical formulae whose validity has been proven. In Korea, the Kraven (II) formula (JSCE 1999) is used for the concentration time, which estimates the concentration time considering both the channel length and the average flow velocity. Here, the flow velocity is assigned differently according to the slope of the channel. While the Sabol formula (Sabol 1988) and the modified Sabol formula (Jeong 2006) are generally used for estimating the storage coefficient (MLTM 2012). These formulas link the concentration time with the storage coefficient by considering the basin shape factor.

As most empirical formulae are generally developed under the average runoff conditions, its application may bias or underestimate the design condition of particularly large-scale hydraulic structures. The upper limit for the application of those estimated parameters by the empirical formulae and the resulting UH is assumed to be 100–200 years of the return period (KISTEC 2013). Several studies have already proven that in a more severe situation with a longer return period than that under the average condition, the runoff response is faster (KDI 2007; K-water 2008; Kjeldsen et al. 2016). That is, the UH model parameters like the concentration time and storage coefficient tend to decrease as the scale of rainfall events increases.

Among the many causes of the nonlinear behavior of the rainfall–runoff process in a basin, the flow velocity must play the key role, as it varies considerably depending on the rainfall characteristics, even under the same basin condition. Simply put, high rainfall intensity leads to faster flow velocity. The shape of the UH also varies considerably depending on the flow velocity or the characteristics of the rainfall event. There are numerous computational definitions for the concentration time (McCuen 2009). However, in this study, the concentration time of the Clark UH is defined as the time that a raindrop spends from the hydrologically farthest point of the basin to reach the basin outlet (WMO 1974). Based on this definition, the concentration time can be directly related to the flow velocity on the hillslope, as well as in the channel. Many studies attempted to estimate the concentration time using the flow velocity (NRCS 1985; JSCE 1999; Du et al. 2009; Perdikaris et al. 2018). It is also possible to find some studies that considered the wave celerity (Saghafian et al. 2002; Wong 2003). On the other hand, the storage coefficient is generally assumed to be proportional to the concentration time (Russell et al. 1979; Hoggan 1989; Yoo 2009). Thus, the storage coefficient is also likely to be closely related to the flow velocity.

Those previous studies mostly focused on estimating the representative values under the ordinary condition. In the case of considering more extreme conditions, it should be possible to modify or re-estimate the concentration time and storage coefficient. Two issues are raised in this problem; under the somewhat extreme condition, the first is whether the concentration time is still fully expressed by the flow velocity, while the other is whether the storage coefficient is still linearly proportional to the concentration time. To resolve these two questions, this study investigates whether when the flow velocity information is available, these two Clark UH parameters can adequately be determined. In this study, first, the relationship between the concentration time and the flow velocity is theoretically explored. Also, by evaluating the relationship between the concentration time and the storage coefficient, this study tries to show that the storage coefficient is also dependent upon the flow velocity.

As an application example, this study considers the major rainfall events that have occurred in the Chungju Dam basin, Korea. Both the reliable rainfall and runoff measurements are available for this dam basin; also, very regular measurements of channel flow velocity are available. By analyzing the rainfall–runoff measurements, the two parameters of the Clark UH are estimated. These estimated parameters are then used to evaluate their relationships, their dependency on the flow velocity, and the empirical formulae generally used in Korea. Finally, to prove the applicability of the proposed method, the study results are compared with the observed.

A total of 57 rain gauge stations, managed by the Ministry of Environment, Meteorological Administration, and K-water were considered to derive the areal average rainfall data. The runoff data were derived by applying the corresponding rating curve to each water level data. It was possible to find the water level data and rating curves from the water resources information system of Korea, the WAMIS (www.wamis.go.kr). The hourly data collected from the period of 1979–2018 (40 years total) were used in this study.

Independent rainfall events were first separated using the basin-average rainfall time series constructed for each basin. An appropriate minimum no rain period, the so-called inter-event time definition (IETD), should be determined to separate independent rainfall events. Referring to Park & Yoo (2012), this study applied 12 h as IETD and 1 mm as a threshold. However, as many events did not show sufficient direct runoff, especially when the total rainfall was less than 30 mm, only the rainfall events with total rainfall of 30 mm or more were considered in this study.

The peak velocity was selected as the velocity close to the peak time for each rainfall event. In case no observed velocity was available at the exact peak time, the nearest value in the rising limb was selected as the peak velocity. This velocity data could also be obtained from the WAMIS (www.wamis.go.kr), as well as from the Korea Annual Hydrological Report. The number of events considered in this study varies greatly by basin from 3 to 22, and the total number of events considered is 162. Table 1 compares the number of rainfall events considered for each basin, including some basic information about the study basins.

The concentration time and storage coefficient for each event were estimated by applying the method proposed by Yoo & Shin (2010). This method recursively searches the convergent concentration time and storage coefficient using the relationship between the number of linear reservoirs and the storage coefficient of the linear reservoir of the Nash model. This is to overcome the problem of applying the optimization technique which selects optimal parameters separately. In addition, this study considered the N-day method and the straight-line method, to separate the direct runoff hydrograph from the total hydrograph and the NRCS-CN method and the -Index method, for effective rainfall estimation. That is, a total of four combinations were considered in the parameter estimation, and the result with the minimum root mean square error was selected as the optimal one.

This study also tried to apply a regression line or curve for each case. For example, the regression model of the form y = c/x was applied to Figure 3(a) and estimated the constant to be 84.5 km. In fact, this value is quite close to the channel length of the basin (i.e., 90.3 km). This result indicates that the concentration time could be expressed as the travel time from the uppermost point of the basin to the basin outlet (i.e., ⁠). The same regression model was also applied to Figure 3(b) to quantify the relationship between the storage coefficient and the peak velocity. The constant was determined to be 118.5 km, slightly larger than the previous case. That is, the storage effect in this basin seems to be higher than the concentration time.

The linear relationship between the concentration time and the storage coefficient in Figure 3(c) could also be quantified by the regression model of the form, ⁠. In this case, the constant was estimated to be 1.4. Simply put, the storage coefficient was found to be 1.4 times larger than the concentration time, over all ranges of observed peak discharges or peak velocities. Based on the very high coefficient of determination of 0.95 or higher, this linear relationship could be assumed to be reliable. That is, even though the peak discharge and the peak velocity increase, the ratio between the two parameters does not change significantly. This strong linear relationship could also be confirmed in other basins, whose results are summarized in Table 3. In this table, basin #1 is the smallest one, while basin #21 is the largest one. As the basin becomes larger in the area, the ratio between the two parameters becomes larger to indicate a greater storage effect.

Table 3

Regression coefficient and the coefficient of determination for each regression model in Figure 4 

Basin # .	(m/s) vs. (h) .		(m/s) vs. K (h) .		(h) vs. K (h) .
	(km) .	R2 .	(km) .	R2 .	(h/h) .	R2 .
1	20.8	0.881	21.7	0.876	1.0	0.981
2	20.1	0.986	21.6	0.980	1.1	0.999
3	27.8	0.897	30.6	0.772	1.1	0.960
4	26.9	0.997	24.3	0.963	0.9	0.944
5	15.2	0.998	12.8	0.999	0.8	0.999
6	31.4	0.998	37.3	0.997	1.2	0.999
7	56.4	0.975	81.9	0.934	1.5	0.961
8	48.5	0.943	46.2	0.800	1.0	0.932
9	56.1	0.998	65.4	0.996	1.2	0.999
10	49.2	0.981	49.4	0.993	1.0	0.992
11	55.0	0.968	56.0	0.955	1.0	0.997
12	65.6	0.990	86.8	0.993	1.3	0.991
13	79.2	0.964	107.4	0.931	1.4	0.978
14	70.5	0.963	91.0	0.979	1.3	0.991
15	84.5	0.942	118.5	0.926	1.4	0.985
16	115.2	0.958	163.3	0.930	1.4	0.974
17	100.0	0.991	105.8	0.993	1.1	0.995
18	123.6	0.793	169.3	0.887	1.4	0.988
19	156.9	0.878	215.9	0.841	1.4	0.998
20	178.8	0.939	283.4	0.953	1.6	0.988
21	166.3	0.945	204.2	0.922	1.2	0.981

Basin # .	(m/s) vs. (h) .		(m/s) vs. K (h) .		(h) vs. K (h) .
	(km) .	R2 .	(km) .	R2 .	(h/h) .	R2 .
1	20.8	0.881	21.7	0.876	1.0	0.981
2	20.1	0.986	21.6	0.980	1.1	0.999
3	27.8	0.897	30.6	0.772	1.1	0.960
4	26.9	0.997	24.3	0.963	0.9	0.944
5	15.2	0.998	12.8	0.999	0.8	0.999
6	31.4	0.998	37.3	0.997	1.2	0.999
7	56.4	0.975	81.9	0.934	1.5	0.961
8	48.5	0.943	46.2	0.800	1.0	0.932
9	56.1	0.998	65.4	0.996	1.2	0.999
10	49.2	0.981	49.4	0.993	1.0	0.992
11	55.0	0.968	56.0	0.955	1.0	0.997
12	65.6	0.990	86.8	0.993	1.3	0.991
13	79.2	0.964	107.4	0.931	1.4	0.978
14	70.5	0.963	91.0	0.979	1.3	0.991
15	84.5	0.942	118.5	0.926	1.4	0.985
16	115.2	0.958	163.3	0.930	1.4	0.974
17	100.0	0.991	105.8	0.993	1.1	0.995
18	123.6	0.793	169.3	0.887	1.4	0.988
19	156.9	0.878	215.9	0.841	1.4	0.998
20	178.8	0.939	283.4	0.953	1.6	0.988
21	166.3	0.945	204.2	0.922	1.2	0.981

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Table 4 summarizes the results of this hypothesis test. As shown in this table, it was confirmed that the ratio data (⁠⁠) between the concentration time and the storage coefficient could be assumed constant in all basins. In all basins considered in this study, the p-value was estimated to be far larger than 0.05; as a result, the null hypothesis could not be rejected.

Table 4

Hypothesis test result on the slope of the regression line for each basin

Basin # .	Slope .	-statistic .	-value .
1	0.004	0.119	0.910
2	−0.027	−4.907	0.128
3	0.028	0.566	0.596
4	0.041	1.052	0.352
5	−0.009	−1.817	0.320
6	0.001	0.025	0.984
7	−0.019	−0.448	0.673
8	0.042	1.703	0.127
9	−0.004	−0.604	0.654
10	0.017	0.457	0.693
11	0.006	0.731	0.483
12	0.043	1.265	0.246
13	0.050	0.660	0.577
14	0.003	0.113	0.912
15	0.004	0.240	0.813
16	0.033	0.858	0.406
17	0.029	1.632	0.154
18	−0.054	−0.738	0.538
19	0.015	0.853	0.483
20	−0.016	−0.584	0.575
21	0.023	1.186	0.266

Basin # .	Slope .	-statistic .	-value .
1	0.004	0.119	0.910
2	−0.027	−4.907	0.128
3	0.028	0.566	0.596
4	0.041	1.052	0.352
5	−0.009	−1.817	0.320
6	0.001	0.025	0.984
7	−0.019	−0.448	0.673
8	0.042	1.703	0.127
9	−0.004	−0.604	0.654
10	0.017	0.457	0.693
11	0.006	0.731	0.483
12	0.043	1.265	0.246
13	0.050	0.660	0.577
14	0.003	0.113	0.912
15	0.004	0.240	0.813
16	0.033	0.858	0.406
17	0.029	1.632	0.154
18	−0.054	−0.738	0.538
19	0.015	0.853	0.483
20	−0.016	−0.584	0.575
21	0.023	1.186	0.266

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Even in a given basin, different values of parameters were derived whenever analyzing a different rainfall event. In general, their representative values are determined as their means. In particular, when sufficient rainfall events are available, the mean value could be acceptable. However, when only a few events are available, the mean values can be somewhat biased (Yoo et al. 2007a). Based on Yoo et al. (2007b), at least 20 different rainfall events are required to estimate the acceptable mean values. In the case that the number of rainfall events is far smaller than 20, they proposed to use the mode, rather than the mean value. This study also considered this direction to determine the representative values of concentration time and storage coefficient for each basin. Table 5 summarizes the representative parameters determined for each basin and also provides their ratio (⁠⁠) for reference:

Table 5

Representative UH parameters estimated by analyzing the observed data for each basin

Basin # .	(h) .	(h) .	(h/h) .
1	3.0	3.3	1.1
2	4.1	4.4	1.1
3	6.2	6.8	1.1
4	3.9	3.7	1.0
5	2.5	2.1	0.8
6	3.5	4.0	1.1
7	10.5	15.5	1.5
8	7.3	7.1	1.0
9	6.8	7.8	1.1
10	10.3	9.9	1.0
11	6.0	6.3	1.1
12	8.8	11.8	1.3
13	11.1	15.2	1.4
14	7.4	9.3	1.3
15	14.9	21.0	1.4
16	19.8	28.0	1.4
17	11.0	11.9	1.1
18	27.5	36.9	1.3
19	20.2	28.0	1.4
20	19.1	30.1	1.6
21	22.0	28.1	1.3

Basin # .	(h) .	(h) .	(h/h) .
1	3.0	3.3	1.1
2	4.1	4.4	1.1
3	6.2	6.8	1.1
4	3.9	3.7	1.0
5	2.5	2.1	0.8
6	3.5	4.0	1.1
7	10.5	15.5	1.5
8	7.3	7.1	1.0
9	6.8	7.8	1.1
10	10.3	9.9	1.0
11	6.0	6.3	1.1
12	8.8	11.8	1.3
13	11.1	15.2	1.4
14	7.4	9.3	1.3
15	14.9	21.0	1.4
16	19.8	28.0	1.4
17	11.0	11.9	1.1
18	27.5	36.9	1.3
19	20.2	28.0	1.4
20	19.1	30.1	1.6
21	22.0	28.1	1.3

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The above equation indicates that the ratio between the two parameters (⁠⁠) is constant.

The results derived in the previous section indicate that, under the condition that the only available information is the peak velocity, it is possible to determine the concentration time and the storage coefficient. The information of the peak velocity could be derived by analyzing the runoff data, if available, or by applying some appropriate empirical formulae. In this part of the study, the two parameters, the concentration time, and the storage coefficient of the Clark UH were derived using the available information of peak velocity, after assuming that the study basin is ungauged. The derived parameters were then compared with those determined by analyzing the observed data.

For basin #15, as an example, the observed peak velocity ranges from 0.9 to 3.3 m/s. With the observed peak velocity, the concentration time could easily be determined by dividing the channel length by the corresponding peak velocity. The determined concentration time ranged from 7.6 to 28.2 h. Then to determine the storage coefficient, it was required to determine the ratio between the storage coefficient and the concentration time. As the Kraven (II) formula for the concentration time and the modified Sabol formula for the storage coefficient were known to be valid in this basin, this study also considered these formulae. By applying these formulae, the concentration time was estimated to be 11.9 h, and the storage coefficient was 17.1 h. Thus, the ratio was determined to be 1.44. This ratio was also assumed to be unchanged, regardless of the peak velocity. Now the storage coefficient was determined to be within the range of 10.8–40.4 h.

As the size of the basin increases, the mean difference of the parameters shows a tendency to lean toward the positive. Basin #1 showed the smallest difference of −14.8%, and basin #21 showed the largest of 13.8% for the concentration time. Similarly, basin #4 showed the smallest difference of −22.9%, and basin #13 showed the largest of 19.0% for the storage coefficient. That is, the method using the peak velocity might overestimate the UH parameters for those large basins, which could underestimate the peak discharge. However, this problem is also related to the applicable limit of the UH. The appropriate sub-basin division could alleviate the possible problem of the proposed method.

In general, most empirical formulae on the basin concentration time and storage coefficient focus on estimating the representative values under the ordinary condition. Under more extreme conditions, those parameters should be modified to consider faster velocity conditions on both the channel and the hillslope. The main objective of this study was to examine the possibility of determining the concentration time and storage coefficient corresponding to the given peak velocity at the basin outlet. Two issues were involved in this problem under somewhat extreme conditions: one was whether the concentration time could be fully expressed by the peak velocity; the other was whether the storage coefficient was still linearly proportional to the concentration time. In this study, first, the relationship between the concentration time and the peak velocity was theoretically explored. Also, by evaluating the relationship between the concentration time and the storage coefficient, this study tried to show that the storage coefficient is also dependent upon the peak velocity. This study also analyzed the observed rainfall–runoff events collected at the Chungju Dam basin, Korea. After estimating the representative parameters by analyzing the observed rainfall–runoff data, this study compared them with the values estimated by applying empirical formulae. Finally, this study determined the two parameters using the given peak velocity information, which were then evaluated by comparing them with the observed parameters, as well as with those estimated by applying the empirical formulae. The results of this study are summarized as follows:

First, the analysis of observed data at the Chungju Dam basin confirmed strong linear relationships among the basin concentration time, storage coefficient, and peak velocity at the basin outlet. It was also observed that as the peak velocity increased, the concentration time and storage coefficient decreased. Their relationship was close to inverse but linear proportion. The concentration time appeared very similar to the value obtained by dividing the channel length by the peak velocity. The relationship between the concentration time and the storage coefficient was also found to be very strongly linear.

Second, the ratio of the storage coefficient to the concentration time was found to be almost identical, regardless of the peak velocity. This result was also confirmed by the hypothesis test. That is, this ratio at a basin could be assumed as a basin characteristic that, regardless of the size of rainfall events, was unchanged. As a result, it could be assumed that with the given concentration time, the storage coefficient of a basin could be estimated very accurately.

Finally, the above results could also be verified in application to several stage-gauging stations within the Chungju Dam basin. The concentration time and the storage coefficient determined using the peak velocity showed a mean difference of about 9.8 and 13.2%, respectively, from the observed values. However, the difference was found to be large for the larger basin. The determined parameters using the peak velocity also showed the tendency to be slightly higher than that of the observed parameters, especially for those large basins, which resulted in slightly low estimates of peak flow and peak time.

It is now clear that using the given peak velocity information, the basin concentration time and storage coefficient can be reasonably estimated. However, it is not yet clear how to obtain the peak velocity information corresponding to the given rainfall event in an ungauged basin. The information of peak flow should be derived by applying the peak flow to the given basin outlet. It may be possible to obtain the information of channel shape and bed materials through a field survey, but the peak flow at the basin outlet still needs to be estimated. To use the results in this study, how to reasonably estimate the peak flow for the given design rainfall event must be given. This issue is especially important in hydrologic practices for designing hydraulic structures. This issue must be an important future research area for the authors of this study.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2C200871411) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2021R1A5A1032433).

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

The authors declare there is no conflict.