Evaluation of the Time of Concentration Models for Enhanced Peak Flood Estimation in Arid Regions

Abstract

The uncertainties in the time of concentration (T_c) model estimate from contrasting environments constitute a setback, as errors in T_c lead to errors in peak discharge. Analysis of such uncertainties in model prediction in arid watersheds is unavailable. This study tests the performance and variability of T_c model estimates. Further, the probability distribution that best fits observed T_c is determined. Lastly, a new T_c model is proposed, relying on data from arid watersheds. A total of 161 storm events from 19 gauged watersheds in Southwest Saudi Arabia were studied. Several indicators of model performance were applied. The Dooge model showed the best correlation, with r equal to 0.60. The Jung model exhibited the best predictive capability, with normalized Nash–Sutcliffe efficiency (NNSE) of 0.60, the lowest root mean square error (RMSE) of 4.72 h, and the least underestimation of T_c by 1%. The Kirpich model demonstrated the least overestimation of T_c by 4%. Log-normal distribution best fits the observed T_c variability. The proposed model shows improved performance with r and NNSE of 0.62, RMSE of 4.53 h, and percent bias (PBIAS) of 0.9%. This model offers a useful alternative for T_c estimation in the Saudi arid environment and improves peak flood forecasting.

1. Introduction

Timing parameters are extensively used in hydrograph, peak discharge, and flood risk analyses. The time of concentration parameter, T_c, is regarded by many researchers as the most important timing parameter. The definitions of T_c are numerous in the literature. It is the time required for runoff to travel from the hydraulically most distant point in a watershed to the outlet of the watershed [1] or a designated location downstream [2]. A more commonly held definition is “the time required for storm runoff to flow from the most remote part of a drainage basin to the outlet” [3]. A more practical definition of T_c, often regarded as the graphical definition, is the time between the centroid of effective rainfall to the point of inflection on the recession limb of the total runoff hydrograph (TRH) or the end of direct runoff hydrograph (DRH), as proposed by Clark [4]. This graphical definition of T_c [5] is adopted for the observed T_c calculation in this study (Figure 1).

T_c is an important variable in runoff estimation in the analyses of the hydrograph timing parameters and response to effective rainfall. While the T_c does not seem explicitly stated in the Rational Method equation (given as Q = CiA, where C is the runoff coefficient, i is the rainfall intensity, and A is the basin area), the Rational Method calculation requires a T_c estimate to determine the actual rainfall intensity [6]. In particular, T_c is equivalent to the duration of the storm’s effective rainfall. When the unit hydrograph method is applied in the design of storm hydrographs, particularly runoff peak discharge, the T_c value becomes a yardstick in determining the area under the hydrograph [7]. Therefore, T_c influences, to a large extent, the shape and consequently the peak of the discharge hydrograph, and understanding peak discharge characteristics in a hydrological basin [8]. Inaccurate T_c evaluation can increase the risk of failure of flood protection structures and can have a severe socio-economic impact in downstream areas of a watershed.

Estimating T_c has received widespread attention among researchers due to the increasing need for researchers and policy makers to predict uncertainties in rainfall and runoff. This is particularly true with widespread flash floods and their associated risks. Extensive work has been conducted on the influences of climatic variability on rainfall trends [9] and streamflow patterns [10]. There is also an increasing need to account for deficiencies in runoff monitoring in ungauged watersheds. McEnroe et al. [11] investigated T_c and lag time in a watershed using thirty gauged sites in Kansas, USA. From the data analysis of this study, a new region-based equation was derived for the lag time after data calibration in a flash-flood warning system, Automated Local Evaluation in Real Time (ALERT). However, the T_c was not determinable from the gauging data. De Almeida et al. [5] calculated T_c using the graphical method in a Brazilian watershed to examine seventeen rainfall–runoff events. They compared the graphical method results with results obtained from twenty equations formulated from non-urban watersheds. The research proved the graphical method to be reliable for T_c determination. Michailidi et al. [8] simulated the travel time across thirty basins around the Mediterranean region in the GIS environment drawing from the popular kinematic and Rational Method approach. T_c was estimated in terms of the intensity of the excess rainfall as a power function using the appropriate procedure to determine the exponents and multipliers of the geomorphological data available. Then, the parameters were compared between both methods with the observed data. The analysis from this study includes not only a GIS approach to T_c evaluation but also highlights the role of T_c in hydrological modeling. Yoo et al. [12] evaluated both the storage coefficient and T_c over basins of interest using globally available theoretical equations and correlated the results with the longest channel length in the dam basins in Korea. They reported proportionality between the main channel and the time of concentration but non-proportionality between the time of concentration and the slope of the channel, and the study showed no correlation between the storage coefficient and time of concentration.

In developing T_c models, researchers combine one or more parameters from the basin geomorphology, rainfall characteristics, and basin land use factors. The relative contributions of these rainfall–runoff determinants are documented in the works of many authors [13]. However, all the empirically based methods are bound to exhibit some degree of variability in model output due to the spatiotemporal variability of the study area from which the models are developed. Other expected variability will include models’ predictive errors due to data uncertainties, boundary conditions, and model structure or model parameter uncertainties inherent in hydrological processes. These diversities in the model derivation also come with some challenges. Large errors in the peak discharge estimates often occur due to errors from T_c prediction. This renders the model inadequate to represent the physical data, especially when these methods are applied outside their original pilot set-up [14]. Even between areas of a similar set of drainage parameters, a −38% to 207% prediction bias range was reported by Fang et al. [15]. Further, González-Álvarez [16] assessed several T_c equations by comparing their statistical viability, including the popular Natural Resources Conservation Service (NRCS) velocity method in watersheds of different sizes and areas. The result showed that one of the studied equations has a high level of reliability in all watersheds, while four other equations showed up to fifty percent reliability levels based on the Nash–Sutcliffe efficiency (NSE) index. More notably, all the empirical equations examined in the study showed errors of underestimation or overestimation [17]. This indicates that T_c model utilization still poses the challenge of estimation bias and poor predictive capability, as well as their suitability for use in hydrological model studies. Consequently, a lesser magnitude of such estimation uncertainty and prediction bias cannot be expected if such models developed from mostly non-arid geomorphological and climatic regions are applied in arid environments with more unpredictable hydrological extremes.

To the authors’ best knowledge, no study has been conducted in arid regions that evaluated the existing T_c models developed in non-arid regions to arid watersheds. In addition, many studies have eluded evaluation of their performance based on data from arid environments. It is also observed that T_c models suitable for arid areas’ watersheds are scarce in the literature. To address these gaps, this study compares the performance of different T_c models developed from non-arid environments to assess their suitability for arid regions. Further, a new T_c model is statistically developed using data from the arid regions. The results are of great importance to engineers and policy makers involved in the design of mitigation structures in arid regions, especially Saudi Arabia.

2. Materials and Methods

The study area is located in the Asir province of southwest Saudi Arabia (Figure 2). The eastern face of the province rises to an altitude of 3000 m, extending westwards before decreasing close to the Red Sea.

The 19 sub-catchments selected are found within 5 major catchments, namely Liyyah, Yiba, Al-Lith, Tabalah, and Habawnah catchments. Liyyah and Tabalah catchments both have 3 sub-catchments each. Yiba catchment and Al-Lith catchment both have 4 sub-catchments each. Habawnah has 5 sub-catchments. Liyyah, Yiba, and Al-Lith all flow to the Red Sea. The Tabalah and Habawnah wadis, located close to the mountain range interior, run into the depression around the inland plains. As for the Tabalah wadi, the altitude rises southwestward starting from the northeast, while in the Habawnah wadi, the altitude ascends from east to west [18].

Table 1. Summary of geomorphological, rainfall, and runoff parameters of the study area.

Parameter	Definition	Range
Basin Area, A (km2)	-	107–4.944
Basin Slope, Sb (m/m)	The average slope of the entire basin	0.09–0.33
Basin length, Lb (km)	Length in a straight line from the mouth of a stream to the farthest point on the drainage divide of its basin	19–112
Basin elevation, Hm (m)	Difference between the elevation of the highest basin divide and the elevation at the basin outlet	234.72–2143.95
Flow length, L (km)	Downslope distance from the hydraulically most distant point to the outlet point	26–158
Flow length slope, S (m/m)	Mean steepness, i.e., the ratio between the mean fall and the L length of the basin’s hydraulically most distant points	0.01–0.08
Average rainfall intensity, i (mm/h)	-	0.19–41.13
Roughness coefficient, n	-	0.04
Curve number, CN	-	44–98
Runoff coefficient, C	-	0.0014–0.3724

presents the physical properties of these basins. The runoff gauging stations in each sub-catchment are also highlighted. Dames and Moore Company of Saudi Arabia documented a detailed study of these watersheds [19].

Data were collected in the five basins consisting of 19 sub-basins for variables such as basin area, watershed length along the main channel, mean basin slope, NRCS curve number, runoff coefficient, Manning’s roughness coefficient, and rainfall intensity (Table 1). These data were input into the selected T_c estimation models, and the model outputs were compared with observed T_c thereafter.

Many models are available for estimating T_c emerging in studies from several parts of the world, as found in the literature. The models used in this study are listed in

Table 2. List of models selected for the estimation of T_c.

S/N	Model	Equation for Tc	Definition of Parameters
1	Kirpich (1940) [1]	$T_c=0.0663S^{-0.385}{L}^{0.77}$	Tc (h) = time of concentration; L (km) = main water line length; and S (m/m) = mean channel slope
2	Soil Conservation Service Lag (1972) [20]	$T_c=0.057{\left(\frac{1000}{CN}-9\right)}^{0.7}\ast L^{0.8}\ast S^{-0.5}$	Tc (h) = time of concentration; CN: SCS curve number; L (km) = flow path length; S (m/m) = mean slope of channel
3	Federal Aviation Administration (1970) [21]	$T_c=0.3788\left(1.1-C\right)L^{0.5}{S}^{-0.333}$	Tc (h) = time of concentration; C: runoff coefficient; L (km) = length of flow path; S (m/m) = mean channel slope
4	Carter (1961) [21]	$T_c=0.0977L^{0.6}{S}^{-0.2}$	Tc (h) = concentration time; L (km) = main water line length; S (m/m) = mean channel slope/mean steepness
5	Kinematic Wave Formula (1964) [21]	$T_c=7.35L^{0.6}{n}^{0.6}{i}^{-0.4}{S}^{-0.2}$	Tc (h) = concentration time; L (km) = length of the main water line or flow path length; S (m/m) = mean slope of channel/mean steepness; n = Manning’s roughness coefficient; i (mm/h) = rainfall intensity
6	Jung (2005) [12]	$T_c=0.119 (\frac{L^{0.777}}{S^{0.212}}$)	Tc (h) = concentration time; L (km) = channel length; S = channel slope
7	Espey (1966) [22]	$T_c=6.89 {\left(\frac{L}{\sqrt{S}}\right)}^{0.36}$	Tc (h) = concentration time; L (km) = channel length; S = slope of basin or channel slope
8	Kraven I (1999) [12]	$T_c=\frac{0.0074 L}{S^{0.515}}$	Tc (h) = concentration time; L (km) = channel length; S = slope of basin or channel slope
9	United States Geological Survey (2000) [12]	$T_c=1.54 (\frac{L^{0.875}}{S^{0.181}}$)	Tc (h) = concentration time; L (km) = channel length; S = slope of basin or channel slope
10	Ven Te Chow (1962) [23]	$T_c=0.1602L^{0.64 }{S}^{-0.32}$	Tc (h) = time of concentration; L (km) = main water line length or flow path length; and S (m/m) = average channel steepness
11	United States Army Corps of Engineers (1954) [4]	$T_c=0.191L^{0.76 }{S}^{-0.19}$	Tc (h) = time of concentration; L (km) = length of the main water line or flow path length; and S (m/m) = average channel steepness
12.	Albishi et al. (2017) [19]	$T_c=\frac{L^{0.09}}{S^{0.11}}$	Tc (h) = time of concentration; L (km) = basin length; and S (m/m) = average basin slope
13	Morgali and Linsley (1965) [22]	$T_c=7.354 \left(\frac{n^{0.6}{L}^{0.6}}{S^{0.3}}\right)i^{-0.4}$	Tc (h) = time of concentration; L (km) = main water line length; S (m/m) = mean channel slope/mean steepness; n = Manning’s roughness coefficient; i (mm/h) = rainfall intensity
14	Izzard and Hicks (1946) [8]	$T_c=\frac{3.46 \left(0.0007i+C\right)L^{0.33}}{S^{0.333}{i}^{0.667}}$	Tc (h) = time of concentration; L (km) = channel length; S (m/m) = basin/channel slope; n = Manning’s roughness coefficient; i = in/h
15	McCuen et al. (1984) [22]	$T_c=2.2535i^{-0.7164 }{L}^{0.5552}{S}^{-0.207}$	Tc (h) = time of concentration; L (km) = main water line length; S (m/m) = mean channel slope/mean steepness; i (mm/h) = rainfall intensity
16	Johnstone (1949) [5]	$T_c=0.4623L^{0.5 }{S}^{-0.25}$	Tc (h) = time of concentration; L (km) = main water line length; S (m/m) = mean channel slope/mean channel steepness; i (mm/h) = rainfall intensity
17	Dooge (1973) [24]	$T_c=0.365A^{0.41 }{S}^{-0.17}$	Tc (h) = time of concentration; S (m/m) = mean channel slope/mean channel steepness; A (km2) = area of the basin
18	Giandotti (1934) [8]	$T_c= \frac{4\sqrt{A}+\frac{3}{2}L}{0.8\sqrt{H_m}}$	Tc (h) = time of concentration; L (km) = main water line length; A (km2) = area of the basin; Hm (m) = mean altitude in the basin (i.e., mean elevation starting from the mouth)
19	Haktanir and Sezen (1990) [4]	$T_c=0.7473L^{0.841}$	Tc (h) = time of concentration; L (km) = main water line length
20	Sheridan (1994) [17]	$T_c=2.20L^{0.92}$	Tc (h) = time of concentration; L (km) = main water line length

. These models were used to calculate the T_c using the data collected in each sub-basin. The models are Kirpich [1], Soil Conservation Service (SCS) Lag [20], FAA [21], Carter [21], Kinematic Wave [21], Jung [12], Espey [22], Kraven (I) [12], United States Geological Survey (USGS) [12], Ven Te Chow [23], USACE [4], Albishi et al. [19], Morgali and Linsley [22], Izzard and Hicks [8], McCuen et al. [22], Johnstone [5], Dooge [24], Giandotti [8], Haktanir and Sezen [4], and Sheridan [17]. The equations were used to estimate T_c and were further statistically compared to the observed T_c using the direct runoff hydrograph data from events in all basins in the study area. All analyses were performed using an excel spreadsheet.

Figure 3 shows some samples of the observed hydrographs from the study watersheds. Observed T_c was obtained from the excess rainfall hyetograph and direct runoff hydrograph (DRH) for each event in all sub-basins in the study area by measuring the time duration in hours (hrs or h) between the end of effective rainfall and the point of inflection on the recession limb of the total runoff hydrograph (TRH) [25].

The uncertainties and difficulties associated with the exact definition of these two-time points have been discussed extensively in the literature [17]. The end of effective rainfall for each event was marked by the immediate clock hour before the effective rainfall ends. The point of inflection on the recession limb of the runoff hydrograph was marked by the recorded clock time at which the second derivative of the discharge ordinate after the peak discharge changes signs from negative to positive [26]. This calculation was performed for one hundred and sixty-one (161) rainfall and runoff events retrieved from Dames and Moore Company between 1984 and 1986 [27]. The results of the T_c from all models applied were compared with the observed T_c.

Performance indicators were used to calculate the performance of the tested T_c models. These include correlation coefficient (r), mean error (ME), relative bias (PBIAS), root mean square error (RMSE), Nash–Sutcliffe efficiency (NSE), normalized Nash–Sutcliffe efficiency (NNSE), as well as the percentage of outbound data.

Correlation is employed to depict the direction of the relationship (positive and negative) and the strength or magnitude of the particular relationship. A correlation coefficient of 1 depicts a perfect positive relationship, while a correlation of −1 shows a perfect negative relationship.

ME measures the mean difference between the observed versus the estimated values, and a mean error close to zero is regarded as the best. PBIAS is an indicator of the magnitude of and the tendency for a model to underestimate (negative values) or overestimate (positive values) a variable of interest; a value of 0 is a perfect model. RMSE helps to determine model estimation error magnitude.

NSE gives the prediction capability of a model, where NSE equal to 1 indicates a perfect model, by which a model is adjudged as perfectly suitable to predict an observed value, and NSE equal to 0 means a model is not better than an observed value. A negative value means a poor predictive capability of a model, by which the model’s prediction is identified as not representative of the observed values.

NNSE is a normalized version of NSE that rescales NSE from a range between infinity and one (−∞ and 1) to a range between 0 and 1. A value of 1 indicates the perfect predictive capability of a model, while a value of 0 indicates the poorest predictive capability of a model. The percentage of data outbound the upper and lower ninety-five percent (±95%) confidence limits was calculated by estimating the percentage of data above the upper ninety-five percent confidence limit of the pairs of data (+95%) line (green line) and below the lower ninety-five percent confidence limit of the pairs of data (−95%) line (red line) from all models. The equations for calculating each of the performance indicators are as follows:

$$r=\frac{{{\displaystyle \sum }}_{i=1}^n\left(T_{c_{obs}}-{\overline{T}}_{c_{obs}}\right)\left(T_{c_{est}}-{\overline{T}}_{c_{est}}\right)}{\sqrt{{{\displaystyle \sum }}_{i=1}^n{\left(T_{c_{obs}}-{\overline{T}}_{c_{obs}}\right)}^2{{\displaystyle \sum }}_{i=1}^n{\left(T_{c_{est}}-{\overline{T}}_{c_{est}}\right)}^2}}$$

(1)

$$ME \left(h\right)=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(T_{c_{est}}-T_{c_{obs}}\right)}_i$$

(2)

$$PBIAS \left(\%\right)=\left[\frac{{{\displaystyle \sum }}_{i=1}^n{\left(T_{c_{est}}-T_{c_{obs}}\right)}_i}{{{\displaystyle \sum }}_{i=1}^n{\left(T_{c_{obs}}\right)}_i}\right]· 100$$

(3)

$$RMSE=\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(T_{c_{est}}-T_{c_{obs}}\right)}_i^2}$$

(4)

$$NSE=1-\left(\frac{{{\displaystyle \sum }}_{i=1}^n{\left(T_{c_{est}}-T_{c_{obs}}\right)}_i^2}{{{\displaystyle \sum }}_{i=1}^n{\left(T_{c_{obs}}-{\overline{T}}_{c_{obs}}\right)}_i^2}\right)$$

(5)

$$NNSE=\frac{1}{2-\left[1-\left(\frac{{{\displaystyle \sum }}_{i=1}^n{\left(T_{c_{est}}-T_{c_{obs}}\right)}_i^2}{{{\displaystyle \sum }}_{i=1}^n{\left(T_{c_{obs}}-{\overline{T}}_{c_{obs}}\right)}_i^2}\right)\right]}$$

(6)

where

n = total number of events;

Tc_est = T_c value estimated with a T_c model;

Tc_obs = T_c value observed from actual flood hydrograph;

$\overline{T}$c_est = mean of all T_c values estimated with a T_c model for all events;

$\overline{T}$c_obs = mean of T_c values observed from actual flood hydrograph for all events.

An optimization procedure was set up to derive a new model for arid regions based on the study area’s geomorphological parameters. The model equation considers common parameters such as the area, A, the basin length, L_b, and the basin slope, S_b, as defined in Equation (7). The optimization problem is formulated by the objective function given below:

$$obj\left(a,b,c,d\right)=min \left[\left(\frac{1}{n} {{\displaystyle \sum }}_{i=1}^n{\left[{\left(T_{ci}^{obs}-\frac{a{A}_B^b{L}_B^c}{S_B^d}\right)}^2\right]}_{}\right)\right]$$

(7)

where

n = number of events;

$T_{ci}^{obs}$ = the observed time of concentration of event “i”;

A_B (km²) = basin area;

L_B (km) = basin length;

S_B (m/m) = average basin slope;

a, b, c, and d = parameters to be found through the minimization of the sum of the squared difference between the observed and proposed model.

Table 3. PDFs used in the study.

Distribution Type	PDF Formula	Parameters of PDF
		µ	σ2
Gaussian	$f\left(X\right)=\frac{1}{\beta \sqrt{2\Pi }}{e}^{-\frac{1}{2{\beta }^2}{\left(X-\alpha \right)}^2} -\infty <X<\infty$	α	β2
Log-normal	$f\left(X\right)=\frac{1}{X\beta \sqrt{2\Pi }}{e}^{-\frac{1}{2{\beta }^2}{\left(LnX-\alpha \right)}^2} X>0$	α	β2
Exponential	$f\left(X\right)=\frac{1}{\alpha }{e}^{-\alpha X} \alpha >0,X>0$	$\frac{1}{\alpha }$	$\frac{1}{{\alpha }^2}$
Gamma	$f\left(X\right)=\frac{1}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}{X}^{\alpha -1}{e}^{-\left(\frac{X}{\beta }\right)} \alpha ,\beta >0,X>0$	$\frac{\alpha }{\beta }$	$\frac{\alpha }{{\beta }^2}$
Beta	$f\left(X\right)=\frac{\Gamma \left(\alpha \right)+\Gamma \left(\beta \right)}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{X}^{\alpha -1}{\left(1-X\right)}^{\beta -1} \alpha ,\beta >0,X>0$	$\frac{\alpha }{\alpha +\beta }$	αβ/(α + β)2 (α + β + 1)
Gumbel	$f\left(X\right)=\frac{1}{\beta }{e}^{\left[-\left(\frac{X-\alpha }{\beta }\right)e^{-\left(\frac{X-\alpha }{\beta }\right)}\right]} -\infty <X<\infty$	α + 0.5772β	$\frac{{\Pi }^2}{6} {\beta }^2$

µ = mean; σ² = variance of the distribution.

shows the probability distribution functions (PDFs) applied to fit the observed data. Hypothesis testing was performed to determine which of the theoretical probability provides a good fit for the observed data. The empirical distribution of the data were tested using the Kolmogorov–Smirnov (K–S) test [28,29] to provide the PDF that fits the data. The K–S test is given by Equation (8):

$$K–S=ma{x}_{i=1,\dots , n}\left|F\left(x_i, \mathbf{\theta }\right)-F_n\left(x_i\right)\right|$$

(8)

where F(x_i, θ) is the theoretical distribution function with parameters θ, and F_n(x_i) is the empirical cumulative distribution function (CDF).

If the K–S value obtained from Equation (8) exceeds the tabular value at a specified significance level, α, herein taken as 5%, the decision is to reject the null hypothesis. In the situation whereby at least two types of distribution fit the data, however, the best distribution can be determined using the Akaike Information Criterion (AIC). As proposed by Laio et al. [30], AIC is calculated in Equation (9):

$$AIC=-2{\displaystyle \sum }_{i=1}^n\ln f\left(x_i\right)+2N$$

(9)

where f(x_i) is the probability density function of the theoretical model evaluated at x_i and N is the number of parameters of the theoretical model. The minimum value of the AIC states the best distribution to represent the data [31].

3. Results and Discussion

3.1. Evaluation of Model Performance

In this study, 20 empirical equations for estimating T_c that were derived from experiments and studies in a wide range of basin areas and climatic conditions were reviewed from the literature.

Table 4. Performance indicators and indices.

Tc Model	Mean Model Tc (h)	Mean Observed Tc (h)	r	ME (h)	PBIAS (%)	RMSE (h)	NSE	NNSE	Data Outbound ±95% Confidence Limits (%)
Kirpich (1940) [1]	7.64	7.35	0.57	0.29	4	4.83	0.30	0.59	1.9
SCS Lag (1972) [20]	22.94	7.35	0.36	15.59	212	21.54	−12.92	0.07	2.5
FAA (1970) [21]	11.35	7.35	0.57	4.00	54	6.40	−0.23	0.45	0
Carter (1961) [21]	2.62	7.35	0.58	−4.73	−64	7.06	−0.50	0.40	46
Kinematic Wave (1964) [21]	19.14	7.35	0.49	11.79	160	15.68	−6.38	0.12	1.9
Jung (2005) [12]	7.25	7.35	0.58	−0.10	−1	4.72	0.33	0.60	3.1
Espey (1966) [22]	60.91	7.35	0.57	53.56	729	55.36	−91.00	0.01	0.6
Kraven I (1999) [12]	3.95	7.35	0.56	−3.40	−46	5.89	−0.04	0.49	13
USGS (2000) [12]	128.12	7.35	0.58	120.77	1643	135.5	−550.12	0.00	1.9
Ven Te Chow (1962) [23]	8.11	7.35	0.57	0.76	10	4.81	0.30	0.59	1.9
USACE (1954) [4]	9.92	7.35	0.58	2.58	35	5.51	0.09	0.52	0
Albishi et al. (2017) [19]	1.66	7.35	0.52	−5.69	−77	8.07	−0.96	0.34	67.7
Morgali and Linsley (1965) [22]	28.25	7.35	0.50	20.90	284	26.59	−20.22	0.04	2.5
Izzard and Hicks (1946) [8]	9.48	7.35	−0.15	2.13	29	17.15	−7.83	0.10	3.1
McCuen et al. (1984) [22]	31.17	7.35	0.39	23.82	324	34.10	−33.90	0.03	2.5
Johnstone (1949) [5]	9.73	7.35	0.57	2.38	32	5.30	0.16	0.54	0
Dooge (1973) [24]	11.84	7.35	0.60	4.49	61	6.55	−0.29	0.44	0
Giandotti (1934) [8]	9.01	7.35	0.58	1.66	23	5.07	0.23	0.56	0.6
Haktanir and Sezen (1990) [4]	26.83	7.35	0.57	19.48	265	21.56	−12.96	0.07	0
Sheridan (1994) [17]	111.60	7.35	0.57	104.25	1419	114.61	−393.31	0.00	1.2
Proposed Tc Model	7.42	7.35	0.62	0.07	0.9	4.53	0.38	0.62	3.1

and Figure 4 show the average T_c estimates of various models and their performance indicators against the observed T_c in terms of correlation coefficient (r), mean error (ME), root mean square error (RMSE), Nash–Sutcliffe efficiency (NSE), normalized Nash–Sutcliffe efficiency (NNSE), and relative bias (PBIAS).

The average value of T_c estimated from the various models shows that the lowest value of T_c at 1.66 h was obtained by Albishi et al. (2017) [19], and the highest T_c was obtained by USGS, estimated at 128.12 h. These results summarized the T_c model estimate range for arid regions to between 1.66 h and 128.12 h.

A correlation was performed between the twenty equations with observed T_c values. The results show a range of correlation coefficients between −0.15 obtained by Izzard and Hicks and 0.60 by Dooge, indicating the highest strong positive correlation between the Dooge model estimate and observed T_c in arid regions. The correlation between observed T_c and Carter, Jung, USGS, USACE, and Giandotti models is 0.58, which is also close to the Dooge model. The plot between the observed and each estimated T_c are shown in Figure 4. Table 4 summarizes the performance of the different models using various model evaluation criteria (Equations (1)–(6)).

The predictive efficiency of the equations varies. NSE values ranged between −550.12 by USGS and 0.33 by Jung. Jung’s model, therefore, shows the best predictive efficiency, and USGS shows the worst predictive efficiency. This also means the error variance of the Jung model is less than that of the observed T_c. Similarly, the Jung model shows the best NNSE value at 0.60, and USGS and Sheridan obtained the worst NNSE values at 0.00. Kirpich and Ven Te Chow also performed next to Jung, each with an NSE value of 0.3 and an NNSE value of 0.59. Jung’s model performed the best regarding the predictive efficiency of T_c models in arid environments.

Table 4 shows that ME calculated for all the models ranges between −5.69 h by Albishi et al. (2017) [19] and 120.77 h by USGS. Jung obtained the least ME at −0.10 h. RMSE values range between 4.72 h, estimated by the Jung model, and 135.50 h, estimated by the USGS model. The above results support the Jung model since it produces the least RMSE when comparing estimated T_c with observed T_c in arid regions. The relative bias indicator shows the highest underestimation by the Albishi et al. model at −77%, the least underestimation by the Jung model at −1%, the highest overestimation by USGS at 1643%, and the least overestimation by Kirpich at 4%. It is also observed that 80% of the models overestimated T_c while the remaining 20% underestimated T_c in arid regions. This shows that the Jung model, as well as the Kirpich model, produced the least bias in the estimation of T_c in arid regions, while the Jung model had the least bias overall.

Table 4 also shows the percentage of data outbound the ±95% confidence limits. The highest outbound data is found in the Albishi et al. (2017) and Carter (1961), models while the lowest is found in the FAA, USACE, Johnstone, Dooge, and Haktanir and Sezen models. This demonstrates that the latter models perform better by showing no or fewer data outside the confidence limits, while the former experienced more data outside the defined confidence limits.

3.2. Development of a New Model for the Saudi Arid Environment

The new model for the Saudi arid environment is developed based on the geomorphological parameters of the study area and following the optimization procedure explained in the methodology. The following Equation (10) is the result of the optimization:

$$T_c=\frac{0.08A_B{}^{0.38}{L}_B^{0.33}}{S_B^{0.34}}$$

(10)

where

T_c (h) = time of concentration;

A_B (km²) = basin area;

L_B (km) = basin length;

S_B (m/m) = average basin slope.

Figure 5 (top left) shows the scatter plot between the proposed T_c model and the observed T_c. Furthermore, the figure shows the scatter plots and linear relationships between the geomorphological parameters (as shown in Equation (10)) used in deriving the proposed T_c model for arid regions. The basin length and area vary positively with the proposed T_c model. On the other hand, there is a negative relationship between the basin slope and the proposed T_c model.

Geomorphological data are widely available in many arid regions from either topographic maps or digital elevation models. Therefore, geomorphological data are utilized to develop such a model for potential applications in ungauged basins in arid regions.

The performance indicators of the newly developed T_c model were evaluated. The T_c value obtained with the new model is 7.42 h. The correlation coefficient was 0.62. The mean error was 0.07. Relative bias was 0.9%. The RMSE was 4.53 h. Nash–Sutcliffe efficiency was calculated as 0.38, while the normalized Nash–Sutcliffe efficiency was calculated as 0.62. About 3.1% of the data fall outside the ±95% confidence limits of the pairs of data. The correlation coefficient, mean error, relative bias, root mean square error, Nash–Sutcliffe efficiency, and normalized Nash–Sutcliffe efficiency values obtained with the new model show that the new model performs better than the earlier T_c models tested based on the same indicators. Thus, the proposed model is recommended for T_c estimation of watersheds in arid environments.

3.3. Probability Distribution and Hypothesis Testing

Figure 6 presents the frequency histogram (6a) and the probability distribution fitting (6b) of the observed T_c data. These distributions, namely Gaussian, log-normal, exponential, Gamma, Beta, and Gumbel distributions, were applied to fit the data. Hypothesis testing was applied to determine which distributions fit the data well.

Accordingly, the K–S test at a 5% significant level shows that the log-normal, Gamma, and Beta distributions fit the data well, with K–S values of 0.053, 0.052, and 0.095, respectively. Thus, the null hypothesis is accepted for these three distributions. To further determine which of the three fitting distributions is the best distribution, the AIC criterion was applied. AIC values for log-normal, Gamma, and Beta distributions were 449.5, 446.6, and 457.9, respectively. The log-normal distribution, with the lowest AIC value, produced the best distribution to fit the T_c data. This result is useful for quantifying the uncertainty of T_c and, consequently, the uncertainty of peak flow.

4. Conclusions

In evaluating the application of the existing T_c models and their performance in arid regions, the model performance indicators used in this study show that different T_c models perform better regarding some indicators than the others. The following conclusions are derived from our findings:

• The Dooge model shows the highest correlation with observed T_c at an r equal to 0.60, while the Izzard and Hicks model shows the least correlation at an r equal to −0.15.
• With regard to the predictive capability, the Jung model shows the best predictive efficiency, with Nash–Sutcliffe efficiency and normalized Nash–Sutcliffe efficiency values of 0.33 and 0.60, respectively, while USGS shows the poorest predictive efficiency, with Nash–Sutcliffe efficiency and normalized Nash–Sutcliffe efficiency values of −550.12 and 0.00, respectively.
• Based on the mean error and root mean square error, the Jung model produced the least mean error of −0.10 h, while USGS resulted in the largest mean error of 120.77 h. Similarly, the Jung model produced the least root mean square error of 4.72 h, while USGS produced the largest root mean square error of 1643 h.
• According to the relative bias, the highest underestimation is observed with the Albishi et al. (2017) [19] model at −77%, the least underestimation with the Jung model at −1%, the highest overestimation with USGS at 1643%, and the least overestimation with Kirpich at 4%.
• It is observed that 80% of all the models evaluated overestimated observed T_c, while the remaining 20% underestimated observed T_c in arid regions.
• The new T_c model developed from data in arid environments performed better than the models evaluated, with a correlation coefficient of 0.62, mean error of 0.07 h, root mean square error of 4.53 h, relative bias of 0.9%, as well as Nash–Sutcliffe efficiency of 0.38 and normalized Nash–Sutcliffe efficiency of 0.62. This proposed model is recommended to be used in flood studies in the Saudi arid environment.
• Hypothesis testing revealed that log-normal, Gamma, and Beta distributions are a good fit for the T_c data in arid regions at a 5% significance level.
• The AIC test, which was applied to demonstrate the best probability distribution, shows that log-normal provides the best fit for the observed T_c data at a 5% significance level.