Proficiency of probability distributions in unit hydrograph derivation

PDF

Probability distribution function

UH

Unit hydrograph

q_p

Peak discharge

t_p

Time to peak

t_b

Time base

q_v

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

L_C

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

β

Non-dimensional parameter

R_SO

Overall shape-based rank

R_S

Event-wise shape-based rank

Rq_pO

Overall q_p-based rank

Rt_pO

Overall t_p-based rank

Rq_p

Event-wise q_p-based rank

Rt_p

Event-wise t_p-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

R_NSE

Event-wise NSE-based rank

R_R²

Event-wise R²-based rank

R_MSE

Event-wise MSE-based rank

R_RMSE

Event-wise RMSE-based rank

R_MAE

Event-wise MAE-based rank

R_MAPE

Event-wise MAPE-based rank

R_SE

Event-wise SE-based rank

R_RE

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 (q_p), time to peak (t_p), time base (t_b), and runoff volume (q_v), 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 (q_p or t_p) 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 q_p, t_p, t_b, 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.

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

Table 2

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

Sl.No. .	Distribution and reference .	PDF .	tp .	qp .
1	Shifted Gompertz (Torres 2014)
2	Dagum (Kleiber 2008)
3	Rice (Talukdar & Lawing 1991)
4	Rayleigh (Cleveland et al. 2006)
5	Erlang (Mudasir & Ahmad 2017)
6	Incomplete-Gamma (Sade 2001)
7	Kumaraswamy (Nadarajah 2007)
8	Weibull (Nadarajah 2007)
9	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 reference .	PDF .	tp .	qp .
1	Shifted Gompertz (Torres 2014)
2	Dagum (Kleiber 2008)
3	Rice (Talukdar & Lawing 1991)
4	Rayleigh (Cleveland et al. 2006)
5	Erlang (Mudasir & Ahmad 2017)
6	Incomplete-Gamma (Sade 2001)
7	Kumaraswamy (Nadarajah 2007)
8	Weibull (Nadarajah 2007)
9	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)

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Eighteen UHs were employed, each characterized by a unique time step (Δt) and properties such as q_p, t_p, q_v, 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.

A symmetrical, equally fragmented actual UH (t_p = 10 h) was distributed over t_b (=20 h) (Figure 5(b)). Both event-specific-R_S and overall-R_SO showed a good agreement with R² = 0.85 (Table 3). Some PDFs shared the same rank (Rt_p) due to identical RE(t_p), resulting in Rt_pO ranging from 1 to 16 (Table 4). This pattern was similarly observed in shape prediction, as reflected by the assignment of identical rank, R_SO = 19 (Table 3). The discussion here focuses on the top five ranked (R_S) 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 R_SO of 1 (Table 3). Despite its overall performance, Dagum lags in estimating q_p (Rq_pO = 3) and t_p (Rt_pO = 5, as of 2-PGD and Log-Logistic) (Table 4). In the specific event (Table 5), Dagum UH precisely captures t_p [RE(t_p) = 0, Rt_p = 1] but underestimates q_p [RE(q_p) = 5.17%, PBIAS = 2.9%, Rq_p = 4] due to overestimation at the inflection point and tail end, aiming to conserve unit volume (q_v = 0.967) over a given t_b (Figure 5(b)).

Following Dagum, the Log-Logistic PDF maintains a comparable q_v = 0.973, achieving R_S = 2, Rq_p = 5 (RE(q_p) = 5.22%) and Rt_p = 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 R_SO = 3, Rq_pO = 2 and Rt_pO = 5 (Tables 3 and 4).

Underestimating Gumbel Type-I UH at the head and peak results in R_S = 3, Rq_p = 11 (RE(q_p) = 7%), Rt_p = 1, and PBIAS = 2.1% (Table 5). Despite the peak underestimation, overestimation at the inflection point and tail end aids volume conservation (q_v = 0.979) (Figure 5(b)) achieving R_SO = 4, Rq_pO = 7, and Rt_pO = 3 (Tables 3 and 4).

Having R_S = 4, Rq_p = 10 (RE(q_p) = 6.66%), and Rt_p = 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 R_SO = 5, Rq_pO = 8, and Rt_pO = 4 (Tables 3 and 4).

The broad crest of the Generalized Gamma UH reduces the peak (Rq_p = 14, RE(q_p) = 7.43%) precisely at t_p (Rt_p = 1) during a given t_b, resulting in minimal volumetric shortfall (q_v = 0.941) (Table 5). This leads to overestimation before and after the crest, along with underestimation at the tail end, indicated by R_S = 5 and PBIAS = 0.8% (Figure 5(b), Table 5). The Generalized Gamma achieves R_SO = 2, Rq_pO = 12, and Rt_pO = 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, R_SO range (Tables 3 and 5). Subsequent PDFs are ranked based on their respective indices (Tables 3–5). In the case of the positively skewed Bixler Run basin UH (16 April 1961) with t_b = 14 h and t_p = 6 h, R_S and R_SO demonstrate strong agreement with R² = 0.87 (Table 3). However, for the negatively skewed UH (6 March 1963) with t_b = 24 h and t_p = 14 h, R_S and R_SO show poor agreement (R² = 0.10, Table 3).

Among the 12 new PDFs, 5 achieved top 5 rankings, R_SO is discussed (Table 3). The Dagum PDF secured the 1st rank, R_SO 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 (t_b = 14 h and t_p = 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(q_p) = 5.71%, RE(t_p) = 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 q_p − t_p estimation, it exhibits volumetric efficiency (q_v ≈ 1). Notably, for this specific event (21 June 1956), R_S shows poor agreement (R² = 0.53) with R_SO (Table 3).

R_SO = 2 confirms Generalized Gamma's second rank, trailing Dagum, with ranks (R_S) ranging from 1st (over 4 UHs) to 5th (R_S) across 12 out of 18 UHs (Table 3). Notably, its highest (i.e. 10th) rank (R_S) against a positively skewed UH (t_b = 14 h and t_p = 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(q_p) = 6.51%, RE(t_p) = 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 q_p and t_p, as indicated by Rq_pO = 12 and Rt_pO = 6 (Table 4).

The Log-Logistic PDF achieves the second rank (R_S) across seven events, displaying exceptional performance in replicating negatively skewed UH (6 March 1963) [R_S-16, RMSE = 0.009, NSE = 94.5%, R² = 0.946, MSE = 0.00, MAE = 0.007, MAPE = 54.14%, SE = 0.009, RE(q_p) = 2.58%, RE(t_p) = 14.29%]. For other events, its rank ranges up to 9th (Table 3). Despite having the same RE(t_p) as the superior Dagum PDF [R_S-1, RMSE = 0.003, NSE = 99.4%, R² = 0.994, MSE = 0.00, MAE = 0.002, MAPE = 10.54%, SE = 0.003, RE(q_p) = 1.06%, RE(t_p) = 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 q_p (Rq_p = 2) and t_p (Rt_p = 5, equivalent to Dagum and 2-PGD) (Table 4). Consequently, it demonstrates superiority over Dagum and Generalized Gamma in q_p − t_p estimation.

Gumbel Type-I demonstrates performance comparable to Log-Logistic in handling negatively skewed UH [6 March 1963, R_S-20, RMSE = 0.011, NSE = 91.9%, R² = 0.920, MSE = 0.00, MAE = 0.009, MAPE = 64.55%, SE = 0.011, RE(q_p) = 6.59%, RE(t_p) = 14.29%]. While it attains the 1st rank (R_S) 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 q_p and the 3rd rank for t_p (Table 4).

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

The Inverse Gaussian performs satisfactorily, securing ranks from 3rd to 8th (R_S) 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 (R_S = 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 q_p and t_p estimation, the Inverse Gaussian secures the 10th and 6th ranks, respectively (Table 4).

With R_SO = 7, the 2-PGD (Table 3) appears relatively less efficient [R_S-15, RMSE = 0.008, NSE = 95.2%, R² = 0.953, MSE = 0.00, MAE = 0.007, MAPE = 46.50%, SE = 0.008, RE(q_p) = 3.67%, RE(t_p) = 14.29%] compared to Dagum (R_S-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 (q_v = 0.99) than Dagum (q_v = 0.98). Excluding this event, 2-PGD demonstrates its capability across other events, with R_S ranging from 2nd to 11th (Table 3). It shares the same Rt_pO (5th) as Dagum and Log-Logistic while following Generalized Gamma with Rq_pO = 13th (Table 4).

The Log-Normal PDF, with an R_SO of 8th (Table 3), holds ranks, R_S from 3rd to 11th, except an exceptional R_S = 19th [RMSE = 0.010, NSE = 92.7%, R² = 0.928, MSE = 0.001, MAE = 0.001, MAPE = 59.20%, SE = 0.010, RE in q_p = 3.37%, RE in t_p = 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 q_p and 7th for t_p (Table 4).

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

The Negative Binomial, Chi-square, Poisson, and Binomial PDFs exhibit unacceptable performance with negative NSE and the highest RE(q_p), resulting in R_S of 26, 28, 29, and 30, respectively. This is evident in defining a positively skewed UH (t_p = 17 h, Kurtleravsari Creek basin) with the longest t_b (=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 t_b (≈40 h). In contrast, by underestimating the peak [RE(q_p) = 44.91%] at the exact t_p over an identical t_b as observed, the 3-Parameter 2-Sided Power-based triangular UH attains identical but unacceptable rank, R_S = 26 as the Negative Binomial, while conserving the approximate volume (q_v = 0.980) with a positive NSE (=54.8%).

The novel Negative Binomial, Poisson, and Binomial PDFs hold ranks (R_SO) of 16th, 21st, and 23rd, respectively (Table 3). Chi-square (R_SO = 26th) and 3-Parameter 2-Sided Power (R_SO = 29th, attributed to triangular UH fitted) follow them. The Simple 3-Parameter PDF secures the last (30th) rank based on shape and q_p (Tables 3 and 4), owing to its underestimated concave peaks against 10 out of 18 observed UHs. In terms of t_p, the Gompertz PDF attains the last (16th) rank among all PDFs (Table 4). When t_b is computed, q_v is either underestimated or overestimated, justifiable by positive or negative PBIAS, respectively. A PBIAS = 0 indicates unit volume conservation over the given t_b. 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.

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 (R_SO = 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 q_p and t_p. However, the widely-used 2-PGD falls short in accurately estimating q_p while it reasonably estimates t_p.

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 ‘q_p and t_p’ 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 t_b.

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.