The probability distribution function (PDF)-based unit hydrographs (UHs) are gaining momentum in an application for more accurate rainfall-runoff transformation. Employing seven statistical performance indices, R2, NSE, MSE, RMSE, MAE, MAPE, and SE in generalized reduced gradient nonlinear programming (GRG-NLP) optimization, 18 known and 12 adaptable PDF-based UHs were assessed against UHs derived from 18 storms in 7 basins across the United States, Turkey, and India. To this end, 27 Maple codes were proposed for UH-application requiring only peak discharge (qp), time to peak (tp), and time base (tb) for derivation. The introduced PDFs, such as Dagum, Generalized Gamma, Log-Logistic, Gumbel Type-I, and Shifted Gompertz, replicated the observed data-derived UHs more closely than did the known PDFs, like Inverse Gaussian, two-parameter gamma distribution (2-PGD), Log-Normal, Inverse-Gamma, and Nagakami. Among the three-parameter (6 nos.), two-parameter (21 nos.), and single-parameter (3 nos.) PDFs, the Dagum, Log-Logistic, and Poisson consistently outperformed their respective counterparts in replication.

  • The present study evaluates the adequacy of 30 (18 old +12 new) probability density functions (PDFs) for unit hydrograph development.

  • The novel PDFs are more promising than existing PDFs in the literature.

  • Proposed PDFs with simple formulaic structures are easily adaptable as executables using a calculator.

  • The incomplete-Gamma distribution has the highest ability for qp and tp.

  • Maple codes are provided for 27 PDFs.

PDF

Probability distribution function

UH

Unit hydrograph

qp

Peak discharge

tp

Time to peak

tb

Time base

qv

Runoff volume

W50

Width of UH measured at 50% of peak discharge ordinates

W75

Width of UH measured at 75% of peak discharge ordinates

WR50

Width of the rising limb of UH measured at 50% of peak discharge ordinate

WR75

Width of the rising limb of UH measured at 75% of peak discharge ordinate

L

Length of the longest water course

LC

Length along the main stream from the outlet to the centroid of the basin

β

Non-dimensional parameter

RSO

Overall shape-based rank

RS

Event-wise shape-based rank

RqpO

Overall qp-based rank

RtpO

Overall tp-based rank

Rqp

Event-wise qp-based rank

Rtp

Event-wise tp-based rank

MSE

Mean square error

RMSE

Root mean square error

MAE

Mean absolute error

MAPE

Mean absolute percent error

SE

Standard error absolute

RE

Relative error

R2

Coefficient of determination

NSE

Nash–Sutcliffe efficiency

RNSE

Event-wise NSE-based rank

RR2

Event-wise R2-based rank

RMSE

Event-wise MSE-based rank

RRMSE

Event-wise RMSE-based rank

RMAE

Event-wise MAE-based rank

RMAPE

Event-wise MAPE-based rank

RSE

Event-wise SE-based rank

RRE

Event-wise RE-based rank

To design and operate effectively the stormwater management facilities, treatment plants, hydraulic structures, flood control projects, and ground water development works, it is essential to accurately measure the complex, dynamic, and nonlinear rainfall-runoff transformation process. This complexity is a result of the considerable variability in rainfall patterns and physical characteristics (Tokar & Markus 2000; Ito et al. 2006; Geetha et al. 2008; Yoshitani et al. 2009; Bhadra et al. 2010; Brooks et al. 2012). However, understanding and quantifying this process remain challenging due to the difficulty in directly measuring various water cycle fluxes, especially over large regions where instrumentation may be limited (McDonnell & Tanaka 2001). Consequently, our understanding of the water cycle relies heavily on mathematical models.

Analyzing the transformation process using the unit hydrograph (UH) and its variants has been a focus in hydrologic studies since Sherman (1932) introduced it. Accurately predicting UH parameters, such as peak discharge (qp), time to peak (tp), time base (tb), and runoff volume (qv), is crucial for hydrologic investigations. The UH, representing a linear system, has been extensively studied by various scholars (Nash 1957; Dooge 1959, 1973; Chow et al. 1988; Singh 1988). Various empirical UH models, ranging from single curves to a specific family of equations exist in the literature (Langbein 1940; Commons 1942; Williams 1945; Mockus 1957; Bender & Roberson 1961; SCS 1972; Singh 2000; Bhunya et al. 2003). Single curves can represent UHs for a region or specific watershed sizes. Fixing the scale (qp or tp) allows a standard non-dimensional curve to define a complete UH under unit volumetric conditions. Alternatively, manually drawing a curve conserving unit volume through parameters like qp, tp, tb, W50, W75, WR50, and WR75 can be error-prone. Flexible 2-parameter models, representing a family of curves, capture UH more accurately than do 1-parameter models. However, empirical equations represent UH shapes more precisely than do the family of curves. The adoption of probability distribution functions (PDFs) as an analytical tool to mimic UHs is inspired by their similarities in shape, unit volume, and positive ordinates. The robust mathematical foundation and enhanced hydrological understanding also advocate for the applicability of PDFs in representing UHs.

Empirical hydrographs have been proposed by many researchers, viz., first empirical UH by Edson (1951); two-parameter gamma distribution (2PGD) or Nash (1957, 1958, 1959, 1960) model, Gray (1961, 1962), Reich (1962), Wu (1963), Lienhard (1964), DeCoursey (1966), Lienhard & Meyer (1967), Lienhard & Davis (1971), Gupta & Moin (1974), Gupta et al. (1974), Sokolov et al. (1976), Haan (1977), Cruise & Contractor (1978), Croley (1980), Aron & White (1982), Singh (1981, 1982), Collins (1983), Rosso (1984), Phien & Jivajirajah (1984), Yevjevich & Obeysekera (1984), Ciepielowski (1987), James et al. (1987), Singh (1987), Koutsoyiannis & Xanthopoulos (1989), Haktanir & Sezen (1990), Meadows & Ramsey (1991a, 1991b), Haan et al. (1994), Gottschalk & Weingartner (1998), Evans et al. (2001), Singh (2000, 2004, 2005, 2006, 2007a, 2007b), Yue et al. (2002), inverse Gaussian instantaneous UH (IUH) by Bardsley (1983), Gamma- and log-normal-UH by Bhattacharjya (2004), Gamma-, Beta- Chi-square- and Weibull-synthetic UHs (SUHs) by Bhunya et al. (2003, 2004, 2005, 2007, 2008, 2009), Sahoo et al. (2006), 11 PDF-based UH along with Maple codes by Nadarajah (2007), Rai et al. (2007, 2008, 2009), Nakagami-m and seven other PDF-based UHs by Rai et al. (2010), entropy-based IUH specialized into Gamma, Lienhard and Nakagami-m PDFs by Singh (2011), Singh et al. (2014), and Gamma, Gumbel, Log-normal, Normal, Weibull, and 3-parameter Pearson PDFs transmuted UH by Ghorbani et al. (2017), etc. which laid the base of hydro-physical perception underlying the integration of PDFs in synthesizing hydrographs.

Nevertheless, the utility of PDFs extends beyond hydrologic studies (Abramowitz et al. 1988; Johnson et al. 1994, 1995; Walck 1996; Forbes et al. 2011; Krishnamoorthy 2016). The presented PDFs find application in various fields, for instance, the Dagum PDF proves valuable in actuarial sciences, survival and reliability analysis, atmospheric modeling, and meteorology (Benjamin et al. 2013; Oluyede & Rajasooriya 2013; Shahzad & Asghar 2016). The Log-Logistic PDF is relevant in survival analysis, finance, economics, networking, hydrology (for fitting long-duration extremes), and diverse human-centric domains such as biology, epidemiology, psychology, technology, and energy (Shoukri et al. 1988; Aryal 2013; Gago-Benítez et al. 2013). The Gumbel PDF finds applications in ocean wave modeling, wind and electrical engineering, earthquakes' thermodynamics, financial risk assessment, probability analysis of floods/rainfall/droughts, survival analysis, meteorology, air pollution, and geology (Kotz & Nadarajah 2000). Gamma family PDFs serve engineering, hydrology, survival and reliability analysis, metallurgy, and Bayesian statistics purposes (Agarwal & Kalla 1996; Aksoy 2000; Barriga et al. 2018). The Shifted Gompertz PDF (Bemmaor 1992) is employed for modeling and forecasting adoption timing and diffusion of innovations in markets, as well as predicting the growth and decline of services or in marketing decision models (Jiménez & Jodrá 2008; Bauckhage & Kersting 2014). Other PDFs have specific applications: binomial PDF is suitable for analyzing repeated independent trials, risk management, and quality assurance (Friedman et al. 1983; Consul 1990). The negative binomial PDF is used in failure analysis, machine reliability, RNA and DNA sequencing, meteorological studies, or as an alternative to the Poisson PDF (Bain & Wright 1982). The Poisson PDF is exercised in ET Engg., internet traffic assessment, mortality rate studies, stock markets, genetics, biotechnology, and physics (Clarke 1946). The Gompertz PDF is used to investigate age-specific mortality (Pollard & Valkovics 1992). The Rice PDF is found worthy for sonar echo- and radar signal-analysis and magnetic resonance imaging (Talukdar & Lawing 1991; Carobbi & Cati 2008). The Erlang PDF has utility in telecommunication traffic load assessment, disease analysis, cell cycle time distribution, and business economics (Gavagnin et al. 2019). Notably, with the exception of a few gamma family PDFs such as Generalized- and Log-Gamma, the introduced PDFs have not been specifically adopted in UH investigations so far. PDFs enhance flood forecasting and agricultural planning by improving risk assessment and resource allocation. However, their effectiveness relies on accurate data, appropriate assumptions, and an understanding of uncertainties. PDFs offer insights into flood probabilities for tailored planning, yet limitations include data dependence, sensitivity to assumptions, and computational complexity. Reliability may be compromised in regions with limited monitoring, but a careful consideration of these factors is crucial for successful real-world application.

The method of moments (MOM) often prioritizes the extremities over the center of a PDF, leading to errors, especially at the peak (Nash 1957). It could be addressed through time-consuming linear programming or least-squares approaches, particularly when dealing with oscillatory UHs where computational effort escalates with the number of storms (Bree 1978; Singh 2007b). Genetic algorithms do not outperform nonlinear optimization methods in enhancing PDF performance (Ghorbani et al. 2017). Therefore, the study aims to (a) evaluate the proposed PDFs for UH fitting; (b) determine optimum PDF parameters using the Excel-based generalized reduced gradient (GRG) nonlinear optimization algorithm (Lasdon et al. 1978); and (3) scrutinize the adequacy of PDFs in defining the UH and its properties. The investigation delves into the suitability of both existing and additional PDFs as UHs for more accurate rainfall-runoff transformation.

Eighteen storm-based UHs of seven basins were used to examine PDFs. The salient UH and watershed characteristics obtained from the literature are given in Table 1.

Table 1

Salient unit hydrograph and watershed characteristics

Sl. No.WatershedCatchment area A (km2)Length of main stream L (km)Length from outlet to centroid LC (km)SlopePeak discharge qp (m3/s/cm)Time to peak tp (h)Time base tb (h)
1. Myntdu-Leska,
Meghalaya, India 
350.0 51.778 27.8 4.2 km/m 11.829 30 
2. Bridge Catchment No. 253
Narmada River,
Madhya Pradesh, India 
114.22 35.42 16.9 2.5 km/m 5.46 21 
3. Karabalcik Creek,
Anatolia, Turkey,
Basin 16 
10.6 2.65 2.7% 12.84 1.75 14.4 
4. Inderesi Creek,
Anatolia, Turkey,
Basin 18 
98.0 17 7.8 2.0% 127.5 2.22 12.5 
5. Gormel, Ermenek Creek,
Anatolia, Turkey, Basin 17 
141.5 23.5 12 1.18% 54 61 
6. Kurtleravsari, Aksu,
Anatolia, Turkey,
Basin 20 
3360 123 60 0.43% 301 18 113 
Sl. No.WatershedCatchment area A (km2)Length of main stream L (km)Length from outlet to centroid LC (km)SlopePeak discharge qp (m3/s/cm)Time to peak tp (h)Time base tb (h)
1. Myntdu-Leska,
Meghalaya, India 
350.0 51.778 27.8 4.2 km/m 11.829 30 
2. Bridge Catchment No. 253
Narmada River,
Madhya Pradesh, India 
114.22 35.42 16.9 2.5 km/m 5.46 21 
3. Karabalcik Creek,
Anatolia, Turkey,
Basin 16 
10.6 2.65 2.7% 12.84 1.75 14.4 
4. Inderesi Creek,
Anatolia, Turkey,
Basin 18 
98.0 17 7.8 2.0% 127.5 2.22 12.5 
5. Gormel, Ermenek Creek,
Anatolia, Turkey, Basin 17 
141.5 23.5 12 1.18% 54 61 
6. Kurtleravsari, Aksu,
Anatolia, Turkey,
Basin 20 
3360 123 60 0.43% 301 18 113 

Indian catchments

The Myntdu-Leska River Basin (Figure 1), originating from Mih Myntdu, spans an area of 350 km2 and is situated in the Jaintia Hills district of Meghalaya (92°05′−92°20′E and 25°10′−25°30′N) in North-eastern India, positioned on the southern slope of the state bordering Bangladesh. This basin experiences a subtropical humid climate, marked by substantial rainfall from May to October, peaking at a monthly high of 715 mm. The annual rainfall in the basin ranges from 3,537 to 13,710 mm. Nestled between the central upland of the Meghalaya hills, the basin is characterized by its narrow and steep topography, with elevations ranging from 595 to 1,372 m. The hills exhibit sudden and sharp descents at various points (1,372–1,220 m within 10 km), resulting in deep gorges and river valleys. The hydrologic data of this region were previously utilized by Mani & Panigrahy (1998) and Bhunya et al. (2004, 2005, 2008, 2009). The climate here is moderate and subtropical, featuring varying vegetation from medium to sparse. Summer temperatures fluctuate between 24 and 32°C while winter temperatures range from 3 to 12°C. The region stands out for its Jhoom cultivation, a practice attributed to the lack of forest cover and steep slopes, causing immediate runoff with minimal groundwater storage.
Figure 1

Myntdu-Leska River Basin (Mani & Panigrahy 1998).

Narmada railway bridge catchment no. 253 (Figure 2, 114.22 km2) defined by the outlet (80°E; 23°5′N) of a tributary ‘Tyria’ of the Narmada River on the Gondia-Jabalpur railway line. Previously, Lohani et al. (2001) and Bhunya et al. (2004, 2005, 2007, 2008, 2009) utilized these data.
Figure 2

Narmada (Gondia-Jabalpur) Railway Bridge Catchment No. 253 (Lohani et al. 2001).

Figure 2

Narmada (Gondia-Jabalpur) Railway Bridge Catchment No. 253 (Lohani et al. 2001).

Close modal

Turkish watersheds

The study utilized storm events UHs from Karabalcik Creek (10.6 km2), Inderesi Creek (98 km2), Gormel Ermenek Creek (141.5 km2), and Kurtleravsari Aksu (3,360 km2) basins in Anatolia, Turkey (Figure 3) (Haktanir and Sezen, 1990). These data are also accessible in reports from the General Directorate of Electric Works Planning (1985), a state office assessing the hydroelectric potential of Turkish rivers. The catchment is located in the southeast of Turkey, exhibiting varying forest covers of 2, 26, 57, and 25%, respectively. These basins showed a significant correlation between L and LC. The rainfall-excess duration and lag time for these basins are 1, 1, 2, and 6 h and 1.25, 1.72, 5 and 15 h, respectively. These watersheds demonstrate diverse surface runoff draining characteristics, some exhibiting rapidly peaking UHs and others displaying delayed responses.
Figure 3

Basin of Anatolia-Turkey (a) Karabalcik Creek, Basin 16 (b) Inderesi Creek, Basin 18 (c) Gormel, Ermenek Creek, Basin 17 (d) Kurtleravsari, Aksu, Basin 20 (Haktanir & Sezen 1990).

Figure 3

Basin of Anatolia-Turkey (a) Karabalcik Creek, Basin 16 (b) Inderesi Creek, Basin 18 (c) Gormel, Ermenek Creek, Basin 17 (d) Kurtleravsari, Aksu, Basin 20 (Haktanir & Sezen 1990).

Close modal

U.S.A. catchment

Bixler Run (Rendon-Herrero 1978) is a small wash-load generating watershed (Figure 4, USGS No. 01567500, 38.85 km2, 77°24′09″ E and 40°22′15″N) situated 6.5 km from Loysville, Perry County, Pennsylvania, USA. The basin, extending from the top of Conococheague Mountain (600 m) in the northwest to a stream-gaging station (180 m) 5 km west of Loysville, features slopes up to 45% on the mountain face, with a mean basin slope of 11%. The basin's average altitude is 265 m. Over an 11-year period (1956–1967), 12 recorded storms (6 summer and 6 winter) of known duration were normalized to derive UHs for this study, emphasizing the scarcity of truly isolated events with negligible antecedent moisture effects. The mean annual precipitation between 1954 and 1969 was 1,060 mm, with streamflow ranging from 0.054 m3/s to a peak of 250 m3/s.
Figure 4

Bixler Run Basin (Reed 1976).

The basic equations for estimating 2-parameter PDF, f(x) parameters with known qp and tp are: Mode = tp, f(Mode) = qp, β = qp.tp. For 3-parameter PDF, one of the scale parameters is assumed as tp or tb. The proposed 27 Maple codes (Appendix-Table A.1) estimate unknown PDF parameters by solving the expressions (Table 2) with known qp, tp, and tb.

Table 2

Probability distribution function and it's time to peak and peak discharge

Sl.No.Distribution and referencePDF tpqp
Shifted Gompertz (Torres 2014 
 
 
Dagum (Kleiber 2008   
Rice (Talukdar & Lawing 1991   
Rayleigh (Cleveland et al. 2006   
Erlang (Mudasir & Ahmad 2017   
Incomplete-Gamma (Sade 2001   
Kumaraswamy (Nadarajah 2007   
Weibull (Nadarajah 2007   
Log-Normal (Nadarajah 2007   
10 Fréchet (Nadarajah 2007   
11 Nagakami (Yacoub et al. 1999   
12 Beta (Nadarajah 2007   
13 Inverse-Gamma (Nadarajah 2007   
14 Hybrid/Nash Modified (Bhunya et al. 2005   
15 3 Parameter 2 Sided Power (Nadarajah 2007   
16 Simple 3 Parameter (Singh 2015   
17 Binomial (Edwards 1960
 
  
18 Negative- Binominal (Fisher 1941   
19 Normal (Ghorbani et al. 2017   
20 Gumbel Type-I (Chakraborty & Chakravarty 2014
 
  
21 Poisson (Sadooghi-Alvandi, 1990   
22 2-PGD (Bhunya et al. 2003   
23 Logistic (Balakrishnan 1991   
24 Log-Logistic/Fisk (Ashkar & Mahdi 2006   
25 Log-Gamma (Consul & Jain 1971   
26 Gumbel Type II (Gumbel 1960   
27 Chi-Square (Nadarajah 2007   
28 Generalized- Gamma (Stacy & Mihram, 1965   
29 Inverse Gaussian (Nadarajah 2007   
30 Gompertz (Dey et al. 2018   
Sl.No.Distribution and referencePDF tpqp
Shifted Gompertz (Torres 2014 
 
 
Dagum (Kleiber 2008   
Rice (Talukdar & Lawing 1991   
Rayleigh (Cleveland et al. 2006   
Erlang (Mudasir & Ahmad 2017   
Incomplete-Gamma (Sade 2001   
Kumaraswamy (Nadarajah 2007   
Weibull (Nadarajah 2007   
Log-Normal (Nadarajah 2007   
10 Fréchet (Nadarajah 2007   
11 Nagakami (Yacoub et al. 1999   
12 Beta (Nadarajah 2007   
13 Inverse-Gamma (Nadarajah 2007   
14 Hybrid/Nash Modified (Bhunya et al. 2005   
15 3 Parameter 2 Sided Power (Nadarajah 2007   
16 Simple 3 Parameter (Singh 2015   
17 Binomial (Edwards 1960
 
  
18 Negative- Binominal (Fisher 1941   
19 Normal (Ghorbani et al. 2017   
20 Gumbel Type-I (Chakraborty & Chakravarty 2014
 
  
21 Poisson (Sadooghi-Alvandi, 1990   
22 2-PGD (Bhunya et al. 2003   
23 Logistic (Balakrishnan 1991   
24 Log-Logistic/Fisk (Ashkar & Mahdi 2006   
25 Log-Gamma (Consul & Jain 1971   
26 Gumbel Type II (Gumbel 1960   
27 Chi-Square (Nadarajah 2007   
28 Generalized- Gamma (Stacy & Mihram, 1965   
29 Inverse Gaussian (Nadarajah 2007   
30 Gompertz (Dey et al. 2018   

The credibility of a model depends on its ability to accurately replicate observations. In this context, 3 one-parameter, 21 two-parameter, and 6 three-parameter PDFs were assessed as UHs (Table 2). Performing a graphical/visual comparison of 30 PDF-based UHs on an event-wise basis proved challenging (Figure 5(a)). To address this, the PDF that best replicated the true UH was identified by utilizing the ‘overall shape-based rank-RSO’. This was obtained by averaging ‘event-wise shape-based ranks (RS)’ assigned to each PDF, calculated as (RNSE+RR2+RMSE+RRMSE+RMAE+RMAPE+RSE)/7 (Table 3). The PDFs were ranked based on indices (, , , , , , , and ) in ascending order (1-excellent to 30-poor). This ranking is based on the arrangement of errors [mean square error (MSE), root MSE (RMSE), mean absolute error (MAE), mean absolute percent error (MAPE), standard error (SE), and absolute relative error (RE)] in ascending order, and goodness-of-fit indices (coefficient of determination-R2 and Nash–Sutcliffe efficiency-NSE) in descending order. The assessment of shape involved indices based on the observed and computed ordinates. Similarly, the PDF that accurately reproduced actual qp and tp was identified using ‘overall qp and tp-based ranks (RqpO and RtpO)’. These ranks were derived by averaging ‘event-wise qp and tp-based ranks (Rqp and Rtp)’ assigned to each PDF (Table 4).
Table 3

Ranking of PDFs based on agreement with observed UH shape (S, symmetric, PS, positively skewed, NS, negatively skewed)

NoCatchment/EventRS for Bixler Run Basin (U.S.A)
RS for Turkish Basin
RS-Indian
Average of rank (RS)Overall rank (RSO)
MM DD YYYY PDF (No. of Parameters)02-07-1965 (PS)02-26-1957 (PS)03-06-1963 (NS)03-14-1956 (S)03-17-1963 (PS)04-16-1961 (PS)05-18-1963 (S)06-06-1964 (PS)06-21-1956 (PS)10-04-1962 (PS)10-18-1967 (PS)12-06-1962 (S)KarabalcikKurtleravsariInderesiGormelMyntduBr. No.253
R2 w.r.t. overall rank →0.860.750.100.800.830.870.820.820.530.790.800.850.50 (PS)0.49 (PS)0.67 (PS)0.82 (PS)0.62 (PS)0.76 (PS)
Shifted Gompertz (2P) 21 5.78 
2-PGD (2P) 10 10 15 10 11 11 7.44 
Dagum (3P) 2.33 
Rice (2P) 14 15 10 11 15 14 14 19 17 17 17 18 17 21 17 22 18 15.56 13 
Rayleigh (1P) 30 30 30 30 30 30 30 30 18 30 27 30 17 18 23 25 19 23 26.11 28 
Erlang (2P) 15 14 17 19 18 12 15 11 14 18 13 19 25 12 28 29 25 29 18.50 19 
Incomplete Gamma (2P) 20 12 27 17 22 23 25 14 12 13 20 10 13 12 17 24 16.28 15 
Kumaraswamy (3P) 18 21 21 22 20 19 21 20 23 19 20 16 15 19 16 20 19 18.67 22 
Log-Normal (2P) 19 11 10 10 7.61 
10 Weibull (2P) 21 23 17 20 23 21 22 16 24 22 23 14 13 17 14 18 15 18.28 17 
11 Fréchet (2P) 19 19 26 25 21 14 23 16 22 12 21 24 18 24 14 21 18.44 18 
12 Nakagami (2P) 11 13 10 11 10 13 10 13 15 10 12 14 16 15 11.28 10 
13 Beta (3P) 12 20 18 15 18 19 20 11 20 16 21 13 14 10 13 11 14.83 12 
14 Inverse-Gamma (2P) 23 16 12 12 12 11 21 10 11 10 10.00 
15 Hybrid/Nash Modified (2P) 28 28 28 28 26 26 28 28 27 21 28 10 21 13 19.89 24 
16 3 Para.2 Sided Power (3P) 26 24 25 23 24 26 25 24 28 25 29 27 30 26 30 30 30 30 26.78 29 
17 Simple 3 Parameter (3P) 29 29 29 29 29 29 29 29 24 29 30 29 21 11 27 28 23 27 26.72 30 
18 Poisson (1P) 22 16 12 20 13 21 10 19 15 15 22 14 24 29 21 18 26 17 18.56 21 
19 Binomial (2P) 17 13 22 23 16 18 18 21 18 20 15 23 30 25 23 28 13 19.50 23 
20 Negative Binominal (2P) 23 22 14 12 16 22 12 14 13 11 24 12 28 26 15 15 12 20 17.28 16 
21 Normal (2P) 16 18 13 14 19 17 17 26 21 18 18 19 22 24 22 24 22 18.50 19 
22 Gumbel Type-I (2P) 20 10 5.61 
23 Logistic (2P) 13 17 15 17 13 16 12 23 16 14 16 15 20 20 19 21 16 16.00 14 
24 Log-Logistic (2P) 16 4.83 
25 Log-Gamma (2P) 11 22 14 19 10 17 27 19 12 12 11 12 12.56 11 
26 Generalized Gamma (3P) 10 4.67 
27 Inverse Gaussian (2P) 18 16 7.39 
28 Gumbel Type-II (2P) 24 25 27 25 24 24 25 29 26 25 24 22 23 26 26 27 26 24.28 25 
29 Gompertz (2P) 25 26 11 26 27 25 25 26 30 28 26 26 26 25 29 27 29 28 25.83 27 
30 Chi-square (1P) 27 26 24 24 28 28 25 27 27 22 28 25 29 28 11 20 16 25 24.44 26 
NoCatchment/EventRS for Bixler Run Basin (U.S.A)
RS for Turkish Basin
RS-Indian
Average of rank (RS)Overall rank (RSO)
MM DD YYYY PDF (No. of Parameters)02-07-1965 (PS)02-26-1957 (PS)03-06-1963 (NS)03-14-1956 (S)03-17-1963 (PS)04-16-1961 (PS)05-18-1963 (S)06-06-1964 (PS)06-21-1956 (PS)10-04-1962 (PS)10-18-1967 (PS)12-06-1962 (S)KarabalcikKurtleravsariInderesiGormelMyntduBr. No.253
R2 w.r.t. overall rank →0.860.750.100.800.830.870.820.820.530.790.800.850.50 (PS)0.49 (PS)0.67 (PS)0.82 (PS)0.62 (PS)0.76 (PS)
Shifted Gompertz (2P) 21 5.78 
2-PGD (2P) 10 10 15 10 11 11 7.44 
Dagum (3P) 2.33 
Rice (2P) 14 15 10 11 15 14 14 19 17 17 17 18 17 21 17 22 18 15.56 13 
Rayleigh (1P) 30 30 30 30 30 30 30 30 18 30 27 30 17 18 23 25 19 23 26.11 28 
Erlang (2P) 15 14 17 19 18 12 15 11 14 18 13 19 25 12 28 29 25 29 18.50 19 
Incomplete Gamma (2P) 20 12 27 17 22 23 25 14 12 13 20 10 13 12 17 24 16.28 15 
Kumaraswamy (3P) 18 21 21 22 20 19 21 20 23 19 20 16 15 19 16 20 19 18.67 22 
Log-Normal (2P) 19 11 10 10 7.61 
10 Weibull (2P) 21 23 17 20 23 21 22 16 24 22 23 14 13 17 14 18 15 18.28 17 
11 Fréchet (2P) 19 19 26 25 21 14 23 16 22 12 21 24 18 24 14 21 18.44 18 
12 Nakagami (2P) 11 13 10 11 10 13 10 13 15 10 12 14 16 15 11.28 10 
13 Beta (3P) 12 20 18 15 18 19 20 11 20 16 21 13 14 10 13 11 14.83 12 
14 Inverse-Gamma (2P) 23 16 12 12 12 11 21 10 11 10 10.00 
15 Hybrid/Nash Modified (2P) 28 28 28 28 26 26 28 28 27 21 28 10 21 13 19.89 24 
16 3 Para.2 Sided Power (3P) 26 24 25 23 24 26 25 24 28 25 29 27 30 26 30 30 30 30 26.78 29 
17 Simple 3 Parameter (3P) 29 29 29 29 29 29 29 29 24 29 30 29 21 11 27 28 23 27 26.72 30 
18 Poisson (1P) 22 16 12 20 13 21 10 19 15 15 22 14 24 29 21 18 26 17 18.56 21 
19 Binomial (2P) 17 13 22 23 16 18 18 21 18 20 15 23 30 25 23 28 13 19.50 23 
20 Negative Binominal (2P) 23 22 14 12 16 22 12 14 13 11 24 12 28 26 15 15 12 20 17.28 16 
21 Normal (2P) 16 18 13 14 19 17 17 26 21 18 18 19 22 24 22 24 22 18.50 19 
22 Gumbel Type-I (2P) 20 10 5.61 
23 Logistic (2P) 13 17 15 17 13 16 12 23 16 14 16 15 20 20 19 21 16 16.00 14 
24 Log-Logistic (2P) 16 4.83 
25 Log-Gamma (2P) 11 22 14 19 10 17 27 19 12 12 11 12 12.56 11 
26 Generalized Gamma (3P) 10 4.67 
27 Inverse Gaussian (2P) 18 16 7.39 
28 Gumbel Type-II (2P) 24 25 27 25 24 24 25 29 26 25 24 22 23 26 26 27 26 24.28 25 
29 Gompertz (2P) 25 26 11 26 27 25 25 26 30 28 26 26 26 25 29 27 29 28 25.83 27 
30 Chi-square (1P) 27 26 24 24 28 28 25 27 27 22 28 25 29 28 11 20 16 25 24.44 26 
Table 4

Ranking of PDFs based on agreement with observed UH peak and time to peak

NoPDF (No. of parameters)Average of event-wise qp-based ranks (Rqp)Overall rank (RqpO)Average of event-wise tp-based ranks (Rtp)Overall rank (RtpO)
Shifted Gompertz (2P) 9.89 1.39 
2-PGD (2P) 14.39 13 1.44 
Dagum (3P) 6.17 1.44 
Rice (2P) 15.33 15 1.78 11 
Rayleigh (1P) 27.00 29 2.00 14 
Erlang (2P) 19.61 21 1.67 
Incomplete Gamma (2P) 2.06 1.06 
Kumaraswamy (3P) 19.83 22 1.78 11 
Log-Normal (2P) 10.83 11 1.56 
10 Weibull (2P) 20.89 24 1.67 
11 Fréchet (2P) 7.39 1.72 10 
12 Nakagami (2P) 15.50 17 1.61 
13 Beta (3P) 16.11 18 1.61 
14 Inverse-Gamma (2P) 6.83 1.50 
15 Hybrid/Nash Modified (2P) 25.39 27 1.94 13 
16 3 Para. 2 Sided Power (3P) 25.22 26 1.11 
17 Simple 3 Parameter (3P) 28.28 30 1.67 
18 Poisson (1P) 19.33 20 1.50 
19 Binomial (2P) 15.06 14 1.78 11 
20 Negative Binominal (2P) 22.28 25 1.61 
21 Normal (2P) 15.44 16 1.89 12 
22 Gumbel Type-I (2P) 9.83 1.33 
23 Logistic (2P) 10.11 1.67 
24 Log-Logistic (2P) 5.94 1.44 
25 Log-Gamma (2P) 6.78 1.50 
26 Generalized Gamma (3P) 14.28 12 1.50 
27 Inverse Gaussian (2P) 10.17 10 1.50 
28 Gumbel Type-II (2P) 20.61 23 2.06 15 
29 Gompertz (2P) 18.50 19 2.17 16 
30 Chi-square (1P) 25.94 28 1.72 10 
NoPDF (No. of parameters)Average of event-wise qp-based ranks (Rqp)Overall rank (RqpO)Average of event-wise tp-based ranks (Rtp)Overall rank (RtpO)
Shifted Gompertz (2P) 9.89 1.39 
2-PGD (2P) 14.39 13 1.44 
Dagum (3P) 6.17 1.44 
Rice (2P) 15.33 15 1.78 11 
Rayleigh (1P) 27.00 29 2.00 14 
Erlang (2P) 19.61 21 1.67 
Incomplete Gamma (2P) 2.06 1.06 
Kumaraswamy (3P) 19.83 22 1.78 11 
Log-Normal (2P) 10.83 11 1.56 
10 Weibull (2P) 20.89 24 1.67 
11 Fréchet (2P) 7.39 1.72 10 
12 Nakagami (2P) 15.50 17 1.61 
13 Beta (3P) 16.11 18 1.61 
14 Inverse-Gamma (2P) 6.83 1.50 
15 Hybrid/Nash Modified (2P) 25.39 27 1.94 13 
16 3 Para. 2 Sided Power (3P) 25.22 26 1.11 
17 Simple 3 Parameter (3P) 28.28 30 1.67 
18 Poisson (1P) 19.33 20 1.50 
19 Binomial (2P) 15.06 14 1.78 11 
20 Negative Binominal (2P) 22.28 25 1.61 
21 Normal (2P) 15.44 16 1.89 12 
22 Gumbel Type-I (2P) 9.83 1.33 
23 Logistic (2P) 10.11 1.67 
24 Log-Logistic (2P) 5.94 1.44 
25 Log-Gamma (2P) 6.78 1.50 
26 Generalized Gamma (3P) 14.28 12 1.50 
27 Inverse Gaussian (2P) 10.17 10 1.50 
28 Gumbel Type-II (2P) 20.61 23 2.06 15 
29 Gompertz (2P) 18.50 19 2.17 16 
30 Chi-square (1P) 25.94 28 1.72 10 
Figure 5

Observed and simulated UHs for Bixler Run Basin event (12-06-1962).

Figure 5

Observed and simulated UHs for Bixler Run Basin event (12-06-1962).

Close modal

Eighteen UHs were employed, each characterized by a unique time step (Δt) and properties such as qp, tp, qv, and shape, including symmetrical/normal (3 UHs), positively skewed (14 UHs), and negatively skewed (01 UH). Utilizing an objective function ‘Minimization of RMSE’ within an Excel Solver routine, optimal PDF parameters were efficiently determined, facilitating the accurate reproduction of actual UHs.

Demonstration of PDF-based UHs against Bixler Run basin UH (6 December 1962)

A symmetrical, equally fragmented actual UH (tp = 10 h) was distributed over tb (=20 h) (Figure 5(b)). Both event-specific-RS and overall-RSO showed a good agreement with R2 = 0.85 (Table 3). Some PDFs shared the same rank (Rtp) due to identical RE(tp), resulting in RtpO ranging from 1 to 16 (Table 4). This pattern was similarly observed in shape prediction, as reflected by the assignment of identical rank, RSO = 19 (Table 3). The discussion here focuses on the top five ranked (RS) PDFs (Figure 5(b)).

Dagum PDF demonstrates superiority as a UH over other PDFs, supported by its indices [RMSE = 0.0048, NSE = 98.9%, R2 = 0.990, MSE = 0.00, MAE = 0.004, MAPE = 24.09%, SE = 0.005] (Table 5). Additionally, it achieves an RSO of 1 (Table 3). Despite its overall performance, Dagum lags in estimating qp (RqpO = 3) and tp (RtpO = 5, as of 2-PGD and Log-Logistic) (Table 4). In the specific event (Table 5), Dagum UH precisely captures tp [RE(tp) = 0, Rtp = 1] but underestimates qp [RE(qp) = 5.17%, PBIAS = 2.9%, Rqp = 4] due to overestimation at the inflection point and tail end, aiming to conserve unit volume (qv = 0.967) over a given tb (Figure 5(b)).

Table 5

Parameter estimation, statistical evaluation and ranking of the distributions for Bixler Run basin event (12 Jun 1962)

PDFRMSENSE (%)R2MSEMAEPBIAS (%)MAPE (%)SEVol. (C)RANK shape (RS)Parameters
Abs RE (qp) (%)Abs RE (tp) (%)RANK peak (Rqp)RANK time to peak (Rtp)
Shifted Gompertz 0.0057 98.4 0.985 0.000 0.005 2.1 29.14 0.0059 0.997 η,b 0.01 30.82  6.99 0.00 10 
2-PGD 0.0062 98.1 0.982 0.000 0.005 0.6 25.36 0.0063 0.994 α,B 11.63 0.94  7.77 0.00 15 
Dagum 0.0048 98.9 0.990 0.000 0.004 2.9 24.09 0.0050 0.967 α,b,p 0.01 5.28 10.4 5.17 0.00 
Rice 0.0091 95.9 0.960 0.000 0.007 0.1 26.62 0.0093 0.999 17 ν,σ 9.89 3.19  9.41 0.00 18 
Rayleigh 0.0309 52.4 0.654 0.001 0.024 6.2 135.80 0.0309 0.938 30 σ 0.03   50.22 10.00 30 
Erlang 0.0093 95.7 0.960 0.000 0.007 0.4 30.64 0.0096 0.996 19 κ,γ 0.01 11.00  0.44 0.00 
Incomplete-Gamma 0.0076 97.1 0.984 0.000 0.006 0.1 31.60 0.0078 0.999 13 α 12.50   0.00 0.00 
Kumaraswamy 0.0129 91.7 0.921 0.002 0.000 0.0 41.32 0.0132 1.000 20 p,q,b 0.01 3.51 6.8 11.38 10.00 22 
Log-Normal 0.0058 98.3 0.985 0.002 0.005 1.4 28.35 0.0059 0.980 μ,σ 0.01 2.37  7.03 0.00 12 
Weibull 0.0108 94.2 0.942 0.002 0.009 0.0 34.29 0.0111 1.000 23 α,b 1.32 0.01  12.58 10.00 24 
Fréchet 0.0101 94.9 0.957 0.002 0.008 7.7 40.06 0.0103 0.923 21 α,c 0.01 1.24  5.39 10.00 
Nakagami 0.0075 97.2 0.973 0.000 0.006 0.2 26.09 0.0077 0.998 10 m, 3.09 123.34  8.72 0.00 16 
Beta 0.0110 94.0 0.945 0.000 0.009 0.0 36.51 0.0113 1.000 21 q,b,p 5.51 20.00 10.6 10.14 0.00 19 
Inverse-Gamma 0.0065 97.9 0.981 0.000 0.005 2.7 31.84 0.0066 0.973 11 α,B 11.24 117.56  6.57 0.00 
Hybrid/Nash Modified 0.0268 64.3 0.758 0.001 0.020 8.2 72.59 0.0275 0.918 28 k1,k2 2.93 2.93  45.65 10.00 28 
3-Para.2-Sided Power 0.0196 80.9 0.875 0.000 0.014 0.0 92.01 0.0201 1.000 27 n,b,m 2.00 20.00 10.0 28.81 0.00 26 
Simple 3-Parameter 0.0275 62.4 0.707 0.001 0.021 0.2 127.65 0.0282 0.998 29 M,N,td 1.37 1.37 20.0 46.37 0.00 29 
Poisson 0.0079 96.9 0.969 0.000 0.006 0.3 24.03 0.0079 0.997 14 μ 10.59   12.42 0.00 23 
Binomial 0.0088 96.1 0.972 0.000 0.007 0.1 29.26 0.0090 0.999 15 n,p 50.00 0.21  2.14 0.00 
Negative Binominal 0.0079 96.9 0.969 0.000 0.006 0.4 22.56 0.0081 0.996 12 r,p 1000.00 0.99  13.27 0.00 25 
Normal 0.0093 95.7 0.958 0.000 0.008 0.1 26.18 0.0095 0.999 18 μ,σ 10.39 3.11  9.39 0.00 17 
Gumbel Type-I 0.0057 98.4 0.985 0.000 0.005 2.1 29.10 0.0059 0.979 B,μ 2.80 9.69  7.00 0.00 11 
Logistic 0.0082 96.6 0.966 0.000 0.007 0.8 29.72 0.0084 0.992 16 s,μ 1.89 10.31  6.49 0.00 
Log-Logistic 0.0048 98.8 0.989 0.000 0.004 2.7 22.88 0.0049 0.973 α,B 10.62 5.47  5.22 0.00 
Log-Gamma 0.0064 98.0 0.982 0.000 0.005 2.4 31.49 0.0065 0.976 α,B 62.44 0.04  6.66 0.00 
Generalized Gamma 0.0059 98.3 0.984 0.000 0.005 0.8 26.37 0.0062 0.942 d,α,p 17.18 0.08 0.7 7.43 0.00 14 
Inverse Gaussian 0.0060 98.2 0.984 0.000 0.005 1.3 28.70 0.0061 0.980 γ,μ 122.41 11.17  7.10 0.00 13 
Gumbel Type-II 0.0170 85.6 0.858 0.000 0.015 2.0 86.83 0.0174 0.999 24 B,α 2.95 11.04  11.33 10.00 21 
Gompertz 0.0171 85.5 0.856 0.000 0.015 −0.4 91.29 0.0175 0.987 26 η,b 0.02 0.03  10.88 10.00 20 
Chi-Square 0.0196 80.9 0.888 0.000 0.014 5.8 42.99 0.0196 1.004 25 k 12.00   37.55 0.00 27 
PDFRMSENSE (%)R2MSEMAEPBIAS (%)MAPE (%)SEVol. (C)RANK shape (RS)Parameters
Abs RE (qp) (%)Abs RE (tp) (%)RANK peak (Rqp)RANK time to peak (Rtp)
Shifted Gompertz 0.0057 98.4 0.985 0.000 0.005 2.1 29.14 0.0059 0.997 η,b 0.01 30.82  6.99 0.00 10 
2-PGD 0.0062 98.1 0.982 0.000 0.005 0.6 25.36 0.0063 0.994 α,B 11.63 0.94  7.77 0.00 15 
Dagum 0.0048 98.9 0.990 0.000 0.004 2.9 24.09 0.0050 0.967 α,b,p 0.01 5.28 10.4 5.17 0.00 
Rice 0.0091 95.9 0.960 0.000 0.007 0.1 26.62 0.0093 0.999 17 ν,σ 9.89 3.19  9.41 0.00 18 
Rayleigh 0.0309 52.4 0.654 0.001 0.024 6.2 135.80 0.0309 0.938 30 σ 0.03   50.22 10.00 30 
Erlang 0.0093 95.7 0.960 0.000 0.007 0.4 30.64 0.0096 0.996 19 κ,γ 0.01 11.00  0.44 0.00 
Incomplete-Gamma 0.0076 97.1 0.984 0.000 0.006 0.1 31.60 0.0078 0.999 13 α 12.50   0.00 0.00 
Kumaraswamy 0.0129 91.7 0.921 0.002 0.000 0.0 41.32 0.0132 1.000 20 p,q,b 0.01 3.51 6.8 11.38 10.00 22 
Log-Normal 0.0058 98.3 0.985 0.002 0.005 1.4 28.35 0.0059 0.980 μ,σ 0.01 2.37  7.03 0.00 12 
Weibull 0.0108 94.2 0.942 0.002 0.009 0.0 34.29 0.0111 1.000 23 α,b 1.32 0.01  12.58 10.00 24 
Fréchet 0.0101 94.9 0.957 0.002 0.008 7.7 40.06 0.0103 0.923 21 α,c 0.01 1.24  5.39 10.00 
Nakagami 0.0075 97.2 0.973 0.000 0.006 0.2 26.09 0.0077 0.998 10 m, 3.09 123.34  8.72 0.00 16 
Beta 0.0110 94.0 0.945 0.000 0.009 0.0 36.51 0.0113 1.000 21 q,b,p 5.51 20.00 10.6 10.14 0.00 19 
Inverse-Gamma 0.0065 97.9 0.981 0.000 0.005 2.7 31.84 0.0066 0.973 11 α,B 11.24 117.56  6.57 0.00 
Hybrid/Nash Modified 0.0268 64.3 0.758 0.001 0.020 8.2 72.59 0.0275 0.918 28 k1,k2 2.93 2.93  45.65 10.00 28 
3-Para.2-Sided Power 0.0196 80.9 0.875 0.000 0.014 0.0 92.01 0.0201 1.000 27 n,b,m 2.00 20.00 10.0 28.81 0.00 26 
Simple 3-Parameter 0.0275 62.4 0.707 0.001 0.021 0.2 127.65 0.0282 0.998 29 M,N,td 1.37 1.37 20.0 46.37 0.00 29 
Poisson 0.0079 96.9 0.969 0.000 0.006 0.3 24.03 0.0079 0.997 14 μ 10.59   12.42 0.00 23 
Binomial 0.0088 96.1 0.972 0.000 0.007 0.1 29.26 0.0090 0.999 15 n,p 50.00 0.21  2.14 0.00 
Negative Binominal 0.0079 96.9 0.969 0.000 0.006 0.4 22.56 0.0081 0.996 12 r,p 1000.00 0.99  13.27 0.00 25 
Normal 0.0093 95.7 0.958 0.000 0.008 0.1 26.18 0.0095 0.999 18 μ,σ 10.39 3.11  9.39 0.00 17 
Gumbel Type-I 0.0057 98.4 0.985 0.000 0.005 2.1 29.10 0.0059 0.979 B,μ 2.80 9.69  7.00 0.00 11 
Logistic 0.0082 96.6 0.966 0.000 0.007 0.8 29.72 0.0084 0.992 16 s,μ 1.89 10.31  6.49 0.00 
Log-Logistic 0.0048 98.8 0.989 0.000 0.004 2.7 22.88 0.0049 0.973 α,B 10.62 5.47  5.22 0.00 
Log-Gamma 0.0064 98.0 0.982 0.000 0.005 2.4 31.49 0.0065 0.976 α,B 62.44 0.04  6.66 0.00 
Generalized Gamma 0.0059 98.3 0.984 0.000 0.005 0.8 26.37 0.0062 0.942 d,α,p 17.18 0.08 0.7 7.43 0.00 14 
Inverse Gaussian 0.0060 98.2 0.984 0.000 0.005 1.3 28.70 0.0061 0.980 γ,μ 122.41 11.17  7.10 0.00 13 
Gumbel Type-II 0.0170 85.6 0.858 0.000 0.015 2.0 86.83 0.0174 0.999 24 B,α 2.95 11.04  11.33 10.00 21 
Gompertz 0.0171 85.5 0.856 0.000 0.015 −0.4 91.29 0.0175 0.987 26 η,b 0.02 0.03  10.88 10.00 20 
Chi-Square 0.0196 80.9 0.888 0.000 0.014 5.8 42.99 0.0196 1.004 25 k 12.00   37.55 0.00 27 

Following Dagum, the Log-Logistic PDF maintains a comparable qv = 0.973, achieving RS = 2, Rqp = 5 (RE(qp) = 5.22%) and Rtp = 1 (Table 5). However, Log-Logistic UH exhibits an earlier rise than Dagum UH at the head end, underestimates at the peak (PBIAS = 2.7%), overestimates near the point of inflection, and lies below Dagum and the actual UH at the tail end (Figure 5(b)). Notably, Log-Logistic attains RSO = 3, RqpO = 2 and RtpO = 5 (Tables 3 and 4).

Underestimating Gumbel Type-I UH at the head and peak results in RS = 3, Rqp = 11 (RE(qp) = 7%), Rtp = 1, and PBIAS = 2.1% (Table 5). Despite the peak underestimation, overestimation at the inflection point and tail end aids volume conservation (qv = 0.979) (Figure 5(b)) achieving RSO = 4, RqpO = 7, and RtpO = 3 (Tables 3 and 4).

Having RS = 4, Rqp = 10 (RE(qp) = 6.66%), and Rtp = 1, the Shifted Gompertz UH follows Gumbel Type-I UH (Table 5). The broader derived UH encompasses the actual UH, preserving unit volume despite the underestimated peak (PBIAS = 2.1%, Figure 5(b)). Overall, the ranks for Shifted Gompertz are RSO = 5, RqpO = 8, and RtpO = 4 (Tables 3 and 4).

The broad crest of the Generalized Gamma UH reduces the peak (Rqp = 14, RE(qp) = 7.43%) precisely at tp (Rtp = 1) during a given tb, resulting in minimal volumetric shortfall (qv = 0.941) (Table 5). This leads to overestimation before and after the crest, along with underestimation at the tail end, indicated by RS = 5 and PBIAS = 0.8% (Figure 5(b), Table 5). The Generalized Gamma achieves RSO = 2, RqpO = 12, and RtpO = 6 (Tables 3 and 4).

Furthermore, the Inverse Gaussian, 2-PGD, Log-Normal, Log-Gamma, Nagakami, and Inverse-Gamma, rank 6th to 11th, exhibiting consistent shape-based ranking, RSO range (Tables 3 and 5). Subsequent PDFs are ranked based on their respective indices (Tables 35). In the case of the positively skewed Bixler Run basin UH (16 April 1961) with tb = 14 h and tp = 6 h, RS and RSO demonstrate strong agreement with R2 = 0.87 (Table 3). However, for the negatively skewed UH (6 March 1963) with tb = 24 h and tp = 14 h, RS and RSO show poor agreement (R2 = 0.10, Table 3).

General discussion

Among the 12 new PDFs, 5 achieved top 5 rankings, RSO is discussed (Table 3). The Dagum PDF secured the 1st rank, RSO for 10 UHs across three regions while obtaining 2nd to 8th ranks for the remaining 08 UHs. Its lowest (i.e. 8th) rank for positively skewed UH (tb = 14 h and tp = 03 h, 21 June 1956) is justified by reasonable indices [RMSE = 0.0086, NSE = 98.3%, R2 = 0.985, MSE = 0.00, MAE = 0.007, MAPE = 61.56%, SE = 0.009, RE(qp) = 5.71%, RE(tp) = 0.00, PBIAS = 2.8% – excess underestimation] compared to the event-specific top-ranked Hybrid–Nash (H-N) PDF (indices: 0.0072, 98.8%, 0.988, 0.00, 0.005, 26.70%, 0.007, 8.09%, 33.33%, 0.2% – negligible underestimation). Despite H-N PDFs erroneous qptp estimation, it exhibits volumetric efficiency (qv ≈ 1). Notably, for this specific event (21 June 1956), RS shows poor agreement (R2 = 0.53) with RSO (Table 3).

RSO = 2 confirms Generalized Gamma's second rank, trailing Dagum, with ranks (RS) ranging from 1st (over 4 UHs) to 5th (RS) across 12 out of 18 UHs (Table 3). Notably, its highest (i.e. 10th) rank (RS) against a positively skewed UH (tb = 14 h and tp = 04 h, 18 October 1967) exhibits indices [RMSE = 0.016, NSE = 95.5%, R2 = 0.963, MSE = 0.00, MAE = 0.013, MAPE = 40.35%, SE = 0.018, RE(qp) = 6.51%, RE(tp) = 0.00]. In comparison to the top-ranked (1st) Dagum with respective indices [0.010, 98.1%, 0.984, 0.00, 0.008, 24.90%, 0.011, 3.49%, 0.00], this deviation is attributed to an attenuated peak expanding UH, resulting in a slight deviation (PBIAS = 0.03%). Excluding this exceptional event, Generalized Gamma demonstrates adaptability in reproducing actual qp and tp, as indicated by RqpO = 12 and RtpO = 6 (Table 4).

The Log-Logistic PDF achieves the second rank (RS) across seven events, displaying exceptional performance in replicating negatively skewed UH (6 March 1963) [RS-16, RMSE = 0.009, NSE = 94.5%, R2 = 0.946, MSE = 0.00, MAE = 0.007, MAPE = 54.14%, SE = 0.009, RE(qp) = 2.58%, RE(tp) = 14.29%]. For other events, its rank ranges up to 9th (Table 3). Despite having the same RE(tp) as the superior Dagum PDF [RS-1, RMSE = 0.003, NSE = 99.4%, R2 = 0.994, MSE = 0.00, MAE = 0.002, MAPE = 10.54%, SE = 0.003, RE(qp) = 1.06%, RE(tp) = 14.29%], the Log-Logistic PDF's smooth peak occurring prior to the actual tapered/negatively skewed peak is attributed to UH deviations (PBIAS = 3.2%). Excluding this event, the Log-logistic PDF justifiably estimates qp (Rqp = 2) and tp (Rtp = 5, equivalent to Dagum and 2-PGD) (Table 4). Consequently, it demonstrates superiority over Dagum and Generalized Gamma in qptp estimation.

Gumbel Type-I demonstrates performance comparable to Log-Logistic in handling negatively skewed UH [6 March 1963, RS-20, RMSE = 0.011, NSE = 91.9%, R2 = 0.920, MSE = 0.00, MAE = 0.009, MAPE = 64.55%, SE = 0.011, RE(qp) = 6.59%, RE(tp) = 14.29%]. While it attains the 1st rank (RS) for 2 events, its rank for the remaining 15 events ranges up to 10th (Table 3). The underestimated peak expands both limbs of the UH, resulting in a PBIAS of 2.7%. In total, Gumbel Type-I secures the 7th rank for qp and the 3rd rank for tp (Table 4).

Shifted Gompertz exhibits variability in RS, ranging from 2nd (for 2 events) to 9th (Table 3), closely mirroring the Gumbel Type-I PDF with a slight difference [RS-21, RMSE = 0.011, NSE = 91.9%, R2 = 0.920, MSE = 0.00, MAE = 0.009, MAPE = 64.63%, SE = 0.011, RE(qp) = 6.55%, RE(tp) = 14.29%] over a negatively skewed exceptional UH (6 March 1963). In qptp fitting, it aligns with Gumbel Type-I, securing the 8th rank for qp and the 4th rank for tp overall (Table 4).

The Inverse Gaussian performs satisfactorily, securing ranks from 3rd to 8th (RS) over 16 events. However, for two specific events (Kurtleravsari UH and Bixler Run Basin UH-6 March 1963), it attains the 16th (RMSE = 0.002, NSE = 97%) and 18th (RMSE = 0.010, NSE = 92.3%) ranks, respectively (Table 3). Notably, Generalized Gamma and Dagum excel in these particular events, achieving the 1st rank (RS = 1). The 16th rank is attributed to deviations in the positively skewed UH limbs (PBIAS = 58.79%) from the actual UH with an oscillatory crest. The 18th rank results from underestimating a smooth symmetric peak that represents the actual UH with a tapered crest. In overall qp and tp estimation, the Inverse Gaussian secures the 10th and 6th ranks, respectively (Table 4).

With RSO = 7, the 2-PGD (Table 3) appears relatively less efficient [RS-15, RMSE = 0.008, NSE = 95.2%, R2 = 0.953, MSE = 0.00, MAE = 0.007, MAPE = 46.50%, SE = 0.008, RE(qp) = 3.67%, RE(tp) = 14.29%] compared to Dagum (RS-1) in the specified event (6 March 1963). This is attributed to peak attenuation, causing UH expansion and deviations/errors (PBIAS = 0.08%). However, its volumetric accuracy is higher (qv = 0.99) than Dagum (qv = 0.98). Excluding this event, 2-PGD demonstrates its capability across other events, with RS ranging from 2nd to 11th (Table 3). It shares the same RtpO (5th) as Dagum and Log-Logistic while following Generalized Gamma with RqpO = 13th (Table 4).

The Log-Normal PDF, with an RSO of 8th (Table 3), holds ranks, RS from 3rd to 11th, except an exceptional RS = 19th [RMSE = 0.010, NSE = 92.7%, R2 = 0.928, MSE = 0.001, MAE = 0.001, MAPE = 59.20%, SE = 0.010, RE in qp = 3.37%, RE in tp = 21.43%] against the UH (6 March 1963). This deviation is attributed to the early occurrence of an underestimated peak (PBIAS = 1.7%). In the overall assessment, Log-Normal is ranked 11th for qp and 7th for tp (Table 4).

Similarly, the Inverse-Gamma, with an RSO of 9th, exhibits RS ranging from 1st to 23rd (6 March 1963) across 18 events (Table 3). In the overall assessment of qptp, it secures the 5th and 6th ranks, respectively (Table 4). The Japanese Nagakami PDF holds the 10th place in RSO, with event-wise ranks (RS) varying from 4th to 16th (Inderesi Creek basin UH) across 18 events (Table 3). However, in qptp estimation, it attains the 17th and 8th ranks, respectively (Table 4). The new Log-Gamma PDF achieves an RSO of 11th, with RS varying from 3rd to 27th (Karabalcik Creek basin UH) across 18 events (Table 3). Excluding this, it efficiently estimates qp following Dagum with an RqpO of 4th (Table 4). Its performance aligns with Inverse-Gamma, Poisson, Log-Gamma, Generalized Gamma, and Inverse Gaussian, sharing an RtpO of 6th (Table 4)

The Negative Binomial, Chi-square, Poisson, and Binomial PDFs exhibit unacceptable performance with negative NSE and the highest RE(qp), resulting in RS of 26, 28, 29, and 30, respectively. This is evident in defining a positively skewed UH (tp = 17 h, Kurtleravsari Creek basin) with the longest tb (=110 h) among the 18 observed UHs. The computed peaks, occurring either prior to or later than the actual peak, are two to three times in magnitude, aiming to conserve unit volume over a constrained tb (≈40 h). In contrast, by underestimating the peak [RE(qp) = 44.91%] at the exact tp over an identical tb as observed, the 3-Parameter 2-Sided Power-based triangular UH attains identical but unacceptable rank, RS = 26 as the Negative Binomial, while conserving the approximate volume (qv = 0.980) with a positive NSE (=54.8%).

The novel Negative Binomial, Poisson, and Binomial PDFs hold ranks (RSO) of 16th, 21st, and 23rd, respectively (Table 3). Chi-square (RSO = 26th) and 3-Parameter 2-Sided Power (RSO = 29th, attributed to triangular UH fitted) follow them. The Simple 3-Parameter PDF secures the last (30th) rank based on shape and qp (Tables 3 and 4), owing to its underestimated concave peaks against 10 out of 18 observed UHs. In terms of tp, the Gompertz PDF attains the last (16th) rank among all PDFs (Table 4). When tb is computed, qv is either underestimated or overestimated, justifiable by positive or negative PBIAS, respectively. A PBIAS = 0 indicates unit volume conservation over the given tb. Furthermore, the best PDFs for a specific watershed class (based on drainage area) have been identified (Table 6) by averaging the event-wise PDF ranks for that particular watershed class.

Table 6

Drainage area based watershed class-wise best shape-defining PDFs

S. N.Watershed class and events usedTop 10 UH shape-defining PDFs
Karabalcik
Bixler Run
10.6–38.85 km2 (13) 
  • 1.

    Dagum

  • 2.

    Generalized Gamma

  • 3.

    Log-Logistic

  • 4.

    Gumbel Type-I

  • 5.

    Shifted Gompertz

 
  • 6.

    Inverse Gaussian

  • 7.

    2-PGD

  • 8.

    Log-Normal

  • 9.

    Inverse-Gamma

  • 10.

    Nagakami

 
Inderesi
Bridge No. 253
Gormel Ermenek
98–114.22–141.5 km2 (3) 
  • 1.

    Dagum

  • 2.

    Shifted Gompertz

  • 3.

    Log-Logistic, Gumbel Type- I

  • 4.

    Generalized Gamma

  • 5.

    Log-Normal

 
  • 6.

    2-PGD

  • 7.

    Inverse Gaussian

  • 8.

    Inverse-Gamma

  • 9.

    Nagakami

  • 10.

    Beta

 
Myntdu-Leska
350 km2 (1) 
  • 1.

    Dagum

  • 2.

    Log-Logistic

  • 3.

    Log-Normal

  • 4.

    Generalized Gamma

  • 5.

    Inverse Gaussian

 
  • 6.

    Hybrid/Nash Modified

  • 7.

    Shifted Gompertz

  • 8.

    2-PGD

  • 9.

    Inverse-Gamma

  • 10.

    Gumbel Type-I

 
Kurtleravsari Aksu
3360 km2 (1) 
  • 1.

    Generalized Gamma

  • 2.

    Hybrid/Nash Modified

  • 3.

    2-PGD

  • 4.

    Dagum

  • 5.

    Log-Logistic

 
  • 6.

    Shifted Gompertz

  • 7.

    Log-Normal

  • 8.

    Gumbel Type-I

  • 9.

    Incomplete-Gamma

  • 10.

    Beta

 
S. N.Watershed class and events usedTop 10 UH shape-defining PDFs
Karabalcik
Bixler Run
10.6–38.85 km2 (13) 
  • 1.

    Dagum

  • 2.

    Generalized Gamma

  • 3.

    Log-Logistic

  • 4.

    Gumbel Type-I

  • 5.

    Shifted Gompertz

 
  • 6.

    Inverse Gaussian

  • 7.

    2-PGD

  • 8.

    Log-Normal

  • 9.

    Inverse-Gamma

  • 10.

    Nagakami

 
Inderesi
Bridge No. 253
Gormel Ermenek
98–114.22–141.5 km2 (3) 
  • 1.

    Dagum

  • 2.

    Shifted Gompertz

  • 3.

    Log-Logistic, Gumbel Type- I

  • 4.

    Generalized Gamma

  • 5.

    Log-Normal

 
  • 6.

    2-PGD

  • 7.

    Inverse Gaussian

  • 8.

    Inverse-Gamma

  • 9.

    Nagakami

  • 10.

    Beta

 
Myntdu-Leska
350 km2 (1) 
  • 1.

    Dagum

  • 2.

    Log-Logistic

  • 3.

    Log-Normal

  • 4.

    Generalized Gamma

  • 5.

    Inverse Gaussian

 
  • 6.

    Hybrid/Nash Modified

  • 7.

    Shifted Gompertz

  • 8.

    2-PGD

  • 9.

    Inverse-Gamma

  • 10.

    Gumbel Type-I

 
Kurtleravsari Aksu
3360 km2 (1) 
  • 1.

    Generalized Gamma

  • 2.

    Hybrid/Nash Modified

  • 3.

    2-PGD

  • 4.

    Dagum

  • 5.

    Log-Logistic

 
  • 6.

    Shifted Gompertz

  • 7.

    Log-Normal

  • 8.

    Gumbel Type-I

  • 9.

    Incomplete-Gamma

  • 10.

    Beta

 

We examined both established hydrologic literature PDFs and those not yet tested as UH to identify a reliable PDF. However, the PDFs resembling UH are not restricted to those investigated in this study. There is a significant possibility that additional PDFs may exist in the statistical literature, but they require evaluation as UH before being considered. The following conclusions can be derived from the study:

  • 1.

    The diverse selection of available PDFs empowers practitioners to choose the most suitable one for their objectives. While PDFs may resemble UH, a single PDF is insufficient to precisely capture all the characteristics of a UH. Different UHs within a region may be well-fitted by more than one PDF.

  • 2.

    The parameters obtained through the Excel-based GRG-NLP algorithm define smooth PDF-based UHs, automatically addressing volume and non-negativity constraints. The proposed Maple codes enhance and expedite hydrologic simulation.

  • 3.

    The novel PDFs (Dagum, Generalized Gamma, Log-Logistic, Gumbel Type-I, and Shifted Gompertz) accurately replicate UH compared to established PDFs (2-PGD, Chi-Square, Weibull, Beta, Hybrid Model, Simple 3 Parameter, Rayleigh, etc.) recommended in hydrologic literature.

  • 4.

    Among the 18 UHs, 16 have been excellently defined (RSO = 1) by new PDFs (10 by Dagum, 04 by Generalized Gamma, and 02 by Gumbel Type-I) while the remaining 2 were well-fitted by the Hybrid/Modified-Nash model.

  • 5.

    Dagum, Generalized Gamma, and Log-Logistic PDFs excel in preserving the rising and recession limbs across all watershed classes. The Incomplete-Gamma PDF is particularly effective in predicting both qp and tp. However, the widely-used 2-PGD falls short in accurately estimating qp while it reasonably estimates tp.

  • 6.

    Gompertz, among the novel PDFs, is the sole PDF that fits erratic UHs, such as those modeled by the existing PDFs (Rayleigh – 10 events and Simple 3-Parameter – 7 events), both of which rank last across 17 events.

  • 7.

    The novel 3-parameter PDFs, Dagum and Generalized Gamma, can be utilized with known ‘qp and tp’ by fixing the least sensitive shape parameter as unity. In contrast, for the existing 3-parameter Kumaraswamy, Beta, 3-Parameter 2-Sided Power, and Simple 3-Parameter PDFs, the application requires knowledge of tb.

  • 8.

    The proposed Dagum, Generalized Gamma, Log-Logistic, Gumbel Type-I, and Shifted Gompertz PDFs are highly reliable for small-size watersheds. For mid-size watersheds, Dagum, Log-Logistic, Log-Normal, Generalized Gamma, and Inverse Gaussian PDFs are most suitable. Sequentially, Generalized Gamma, Hybrid/Nash Modified, 2-PGD, Dagum, and Log-Logistic PDFs are efficient for large-size watersheds.

The authors express their gratitude to IIT, Roorkee and NIH, Roorkee for furnishing the essential research facilities.

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

The authors declare there is no conflict.

Abramowitz
M.
,
Stegun
I. A.
&
Romer
R. H.
1988
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
.
https://doi.org/10.1119/1.15378
.
Agarwal
S. K.
&
Kalla
S. L.
1996
A generalized gamma distribution and its application in reliability
.
Communi Stat. Theory Methods
25
(
1
),
201
210
.
https://doi.org/10.1080/03610929608831688
.
Aksoy
H.
2000
Use of gamma distribution in hydrological analysis
.
Turkish J. Engg. Env. Sci.
24
(
6
),
419
428
.
Aron
G.
&
White
E. L.
1982
Fitting a gamma distribution over a synthetic unit hydrograph 1
.
J. Am. Water. Resour. As.
18
(
1
),
95
98
.
https://doi.org/10.1111/j.1752-1688.1982.tb04533.x
.
Aryal
G. R.
2013
Transmuted log-logistic distribution
.
J. Stat. Appl. Probab.
2
(
1
),
11
.
https://doi.org/10.12785/jsap/020102
.
Ashkar
F.
&
Mahdi
S.
2006
Fitting the log-logistic distribution by generalized moments
.
J. Hydrol.
328
(
3–4
),
694
703
.
https://doi.org/10.1016/j.jhydrol.2006.01.014
.
Bain
L. J.
&
Wright
F. T.
1982
The negative binomial process with applications to reliability
.
J. Qual. Tech.
14
(
2
),
60
66
.
https://doi.org/10.1080/00224065.1982.11978791
.
Balakrishnan
N.
1991
Handbook of the Logistic Distribution
.
CRC Press
.
https://doi.org/10.1201/9781482277098
.
Bardsley
W. E.
1983
An alternative distribution for describing the instantaneous unit hydrograph
.
J. Hydrol.
62
(
1–4
),
375
378
.
https://doi.org/10.1016/0022-1694(83)90115-4
.
Barriga
G. D.
,
Cordeiro
G. M.
,
Dey
D. K.
,
Cancho
V. G.
,
Louzada
F.
&
Suzuki
A. K.
2018
The Marshall-Olkin generalized gamma distribution
.
Commun. Stat. Appl. Methods
25
(
3
),
245
261
.
https://doi.org/10.29220/CSAM.2018.25.3.245
.
Bauckhage
C.
&
Kersting
K.
2014
Strong regularities in growth and decline of popularity of social media services. arXiv:1406.6529. https://doi.org/10.48550/arXiv.1406.6529
Bemmaor
A. C
.
1992
Modeling the diffusion of new durable goods: Word-of-mouth effect versus consumer heterogeneity
, In:
Research Traditions in Marketing
.
Springer
,
Dordrecht
, pp.
201
229
.
https://doi.org/10.1007/978-94-011-1402-8_6
.
Bender
D. L.
&
Roberson
J. A.
1961
The use of a dimensionless unit hydrograph to derive unit hydrographs for some Pacific Northwest basins
.
J. Geophys. Res.
66
(
2
),
521
528
.
https://doi.org/10.1029/JZ066i002p00521
.
Benjamin
S. M.
,
Humberto
V. H.
&
Barry
C. A.
2013
Use of the Dagum distribution for modeling tropospheric ozone levels
.
J. Env. Stat.
5
(
6
),
1
11
.
Bhadra
A.
,
Bandyopadhyay
A.
,
Singh
R.
&
Raghuwanshi
N. S.
2010
Rainfall-runoff modeling: Comparison of two approaches with different data requirements
.
Water Resour. Manag.
24
(
1
),
37
62
.
https://doi.org/10.1007/s11269-009-9436-z
.
Bhattacharjya
R. K.
2004
Optimal design of unit hydrographs using probability distribution and genetic algorithms
.
Sadhana
29
(
5
),
499
508
.
https://doi.org/10.1007/BF02703257
.
Bhunya
P. K.
,
Mishra
S. K.
&
Berndtsson
R.
2003
Simplified two parameters gamma distribution for derivation of synthetic unit hydrograph
.
J. Hydrol. Eng.
8
(
4
),
226
230
.
https://doi.org/10.1061/(ASCE)1084-0699(2003)8 : 4(226)
.
Bhunya
P. K.
,
Mishra
S. K.
,
Ojha
C. S. P.
&
Berndtsson
R.
2004
Parameter estimation of beta distribution for unit hydrograph derivation
.
J. Hydrol. Eng.
9
(
4
),
325
332
.
https://doi.org/10.1061/(ASCE)1084-0699(2004)9 : 4(325)
.
Bhunya
P. K.
,
Ghosh
N. C.
,
Mishra
S. K.
,
Ojha
C. S. P.
&
Berndtsson
R.
2005
Hybrid model for derivation of synthetic unit hydrograph
.
J. Hydrol. Eng.
10
(
6
),
458
467
.
https://doi.org/10.1061/(ASCE)1084-0699(2005)10 : 6(458)
.
Bhunya
P. K.
,
Berndtsson
R.
,
Ojha
C. S. P.
&
Mishra
S. K.
2007
Suitability of Gamma, Chi-square, Weibull, and Beta distributions as synthetic unit hydrographs
.
J. Hydrol.
334
(
1–2
),
28
38
.
https://doi.org/10.1016/j.jhydrol.2006.09.022
.
Bhunya
P. K.
,
Berndtsson
R.
,
Singh
P. K.
&
Hubert
P.
2008
Comparison between Weibull and gamma distributions to derive synthetic unit hydrograph using Horton ratios
.
Water Resour. Res.
44
(
4
).
https://doi.org/10.1029/2007WR006031
.
Bhunya
P. K.
,
Singh
P. K.
&
Mishra
S. K.
2009
Fréchet and chi-square parametric expressions combined with Horton ratios to derive a synthetic unit hydrograph
.
Hydrolog. Sci. J.
54
(
2
),
274
286
.
https://doi.org/10.1623/hysj.54.2.274
.
Bree
T.
1978
The stability of parameter estimation in the general linear model
.
J. Hydrol.
37
(
1–2
),
47
66
.
https://doi.org/10.1016/0022-1694(78)90095-1
.
Brooks
K. N.
,
Ffolliott
P. F.
&
Magner
J. A.
2012
Hydrology and the Management of Watersheds
.
John Wiley & Sons
,
Oxford, UK
.
Chakraborty
S.
&
Chakravarty
D.
2014
A discrete Gumbel distribution.’ arXiv preprint arXiv:1410.7568. https://doi.org/10.48550/arXiv.1410.7568
.
Chow
V. T.
,
Maidment
D. R.
&
Mays
L. W.
1988
Applied Hydrology
.
Macgraw-Hill Inc
,
Singapore
.
Ciepielowski
A
.
1987
Statistical methods of determining typical winter and summer hydrographs for ungauged watersheds
. In
Flood Hydrology
.
Dordrecht
,
Springer
, pp.
117
124
.
https://doi.org/10.1007/978-94-009-3957-8_10
.
Clarke
R. D.
1946
An application of the Poisson distribution
.
J. Inst. Actuaries.
72
(
3
),
481
481
.
https://doi.org/10.1017/S0020268100035435
.
Cleveland
T. G.
,
He
X.
,
Asquith
W. H.
,
Fang
X.
&
Thompson
D. B.
2006
Instantaneous unit hydrograph evaluation for rainfall-runoff modeling of small watersheds in north and south central Texas
.
J. Irrig. Drain. Engg.
132
(
5
),
479
485
.
https://doi.org/10.1061/(ASCE)0733-9437(2006)132 : 5(479)
Commons
C. G.
1942
Flood hydrographs
.
Civ. Engin.
12
,
571
572
.
Consul
P. C.
1990
On some properties and applications of quasi-binomial distribution
.
Communications Stat.-Theory Methods
19
(
2
),
477
504
.
https://doi.org/10.1080/03610929008830214
Consul
P. C.
&
Jain
G. C.
1971
On the log-gamma distribution and its properties
.
Statistische Hefte.
12
(
2
),
100
106
.
https://doi.org/10.1007/BF02922944
Croley
I. I. T. E.
1980
Gamma synthetic hydrographs
.
J. Hydrol.
47
,
41
52
.
https://doi.org/10.1016/0022-1694(80)90046-3
Cruise
J. F.
&
Contractor
D. N.
1978
Solution of 2-parameter gamma-model to relate unit hydrograph features to basin characteristics
.
Trans. Am. Geophys. Union.
59
(
4
),
273
273
.
DeCoursey
D. G.
1966
A Runoff Hydrograph Equation (Vol. 41)
.
Agricultural Research Service, USDA
,
Washington, DC
.
Dey
S.
,
Moala
F. A.
&
Kumar
D.
2018
Statistical properties and different methods of estimation of Gompertz distribution with application
.
J. Stat. Manag. Syst.
21
(
5
),
839
876
.
https://doi.org/10.1080/09720510.2018.1450197
Dooge
J. C. I.
1959
A general theory of the unit hydrograph
.
J. Geophys. Res.
64
(
2
),
241
256
.
https://doi.org/10.1029/JZ064i002p00241
Dooge
J. C. I.
1973
Linear Theory of Hydrologic Systems
.
Agricultural Research Service, USDA, No. 1468
,
Washington, DC
.
Edson
C. G.
1951
Parameters for relating unit hydrographs to watershed characteristics
.
Trans. Am. Geophys. Union.
32
(
4
),
591
596
.
https://doi.org/10.1029/TR032i004p00591
Edwards
A. W. F.
1960
The meaning of binomial distribution
.
Nature
186
(
4730
),
1074
.
http://dx.doi.org/10.1038/1861074a0
Evans
M.
,
Hastings
N.
&
Peacock
B.
2001
Statistical Distributions
, 3rd edn.
Wiley
,
New York
.
http://dx.doi.org/10.1088/0957-0233/12/1/702
Fisher
R. A.
1941
The negative binomial distribution
.
Ann. Eugen.
11
(
1
),
182
187
.
https://doi.org/10.1111/j.1469-1809.1941.tb02284.x
Forbes
C.
,
Evans
M.
,
Hastings
N.
&
Peacock
B.
2011
Statistical Distributions
.
John Wiley & Sons
,
Hoboken, NJ
.
Friedman
L. C.
,
Bradford
W. L.
&
Peart
D. B.
1983
Application of binomial distributions to quality assurance of quantitative chemical analyses
.
J. Env. Sci. Health Part A
18
(
4
),
561
570
.
https://doi.org/10.1080/10934528309375123
Gago-Benítez
A.
,
Fernández-Madrigal
J. A.
&
Cruz-Martín
A.
2013
Log-logistic modeling of sensory flow delays in networked telerobots
.
IEEE Sens. J.
13
(
8
),
2944
2953
.
https://doi.org/10.1109/JSEN.2013.2263381
Gavagnin
E.
,
Ford
M. J.
,
Mort
R. L.
,
Rogers
T.
&
Yates
C. A.
2019
The invasion speed of cell migration models with realistic cell cycle time distributions
.
J. Theo. Biol.
481
,
91
99
.
https://doi.org/10.1016/j.jtbi.2018.09.010
Geetha
K.
,
Mishra
S. K.
,
Eldho
T. I.
,
Rastogi
A. K.
&
Pandey
R. P.
2008
SCS-CN-based continuous simulation model for hydrologic forecasting
.
Water Resour. Manag.
22
(
2
),
165
190
.
https://doi.org/10.1007/s11269-006-9149-5
Ghorbani
M. A.
,
Singh
V. P.
,
Sivakumar
B.
,
Kashani
M. H.
,
Atre
A. A.
&
Asadi
H.
2017
Probability distribution functions for unit hydrographs with optimization using genetic algorithm
.
Appl. Water Sci.
7
(
2
),
663
676
.
https://doi.org/10.1007/s13201-015-0278-y
Gottschalk
L.
&
Weingartner
R.
1998
Distribution of peak flow derived from a distribution of rainfall volume and runoff coefficient, and a unit hydrograph
.
J. Hydrol.
208
(
3–4
),
148
162
.
https://doi.org/10.1016/S0022-1694(98)00152-8
Gray
D. M.
1961
Synthetic hydrographs for small drainage areas
.
J. Hydraul. Div. ASCE
87
(
4
),
33
54
.
https://doi.org/10.1061/JYCEAJ.0000631
Gray
D. M.
1962
Derivation of hydrographs for small watersheds from measurable physical characteristics
.
Iowa State Univ. Agric. Home Econ. Exp. Stn. Res. Bull.
506
,
514
570
.
https://doi.org/10.31274/rtd-180813-3274
Gumbel
E. J.
1960
Multivariate extreme distributions
.
Bull. Int. Stat. Inst.
39
(
2
),
471
475
.
Gupta
V. L.
&
Moin
S. A.
1974
Surface runoff hydrograph equation
.
J. Hydraul. Div.
100
(
10
),
1353
1368
.
https://doi.org/10.1061/JYCEAJ.0004075
Gupta
V. L.
,
Thongchareon
V.
&
Moin
S. A.
1974
‘Analytical modeling of surface runoff hydrographs for major streams in northeast Thailand
.
Hydrolog. Sci. Bulletin
XIX
,
523
540
.
Haan
C. T.
1977
Statistical Methods in Hydrology
.
The Iowa State University Press
,
Annes, IA
.
Haan
C. T.
,
Barfield
B. J.
&
Hayes
J. C.
1994
Design Hydrology and Sedimentology for Small Catchments
.
Elsevier
,
San Diego, CA
.
Haktanir
T.
&
Sezen
N.
1990
Suitability of two-parameter gamma and three-parameter beta distributions as synthetic unit hydrographs in Anatolia
.
Hydrolog. Sci. J.
35
(
2
),
167
184
.
https://doi.org/10.1080/02626669009492416
Ito
Y.
,
Momii
K.
,
Nakagawa
K.
&
Takagi
A.
2006
Estimation Model of Lake Level in Lake Ikeda-Hydrologic budget of a lake as water resources
.
Trans. Jpn. Soc. Irri., Drain., Recla. Eng.
74
(
3
),
65
72
.
https://doi.org/10.11408/jsidre1965.2006.341
James
W. P.
,
Winsor
P. W.
&
Williams
J. R.
1987
Synthetic unit hydrograph
.
J. Water Res. Pl.-ASCE
113
(
1
),
70
81
.
https://doi.org/10.1061/(ASCE)0733-9496(1987)113:1(70)
Jiménez
F.
&
Jodrá
P.
2008
A note on the moments and computer generation of the shifted Gompertz distribution
.
Commun. Stat. Theory Methods
38
(
1
),
75
89
.
https://doi.org/10.1080/03610920802155502
Johnson
N. L.
,
Kotz
S.
&
Balakrishnan
N.
1994
Continuous Univariate Distributions
, 2nd edn, Vol.
I
.
John Wiley & Sons
,
New York
.
Johnson
N. L.
,
Kotz
S.
&
Balakrishnan
N.
1995
Continuous Univariate Distributions
, 2nd edn, Vol.
II
.
John Wiley & Sons
,
New York
.
Kleiber
C.
2008
A guide to the Dagum distributions
. In
Modeling Income Distributions and Lorenz Curves
.
Springer
,
New York
. pp.
97
117
.
https://doi.org/10.1007/978-0-387-72796-7_6
Kotz
S.
&
Nadarajah
S.
2000
Extreme Value Distributions: Theory and Applications
.
Imperial College Press
,
London, UK
.
Koutsoyiannis
D.
&
Xanthopoulos
T.
1989
On the parametric approach to unit hydrograph identification
.
Water Resour. Manag.
3
(
2
),
107
128
.
https://doi.org/10.1007/BF00872467
Krishnamoorthy
K.
2016
Handbook of Statistical Distributions with Applications
.
CRC Press
,
Boca Raton, FL
.
Langbein
W. B.
1940
Channel-storage and unit-hydrograph studies
.
Trans. Am. Geophys. Union.
21
(
2
),
620
627
.
https://doi.org/10.1029/TR021i002p00620
Lasdon
L.
,
Waren
A. D.
,
Jain
A.
&
Ratner
M.
1978
Design and testing of a generalized reduced gradient code for nonlinear programming
.
ACM Trans. Math. Softw. (TOMS)
4
(
1
),
34
50
.
https://doi.org/10.1145/355769.355773
Lienhard
J. H.
1964
A statistical mechanical prediction of the dimensionless unit hydrograph
.
J. Geophys. Res.
69
(
24
),
5231
5238
.
https://doi.org/10.1029/JZ069i024p05231
Lienhard
J. H.
&
Meyer
P. L.
1967
A physical basis for the generalized gamma distribution
.
Q. Appl. Math.
25
(
3
),
330
334
.
https://doi.org/ 10.1090/qam/99884
Lienhard
J. H.
&
Davis
L. B.
1971
An extension of statistical mechanics to the description of a broad class of macroscopic systems
.
Zeitschrift für Angewandte Mathematik und Physik ZAMP
22
(
1
),
85
96
.
https://doi.org/10.1007/BF01624055
Lohani
A. K.
,
Singh
R. D.
&
Nema
R. K.
2001
Comparison of Geomorphological Based Rainfall-Runoff Models
.
CS(AR) Technical Report
,
National Institute of Hydrology
,
Roorkee, India
.
Mani
P.
&
Panigrahy
N.
1998
Geomorphological Study of Myntdu-Leska River Basin
.
Tech. Report CS(AR)7/97-98
.
National Institute of Hydrology
,
Roorkee, India
.
McDonnell
J. J.
&
Tanaka
T.
2001
On the future of forest hydrology and biogeochemistry
.
Hydrol. Process.
15
(
10
),
2053
2055
.
https://doi.org/10.1002/hyp.493
Meadows
M. E.
&
Ramsey
E. W.
1991a
User's Manual for a Unit Hydrograph Optimization Program. U.S. Geological Survey Project Completion Report Volume I, 64
.
US Geological Survey
,
Reston, VA
.
Meadows
M. E.
&
Ramsey
E. W.
1991b
South Carolina Regional Synthetic Unit Hydrograph Study: Methodology and Results. U.S. Geological Survey Project Completion Report Volume. 2, 33
.
US Geological Survey
.
Reston, VA
.
Mockus
V.
1957
Use of Storm and Watershed Characteristics in Synthetic Hydrograph Analysis and Application
.
US Department of Agriculture, Soil Conservation Service
,
Washington, DC
.
Mudasir
S.
&
Ahmad
S. P.
2017
Characterization and information measures of weighted Erlang distribution
.
J. Stat. Appl. Probab. Lett.
4
,
91
95
.
https://doi.org/10.18576/jsapl/040302
Nadarajah
S.
2007
Probability models for unit hydrograph derivation
.
J. Hydrol.
344
(
3–4
),
185
189
.
https://doi.org/10.1016/j.jhydrol.2007.07.004
Nash
J. E.
1957
The form of the instantaneous unit hydrograph
.
Int. Assoc. Hydrol. Sci.
3
,
114
121
.
Nash
J. E.
1958
Determining run-off from rainfall
.
Proc. Inst. Civil Eng.
10
(
2
),
163
184
.
https://doi.org/10.1680/iicep.1958.2025
Nash
J. E.
1959
The effect of flood-elimination works on the flood frequency of the river Wandle
.
Proc. Inst. Civil Eng.
13
(
3
),
317
338
.
https://doi.org/10.1680/iicep.1959.12059
Nash
J. E.
1960
A unit hydrograph study, with particular reference to British catchments
.
Proc. Inst. Civil Eng.
17
(
3
),
249
282
.
https://doi.org/10.1680/iicep.1960.11649
Oluyede
B. O.
&
Rajasooriya
S.
2013
The Mc-Dagum distribution and its statistical properties with applications
.
Asian J. Math. Appl.
1
.
https://doi.org/10.1.1.925.7006
Phien
H. N.
&
Jivajirajah
T.
1984
The transformed gamma distribution for annual streamflow frequency analysis
. In
Proc. Fourth Congress of IAHR-APD on WRD&M
,
Chiang Mai, Thailand, Vol.
2
, pp.
1151
1166
.
Pollard
J. H.
&
Valkovics
E. J.
1992
The Gompertz distribution and its applications
.
Genus
48
(
3–4
),
15
28
.
Rai
R. K.
,
Jain
M. K.
,
Mishra
S. K.
,
Ojha
C. S. P.
&
Singh
V. P.
2007
Another look at Z-transform technique for deriving unit impulse response function
.
Water Resour. Manag.
21
(
11
),
1829
1848
.
https://doi.org/10.1007/s11269-006-9132-1
Rai
R. K.
,
Sarkar
S.
&
Gundekar
H. G.
2008
Adequacy of two-parameter beta distribution for deriving the unit hydrograph
.
Hydrol. Res.
39
(
3
),
201
208
.
https://doi.org/10.2166/nh.2008.038
Rai
R. K.
,
Sarkar
S.
&
Singh
V. P.
2009
Evaluation of the adequacy of statistical distribution functions for deriving unit hydrograph
.
Water Resour. Manag.
23
(
5
),
899
929
.
https://doi.org/10.1007/s11269-008-9306-0
Rai
R. K.
,
Sarkar
S.
,
Upadhyay
A.
&
Singh
V. P.
2010
Efficacy of Nakagami-m distribution function for deriving unit hydrograph
.
Water Resour. Manag.
24
(
3
),
563
575
.
https://doi.org/10.1007/s11269-009-9459-5
Reed
L. A.
1976
Hydrology and Sedimentation of Bixler Run Basin, Central Pennsylvania (No. 1798-N)
.
US Government Printing Office
,
Washington, DC
.
Reich
B. M.
1962
Design hydrographs for very small watersheds from rainfall
.
Doctoral dissertation, Colorado State University, Fort Collins, CO
.
Rendon-Herrero
O.
1978
Unit sediment graph
.
Water Resour. Res.
14
(
5
),
889
901
.
https://doi.org/10.1029/WR014i005p00889
Rosso
R.
1984
Nash model relation to Horton order ratios
.
Water Resour. Res.
20
(
7
),
914
920
.
https://doi.org/10.1029/WR020i007p00914
Sade
M.
2001
A gamma distribution unit hydrograph for flat terrain watersheds. In: Bridging the Gap: Meeting the World's Water and Environmental Resources Challenges
. In
Proceedings of the World Water and Environmental Resource Congress
, pp.
1
7
.
Sadooghi-Alvandi
S. M.
1990
Estimation of the parameter of a Poisson distribution using a LINEX loss function
.
Aust. J. Stat.
32
(
3
),
393
398
.
https://doi.org/10.1111/j.1467-842X.1990.tb01033.x
Sahoo
B.
,
Chatterjee
C.
,
Raghuwanshi
N. S.
,
Singh
R.
&
Kumar
R.
2006
Flood estimation by GIUH-based Clark and Nash models
.
J. Hydrol. Eng.
11
(
6
),
515
525
.
https://doi.org/10.1061/(ASCE)1084-0699(2006)11 : 6(515)
SCS
.
1972
Soil Conservation Service-National Engineering Handbook
.
U.S. Department of Agriculture
,
Washington, DC
.
Shahzad
M. N.
&
Asghar
Z.
2016
Transmuted Dagum distribution: A more flexible and broad shaped hazard function model
.
Hacettepe J. Math. Stat.
45
(
1
),
1
18
.
https://doi.org/10.15672/HJMS.2015529452
Sherman
L. K.
1932
Streamflow from rainfall by the unit-graph method
.
Eng. News Record.
108
,
501
505
.
Shoukri
M. M.
,
Mian
I. U. H.
&
Tracy
D. S.
1988
Sampling properties of estimators of the log-logistic distribution with application to Canadian precipitation data
.
Can. J. Stat.
16
(
3
),
223
236
.
https://doi.org/10.2307/3314729
Singh
V. P.
1981
Rainfall-runoff relationship
. In
Proceedings of the International Symposium on Rainfall-Runoff Modeling, Mississippi State University
,
May 1981
.
Water Resources Publications
.
Littleton, CO
.
Singh
V. P.
1982
Statistical Analysis of Rainfall and Runoff
.
Water Resources Publications
,
Littleton, CO
.
Singh
V. P.
1987
On application of the Weibull distribution in hydrology
.
Water Resour. Manag.
1
(
1
),
33
43
.
https://doi.org/10.1007/BF00421796
Singh
V. P.
1988
Hydrologic Systems. Volume I: Rainfall-Runoff Modeling
.
Prentice Hall
,
Englewood Cliffs, NJ
, p.
480
.
Singh
S. K.
2000
Transmuting synthetic unit hydrographs into gamma distribution
.
J. Hydrol. Eng.
5
(
4
),
380
385
.
https://doi.org/10.1061/(ASCE)1084-0699(2000)5 : 4(380)
Singh
S. K.
2004
Simplified use of gamma-distribution/Nash model for runoff modeling
.
J. Hydrol. Eng.
9
(
3
),
240
243
.
https://doi.org/10.1061/(ASCE)1084-0699(2004)9 : 3(240)
Singh
S. K.
2005
Clark's and Espey's unit hydrographs vs the gamma unit hydrograph
.
Hydrolog. Sci. J.
50
(
6
).
https://doi.org/10.1623/hysj.2005.50.6.1053
Singh
S. K.
2006
Optimal instantaneous unit hydrograph from multistorm data
.
J. Irrig. Drain. E.-ASCE.
132
(
3
),
298
302
.
https://doi.org/10.1061/(ASCE)0733-9437(2006)132 : 3(298)
Singh
S. K.
2007a
Use of gamma distribution/Nash model further simplified for runoff modeling
.
J. Hydrol. Eng.
12
(
2
),
222
224
.
https://doi.org/10.1061/(ASCE)1084-0699(2007)12 : 2(222)
Singh
S. K.
2007b
Identifying representative parameters of IUH
.
J. Irrig. Drain. E-ASCE.
133
(
6
),
602
608
.
https://doi.org/10.1061/(ASCE)0733-9437(2007)133 : 6(602)
Singh
V. P.
2011
An IUH equation based on entropy theory
.
Trans. ASABE.
54
(
1
),
131
140
.
https://doi.org/10.13031/2013.36267
Singh
S. K.
2015
Simple parametric instantaneous unit hydrograph
.
J. Irrig. Drain. E-ASCE.
141
(
5
),
04014066
.
https://doi.org/10.1061/(ASCE)IR.1943-4774.0000830
Singh
P. K.
,
Mishra
S. K.
&
Jain
M. K.
2014
A review of the synthetic unit hydrograph: From the empirical UH to advanced geomorphological methods
.
Hydrolog. Sci. J.
59
(
2
),
239
261
.
https://doi.org/10.1080/02626667.2013.870664
Sokolov
A. A.
,
Rantz
S. E.
&
Roche
M.
1976
Methods of Developing Design-Flood Hydrographs. Flood Computation Methods Compiled From World Experience
.
UNESCO
,
Paris
.
Stacy
E. W.
&
Mihram
G. A.
1965
Parameter estimation for a generalized gamma distribution
.
Technometrics.
7
(
3
),
349
358
.
https://doi.org/10.1080/00401706.1965.10490268
.
Talukdar
K. K.
&
Lawing
W. D.
1991
Estimation of the parameters of the rice distribution
.
J. Acoust. Soc. Am. JASA.
89
(
3
),
1193
1197
.
https://doi.org/ 10.1121/1.400532
Tokar
A. S.
&
Markus
M.
2000
Precipitation-runoff modeling using artificial neural networks and conceptual models
.
J. Hydrol. Eng.
5
(
2
),
156
161
.
https://doi.org/ 10.1061/(ASCE)1084-0699(2000)5:2(156)
Torres
F. J.
2014
Estimation of parameters of the shifted Gompertz distribution using least squares, maximum likelihood and moments methods
.
J. Comput. Appl. Math.
255
,
867
877
.
https://doi.org/10.1016/j.cam.2013.07.004
Williams
H. M.
1945
‘Discussion, military airfields: Design of drainage facilities, by Hathaway G. A
.
Amer. Soc. Civ. Engin. Trans.
110
,
820
826
.
Wu
I. P.
1963
Design hydrographs for small watersheds in Indiana
.
J. Hydraul. Div.
89
(
6
),
35
66
.
https://doi.org/10.1061/JYCEAJ.0000968
Yacoub
M. D.
,
Bautista
J. V.
&
de Rezende Guedes
L. G.
1999
On higher order statistics of the Nakagami-m distribution
.
IEEE Trans. Vehicular Technol.
48
(
3
),
790
794
.
https://doi.org/10.1109/25.764995
Yevjevich
V.
&
Obeysekera
J. T.
1984
Estimation of skewness of hydrologic variables
.
Water Resour. Res.
20
(
7
),
935
943
.
https://doi.org/10.1029/WR020i007p00935
.
Yoshitani
J.
,
Chen
Z. Q.
,
Kavvas
M. L.
&
Fukami
K.
2009
Atmospheric model-based streamflow forecasting at small, mountainous watersheds by a distributed hydrologic model: Application to a watershed in Japan
.
J. Hydrol. Eng.
14
(
10
),
1107
1118
.
https://doi.org/10.1061/(ASCE)HE.1943-5584.0000111
Yue
S.
,
Ouarda
T. B.
,
Bobée
B.
,
Legendre
P.
&
Bruneau
P.
2002
Approach for describing statistical properties of flood hydrograph
.
J. Hydrol. Eng.
7
(
2
),
147
153
.
https://doi.org/10.1061/(ASCE)1084-0699(2002)7 : 2(147)
Walck
C.
1996
Hand-book on Statistical Distributions for Experimentalists
.
(No. SUF-PFY/96–01). University of Stockholm, Stockholm
.
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