Review of methods used to estimate catchment response time for the purpose of peak discharge estimation

Figures & data

Fig. 1 Location of the pilot study area (C5 secondary drainage region).

Table 1 Overland flow distances associated with different slope classes (DAWS 1986).

Slope class, SO (%)	Distance, LO (m)
0–3	110
3.1–5	95
5.1–10	80
10.1–15	65
15.1–20	50
20.1–25	35
25.1–30	20

Fig. 2 Schematic diagram illustrative of the different time parameter definitions and relationships (after Heggen 2003, McCuen 2009).

Table 2 Correction factors (τ) for T_C (Van der Spuy and Rademeyer 2010).

Area, A (km^2)	Correction factor, τ
<1	2
1–100	2−0.5 logA
100–5 000	1
5 000–100 000	2.42−0.385 logA
>100 000	0.5

Fig. 3 Conceptual travel time from the centroid of each sub-area to the catchment outlet (USDA NRCS 2010).

Table 3 Generalized regional storage coefficients (HRU 1972).

Veld region	Veld-type description	CT1
1	Coastal tropical forest	0.99
2	Schlerophyllous bush	0.62
3	Mountain sourveld	0.35
4	Grassland of interior plateau	0.32
5	Highland sourveld and Dohne sourveld	0.21
5A	Zone 5, soils weakly developed	0.53
6	Karoo	0.19
7	False Karoo	0.19
8	Bushveld	0.19
9	Tall sourveld	0.13

Table 4 General catchment information. P₂: 2-year return period 24-h rainfall depth (mm); P_D: unit hydrograph critical storm duration (h); A: area (km²); A_C: circle-area (km²) with perimeter = catchment perimeter; P: perimeter (km); W: width (km); L_C: centroid distance (km); L_H: hydraulic length of catchment (km); L_M: max. length parallel to principal drainage line (km); L_S: max. straight-line catchment length (km); S: average catchment slope (m m^-1); F_S1: shape parameter; R_C1: circularity ratio; R_E: elongation ratio; i_p: imperviousness/urbanization factor (%); CN: weighted runoff curve number; C: weighted rational runoff coefficient; C_SDF: regional SDF runoff coefficient; C_T1: HRU regional storage coefficient; C_T2: Snyder’s storage coefficient; C_T3: USACE storage coefficient ; C_T4: Bell-Kar storage coefficient; L_CH: length of channel flow path (km); S_CH: average slope of channel flow path (m m^-1); ϕ_CH: channel conveyance factor; τ: USBR channel flow correction factor.

Catchment descriptors	C5R001	C5R002	C5R003	C5R004	C5R005	C5H003	C5H012	C5H015	C5H016	C5H018	C5H022	C5H054
Climatological variables
P2 (mm)	50	48	54	54	54	54	48	54	50	52	54	54
PD (h)	10	17	7	16	2	7	8	12	40	34	1.5	7
Catchment geomorphology
A (km^2)	922	10 260	937	6 331	116	1 650	2 366	6 009	33 277	17 360	38	688
AC (km^2)	2063	22 269	1 743	13 377	168	2 903	4 210	10 029	77 208	50 930	134	1 696
P (km)	161	529	148	410	46	191	230	355	985	800	41	146
W (km)	17	98	23	66	10	32	47	66	125	64	11	12
LC (km)	53	97	31	113	8	41	48	101	237	233	4	33
LH (km)	86	202	54	187	16	71	87	167	431	375	8	68
LM (km)	55	136	42	141	14	54	60	125	301	272	7	55
LS (km)	49	132	43	118	14	54	59	118	250	225	7	51
S (m m^-1)	0.03054	0.04369	0.05044	0.04186	0.05501	0.05044	0.04771	0.04186	0.03598	0.03211	0.05501	0.03659
FS1	2.6	1.7	2.0	2.2	1.7	1.8	1.5	2.3	1.9	2.9	1.3	3.8
RC1	1.5	1.5	1.4	1.5	1.2	1.3	1.3	1.3	1.5	1.7	1.9	1.6
RE	0.6	0.8	0.8	0.6	0.9	0.8	0.9	0.7	0.7	0.5	1.0	0.5
Catchment variables
ip (%)	5	8	5	5	8	5	10	5	5	5	8	5
CN	78	77.6	76.3	74.4	76.2	76.3	78.3	74.4	69.8	69.8	76.2	77.6
C (T = 2-year)	0.368	0.365	0.358	0.319	0.491	0.358	0.417	0.319	0.283	0.283	0.491	0.283
CSDF (T = 100-year)	0.600	0.600	0.600	0.600	0.600	0.600	0.600	0.600	0.600	0.600	0.600	0.600
CT1	0.268	0.221	0.320	0.317	0.320	0.320	0.194	0.317	0.246	0.246	0.320	0.291
CT2	1.350	1.350	1.500	1.600	1.500	1.500	1.350	1.600	1.600	1.600	1.500	1.500
CT3	0.249	0.268	0.278	0.266	0.327	0.278	0.273	0.266	0.254	0.254	0.327	0.259
CT4	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
Channel geomorphology
LCH (km)	86	202	54	187	16	71	87	167	431	375	8	68
SCH (m m^-1)	0.00229	0.00133	0.00273	0.00131	0.00895	0.00232	0.00269	0.00139	0.00078	0.00079	0.01687	0.00261
ϕCH	1.3	1.3	1.3	1.3	1.3	1.3	1.3	1.3	1.3	1.3	1.3	1.3
τ	1	0.876	1	0.956	1	1	1	0.965	0.679	0.788	1.210	1

Table 5 Consistency measures for the test of overland flow T_C estimation methods compared to the “recommended method” (equation (1)).

Method	Consistency measure
	Mean recommended TC (min)	Mean estimated TC (min)	Standard bias statistic (%)	Mean error (min)	Maximum error (min)	Standard error (min)
SCS, equation (2)	5.3	3.8	−30.6	−1.5	4.7	1.8
NRCS, equation (A2)	5.3	8.4	32.7	3.1	−17.6	0.5
Miller, equation (A3)	5.3	2.4	−57.3	−2.9	−6.0	1.1
FAA, equation (A4)	5.3	9.7	97.4	4.4	14.0	1.7
Espey-Winslow, equation (A6)	5.3	31.1	469.2	25.8	−81.5	1.8

Table 6 Comparison of maximum overland flow length criteria.

Average overland slope class (SO, m m^-1)		0.03	0.05	0.10	0.15	0.20	0.25	0.30
NSCM flow length criteria (LO, m)		110	95	80	65	50	35	20
McCuen-Spiess flow length criteria (LO, m)
Roughness parameters	0.02	264	341	482	590	682	762	835
	0.06	88	114	161	197	227	254	278
	0.09	59	76	107	131	151	169	185
	0.13	41	52	74	91	105	117	128
	0.15	35	45	64	79	91	102	111

Fig. 4 T_C estimation results.

Table 7 Consistency measures for the test of channel flow T_C estimation methods compared to the “recommended method”, equation (4).

Method	Consistency measure
	Mean recommended TC (h)	Mean estimated TC (h)	Standard bias statistic (%)	Mean error (h)	Maximum error (h)	Standard error (h)
USBR correction, equation (4a)	37.3	31.8	−4.4	−5.5	−35.7	6.4
Bransby-Williams, equation (A8)	37.3	54.9	57.8	17.6	43.5	1.4
Kirpich, equation (A9)	37.3	37.3	0.0	0.0	−0.1	0.0
Johnstone-Cross, equation (A10)	37.3	15.6	−5.0	−21.7	−71.0	3.0
Sheridan, equation (A13)	37.3	209.6	537.9	172.3	472.0	1.8
Colorado-Sabol, equation (A15b)	37.3	124.0	315.4	86.7	205.4	3.5

Fig. 5 T_L estimation results.

Table 8 Consistency measures for the test of T_L estimation methods compared to the “recommended method”, equation (7).

Method	Consistency measure
	Mean recommended TL (h)	Mean estimated TL (h)	Standard bias statistic (%)	Mean error (h)	Maximum error (h)	Standard error (h)
SCS, equation (8)	23.9	25.6	–0.5	1.7	17.8	5.0
Snyder, equation (A16)	23.9	23.1	12.1	–0.8	–6.0	2.2
Taylor-Schwarz, equation (A17)	23.9	4.6	–78.3	–19.3	–46.6	4.2
USACE, equation (A18)	23.9	30.6	25.4	6.8	22.5	3.7
Bell-Kar, equation (A21)	23.9	29.1	5.2	5.2	30.3	4.7
Putnam, equation (A23)	23.9	23.7	4.4	–0.2	–5.2	2.3
Rao-Delleur, equation (A24c)	23.9	41.1	56.1	17.2	72.4	6.1
NERC, equation (A25)	23.9	23.8	15.0	–0.1	–7.0	4.0
Mimikou, equation (A27)	23.9	13.3	–38.3	–10.6	–28.1	6.1
Watt-Chow, equation (A28)	23.9	51.2	82.7	27.4	98.8	4.8
Haktanir-Sezen, equation (A29)	23.9	16.9	–29.8	–7.0	–15.9	4.4
McEnroe-Zhao, equation (A31a)	23.9	20.7	–24.8	–3.2	–10.5	4.2
Simas-Hawkins, equation (A32)	23.9	10.2	–40.0	–13.7	–37.4	7.3
Folmar-Miller, equation (A33)	23.9	24.9	20.2	1.0	8.2	4.3

Fig. 6 T_P estimation results.

Table 9 Consistency measures for the test of T_P estimation methods compared to the “recommended method” (equation (10)).

Method	Consistency measure
	Mean recommended TP (h)	Mean estimated TP (h)	Standard bias statistic (%)	Mean error (h)	Maximum error (h)	Standard error (h)
Espey-Morgan, equation (A34)	32.3	5.4	–75.7	–26.9	–84.5	9.8
Williams-Hann, equation (A35)	32.3	143.5	295.9	111.1	365.6	4.3
Espey-Altman, equation (A37)	32.3	5.2	–74.9	–27.1	–85.4	10.5
James-Winsor, equation (A38)	32.3	42.8	9.1	10.4	52.2	11.5

Table A1 Summary of T_C estimation methods used internationally.

Approach (Flow regime)	Method	Mathematical relationship	Comments
Hydraulic (Sheet overland flow)	Kinematic wave method (Morgali and Linsley 1965)	TC5 = (A1)	• This method is based on a combination of Manning’s equation and a kinematic wave approximation
		where: TC5 = time of concentration (min), i = critical rainfall intensity of duration TC (mm h^-1), LO = length of overland flow path (m), n = Manning’s roughness parameter for sheet flow (between 0.01 and 0.8), and SO = average overland slope (m m^-1).	• Assumes that the hydraulic radius of the flow path is equal to the product of travel time and rainfall intensity
			• The iterative use of this method is limited to paved areas
Hydraulic (Sheet overland flow)	NRCS kinematic wave method (Welle and Woodward 1986)	TC6 = (A2)	• This method was originally developed to avoid the iterative use of the original Kinematic wave method (equation (A1))
		where: TC6 = time of concentration (min), LO = length of overland flow path (m), n = Manning’s roughness parameter for sheet flow, P2 = 2-year return period 24-h design rainfall depth (mm), and SO = average overland slope (m m^-1).	• It is based on a power–law relationship between design rainfall intensity and duration
Empirical/Semi-analytical(Sheet overland flow)	Miller’s method (Miller 1951, ADNRW 2007)	TC7 = (A3)	• This method is based on a nomograph for shallow sheet overland flow as published by the Institution of Engineers, Australia (IEA 1977)
		where: TC7 = time of concentration (min), LO = length of overland flow path (m), n = Manning’s roughness parameter for overland flow, and SO = average overland slope (m m^-1).
Empirical/Semi-analytical (Mixed sheet/concentrated overland flow)	Federal Aviation Agency (FAA) method (FAA 1970, McCuen et al. 1984)	TC8 = (A4)	• Commonly used in urban overland flow estimations, since the Rational method’s runoff coefficient (C) is included
		where: TC8 = time of concentration (min), LO = length of overland flow path (m), C = Rational method runoff coefficient, and SO = average overland slope (m m^-1).
Empirical/Semi-analytical (Concentrated overland/ channel flow)	Eagleson’s method (Eagleson 1962, McCuen et al. 1984)	TC9 = (A5)	• This method provides an estimation of TL, i.e. the time between the centroid of effective rainfall and the peak flow rate of a direct runoff hydrograph
		where: TC9 = time of concentration (min), LO,CH = length of flow path, either overland or channel flow (m), n = Manning’s roughness parameter, R = hydraulic radius which equals the flow depth (m), and SO,CH = average overland or channel slope (m m^-1).	• A conversion factor of 1.67 was introduced to estimate TC in catchment areas smaller than ±20 km^2
			• The variables that were used during the development and calibration were based on the characteristics of a sewer system
Empirical/Semi-analytical(Concentrated overland/ channel flow)	Espey-Winslow method (Espey and Winslow 1968)	TC10 = (A6)	• According to Schulz and Lopez (1974, cited by Fang et al. 2005), this method was developed by Espey and Winslow (1968) for 17 catchments in Houston, USA
		where: TC10 = time of concentration (min), ip = imperviousness factor (%), LO,CH = length of flow path, either overland or channel flow (m), ϕ = conveyance factor, and SO,CH = average overland or channel slope (m m^-1).	• The catchment areas varied between 2.6 and 90.7 km^2 while 35% of the catchments were predominantly rural
			• Imperviousness (ip) and conveyance (ϕ) factors were introduced
			• The imperviousness factor (ip) represents overland flow retardance, while the conveyance factor (ϕ) measures subjectively the hydraulic efficiency of a watercourse/channel, taking both the condition of channel vegetation and degree of channel improvement into consideration
			• Typical ϕ values vary between 0.8 (concrete lined channels) to 1.3 (natural channels) (Heggen 2003)
Empirical/Semi-analytical (Concentrated overland/ channel flow)	Kadoya-Fukushima method (Kadoya and Fukushima 1979, Su 1995)	TC11 = (A7)	• This method is based on the kinematic wave theory and geomorphological characteristics of the slope-channel network in catchment areas between 0.5 and 143 km^2
		where TC11 = time of concentration (h), A = catchment area (km^2), CT = catchment storage coefficient (typically between 190 and 290), and iE = effective rainfall intensity (mm h^-1).	• It is physically-based with the catchment area and effective rainfall intensity incorporated to estimate TC
Empirical (Channel flow)	Bransby-Williams method(Williams 1922, Li and Chibber 2008)	TC12 = (A8)	• The use of this method is limited to rural catchment areas less than ±130 km^2 (Fang et al. 2005, Li and Chibber 2008)
		where: TC12 = time of concentration (h), A = catchment area (km^2), LCH = length of main watercourse/channel (km), and SCH = average main watercourse slope (m m^-1).	• The Australian Department of Natural Resources and Water (ADNRW 2007) highlighted that the initial overland flow travel time is already incorporated; therefore an overland flow or standard inlet time should not be added
Empirical (Channel flow)	Kirpich method (Kirpich 1940)	TC13 = (A9)	• Kirpich (1940) calibrated two empirical equations to estimate TC in small, agricultural catchments in Pennsylvania and Tennessee, USA
		where: TC13 = time of concentration (h), LCH = length of longest watercourse (km), and SCH = average main watercourse slope (m m^-1).	• The catchment areas ranged from 0.4 to 45.3 ha, with average catchment slopes between 3% and 10%
			• The estimated TC values should be multiplied by 0.4 (overland flow) and 0.2 (channel flow) respectively where the flow paths in a catchment are lined with concrete/asphalt
			• Although this method is proposed to estimate TC in main watercourses as channel flow, McCuen et al. (1984) highlighted that the coefficients used probably reflect significant portions of overland flow travel time, especially if the relatively small catchment areas used during the calibration are taken into consideration
			• The empirically-based coefficients represent regional effects, therefore the use thereof outside the calibration catchments must be limited
			• McCuen et al. (1984) also showed that this method had a tendency to underestimate TC values in 75% of the urbanized catchment areas smaller than 8 km^2, while in 25% of the catchments (8 km^2 < A ≤ 16) with substantial channel flow, it had the smallest bias
			• Pilgrim and Cordery (1993) also confirmed that the latter was also evident from studies conducted in Australia
Empirical (Channel flow)	Johnstone-Cross method (Johnstone and Cross 1949, Fang et al.  2008)	TC14 = (A10)	• This method was developed to estimate TC in the Scioto and Sandusky River catchments (Ohio Basin)
		where: TC14 = time of concentration (h), LCH = length of longest watercourse (km), and SCH = average main watercourse slope (m m^-1).	• The catchment areas ranged from 65 to 4206 km^2
			• It is primarily a function of the main watercourse length and average main watercourse slope
Empirical/Semi-analytical (Channel flow)	McCuen-Wong method (McCuen et al. 1984)	TC15 = (A11a)	• Two empirical equations were developed to estimate TC in 48 urban catchment areas less than 16 km^2
		TC15 = (A11b)	• Stepwise multiple regression analyses were used to select the predictor variables
		where: TC15 = time of concentration (h), i2 = 2-year critical rainfall intensity of duration TC (mm h^-1), LCH = length of longest watercourse (km), ϕ = conveyance factor, and SCH = average main watercourse slope (m m^-1).	• There was not a substantial difference in the goodness-of-fit (GOF) statistics of these equations
			• Equation (A11a) is preferred to estimate TC, except when the hydraulic characteristics of a main watercourse/channel differ substantially from reach to reach
			• In such cases, the conveyance factor (ϕ) should be estimated and used as input to equation (A11b)
Empirical/Semi-analytical (Channel flow)	Papadakis-Kazan method (Papadakis and Kazan 1987, USDA NRCS 2010)	TC16 = (A12)	• Data from 84 rural catchment areas smaller than ±12.4 km^2, as well as experimental data from the United States Army Corps of Engineers (USACE), Colorado State University and the University of Illinois, USA were analysed
		where: TC16 = time of concentration (h), i = critical rainfall intensity of duration TC (mm h^-1), LCH = length of longest watercourse (km), n = Manning’s roughness parameter, and SCH = average main watercourse slope (m m^-1).	• Stepwise multiple regression analyses were used to select the predictor variables from a total of 375 data points to estimate TC
Empirical (Channel flow)	Sheridan’s method (Sheridan 1994, USDA NRCS 2010)	TC17 = (A13)	• Sheridan (1994) performed a study on nine catchment areas of between 2.6 and 334.4 km^2 in Georgia and Florida, USA
		where: TC17 = time of concentration (h), and LCH = length of longest watercourse (km).	• Multiple regression analyses were performed using geomorphological catchment parameters to estimate TC
			• The main watercourse/channel length proved to be the overwhelming characteristic that correlated with TC
			• On average, the coefficient of determination (r^2) equalled 0.96
Empirical (Channel flow)	Thomas-Monde method (Thomas et al. 2000)	TC18 = (A14)	• Thomas et al. (2000) estimated average TC values for 78 rural and urban catchment areas of 4 to 1280 km^2 in three distinctive climatic regions (Appalachian Plateau, Coastal Plain and Piedmont) of Maryland, USA
		where: TC18 = time of concentration (h), AP = 1 if the catchment is in the Appalachian Plateau, otherwise 0, CP = 1 if the catchment is in the Coastal Plain, otherwise 0, FR = forest areas (%), ip = imperviousness factor (%), LCH = length of longest watercourse (km), SCH = average main watercourse slope (m km^-1), and WB = waterbodies (lakes and ponds) (%).	• It was developed by using stepwise multiple regression analyses, i.e. transforming TC and the catchment characteristics (area, main watercourse length and average slope, %-distribution of land use and vegetation, water bodies and impervious areas) to logarithms and fitting a linear regression model to the transformed data
			• This method was compared with the catchment lag times observed by the USGS and estimated with the SCS and Kirpich methods. It overestimated the USGS values by 5%, while the two other methods were consistently lower
Empirical (Channel flow)	Colorado-Sabol method (Sabol 2008)	Rocky Mountain/Great Plains/Colorado Plateau: TC19 = (A15a)	• Sabol (2008) proposed three different empirical TC methods to be used in drainage regions with distinctive geomorphological and land-use characteristics in the State of Colorado, USA
		Rural: TC19 = (A15b)	• Stepwise multiple regression analyses were used to select the predictor variables based on the catchment geomorphology and developmental variables
		Urban: TC19 = (A15c)	• Thereafter, the catchments were grouped as: (i) Rocky Mountains, Great Plains and Colorado Plateau, (ii) rural, and (iii) urban
		where: TC = time of concentration (h), A = catchment area (km^2), ip = imperviousness factor (%), LC = centroid distance (km), LCH = length of longest watercourse (km), and SCH = average main watercourse slope (m m^-1).

Table A2 Summary of T_L estimation methods used internationally.

Approach	Method	Mathematical relationship	Comments
Empirical/Semi-analytical	Snyder’s method (Snyder 1938)	(A16)	• Snyder (1938, cited by Viessman et al. 1989; Pilgrim and Cordery 1993, McCuen 2005) developed a SUH method derived from the relationships between standard unit hydrographs and geomorphological catchment descriptors
		where: TL4 = lag time (h), CT2 = catchment storage coefficient (typically between 1.35 and 1.65), LC = centroid distance (km), and LH = hydraulic length (km).	• The catchment areas evaluated varied between 25  and 25 000 km^2 and are located in the Appalachian Highlands, USA
			• The catchment storage coefficients (CT) were established regionally and include the effects of slope and storage
			• TL is defined as the time between the centroid of effective rainfall and the time of peak discharge
Empirical	Taylor-Schwarz method (Taylor and Schwarz 1952)	(A17) where: TL5 = lag time (h), LC = centroid distance (km), LH = hydraulic length of catchment (km), and S = average catchment slope (%).	• Taylor and Schwarz (1952, cited by Chow 1964) proved that the catchment storage coefficient (CT) as used in Snyder’s method (1938) is primarily influenced by the average catchment slope • Subsequently, a revised version of Snyder’s method was proposed
			• A total of 20 catchments in the North and Middle Atlantic States, USA were evaluated• According to Linsley et al. (1988), the United States Army Corps of Engineers (USACE) developed a general expression for TL in 1958 based on the Snyder (1938) and Taylor-Schwarz (1952) methods• In this method, the average catchment slope (S, %) was replaced with the average main watercourse slope (SCH, m m^-1)• Typical CT values proposed were: 0.24 (valleys; 0–10% slopes), 0.50 (foothills; 10–30% slopes) and 0.83 (mountains; >30% slopes)
Empirical/Semi-analytical	USACE method (Linsley et al. 1988)	TL6 = (A18) where: TL6 = lag time (h), CT3 = catchment storage coefficient, LC = centroid distance (km), LH = hydraulic length of catchment (km), and SCH = average main watercourse slope (m m^-1).
Empirical	Hickok-Keppel method (Hickok et al. 1959)	TL7 = (A19)	• Rainfall and runoff records for 14 catchment areas of between 27 and 1952 ha in Arizona, New Mexico and Colorado, USA were analysed
		where: TL7 = lag time (h), D = drainage density of entire catchment (km^-1), LCSA = centroid distance of source area (km), SSA = average slope of source area (%), and WSA = average width of source area (km).	• It was established that the runoff represented by unit hydrographs is related to the spatial distribution of effective rainfall and subsequently controlled the runoff source area by using possible sub-divided catchments
			• It was also found that the slope of the runoff source areas could be useful in TL estimations, while a runoff source area was defined as that portion of the catchment with the highest average catchment slope
			• The TL estimates are significant in relating the influences of catchment variables to the hydrograph shape, with the average catchment slope more correlated than the average main watercourse slope
			• The drainage density parameter reflects the proportion of channel versus overland flow and provided thus a measure of the hydraulic efficiency of a catchment
Empirical	Kennedy-Watt method (Kennedy and Watt 1967, Heggen 2003)	TL8 = (A20)	• This method takes into consideration the distribution and extent of waterbodies (lakes, marshes and ponds) in a catchment • Multiple regression analyses were used to establish the predictor variables from the catchment geomorphology and distribution of waterbodies in the upper two-thirds of the catchments
		where: TL8 = lag time (h), A = catchment area (km^2), AW = area of waterbodies in the upper two-thirds of the catchment (km^2), LH = hydraulic length of catchment (km), and SCH = average main watercourse slope (m m^-1).
Empirical/Semi-analytical	Bell-Kar method (Bell and Kar 1969)	TL9 = (A21)	• TL is primarily dependent on the geomorphological catchment characteristics
		where: TL9 = lag time (h), CT4 = catchment storage coefficient (typically between 1 and 3.4 × 10^-4), LH = hydraulic length of catchment (km), and SCH = average main watercourse slope (m m^-1).	• Critical TL values, which are arguably suitable representatives of the critical storm duration of design rainfall were used
			• This method is a modified version of the Kirpich method
Empirical/Semi-analytical	Askew’s method (Askew 1970)	TL10 = (A22)	• The variable temporal rainfall distributions had a little effect on TL, while TL can only be correlated with the weighted mean runoff rate in a catchment
		where: TL10 = lag time (h), A = catchment area (km^2), and QWM = weighted mean runoff rate (m^3 s^-1).	• The weighted mean runoff rate was defined as the mean ratio of the total runoff rate divided by the time of occurrence of direct runoff, weighted in proportion to the direct runoff discharge rate
			• A constant exponent was used as a fixed regression coefficient to develop a means of predicting the constant term in this method, which reflects a measure of a linear model’s estimation of TL
			• A high degree of association existed between the regression constant and the catchment area
Empirical	Putnam’s method (Putnam 1972)	TL11 = (A23)	• According to Haan et al. (1994), this method was developed by Putnam (1972) for 34 catchments in North Carolina, USA
		where: TL11 = lag time (h), ip = imperviousness factor (fraction), LCH = main watercourse length (km), and SCH = average main watercourse slope (m m^-1).	• Multiple regression analyses were used to establish the predictor variables from the catchment geomorphology and degree of urbanization
			• TL is defined as the time from the centroid of effective rainfall to the centroid of direct runoff
Empirical/Semi-analytical	Rao-Delleur method (Rao and Delleur 1974, Heggen 2003, Fang et al. 2005, ADNRW 2007)	TL12 = (A24a)	• It was established that average TL values (based on the time lapse between the centroid’s of effective rainfall and direct runoff) could not be used alone for runoff estimation, since it’s depending on various geomorphological and meteorological characteristics
		TL12 = (A24b)	• Three equations based on stepwise multiple regression analyses were developed with the predictor variables only related to catchment geomorphology and developmental variables.
		TL12 = (A24c)	• It was establish that equation (A24c), which included only the catchment area and imperviousness factor (ip), is as effective as equations (A24a and A24b), which include both the main watercourse length and average catchment slope
		TL12 = (A24d)	• An additional equation (A24d) was developed to take meteorological parameters (effective rainfall and duration) also into consideration
		where: TL12 = lag time (h), A = catchment area (km^2), DPE = duration of effective rainfall (h), ip = imperviousness factor (fraction), LCH = main watercourse length (km), PE = effective rainfall (mm), and SCH = average main watercourse slope (m m^-1).	• TL is not only a unique catchment characteristic, but varies from storm to storm
Empirical	NERC method (NERC 1975)	(A25)	• The United Kingdom Flood Studies Report (UK FSR) (NERC 1975) proposed the use of this method to estimate TL in ungauged UK catchments
		where: TL13 = lag time (h), LCH = main watercourse length (km), and SCH = average main watercourse slope (m km^-1).	• TL is primarily dependent on the geomorphological catchment characteristics, e.g. main watercourse length and average slope
Empirical/Semi-analytical	CUHP method Urban Drainage and Flood Control District (UDFCD 1984, cited by Heggen 2003)	TL14 = (A26)	• This method (Colorado Urban Hydrograph Procedure) is a modified version of Snyder’s method as used in urban catchment areas between 40 and 80 ha in the State of Colorado, USA
		where: TL14 = lag time (h), CT = aip^2+bip+c, imperviousness storage coefficients, ip = imperviousness factor (%), LC = centroid distance (km), LH = hydraulic length of catchment (km), and SCH = average main watercourse slope (m m^-1).	• This method was also commonly used to derive unit hydrographs for both urban and rural catchment areas ranging from 0.36  to 13 km^2 • In catchment areas larger than 13 km^2, it is recommended that the catchment be subdivided into sub-catchments of 13 km^2 or less
Empirical	Mimikou’s method (Mimikou 1984)	TL15 = (A27)	• This method was developed for catchment areas of between 202 and 5005 km^2 in the western and north western regions of Greece
		where: TL15 = lag time (h), and A = catchment area (km^2).	• TL and unit hydrograph peaks (QP) were estimated at the catchment outlets from unit hydrographs produced by 10 mm effective rainfall and 6-h storm durations
			• Storm durations of 6 h were used in all the catchments in order to avoid the effect of variable storm durations on the variation of TL and QP values from catchment to catchment. In other words, complex areal storms of various durations were delineated in 6-h intervals according to the well known multi-period technique described in the literature (Linsley et al. 1988)
			• It was established that TL and QP associated with specific storm durations, are increasing power functions of the catchment size
			• Mimikou (1984) also emphasised that the developed regional TL relationship is only applicable to the study area
Empirical	Watt-Chow method (Watt and Chow 1985)	TL16 = (A28)	• This method is based on geomorphological data from 44 catchment areas between 0.01 and 5840 km^2 across the USA and Canada
		where: TL16 = lag time (h), LCH = main watercourse length (km), and SCH = average main watercourse slope (m m^-1).	• The main watercourse slopes ranged between 0.00121 and 0.0978 m m^-1
Empirical	Haktanir-Sezen method (Haktanir and Sezen 1990, cited by Fang et al. 2005)	TL17 = (A29)	• SUHs based on two-parameter Gamma and three-parameter Beta distributions for 10 catchments in Anatolia were developed • Regression analyses were used to establish the relationships between TL and the main watercourse length
		where:
		TL17 = lag time (h), and
		LCH = main watercourse length (km).
Analytical	Loukas-Quick method (Loukas and Quick 1996)	TL18 = (A30) where: TL18 = lag time (h), B = catchment shape factor as a ƒ(k, LCH and regressed catchment parameters), iE = effective rainfall intensity (mm h^-1), KAvg = average saturated hydraulic conductivity of soil (mm h^-1), k = main watercourse shape factor, as a ƒ(channel side slopes and bed width), and SCH = average main watercourse slope (m m^-1).	• This method estimates TL in forested mountainous catchments where most of the flow is generated through subsurface pathways
			• The data acquired from field experiments were combined with the kinematic wave equation to describe the flow generation from steep, forested hillslopes
			• The hillslope runoff was used as input to the main watercourses, where the runoff movement in the channels was described by roughness parameters and slopes that vary from point to point along the main watercourse
			• The resulting equations were integrated to obtain this method, which relates the geomorphological characteristics, effective rainfall intensity and average saturated hydraulic conductivity of a catchment to its response time through an analytical mathematical procedure
			• This method provides reliable TL estimates, however, compared to existing empirical methods (Snyder 1938, NERC 1975, Watt-Chow 1985), it underestimated TL significantly in catchment areas ranging from 3 to 9.5 km^2 in Coastal British Columbia, Canada
Empirical	McEnroe-Zhao method (McEnroe and Zhao 2001)	TL19 = (A31a) TL19 = (A31b) where: TL19 = lag time (h), ip = imperviousness factor (fraction), LCH = main watercourse length (km), RD = road density (km^-1), and SCH = average main watercourse slope (m m^-1).	• TL was estimated utilizing geomorphological catchment characteristics
			• Individual and average TL values were estimated in gauged catchments from 85 observed rainfall and runoff events at 14 different sites in Johnson County, Kansas, USA
			• Two regression equations were developed through multiple regression analyses to estimate TL in urban and developing catchments
			• The catchment and channel geomorphology were obtained from DEMs and manipulated in an ArcGIS^TM environment
			• It was established that urbanization has a major impact on TL; in fully developed catchments, TL can be as much as 50% less than in a natural catchment
			• In small urban catchments with curb-and-gutter streets and storm sewers, the TL values can even be shorter
Empirical/Semi-analytical	Simas-Hawkins method (Simas 1996, Simas and Hawkins (2002))	TL20 = (A32) where: TL20 = lag time (h), A = catchment area (km^2), CN = runoff curve number, LH = hydraulic length of catchment (km), and S = average catchment slope (m m^-1).	• TL is defined as the time difference between the centroid of effective rainfall and direct runoff and was estimated from over 50 000 rainfall–runoff events in 168 catchment areas of between 0.1 and 1412.4 ha in the USA
			• The catchments were grouped into different geographical, catchment management practice, land- use and hydrological behaviour regions to explain the variation of TL between catchments
			• Multiple regression analyses were conducted to establish the most representative TL relationship
Empirical	Folmar-Miller method (Folmar and Miller 2008)	TL21 = (A33) where: TL21 = lag time (h), and LH = hydraulic length of catchment (km).	• Multiple regression analyses were performed on TL values obtained from 10 000 direct runoff events in 52 gauged catchment areas of between 1 and 4991 ha in eight different states throughout the USA
			• It was established that TL correlates strongly (r^2 = 0.89; N = 52) with the catchment hydraulic length (LH) and therefore only this parameter was used to develop this method
			• The inclusion of any other geomorphological catchment characteristics in the method did not improve its ability to predict TL
			• This method, as well as the NRCS methods, were used to estimate TL in all the catchments, after which the results were compared with the TL values obtained from observed hyetographs and hydrographs
			• Overall, this method and the NRCS methods underestimated the TL values by 65% and 62%, respectively

Table A3 Summary of T_P estimation methods used internationally.

Approach	Method	Mathematical relationship	Comments
Empirical	Espey-Morgan method (Espey et al. 1966, cited by Fang et al. 2005)	TP2 = (A34)	• Multiple regression analyses were used to establish TP for 11 rural and 24 urban catchments in Texas, New Mexico and Oklahoma, USA
		where: TP2 = time to peak (h), LCH = main watercourse length (km), and SCH = average main watercourse slope (m m^-1).	• This method is only applicable to the large, rural catchments used during this study
Empirical	Williams-Hann method (Williams and Hann 1973, cited by Viessman et al. 1989)	TP3 = (A35)	• This method is incorporated in the problem-oriented computer language for hydrological modelling (HYMO) to simulate surface runoff from catchments
		where: TP3 = time to peak (h), A = catchment area (km^2), LH = hydraulic length of catchment (km), SCH = average main watercourse slope (m m^-1), and W = width of catchment (km).	• Regional regression analyses were used to establish TP for 34 catchment areas between 1.3 and 65 km^2 in Texas, Oklahoma, Arkansas, Louisiana, Mississippi and Tennessee, USA
Empirical/Semi-analytical	NERC method (NERC 1975)	TP4 = (A36)	• TP was related to the climate, catchment and channel geomorphology and developmental variables by using stepwise multiple regression analyses
		where: TP4 = time to peak (h), Ci = climatic index of the flood runoff potential, ip = imperviousness factor (%), LCH = main watercourse length (km), and SCH = average main watercourse slope (m km^-1).	• The average main watercourse slope and degree of imperviousness were identified as the most important variables explaining the variance of TP
			• The main watercourse length was surprisingly less critical than the degree of imperviousness due to the significant inverse correlation of main watercourse length with average slope
			• The degree of imperviousness had a direct influence on the efficiency of drainage networks, flow velocities and the proportion of total runoff due to surface runoff
Empirical/Semi-analytical	Espey-Altman method (Espey and Altman 1978)	TP5 = (A37)	• A set of regional regression equations to represent 10-min SUHs from a series of effective rainfall events were developed
		where: TP5 = time to peak (h), ip = imperviousness factor (%), LH = hydraulic length of catchment (km), ϕ = conveyance factor, and SCH = average main watercourse slope (m m^-1).	• Forty-one catchment areas of between 4 and 3885 ha were analysed
			• This method is based on the concept of Snyder’s UHs (1938)
Empirical	James-Winsor method (James et al. 1987, cited by Fang et al. 2005)	Mild slope (<5%): TP6 = (A38a) Medium slope (5–10%): TP6 = (A38b)	• 283 rainfall events were analysed in catchment areas of between 0.7 and 62 km^2 in 13 states in the USA • The climate and geomorphology in these catchments were highly variable • Only 48 catchments (31 calibration catchments and 17 verification catchments) were used in the multiple regression analyses to relate the physical catchment characteristics to TP • Three empirical equations were developed for three distinctive slope classes: mild, medium and steep
		Steep slope (>10%): TP6 = (A38c)
		where: ip = imperviousness factor (%), LH = hydraulic length of catchment (km), ϕ = conveyance factor, and SCH = average main watercourse slope (m m^-1).
Empirical	Jena-Tiwari method (Jena and Tiwari 2006)	1-hour SUH: TP7 = (A39a)	• 1-h and 2-h SUHs were developed for two catchments (158 and 69 km^2) in India based on SUH parameters such as TP, QP and TB, which are all related to the catchment and channel geomorphology
		2-hour SUH: TP7 = (A39b)	• A correlation matrix between the SUH parameters and geomorphological parameters was generated to identify the most suitable geomorphological parameters
		where: TP7 = time to peak (h), LC = centroid distance (km), and LM = maximum catchment length parallel to the principal drainage line (km).	• The best single predictor for TP was found to be the catchment hydraulic length, followed by the main watercourse length and centroid distance
			• Regression equations were developed between the individual SUH parameters and the selected geomorphological parameters

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