# Sensitivity of Empirical Equation Parameters for the Calculation of Time of Concentration in Urbanized Watersheds

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{l}) occurs at the headwaters of a basin, concentrated flows (F

_{c}) arise immediately after the sheet flow, and channel flow (F

_{ca}) takes place in the drainage channel. Figure 2 displays the differences between the concentrated flow and the channel-type flow, using as reference the behavior of one of the small watersheds of the study following a rainfall event.

## 2. Case Study

^{2}and its topography is basically flat with a few elevations. The city is surrounded by numerous creeks and streams, and the city’s main water source is the Sinú River. The region has a rainy season between April and September and a dry season between December and April. The city of Montería has an average slope of 0.2%, and a rainfall drainage system that starts out on the streets as a concentrated flow, and whose superficial runoff is subsequently fed into a drainage channel.

## 3. Materials and Methods

_{c}) are between 0.53 and 5.52 km

^{2}, with a minimum watershed slope (S) of 0.00060 m/m and a maximum slope of 0.00225 m/m, which are consistent with the local topographic conditions. The review of ground cover results indicates that urbanized cover is predominant in terms of the runoff coefficient (C) and the weighted curve number (CN). ΔH is a different elevation in a main watercourse. P

_{2}is the maximum daily precipitation associated to a return period (Tr) of 2 yr. Lastly, Manning’s roughness coefficient (n) highlights that most watercourses are covered in concrete, which increases the runoff flow rate.

_{l}) was calculated based on Manning’s roughness coefficient (n), the length and slope of the watercourse and the maximum precipitation of design associated to a two-year return period. The following is the expression used for the calculation:

_{fl}= Sheet flow travel time (h), L

_{fl}= Sheet flow length (m), n = Manning’s roughness coefficient, P

_{2}= Maximum precipitation in 24 h for a 2-year return period (mm), and S

_{c}= Average watercourse slope (m/m).

_{c}) assumes that the sheet flow becomes a superficial concentrated flow. The average velocity of this flow is determined based on the following expressions:

_{fc}= Concentrated flow travel time (h), L

_{fc}= Concentrated flow length (m), and V = Average velocity (m/s).

_{ca}) is calculated for open channels with defined hydraulic characteristics in the cross-section. Manning’s equation or the information of the profile of the water surface is used to estimate the average flow velocity.

_{c}= average watercourse slope (in this case corresponds to the hydraulic slope of the channel) (m/m), and n = Manning’s roughness coefficient.

_{fca}= Channel flow travel time (h) and L

_{fca}= Channel flow length (m).

^{2}) and the mean square error (MSE).

- The first sensitivity analysis focused on finding the variability of the Tc value calculated by means of the baseline equation by calculating the maximum precipitation in 24 h associated with the 2-year return period, as proposed by the different authors cited in [14,30], who recommend the use of distribution methods, from among which GEV, Log Pearson Type III and Pearson Type III were selected. The result was compared to the value used as reference (Gumbel) [31]. This value is necessary in order to calculate the sheet flow travel time.
- The second sensitivity analysis focused on the variability of the roughness coefficient used in the baseline equation to calculate sheet flow and concentrated flow travel time. To this effect, analysis was performed using the values defined as the minimum, normal and maximum values depending on the type of channel and its description. After recalculating the travel times and the time of concentration, statistical analysis was performed to assess the sensitivity of this variable compared to the empirical equations.
- The third sensitivity analysis was performed by calculating the time of concentration value using the baseline equation, initially without considering the sheet flow travel time, and afterwards without considering the concentrated flow travel time. The obtained values were compared to the values calculated by the different equations to estimate Tc, using MSE and R
^{2}as the criteria for comparison. The empirical equations used for the comparison were those that did not display significant differences in the statistical analysis. - The fourth and last sensitivity analysis focused on verifying the behavior of the empirical equations as a function of variations in the length of the main watercourse and the ground cover of the different urban watersheds. The equations selected for this analysis were those that did not display significant differences compared to the baseline equation according to Dunnet’s test. It should be noted that sensitivity to the two variables mentioned above was not assessed for all the selected equations, either because such variables were not included or were not relevant in the equations. Lastly, time of concentration was calculated using the selected equations, and the results obtained were compared to the values of the baseline equation. The variation found in the results obtained in this analysis was assessed by means of MSE.

## 4. Results

#### 4.1. Determination of the Time of Concentration

#### 4.2. Statistical Analysis

- Firstly, the median time of concentration value found using the baseline equation and the Carter equation are equal (blue box), which tentatively leads to believe that the Carter equation is the method with the best fit.
- Secondly, it can be concluded that the median Tc of the equations of Kirpich, Simas-Hawkins, TxDOT, California Culvert Practice and Ventura are within the same interquartile rage (red line) and may consequently be considered as a second group for the assessment of Tc compared to the baseline equation, based on the sensitivity analysis.

^{2}) and the Mean Square Error (MSE), which acted as criteria for the selection of the model that best fits the basins of the urban area of the city of Montería. The models and the results are displayed in Table 7. Figure 6 provides a graphical representation of the behavior of the results of the empirical equations compared to the baseline equation.

^{2}=0.77 and MSE of 0.32 h (see Table 7).

#### 4.3. Sensitivity Analysis of the Studied Parameters

#### 4.3.1. Variation of the Tc in the Baseline Equation

#### 4.3.2. Variation of the Roughness Coefficient

#### 4.3.3. Variation of the Sum of Travel Times for Calculation of the Tc Using the Baseline Equation

#### 4.3.4. Sensitivity of the Parameters of the Empirical Equations

- A first group includes the equations that are formulated only as a function of the length and slope of the main watercourse, namely those by Carter, Kirpich, California Culvert Practice and Ventura. In this group, the sensitized variable was the length of the main watercourse: a percentage of the length of each of the small watercourses of the study was subtracted in different amounts.
- Afterwards, the Tc was recalculated and the statistical analysis was performed, finding that in the Carter equation as the Ls decreases, SME increases. This can be seen in Figure 7 in the case in which the Ls was reduced by 25%, and MSE began to increase with a steeper slope on the trend line, reaching MSE of 0.39 h.
- With the equations by Kirpich and California Culvert Practice, as Ls decreases, MSE decreases. Using the same criteria of analysis used for Carter’s equation, it was found that when Lsi = 0.75 Ls, MSE = 0.48 h for both equations. Similar results were found with the Ventura equation: when Lsi = 0.75 Ls, MSE = 0.49 h.
- The decision not to sensitize the la slope of the main watercourse in this section is because this variable depends on the topographic conditions of the channel, and in the context of the urban development in the area of the study, which as mentioned earlier features a flat topography, this value cannot be modified in practice.
- The second group included the equations that do not depend only on the length and slope of the main watercourse, but whose structure also involves the ground cover, which is a parameter that can produce considerable variation in the calculated Tc.
- When Tc was calculated with the equation by Simas-Hawkins, the value used for the weighted curve number (CN) was calculated in the background of a normal moisture condition (AMC II). For this sensitivity analysis, CN was recalculated for a background of dry conditions (AMC I) and wet conditions (AMC III). According to the results consolidated in Figure 8a, MSE decreased in moisture background conditions (MSE = 0.62 h), and increased considerably in dry conditions (MSE = 1.27 h).
- When the weighted runoff coefficient (C) was sensitized in the equation by TxDOT, it was found that as the C value increases as a function of the return period [19], MSE decreases, in which the value found for a return period of 100 years is the one with best goodness of fit (MSE = 0.44 h), see Figure 8b.

## 5. Discussion

^{2}and lengths of main watercourses between 1.19 and 6.81 km, approximately. Consequently, in order to delimit the small watersheds and define their morphometric parameters, as a minimum, it is necessary to have geographic information available in the form of cartographic charts or satellite images. This methodology is supported by Kobiyama, M., Grison, F., Lino, J.F., & Silva, R.V. (2006) [32], who used for their study a map with scale of 1:10,000 to establish the morphometric parameters of the basin studied in the campus of Universidad Federal de Santa Catarina (UFSC) (BC). Based on the above and an analysis of the methods used to characterize water basins, it is observed that the definition and delimitation of the basin’s area and other morphometric parameters can be performed by means of geographic information; however, for the assessment of small basins with primarily flat topography, aerial photographs, cartographic maps or a DEM will not suffice, because their precision is up to one meter, and therefore miss elevated points on the surface that are not within that range, in this case one meter, which can produce inconsistencies in the delimitation of the basins. Consequently, more precise information is required such as cartographic drawings that contain field measurements using high-precision equipment, in order to improve reliability at the time of defining the area and other morphometric parameters of the basin.

^{2}and MSE, the results of the formal hypothesis test of the Tc (treatments), Fo = 17.15, yielded a p value < 0.0001, and additional hypothesis testing was performed for the micro-basins (blocks), Fo = 23.26, with a p value < 0.0001. The results obtained in the comparison test are displayed in Table 6, and Table 7 displays the goodness of fit results. These results lead to the belief that of the 12 empirical equations selected, 6 did not display significant differences when the mean values are compared to those of the baseline equation, among which the Carter equation was the equation with the best goodness of fit according to the selection criteria that were adopted. However, the results of the other equations cannot be ruled out because the decision criteria are not rejected for these cases either. The results of the means comparison test can be compared to those obtained by Vahabzadeh, G., Saleh, I., Safari, A., & Khosravi, K. (2013) [2], who used the Tukey means comparison test to categorize the best equations. Additionally, the comparison criteria used in this study are supported by Ravazzani, G., Boscarello, L., Cislaghi, A., & Mancini, M. (2019) [19], who based their goodness of fit of the methods on MSE and R

^{2}. On the above, it is important to mention that the use of a means comparison test plays an important role in the categorization of the equations of the study, as it enables the researcher the rule out many of the equations that were initially selected that do meet the expected objectives. Afterwards, assessing which of the resulting methods is most accurate compared to the baseline equation brings the researcher closer to the desired objective, providing backing for the methodological approach and validation that an empirical equation may be highly correlated with an equation almost fully based on physics. Lastly, it can be said that these statistical methods leave a window open to study the remaining equations from a different perspective (sensitivity analysis). The curve number was estimated for each urbanized watershed and extreme scenarios of antecedent moisture conditions were considered [33].

## 6. Conclusions

^{2}). These values were found taking into consideration the total length of the main watercourse using the Carter equation, which is equivalent to the sum of the different travel times of the baseline equation using laminar, concentrate, and channel flow. Complementarily, it can be concluded that the Carter equation was found to have highly significant similarity with the baseline equation for the effects of calculating the Tc, which is validated by the value found in the means in the statistical analysis, in which values were reported of 1.34 h for the baseline equation and 1.28 h for the Carter equation (see Figure 5).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

A_{c} | Basin area |

C | Runoff coefficient rational method |

CN | Curve number |

MSE | Mean square error |

F_{c} | Concentrated flow |

F_{ca} | Channel flow |

F_{l} | Sheet flow |

L_{fc} | Length of concentrated flow |

L_{fca} | Length of channel flow |

L_{fl} | Length of sheet flow |

L_{s} | Length of the main watercourse |

n | Manning’s roughness coefficient |

m | Number of branches |

P_{2} | Maximum precipitation in 24 h for a return period (Tr) of 2 years |

R | Hydraulic radius |

R^{2} | Coefficient of determination |

S | Average slope of a watershed area |

S_{c} | Average slope of a main watercourse |

Tc | Time of concentration |

T_{fc} | Concentrated flow travel time |

T_{fca} | Channel flow travel time |

T_{fl} | Sheet flow travel time |

Tr | Return period |

V | Average velocity |

ΔH | Difference of level between the beginning and the end of the main watercourse |

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**Figure 1.**Illustration drawing of the three time of concentration flows of the equation proposed by the NRCS.

**Figure 2.**Identification of concentrated flow and channel flow in one of the small watersheds of the case study (city of Montería, Colombia).

**Figure 6.**Tc graphs estimated with empirical methods versus Tc calculated with the baseline equation: (

**a**) Carter; (

**b**) Kirpich; (

**c**) California Culvert Practice; (

**d**) Ventura; (

**e**) Simas-Hawkins; and (

**f**) TxDOT.

**Table 1.**Morphometric parameters of the small watersheds of the study and hydraulic parameters of the main watercourse.

Item | Basin | Ac (km^{2}) | Ls (km) | S (m/m) | ΔH (m) | n | P_{2} | C | CN |
---|---|---|---|---|---|---|---|---|---|

1 | Principal Margen Izquierda | 2.83 | 2.73 | 0.00060 | 1.65 | 0.021 | 78.19 | 0.68 | 84 |

2 | Panamá-La Ribera | 0.53 | 1.56 | 0.00112 | 1.75 | 0.015 | 78.19 | 0.88 | 98 |

3 | Centenario | 1.00 | 1.79 | 0.00087 | 1.57 | 0.015 | 78.19 | 0.71 | 86 |

4 | Los Araujos | 1.12 | 2.41 | 0.00122 | 2.94 | 0.015 | 78.19 | 0.88 | 98 |

5 | La Granja | 5.52 | 6.81 | 0.00083 | 5.62 | 0.023 | 78.19 | 0.76 | 90 |

6 | Cantaclaro | 2.52 | 2.91 | 0.00110 | 3.21 | 0.021 | 78.19 | 0.77 | 90 |

7 | El Mora | 0.75 | 1.19 | 0.00199 | 2.36 | 0.017 | 78.19 | 0.88 | 98 |

8 | Av Circunvalar Sur | 3.00 | 3.56 | 0.00135 | 4.80 | 0.015 | 78.19 | 0.88 | 98 |

9 | Calle 44 | 0.77 | 1.78 | 0.00225 | 3.99 | 0.015 | 78.19 | 0.88 | 98 |

10 | Av Circunvalar Norte | 1.72 | 2.18 | 0.00164 | 3.57 | 0.021 | 78.19 | 0.64 | 82 |

11 | La Pradera | 0.74 | 1.44 | 0.00130 | 1.86 | 0.015 | 78.19 | 0.88 | 98 |

No. | Equation | Formula | Description and Reference |
---|---|---|---|

1 | Kirpich Equation [15] | $Tc=0.0663{\left(\frac{{L}_{s}^{2}}{{S}_{c}}\right)}^{0.385}$ | Kirpich (1940), calibrated two empirical models to estimate the time of concentration in small basins in Pennsylvania and Tennessee, with areas between 0.4 and 45.3 ha and average slopes between 3% and 10%. Researchers [6] demonstrated that this method tended to under-estimate Tc by 75% in urbanized basins with areas between 8 and 16 km^{2}, predominantly with channel flows. Researchers [16] demonstrated that using this equation in basins with areas between 17.35 and 598 km^{2}, and average slope between 0.0173 and 0.1029 m/m produced smaller positive biases (mean error of 16.8 h and standard deviation of 37.1%). |

2 | Millers Equation [17] | $Tc=1.7833\left[\frac{n1000{L}_{s}^{0.333}}{{\left(100{S}_{c}\right)}^{0.2}}\right]$ | Method developed from the nomogram of sheet, concentrated and channel flows published by the Institute of Engineers of Australia (IEA, 1977), [12]. The authors demonstrated that predictive variables with most influence in the calculation of time of concentration were the length of the main watercourse, with values between 8 and 431 km, and the average slope of the main watercourse, with values ranging between 0.00078 and 0.01687 m/m, finding the smallest biases at standard deviation values of less than 20% and mean errors of less than 2 h. |

3 | California Culvert Practice, [18] | $Tc=0.951{L}_{s}^{1.155}\Delta {H}^{-0.385}$ | Method developed by the California Roads Division (1960) for small mountainous basins in California [13] and data obtained from small basins in the USA with areas of less than 40.47 km^{2} [11]. This is consistent with the results obtained by [19], which estimated the time of concentration in 46 basins of the Po River with areas between 56 and 1.588 km^{2} and slopes in mountainous terrains ranging between 0.022 and 0.268 m/m. The results displayed values of MSE = 8.21 h, which in general moderately under-estimate the values of Tc, which is neither the best nor the worst result. |

4 | Carter Method [20] | $Tc=0.0977{L}_{s}^{0.6}{S}_{c}^{-0.3}$ | Method developed for urban watersheds with areas of less than 20.8 km^{2} and channels of lengths of less than 11.3 km (Sharifi & Hosseini, 2011). It is recommended for basins whose main watercourse has natural channel flows between 0.013 and 0.025, [5]. Ref. [11] indicates that this equation was developed based on data from urban watersheds. |

5 | Federal Aviation Agency method [21] | $Tc=\frac{0.0165626\left(1.1-\mathrm{C}\right)1000{L}_{s}^{0.5}}{{\left(100{S}_{c}\right)}^{0.333}}$ | Developed by engineers in the US based on drainage of air fields [5]. It is widely used for urban sheet flow [12]. Developed primarily for sheet and concentrated flows, [13] used this equation in five water basins that include several sub-basins within them, with slopes ranging between 0.044 and 0.091 m/m. It was found that this and several other equations has lower values in all the assessment criteria, producing inaccurate estimates (MSE = 0.499 h). |

6 | NRCS equation, kinematic wave method [22] | $Tc=0.0015476\frac{{L}_{S}{}^{0.6}}{{S}_{c}{}^{0.3}}$ | This method is widely used in paved areas, although it was initially used for concentrated flow and channel flow, and is based on the ratio between the intensity and duration of the rainfall associated with a 2-year return period. Additionally, the method was developed to avoid the iterative process of the original formula for the kinematic wave method. When [16] used this equation in basins with average slope between 0.0173 and 0.1029 m/m, they found a bias of less than 10%, with mean error of 1 min. |

7 | TxDOT method [23] | $Tc=0.369986\left(1.1-C\right){L}_{s}^{0.5}{S}_{c}{}^{-0.333}$ | It is the result of a modification to the FAA’s method (FAA, 1970) [24]. The Texas Department of Transportation (TxDOT) adopted the above methodology in its hydraulic design manual to estimate the Tc in basins in the Texas region [10]. Ref. [25] used this equation to compare the value of Tc to the travel time results in a testing strip for runoff by thrust, from which they developed an equation. They concluded that the TxDOT method tends to over-estimate the value of Tc, producing lower results in clay, asphalt and concrete surfaces, with the highest result in grass cover. |

8 | Chow’s Model [26] | $Tc=0.1602\frac{{L}_{s}^{0.64}}{{S}_{c}{}^{0.32}}$ | Equation developed for 20 rural basins rurales in the United States in which the drainage area ranged between 0.01 and 18.5 km^{2} and the slope of the main watercourse was between 0.0051 and 0.09 m/m. In the same basin of the study, [19] found similar performance between the equation of the California Culvert Practice and the equation of Chow’s Model, with a value of MSE = 7.21 h. |

9 | Bransby-Williams method [27] | $Tc=0.605\frac{{L}_{s}}{{\left(100{S}_{c}\right)}^{0.2}A{c}^{0.1}}$ | Williams (1922) conducted a study on flood discharges in India and Haktanir and Sezen (1990) developed his methodology by means of regression analysis using data from basins located in Turkey [11]. This method is based on experimental use for water basins with drainage areas of less than 129.5 km^{2} and dominated by channel flow [19]. Ref. [1] used this equation in basins that were similar in terms of total area and urbanization, but where the slope of the different types of flow was different, which apparently affects the equation’s performance. |

10 | Simas-Hawkins, [28] | $Tc=\frac{0.322{A}^{0.594}}{{L}_{s}^{0.594}{S}_{c}{}^{0.15}}{\left[\frac{\mathrm{25,400}}{CN}-254\right]}^{0.313}$ | Method developed in 168 basins in the United States with areas between 0.001 and 14 km^{2}. A study by [11] used 30 empirical methodologies in one water basin to calculate Tc. Of these equations, the Simas-Hawkins was classified in the group of appropriate equations for natural basins. |

11 | Ventura-HEC-RAS equation [2] | $Tc=0.067\frac{{L}_{s}^{1.155}}{{\left(\frac{\Delta H}{1000}\right)}^{0.385}}$ | This equation applies to small basins [2]. The authors found that in one of the basins of the study, which was sub-divided into 6 sub-basins with slopes ranging between 0.002627 and 0.024079 m/m and length of the main watercourse between 52.389 and 13.345 km, the Venturas method yielded the second-best results with a value of MSE of 2 h. This equation is used for channel flows. |

12 | Kerby Equation [29] | $Tc=0.02399{\left(\frac{1000n{L}_{s}}{{S}_{c}^{0.5}}\right)}^{0.467}$ | This equation can be used for urban drainage lower than 4 ha and slopes under 1%. Commontly, this equation can be used to compute concentrate flow type. |

Watercourse | Travel Time (h) | Tc (h) | ||
---|---|---|---|---|

Sheet Flow | Concentrated Flow | Channel Flow | ||

Principal Margen Izquierda | 0.09 | 1.16 | 0.60 | 1.85 |

Panamá-La Ribera | 0.16 | 0.78 | 0.18 | 1.12 |

Centenario | 0.18 | 0.86 | 0.26 | 1.30 |

Los Araujos | 0.09 | 0.45 | 0.29 | 0.83 |

La Granja | 0.06 | 1.78 | 1.35 | 3.19 |

Cantaclaro | 0.13 | 0.47 | 0.42 | 1.01 |

El Mora | 0.16 | 0.52 | 0.08 | 0.76 |

Av Circunvalar Sur | 0.10 | 0.69 | 0.36 | 1.15 |

Calle 44 | 0.13 | 0.67 | 0.10 | 0.89 |

Av Circunvalar Norte | 0.12 | 1.02 | 0.18 | 1.32 |

La Pradera | 0.15 | 1.08 | 0.07 | 1.30 |

Item Basin | Empirical Equations | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 * | 2 * | 3 * | 4 * | 5 * | 6 * | 7 * | 8 * | 9 * | 10 * | 11 * | 12 * | ||

1 | Principal Margen Izquierda | 2.50 | 0.89 | 2.50 | 1.65 | 0.93 | 5.02 | 3.05 | 3.27 | 2.61 | 3.36 | 2.52 | 0.89 |

2 | Panamá-La Ribera | 1.27 | 0.48 | 1.28 | 0.98 | 0.30 | 1.94 | 0.97 | 1.87 | 1.55 | 0.79 | 1.29 | 0.51 |

3 | Centenario | 1.56 | 0.53 | 1.57 | 1.15 | 0.62 | 2.40 | 2.03 | 2.22 | 1.77 | 2.09 | 1.58 | 0.58 |

4 | Los Araujos | 1.73 | 0.55 | 1.73 | 1.24 | 0.36 | 2.67 | 1.18 | 2.41 | 2.20 | 0.93 | 1.75 | 0.61 |

5 | La Granja | 4.47 | 1.29 | 4.48 | 2.60 | 1.05 | 10.18 | 3.44 | 5.30 | 5.72 | 2.34 | 4.51 | 1.34 |

6 | Cantaclaro | 2.08 | 0.82 | 2.08 | 1.43 | 0.62 | 4.18 | 2.02 | 2.81 | 2.50 | 2.30 | 2.10 | 0.80 |

7 | El Mora | 0.83 | 0.46 | 0.83 | 0.70 | 0.21 | 1.41 | 0.70 | 1.31 | 1.02 | 1.04 | 0.84 | 0.42 |

8 | Av Circunvalar Sur | 2.24 | 0.61 | 2.25 | 1.52 | 0.42 | 3.49 | 1.39 | 2.99 | 2.88 | 1.31 | 2.27 | 0.72 |

9 | Calle 44 | 1.08 | 0.44 | 1.08 | 0.86 | 0.25 | 1.63 | 0.83 | 1.63 | 1.49 | 0.82 | 1.09 | 0.46 |

10 | Av Circunvalar Norte | 1.43 | 0.68 | 1.43 | 1.07 | 0.64 | 2.80 | 2.10 | 2.05 | 1.79 | 2.59 | 1.44 | 0.63 |

11 | La Pradera | 1.13 | 0.45 | 1.14 | 0.89 | 0.27 | 1.72 | 0.89 | 1.69 | 1.35 | 0.99 | 1.14 | 0.48 |

Source | GL | SS | MS | Fo | Fc | p-Value | Decision |
---|---|---|---|---|---|---|---|

Times of concentration | 12 | 87.45 | 7.29 | 17.15 | 1.8337 | 8.1295 × 10^{−21} | Reject H0 |

Micro-basins | 10 | 98.82 | 9.88 | 23.26 | 1.9105 | 1.0166 × 10^{−23} | Reject H0 |

Error | 120 | 50.98 | 0.42 | ||||

Total | 142 | 237.25 |

Hypothesis | Estimate | Standard Error | t Value | Pr (>|t|) | Decision |
---|---|---|---|---|---|

Carter vs. NRCS = 0 | −0.06 | 0.28 | −0.21 | 0.83 | Do not reject H0 |

Chow’s vs. NRCS = 0 | 1.17 | 0.28 | 4.19 | 0.00 | Reject H0 |

California Culvert Practice vs. NRCS = 0 | 0.51 | 0.28 | 1.84 | 0.07 | Do not reject H0 |

FAA vs. NRCS = 0 | −0.74 | 0.28 | −2.95 | 0.01 | Reject H0 |

Simas-Hawkins vs. NRCS = 0 | 0.35 | 0.28 | 1.25 | 0.21 | Do not reject H0 |

Kerby vs. NRCS = 0 | −0.66 | 0.28 | −2.38 | 0.02 | Reject H0 |

Kirpich vs. NRCS = 0 | 0.51 | 0.28 | 1.82 | 0.07 | Do not reject H0 |

Miller vs. NRCS = 0 | −0.77 | 0.28 | −2.75 | 0.01 | Reject H0 |

NRCS vs. NRCS = 0 | 2.07 | 0.28 | 7.45 | 0.00 | Reject H0 |

TxDOT vs. NRCS = 0 | 0.35 | 0.28 | 1.27 | 0.21 | Do not reject H0 |

Ventura vs. NRCS = 0 | 0.53 | 0.28 | 1.89 | 0.06 | Do not reject H0 |

Williams vs. NRCS = 0 | 0.92 | 0.28 | 3.31 | 0.00 | Reject H0 |

Equation | MSE (h) | R^{2} |
---|---|---|

Carter | 0.32 | 0.77 |

Kirpich | 0.70 | 0.80 |

California Culvert Practice | 0.70 | 0.80 |

Ventura | 0.72 | 0.80 |

Simas-Hawkins | 0.81 | 0.29 |

TxDOT | 0.61 | 0.70 |

Watercourse | Hydrological Probability Distribution | |||
---|---|---|---|---|

Gumbel | GEV | Log Pearson Type III | Pearson Type III | |

Principal Margen Izquierda | 1.85 | 1.85 | 1.85 | 1.85 |

Panamá-La Ribera | 1.12 | 1.12 | 1.12 | 1.13 |

Centenario | 1.30 | 1.30 | 1.30 | 1.30 |

Los Araujos | 0.83 | 0.83 | 0.83 | 0.83 |

La Granja | 3.19 | 3.19 | 3.19 | 3.19 |

Cantaclaro | 1.01 | 1.01 | 1.01 | 1.01 |

El Mora | 0.76 | 0.76 | 0.76 | 0.76 |

Av Circunvalar Sur | 1.15 | 1.16 | 1.16 | 1.16 |

Calle 44 | 0.89 | 0.89 | 0.89 | 0.90 |

Av Circunvalar Norte | 1.32 | 1.32 | 1.32 | 1.32 |

La Pradera | 1.30 | 1.30 | 1.30 | 1.30 |

Equation | MSE (h) | ||
---|---|---|---|

n_{minimum} | n_{average} | n_{maximum} | |

Carter | 0.30 | 0.32 | 0.37 |

Kirpich | 0.78 | 0.70 | 0.63 |

California Culvert Practice | 0.78 | 0.70 | 0.63 |

Ventura | 0.80 | 0.72 | 0.65 |

Simas-Hawkins | 0.83 | 0.81 | 0.80 |

TxDOT | 0.67 | 0.61 | 0.56 |

Equation | Calculation of MSE Changing the Sum of Travel Times | ||
---|---|---|---|

F_{l} + F_{c} + F_{ca} | F_{c} + F_{ca} | F_{ca} | |

Carter | 0.32 | 0.32 | 0.72 |

Kirpich | 0.70 | 0.76 | 1.26 |

California Culvert Practice | 0.70 | 0.77 | 1.27 |

Ventura HEC-RAS | 0.72 | 0.78 | 1.28 |

Simas-Hawkins | 0.81 | 0.92 | 2.32 |

TxDOT | 0.61 | 0.67 | 2.25 |

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**MDPI and ACS Style**

Echeverri-Díaz, J.; Coronado-Hernández, Ó.E.; Gatica, G.; Linfati, R.; Méndez-Anillo, R.D.; Coronado-Hernández, J.R.
Sensitivity of Empirical Equation Parameters for the Calculation of Time of Concentration in Urbanized Watersheds. *Water* **2022**, *14*, 2847.
https://doi.org/10.3390/w14182847

**AMA Style**

Echeverri-Díaz J, Coronado-Hernández ÓE, Gatica G, Linfati R, Méndez-Anillo RD, Coronado-Hernández JR.
Sensitivity of Empirical Equation Parameters for the Calculation of Time of Concentration in Urbanized Watersheds. *Water*. 2022; 14(18):2847.
https://doi.org/10.3390/w14182847

**Chicago/Turabian Style**

Echeverri-Díaz, Jamilton, Óscar E. Coronado-Hernández, Gustavo Gatica, Rodrigo Linfati, Rafael D. Méndez-Anillo, and Jairo R. Coronado-Hernández.
2022. "Sensitivity of Empirical Equation Parameters for the Calculation of Time of Concentration in Urbanized Watersheds" *Water* 14, no. 18: 2847.
https://doi.org/10.3390/w14182847