Time of Concentration Estimates for Sub-Catchments
Each of the selected equation listed in Table 1 was applied in order to estimate the time of concentration (Tc) for the eleven catchments delineated. The estimates are displayed in Table 3. The value of Tc computed for the eleven sub-catchments ranges as follows: 0.166 to 3.37, 0.147 to 17.318, 0.366 to 5.283, 0.19 to 10.03, 0.441 to 6.393, 0.12 to 15.227, 0.359 to 5.684, 0.186 to 7.88, 0.108 to 2.722, 0.218 to 4.213 and 0.206 to 7.031 hours for sub-catchments 1 to 11 respectively (Table 4). The Watt & Chow, Kirpich and Carter equations consistently gave the lowest estimates of Tc for all sub-catchments investigated. Perdikaris et al (2018) observed that Watt & Chow equation was developed upon existing studies and data derived from Kirpichs’ and thus could prove a viable explanation for the similarity in their results. These three equations have some commonalities such as low coefficients and similarity of parameters. All three equations use only the longest of flow path and slope to estimate the time of concentration. Besides, their coefficients are about the lowest among all the equations used in this study. Further scrutiny also reveals that all three equations were derived for catchments located in the USA.
The largest Tc values of 17.318 hours (sub-catchment 2) and 17.1 hours (sub-catchment 2) using the equations of Pasini (1914) and Sheridan (1994). The same equations also yielded the next highest values of 15.22 hours (sub-catchment 6) and 11.309 hours (sub-catchment 6). These two models were developed in basins located in rural Italy and the United States of America respectively with entirely different morphological and land management conditions. Greppi (2005) suggest that Pasinis’ equation is applicable to catchments observed with smooth steepness. These values are obvious outliers and cannot be relied upon for any practical application. A closer examination shows that the two sub-catchments (2 & 6) for which these ridiculous values were obtained were the largest in the entire catchment with areas of 20.72 and 20.97 km2 respectively. These outlier values suggest that an increase in catchment area seems to exaggerate the resultant value of time of concentration. Other Tc estimation methods gave much lower values than these the Pasini (1914) and Sheridan (1994) methods the same sub-catchments. The lowest Tc values for the same sub-catchments (2 & 6) were generated by the models of Kirpich (1940) developed for Tennessee, USA and Watt & Chow (1985) derived for the Midwest of the USA and Quebec areas of Canada. The corresponding values for catchments 2 two models are 0.147 & 0.405 and 1.127 & 0.751 hours respectively. The dilemma posed by the largely disparate values of Tc for the same sub-catchment is further exacerbated by considering the scale of difference. The highest degree of disparity was found between the Sheridan and Kirpich models. For sub-catchment 2, the Tc value from Sheridan’s model is 126 times higher than the value from Kirpich model for sub-catchment 2 and 115 times higher the value from Kirpich model for sub-catchment 6. Nearly the same magnitude of disparity was observed between Pasini and Kirpich models for both sub-catchments. For Sheridan versus Watt and Chow models as well as the Pasini versus Watt and Chow models, the disparity ranges from 15.1 to 20.3 hours. In the light of the above, the question then arises - which value of Tc should be adopted for the study area for practical purposes?
Examination of Variabilities
Analysis of variance (ANOVA) was employed to test the equality of the means of the selected empirical equations. Through ANOVA, it would be established whether or not there was any statistical difference between the means of the selected equations. An attempt was also made to select the most appropriate models for estimation Tc in the study area. The ANOVA analysis showed that there was considerable difference between empirical equations used in the study. The p-value obtained was less than 0.001 and F value was greater than Fcrit. Therefore, the conclusion showed that there is no equality among the means of the selected equations. Using the Tukey procedure, a pairwise comparison of the fifteen equations was undertaken to examine the extent and degree of significance of observed discrepancies.
Table 6 displays the values of qstat for pairwise comparison of the equations based on a critical value of q (qcritic = 4.878). This q value is a benchmark to ascertain the degree of difference between any two particular equations, which are to be regarded as significant. The bold face values represent values that were greater than qcritic, and so H0 is rejected for these values indicating that a significant difference in Tc value exists between this specific equation and the others. A cursory glance at Table 6 clearly shows that at a confidence level of 0.05 the equations of Pasini and Sheridan exhibit markedly significant difference from the generality of other equations with respect to the values of Tc. The consistent departure of these two equations from the other selected thirteen equations indicates that they are not suitable for application in the study area. The first plausible explanation for this anomaly is that these equations were derived for catchments that differ significantly from the one under study. But the same allusion can be made with respect to all the other equations. However, a more careful examination of these equations reveals further provides more specific explanations. First, Pasini’s equation presents a fundamental flaw which limits its applicability to catchments order than the one for which it was derived, apart from coupling catchment area into the equation, it further assigns equal weights to both catchment area and flow length (L). This is highly problematic because the effect of catchment area on time of concentration is already implicitly accounted for by L for elongated catchments where area is directly proportional to length of flow. For circular catchments, the use of both parameters in the estimation of Tc will invariably result in gross overestimations of Tc. Similarly, Sheridan’s equation also presents some fundamental flaws which can be readily traced to the large coefficient of 2.2 used. Besides, the length of flow is the only independent variable in the equation and it was assigned an exponent of 0.96 which is quite close to unity. The simplest conceptualization of the Sheridan’s equation is that it estimates Tc as twice the length of flow. Obviously, this cannot be true. Furthermore, it ignores the role of surface gradient (S) in driving the flow. Neglecting S in the time of concentration equations is tantamount to neutralizing the effect of gravity on channel flow.
Hence, eliminating the Pasini and Sheridan models and scaling down to the remaining thirteen, it can be seen that eight out of the thirteen equations do not differ from any of the equations in terms of the value of Tc. The eight equations that do not vary significantly from the others are: Bransby-William (1922), Chow (1988), Haktanir and Sezen (1990), California Culvert Practice (1960), Temez (1987), Dooge (1973), Corps of Engineers (1977) and Pilgrim & McDermont (1982).
Performance Evaluation Based on Representative Tc Values
The fifteen equations of interest gave widely varying results of TC with range up to 17 hours in some cases. The discordance in result was further betrayed by the values of coefficient of variation of Tc calculated for each sub-catchment which ranged from 71% to 85%. And since there are no reference values for Tc of the sub-catchments with which to evaluate the performance of each selected models, there is need to fix a relative benchmark to this effect. A skewness analysis shows that the distribution of Tc values obtained for each sub-catchment using the values fifteen selected equations were consistently greater than 0 (positive skew), ranging from 0.8 for sub- catchment 9 to 1.69 for sub-catchment 1. These values indicate a high degree of skewness and also give an indication of positive outliers, thus suggesting that the higher values of Tc obtained in this study are overestimates of the actual times of concentration. The skew values clearly indicate that the mean values of Tc will not be good representatives of the distribution. Hence, the median (positional average) values for the Tc obtained for each sub-catchment were designated as the representative Tc value for the sub-catchment and are therefore considered the “best fit” value of Tc in each of the sub catchment (Table 7).
Temez equation provided the best estimates values for three sub-catchments (1, 9 and 10). These are small catchments of catchment ranging from 1.65 to 6.0 km2 and longest flow paths ranging from 0.88 to 1.5 km as shown in Table 7. Temez equation uses only flow length and slope to estimate Tc. For larger sub-catchments, the Chow and Hakatanir/Sezen equations yielded the median values of Tc. Both equations yielded the median Tc values for two sub-catchments each (6 & 11 and 2 & 4 respectively) when the computed values are rendered in three decimal places. However, when rendered in two decimal places, both equations yield the median value for an additional sub-catchment each - 8 and 11 respectively. Hence, Chow gave the median Tc values for sub-catchments 6, 8 and 11 with areas of 20.97, 9.1 and 10.58 km2 respectively while Hakatanir and Sezen gave the median Tc values for sub-catchments 2, 4 and 11. Interestingly the sub-catchments for which these equations yielded the median Tc+ values have the largest flow path (3.5 - 9.3 km). It is important to note that 9 (over 80%) of the equations that yielded the representative Tc values are those that use only longest flow path and slope (Temez; Bransby and Williams; California Culvert Practice; Chow; Hakatanir and Sezen). The remaining two equations (Dooge; Pilgrim and McDermont) do not consider the longest flow path in the determination of Tc. and also, relatively high coefficients of 0.365 and 0.76. The two equations tend to assign more weighting to catchment area than any other catchment parameter. The fact that one or two equations that use catchment area while disregarding the role of the longest flow path in determining the time of concentration could yield representative Tc values suggest that these equations might have some merit in specific catchment conditions but not to be generalized. Other equations that yielded the median Tc values at least once are: Pilgrim and Mc Dermont (sub-catchment 7), Bransby Williams (sub-catchment 3), California Culvert Practice (sub-catchment 5) and Dooge (sub-catchment 8). The other equations did not feature at all in the scenario under consideration.
Table 7 | Representative Tc for each sub catchment
Catchment
|
A (km2)
|
L (km)
|
Empirical Equations
|
Major Catchment Parameter Used
|
|
Tc (h)
|
1
|
1.65
|
0.81
|
Temez
|
L
|
1.59
|
0.88
|
9
|
4.45
|
0.81
|
Temez
|
L
|
2.60
|
0.74
|
10
|
5.98
|
2.03
|
Temez
|
L
|
1.21
|
1.47
|
7
|
7.32
|
2.19
|
Pilgrim & McDermont
|
A
|
-
|
1.62
|
3
|
4.28
|
2.59
|
Bransby Williams
|
L
|
0.80
|
1.97
|
5
|
2.98
|
2.65
|
California Culvert Practice
|
L
|
0.65
|
1.92
|
11
|
10.58
|
3.54
|
Chow; Hakatanir & Sezen
|
L
|
0.92
|
2.16
|
8
|
9.11
|
4.00
|
Dooge; Chow
|
A; L
|
0.75-
|
2.33
|
4
|
4.49
|
5.20
|
Hakatanir & Sezen
|
L
|
0.41
|
2.99
|
6
|
20.97
|
8.19
|
Chow
|
L
|
0.56
|
4.03
|
2
|
20.72
|
9.29
|
Hakatanir & Sezen
|
L
|
0.49
|
4.87
|
As a confirmatory test in the selection of equation(s) that yielded the most representative values of time of concentration, further exploratory analyses were performed on the output results. This was necessary because some equations yielded values very close to the median values, though they did not produce the median values. Hence, it was necessary to establish a cut-off range for accepting any equation. This was done by specifying three criteria as follows:
Based on the first criterion, only 7 (46.7%) of the fifteen selected equations produced the median Tc value at least once. The equations ranked from the highest to the least in terms of the number of times they returned the median value are: Temez, Haktanir & Sezen, Chow, Bransby & Williams, California Culvert Practice, Dooge and Pilgrim & Dermont. Ten of the equations gave values that entirely fall within one standard deviation of the median, while six of them gave values that entirely fall within 0.5 standard deviation of the median. The worst performing in terms of the inclusion range are Sheridan and Pasini whose values entirely fall outside the ± SD and ± 0.5 SD of the median values for each sub-catchment. Though the Watt & Chow and Carter equations produced some Tc values that fall within the ± SD of the median, they did not produce any values that fall within the ± 0.5 SD of the median. Based on the foregoing, the equations were partitioned into three groups in terms of performance. All equations that produced the median value for at least one of the catchments and also yielded Tc values that entirely fall within ± SD and ± 0.5 SD of the median value for each sub-catchment are hereby considered the best performing; all equations that produced some Tc values that fall within ± SD and ± 0.5 SD are considered fair while those that did not produce any Tc values within the ± SD and ± 0.5 SD of the median range are considered unacceptable for application in the study area. Interestingly, all the best performing equations are those that assign higher weighting to the longest flow path. Of the five best-performing equations (Temez, Haktanir & Sezen, Chow, Bransby & Williams, and California Culvert Practice), only the equation of Bransby & Williams incorporates catchment area in determining Tc. However, the equation recognizes the overriding role of the longest flow path by assigning it an exponent of 1.0 while assigning an exponent of 0.1 to catchment area.
Sensitivity of Input Parameters
Sensitivity of input parameters is the last part of this methodological framework. It seeks to evaluate changes in the value of time of concentration for the fifteen equations given a change in the magnitude of the value of one or more input parameters for each equation. To this end, the behaviour of each equation can be further evaluated in the process (Jamilton Echeverri-Diaz et al., 2022).
By keeping all other parameters constant and varying only one parameter, it was possible to examine the effect of various catchment parameters used in estimating the time of concentration. Generally, all the equations that use L in the estimation of time of concentration agree that a doubling of L leads to some sort of increase in the estimate time of concentration but not necessarily a commensurate amount of increase. Figure 6 shows that the California Culvert Practice model is the most sensitive the longest flow path such that the value of Tc increases by a factor of 2.26 when the value of L is doubled. This is followed by Bransby Williams and Sheridan models which produce an increase in Tc by a factor of about 2.0 when the value of L is doubled. This hyper-response emanates from the fact that these equations assign very high weighting to L while at the same time having high coefficients very close to 1.0. The exponents of L in the California Culvert Practice, Bransby Williams and Sheridan equations are 1.155, 1.0 and 0.96 respectively. On the other hand, all the equations respond moderately when the slope is doubled. The response of Tc to doubling of slope ranges between 0.71 and 0.97. Results further show all the equations that use catchment area as an independent variable for estimating Tc consider it more important than the slope.
This study has gone to a great extent to show that arbitrary and random use of Tc models without proper scrutiny and evaluation could result in highly unrealistic and impracticable results. One of the most important things researchers should do when publishing newly developed models for Tc is to undertake a comprehensive hydro-morphological characteristics of the catchment of interest and publish same along with the model. This will help users to ascertain apriori if the equation is suitable for the study area in question. Beven (2020) also opined that the variability among methods used in the estimation of Tc in inherent in its definition as “a drop of water”, which is a salient point in defining Tc. According to Beven (2020), the drop of water concept does not describe catchment responses in terms of surface or subsurface flow and further explains that the concept of Tc should be taught based on “celerities”. As mentioned earlier, many of these equations were from other locations with different geographical, climatological and hydrological characteristics which differ from that of the location in this study. With proper understanding of the equation(s), certain modifications can be made to in to fit an already existing equation to suit the catchment features.