Introduction

Surface runoff is a function of many variables such as rainfall duration and intensity, soil moisture, land use/land cover, soil infiltration capacity, watershed slope etc. A number of models exist in the literature that consider the effect of different variables on surface runoff. Among them, the lumped conceptual models are quite useful for simple yet realistic analyses (Mishra and Singh 2003). The Soil Conservation Service curve number (SCS-CN) method (presently also known as the Natural Resources Conservation Service curve number (NRCS-CN) method) is widely used for predicting surface runoff from small agricultural watersheds, primarily because of its simplicity and the requirement of only two parameters for runoff prediction (Ponce and Hawkins 1996), which are the initial abstraction coefficient (λ) and the potential maximum retention (S) expressed in terms of curve number (CN).

In the course of continuous use of the SCS-CN model worldwide, several modifications have been proposed in the literature (Hawkins et al. 1985; Jain et al. 2006a; Mishra and Singh 2003; Mishra et al. 2006a; Sahu et al. 2010b, 2012; Suresh Babu and Mishra 2012; Woodward et al. 2002). These include the effect of slope (Huang et al. 2006; Lal et al. 2015; Sharpley and Williams 1990); improvement in λ (Hawkins et al. 2002; Jain et al. 2006b; Mishra and Singh 2004; Mishra et al. 2006b; Woodward et al. 2004; Yuan et al. 2014); the antecedent moisture on a continuous basis (Ajmal et al. 2015d, 2016; Durbude et al. 2011; Michel et al. 2005; Sahu et al. 2007; Singh et al. 2015); and the antecedent moisture for estimation of initial abstraction, I a (Mishra and Singh 2002; Mishra et al. 2006b; Sahu et al. 2012).

In practice, for ungauged watersheds, CNs are derived from the well-known National Engineering Handbook (NEH) tables using watershed characteristics such as hydrologic soil group (HSG), land use and land condition, and antecedent moisture condition (AMC). Empirical evidence however shows that the use of CN values from the handbook’s chapter 4 (NEH-4) tables normally over-designs the hydrological systems (Schneider and McCuen 2005) and, therefore, use of CN values based on observed rainfall (P) and runoff (Q) data (hereafter termed “P–Q” data) is recommended (Ajmal et al. 2015a; Hawkins 1993). It has been established that CN is not constant for a watershed, but rather has a variable identity which varies with rainfall (Hjelmfelt et al. 1982; McCuen 2002). For a set of observed P–Q data, various approaches for determining CN have been reported in the literature (Bonta 1997; Hjelmfelt 1980; Hauser and Jones 1991; Hawkins 1993; Hawkins et al. 2002, 2009; Sneller 1985; Van Mullem et al. 2002; Woodward et al. 2006; Yuan 1933). Of late, some studies have examined the accuracy of such methods (Ali and Sharda 2008; D’Asaro and Grillone 2012; D’Asaro et al. 2014; Feyereisen et al. 2008; Schneider and McCuen 2005; Stewart et al. 2012; Tedela et al. 2012) relative to CN values in the NEH-4 tables (D’Asaro et al. 2014; Fennessey 2000; Feyereisen et al. 2008; Hawkins 1984; Hawkins and Ward 1998; Sartori et al. 2011; Stewart et al. 2012; Titmarsh et al. 1989, 1995, 1996; Tedela et al. 2012; Taguas et al. 2015). However, in spite of wide-spread use of all approaches, there is no agreed procedure for estimating CN from observed P–Q data (Soulis and Valiantzas 2013) because none shows any particular advantage (Ali and Sharda 2008; Tedela et al. 2008).

An accurate assessment of the initial abstraction coefficient (λ) is essential as it is one of the crucial parameters used in watershed P–Q estimation. It largely depends on regional (i.e. geologic and climatic factors) conditions of the watershed (Mishra and Singh 2003; Ponce and Hawkins 1996); and consists mainly of interception, infiltration, and surface depression storage during the early parts of a storm (Taguas et al. 2015). The standard assumption of λ = 0.2 in the original SCS-CN equation has been frequently questioned by various researchers since its inception (Aron et al. 1977; Baltas et al. 2007; Cazier and Hawkins 1984; D’Asaro and Grillone 2012; D’Asaro et al. 2014; Elhakeem and Papanicolaou 2009; Fu et al. 2011; Hawkins and Khojeini 2000; Hawkins et al. 2002; Mishra and Singh 2004; Menberu et al. 2015; Shi et al. 2009; Woodward et al. 2002, 2004; Yuan et al. 2014; Zhou and Lei 2011) for its validity and applicability, invoking its critical examination for practical applications. Many studies have indicated λ to be variable from watershed to watershed and event to event—see Table SI of the electronic supplementary material (ESM). Its value of about 0.05 or less is said to be more practical for various other parts of the world including the United States. Of late, nonlinear I a–S relations have also been suggested (Elhakeem and Papanicolaou 2009; Jiang 2001; Jain et al. 2006a; Mishra et al. 2004, 2006a). It is however of common experience that the value of λ loses its significance as rainfall increases by a magnitude significantly higher than I a, for which the existing SCS-CN method was developed, which is because of generally high CN (and low S) values for high and low rain events, respectively. Alternatively, I a is insignificant if P is high enough.

Evidently, only a few experimental studies have investigated (1) CN values from NEH-4 tables compared to those based on observed data and (2) the effect of λ on runoff prediction. No systematic experimental effort appears to have been made for Indian watersheds, which invokes a need for such study. Thus, the objectives of this study are to (1) assess the rainfall−runoff behaviour in study plots; (2) compare CN values from NEH-4 tables with those derived from observed data; (3) determine the optimal λ and S (or CN) values by analyzing data from 27 plots; (4) assess the performance of the traditional (λ = 0.2) SCS-CN method; and (5) study λ sensitivity to runoff estimates.

SCS-CN method

The SCS-CN method consists of the following equations:

$$ Q=\frac{{\left(P-{\displaystyle {I}_{\mathrm{a}}}\right)}^2}{\left(P+S-{\displaystyle {I}_{\mathrm{a}}}\right)}\ \mathrm{f}\mathrm{o}\mathrm{r}\ P>{I}_{\mathrm{a}};\ \mathrm{o}\mathrm{therwise}\ Q=0 $$
(1)

where Q (mm) is the direct surface runoff, P (mm) is the rainfall, I a is the initial abstraction (mm), and S (mm) is the potential maximum retention. In Eq. (1), I a is a fraction of S (i.e. I a = λS). Here, λ is the initial abstraction coefficient (λ = 0.2, a standard value). The use of I a = λS in Eq. (1) amplifies it as:

$$ Q=\frac{{\left(P-\lambda S\right)}^2}{\left(P+S-\lambda S\right)}\ \mathrm{f}\mathrm{o}\mathrm{r}\ P>\lambda S;\kern0.5em \mathrm{o}\mathrm{therwise}\ Q=0 $$
(2)

S can be calculated from observed P–Q data as follows (Hawkins 1973):

$$ S=\frac{\left(\left\{2\lambda P+\left(1-\lambda \right)Q\right\}-\sqrt{{\displaystyle {\left\{2\lambda P+\left(1-\lambda \right)Q\right\}}^2}-4{\displaystyle {\left(\lambda P\right)}^2}+4{\displaystyle {\lambda}^2}QP}\right)}{2{\displaystyle {\lambda}^2}} $$
(3)

S can be transformed into CN, and vice versa by using the following equation:

$$ \mathrm{C}\mathrm{N}=\frac{25400}{\left(S+254\right)} $$
(4)

In Eq. (4), S is in mm and CN is dimensionless.

Materials and methods

Site description

The study was conducted in an experimental field located at 29°50′09″ N and 77°55′21″ E, in Roorkee, district Haridwar, Uttarakhand (India) (Fig. 1). This field is located in the River Solani watershed, which is a sub-watershed of the River Ganga. River Solani emerges from the Shivalik range of the great Himalayas, which has three main topographic zones—hills, piedmont, and flat terrain. The study site is located in the flat terrain of the Solani watershed at about 30–60 km south of the foothills of the Himalayas and about 180 km north of New Delhi. The average topographic elevation of the site is about 266 m above mean sea level (amsl). The climate is humid sub-tropical type with three pronounced seasons, summer, monsoon and winter. In summer, the minimum and maximum monthly temperature values are generally 20 and 45 °C, respectively, whereas these are 10 and 27 °C, respectively, in winter. Annual rainfall varies from 1,120 to 1,500 mm and mostly concentrates between mid-June and mid-September, which is the monsoon season. The average annual potential evapotranspiration (PET) is of the order of 1,340 mm and average humidity varies from 30 to 99 %.

Fig. 1
figure 1

Layout of experimental plots located near Roorkee, Uttarakhand, India

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The soil in Solani watershed is mainly comprised of loam, loamy sand, sandy loams, and sandy clay. The upper hilly area consists of sandy loam, whereas lower flat terrain (where the study site is located) is dominated by loam and loamy sand (Garg et al. 2013; Kumar et al. 2012). Forestland, bare soil, and vegetated land are the main classes of land cover in the study area. Forest cover is around 30 % of the total area especially in the hilly part of the watershed and more than 50 % is agricultural land in the lower flat terrain. A significant portion of the land lies in an agricultural area with more than 35 % of vegetal cover. Forests cover around 30 % of the total area, especially in the hilly part of the watershed, and more than 17 % of the total land is fallow. Sugarcane is the perennial crop and wheat, maize, potato and pulses are the seasonal crops (Garg et al. 2013).

Experiment setup and data collection

The selected agricultural field for the experimental work was divided into plots of 22 m length and 5 m width. Four different land uses were selected: sugarcane, maize, black gram and fallow land. The plots were constructed in such a way that each land use was represented with three different slopes (5, 3 and 1 %). The experimental work was conducted during August 2012–April 2015 in which rainfall (P) and runoff (Q) were monitored for a total of 27 experimental plots of various slopes, land uses, and hydrologic soil groups (HSGs; i.e. infiltration capacity). It is worth emphasizing that, for cultivation of crops, normal agricultural practices of mixing soil, seed selection etc. were followed throughout the study period.

The surface runoff generated from each plot was collected in collection chambers of 1 m × 1 m × 1 m in size and constructed at the outlet of each plot followed by a 3-m-long conveyance channel intercepted by a multi-slot divisor with five slots. The multi-slot devisors were used to reduce the volume of runoff to be measured in the collection chamber—in other words, it reduces the frequency of chamber filling. The volume of flow collected in these tanks when multiplied by 5 yielded the plot runoff for a storm-event (during the past 24 hr). Rainfall was recorded with the help of both a tipping bucket rain gauge and a non-recording rain gauge installed at the study site. The distribution of rainfall measured during the study period is shown in Table 1. As seen from this table, 101 rainfall events were captured with the rainfall amount varying from 0.5 to 93.8 mm, and only 42 events produced a significant amount of runoff for measurement. Infiltration tests were conducted for each plot using a double-ring infiltrometer (45/30) for identification of HSGs (SCS 1972). The resulting infiltration capacity (fc) and corresponding HSGs for different plots are shown in Table 2.

Table 1 Rainfall characteristics during the study period (August 2012–April 2015)
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Table 2 Summary of runoff plot characteristics and CN values derived using the least-square method (LSM) and National Engineering Handbook tables (used partial dataset excluding P < 15 mm)
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Estimation of the curve number

NEH-4 table curve number

These CN values of a plot are designated as CNHT (HT refers to the handbook tables). The representative class II (average) antecedent moisture condition curve number, AMC II CN (or CN2), values were derived from NEH-4 tables (SCS 1972) for all the plots based on their land use, HSG, and vegetation (Table 2).

Rainfall–runoff data based curve number

Firstly, event-wise CNs were derived for each plot using Eqs. (3) and (4) (λ = 0.2). Secondly, S (or CN) (with λ = 0.2) was estimated from observed P–Q data using least square (LS) fit, i.e. by minimizing the sum of the squares of residuals (Eq. 5; NRCS 1997) employing Microsoft Excel (Solver):

$$ {\displaystyle \sum_i^n{\displaystyle {\left({\displaystyle {Q}_i}-{\displaystyle {Q}_{\mathrm{c}i}}\right)}^2}}={\displaystyle \sum {\left\{{\displaystyle {Q}_i}-\left[\frac{{\left(P-\uplambda S\right)}^2}{\left(\mathrm{P}+\left(1-\uplambda \right)S\right)}\right]\right\}}^2}\Rightarrow \mathrm{Minimum} $$
(5)

Here, Q i (mm) and Q ci (mm) are respectively the observed and predicted runoff for storm event i and n is the total number of storm events. These CN values of a plot are designated as CNLSn and CNLSo for natural and ordered datasets, respectively. The natural P–Q data consist of the actual observed dataset. In ordered data series, the observed P and Q values were first sorted separately and then realigned using a common rank-order basis to form a new set of P–Q pairs of an equal return period, in which runoff Q is not necessarily matched with that due to original rainfall P (Ajmal et al. 2015a; D’Asaro and Grillone 2012; Hawkins 1993; Hawkins et al. 2009; Lal et al. 2015; Soulis and Valiantzas 2013).

To derive λ values, both S and λ were optimized as before, consistent with the work of Hawkins et al. (2002), using both natural and ordered data consisting of only large storm events with a (arbitrary) P >15 mm criterion to avoid a biasing effect, but to retain a sufficient number of P–Q data for analysis. Only plots having at least 10 observed P–Q events were considered for the optimization study. Notably, model fitting yields only one value of λ from all P–Q events of the plot. The CN values (CNHT, CNLSn, and CNLSo) thus estimated are taken to correspond to the average antecedent moisture condition (AMC-II) of the plot. For wet (AMC-III) and dry (AMC-I) conditions, these CN values were adjusted using Eqs. (6) and (7), respectively, as follows (Hawkins et al. 1985):

$$ {\displaystyle {\mathrm{CN}}_{\mathrm{II}\mathrm{I}}}=\frac{{\displaystyle {\mathrm{CN}}_{\mathrm{II}}}}{0.427+0.00573{\displaystyle {\mathrm{CN}}_{\mathrm{II}}}} $$
(6)
$$ {\displaystyle {\mathrm{CN}}_{\mathrm{I}}}=\frac{{\displaystyle {\mathrm{CN}}_{\mathrm{I}\mathrm{I}}}}{2.281-0.01281{\displaystyle {\mathrm{CN}}_{\mathrm{I}\mathrm{I}}}} $$
(7)

In order to determine AMC of a rainfall event used in a runoff prediction, 5-day antecedent rainfall (P 5) was used as follows: AMC-I if P 5 < 35.56 mm in the growing season or P 5 < 12.7 mm in the dormant season, AMC-II if 35.56 ≤ P 5 ≤ 53.34 mm in the growing season or 12.70 ≤ P 5 ≤ 27.94 mm in the dormant season, and AMC-III if P 5 > 53.34 mm in the growing season or P 5 > 27.94 mm in the dormant season (Ajmal et al. 2015a, b, c; Mays 2005).

Performance evaluation

Performance of the existing SCS-CN model (Eq. 2) with traditional λ = 0.2 was compared with that employing an average λ = 0.030 value derived from the 27 natural P–Q plot-datasets (Table 2). The average is considered instead of the median as the former yielded the smallest standard error (Fu et al. 2011). Here, it is notable that all runoff-producing rainfall events only were used in analyzing the performance of λ = 0.03 over traditionally used λ = 0.20. The effect of variation in λ on CNs (or runoff) has been evaluated using data from five randomly selected plots (i.e. plots 1, 5, 12, 16 and 17 from Table 2). In addition, the relative change in estimated runoff with progressive changes in the λ-value was also analyzed as follows:

$$ \varDelta {\displaystyle {Q}_i}=\frac{\left({\displaystyle {Q}_{ci}}-{\displaystyle {Q}_{ca}}\right)}{{\displaystyle {Q}_{ca}}}\times 100 $$
(8)

where ΔQ i is the relative change of runoff at step i, and Q ci and Q ca are respectively the estimated runoff at step i and step a. Initially, λ = 0.2 was fixed for step a and then reduced by 10 % at each step down to 0.02, and runoff was estimated at each step using Eq. (2). The average CNLSo (=78.92) was estimated from event-based CNs of the 27-plotdata (Table 2) and was used for the S computation in Eq. (4). P = 30 mm was used in Eq. (2) due to its having the highest frequency of occurrence (Table 1).

Statistical analysis

The goodness of fit was evaluated using the coefficient of determination (R 2), root mean square error (RMSE), Nash-Sutcliffe efficiency coefficient (NSE; Nash and Sutcliffe 1970), number of times n t that the observed variability is greater than the mean error, and percent bias (PBIAS). R 2 is expressed as:

$$ {\displaystyle {R}^2}={\displaystyle {\left(\frac{{\displaystyle \sum_{i=1}^n\left({\displaystyle {Q}_i}-\overline{Q}\right)\left({\displaystyle {Q}_{ci}}-\overline{{\displaystyle {Q}_c}}\right)}}{{\displaystyle {\left[{\displaystyle \sum_{i=1}^n{\displaystyle {\left({\displaystyle {Q}_i}-\overline{Q}\right)}^2}}{\displaystyle \sum_{i=1}^n{\displaystyle {\left({\displaystyle {Q}_{ci}}-\overline{{\displaystyle {Q}_c}}\right)}^2}}\right]}^{0.5}}}\right)}^2} $$
(9)

where Q i (mm) and Q ci (mm) are respectively the observed and predicted runoff for storm event i, \( \overline{Q\mathrm{c}} \)(mm) is the average of predicted runoff for all storm events, n is the total number of storm events, and \( \overline{Q} \)(mm) is the average of observed runoff for all storm events. R 2 > 0.6 is considered as acceptable for satisfactory agreement between observed and predicted variables (Moriasi et al. 2007; Santhi et al. 2001; Van Liew et al. 2003).

NSE has been widely used to evaluate hydrological models (Ajmal et al. 2015a, b, c; EI-Sadek et al. 2001; Fentie et al. 2002; Sahu 2007; Sahu et al. 2007, 2010a; Shi et al. 2009; Yuan et al. 2014) and is expressed as follows:

$$ \mathrm{N}\mathrm{S}\mathrm{E}=\left(1-\frac{{\displaystyle \sum_{i=1}^n{\displaystyle {\left({\displaystyle {Q}_i}-{\displaystyle {Q}_{ci}}\right)}^2}}}{{\displaystyle \sum_{i=1}^n{\displaystyle {\left({\displaystyle {Q}_i}-\overline{Q}\right)}^2}}}\right) $$
(10)

According to Motovilov et al. (1999), Moriasi et al. (2007), Lim et al. (2006), Parajuli et al. (2007, 2009), Santhi et al. (2001), 0.75 NSE ≤ 1.0 indicates very good fit; 0.65 ˂ NSE ≤ 0.75, good fit; 0.50 ˂ NSE ≤ 0.65, satisfactory fit; and NSE ≤ 0.50 indicates an unsatisfactory fit. RMSE (Ajmal et al. 2015c; Deshmukh et al. 2013; Jain et al. 2006b; Mishra et al. 2004, 2006a; Sahu et al. 2007, 2010a) is defined as:

$$ \mathrm{RMSE}={\displaystyle {\left(\frac{1}{n}{\displaystyle \sum_i^n{\displaystyle {\left({\displaystyle {Q}_i}-{\displaystyle {Q}_{ci}}\right)}^2}}\right)}^{1/2}} $$
(11)

and n t is expressed as (Ritter and Muñoz-Carpena 2013):

$$ {\displaystyle {n}_{\mathrm{t}}}=\frac{\mathrm{SD}}{\mathrm{RMSE}}-1 $$
(12)

where SD is the standard deviation. n t ≥ 2.2 indicates very good agreement; 1.2 ≤ n t < 2.2 implies good; 0.7 ≤ n t < 1.2 shows satisfactory; and n t < 0.7 indicates an unsatisfactory fit.

Percent bias (PBIAS) measures average tendency of the estimated data to be larger or smaller than their observed data (Ajmal et al. 2015c; Gupta et al. 1999; Moriasi et al. 2007) and is expressed as:

$$ \mathrm{PBIAS} = \left[\frac{{\displaystyle \sum_{i=1}^n\left({\displaystyle {Q}_i}-{\displaystyle {Q}_{ci}}\right)\times 100}}{{\displaystyle \sum_{i=1}^n{\displaystyle {Q}_i}}}\right] $$
(13)

PBIAS indicates whether the method is consistently over-predicting or under-predicting—positive values indicate model underestimation, and negative values overestimation (Gupta et al. 1999; Moriasi et al. 2007; Yuan et al. 2014), while for perfect agreement, PBIAS = 0. According to Archibald et al. 2014; Donigian et al. 1983; Moriasi et al. 2007; Singh et al. 2004; Van Liew et al. 2003, PBIAS < ±10 % indicates very good fit; ±10 % ≤ PBIAS < ±15 %, good; ±15 % ≤ PBIAS < ±25 %, satisfactory; and PBIAS ≥ ±25 %, unsatisfactory fit.

To evaluate the improvement in performance of the modified model over the existing one, the r-statistic (Nash and Sutcliffe 1970; Ajmal et al. 2015d; Ajmal et al. 2016; Senbeta et al. 1999) is used and it is expressed as:

$$ r=\frac{\left({\displaystyle {\mathrm{NSE}}_2}-{\displaystyle {\mathrm{NSE}}_1}\right)}{\left(1-{\displaystyle {\mathrm{NSE}}_1}\right)}\times 100 $$
(14)

where NSE1 and NSE2 are the efficiencies due to the existing model and modified models, respectively. r > 10 % indicates significant improvement of the modified model (Senbeta et al. 1999).

In this study, performance evaluation is primarily based on R 2, NSE, n t, and PBIAS for individual plot data and then the arithmetic means of 27 values are taken as a rough estimate for the overall performance evaluation. The Kolmogorov-Smirnov test was used to assess the normality of data, and the non-parametric Kruskal-Wallis test to assess the significance level. Statistical analysis was carried out using SPSS version 20.0 (IBM 2011), and Microsoft Excel 2007 (Solver) was used for the least square fitting.

Results and discussion

Variation of rainfall threshold for runoff generation (I)

As in the aforementioned, the P–Q analysis is based on 42 natural P–Q events of the plots of different slopes, land uses, and HSGs observed during August 2012–April 2015 (i.e. three crop-growing seasons)—August 2012−May 2013, June 2013−May 2014, and June 2014−April 2015—at the experimental farm in Roorkee, India (Table 1). A total of 11, 18, and 13 runoff-producing events were captured during the first, second, and third years, respectively. In this study, the lowest rainfall value was 5.6 mm, which generated runoff in a year, whereas the highest rainfall of 17.6 mm did not generate runoff in another year, during which the highest storm rainfall was 75.8 mm.

The runoff initiation threshold, also known as rainfall threshold for runoff generation, was determined for each plot from daily P–Q data. Table 3 gives an overview of rainfall threshold (I) values and slope (m/m) of P–Q curves for all plots. As seen, both vary considerably among plots. The highest I was observed for the plots having HSGs A. In contrast, the lowest I was observed for the plots having HSGs C, whereas I for HSG B was in between HSGs A and C; thus, HSG (or indirectly soils infiltration capacity) seems to play a major role in controlling I in the plots.

Table 3 Rainfall threshold (I, mm) and slope of the rainfall–runoff curve along with mean runoff coefficient (Rcm) for each plot
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Variation of mean runoff coefficient (Rcm)

As seen from the preceding, the concept of I is also supported by response of runoff to rainfall, i.e. runoff coefficient which followed a similar pattern as I. The mean runoff coefficient (Rcm) was higher for the plots having HSGs C followed by B and A (Table 3). This pattern for Rcm was followed by nearly all the plots with few exceptions (i.e. plots 12 and 21); Rcm of the plots ranged from 0.093 to 0.473.

Runoff coefficients (Rc) for individual rainfall events also varied considerably from less than 0.005 to over 0.60, depending on the nature of the event and plot type. The Kolmogorov-Smirnov test revealed that event-wise Rc for all individual plots was not normally distributed. The non-parametric Kruskal-Wallis test revealed a statistically significant difference between events Rc of all 27 study plots.

Relation among Q, P and θ

Correlations of the runoff (Q) and Rc with rainfall (P) and previous-day soil moisture (θ) (%) were determined for each plot separately and the results are shown in Table 4. As seen, non−linear variation of Rc with P is similar to the variation of Q with P, but the correlation between Rc and P is much lower than that between Q and P. An example of a non-linear relation between Q and P, and between Rc and P, for plot Nos. 1, 8, and 11 are shown in Fig. 2. The P–Q relationship was statistically significant (p < 0.05) for all the plots. The highest correlation was observed in plot 8 (maize land use), with a coefficient of determination (R 2) of 0.980; the poorest (R 2 = 0.411) was in plot 23 (fallow land use). In contrast, θ did not correlate well with Q as well as did Rc in the study plots (Fig. 3), for R 2 ranging from 0.028 to 0.391. Theoretically, higher θ means higher Q (or Rc); however, in the present study, Q is largely controlled by P, consistent with the findings of Nadal-Romero et al. (2008), Rodríguez-Blanco et al. (2012), Scherrer et al. (2007), and Zhang et al. (2011), rather than θ.

Table 4 Coefficients of determination (R 2) of daily runoff (Q, mm) and runoff coefficients (Rc) with daily rainfall (P, mm) and previous-day soil moisture (θ, %)
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Fig. 2
figure 2

Plots showing relationship of a runoff (Q, mm) and b runoff coefficient (Rc) with rainfall (P, mm) for plot Nos. 1, 8 and 11

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Fig. 3
figure 3

Plots showing relationship of a runoff (Q, mm) and b runoff coefficient (Rc) with previous-day soil moisture (θ) (%) for plot Nos. 1, 8 and 11

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Effect of land use, infiltration capacity, and plot slope on Q (or Rc)

The effects of land use, infiltration capacity (fc), and slope on Q (or Rc) were also tested individually for their significance. To this end, plots located in the same land use, HSG, and slope were grouped separately to check their significance among studied variables. Since the data distribution fails to pass the normality test for all three of the individual groups (i.e. land use, HSG, and slope), the non-parametric Kruskal-Wallis test was used to test the significance level, whereby the results are shown in Table 5. The test revealed that land uses did not show any significant difference in Rc except sugarcane, which produced significantly (p < 0.05) higher Rc than blackgram and fallow land uses. In the case of HSGs, however, HSG C had significantly higher Rc than did B and A, but B and A did not differ from each other. In addition, slope did not show any effect on Rc as all three groups of slope were insignificantly different from each other. Thus, Rc (or Q) is more significantly influenced by infiltration capacity (fc) of soil rather than land uses or slopes. As shown in Fig. 4a, mean runoff (Q m) produced at the study plots was significantly (R 2 = 0.269; p < 0.01) influenced by soil permeability, described by infiltration capacity (fc). With an increase in fc, Q m decreased logarithmically, and vice versa.

Table 5 Mean event runoff coefficient (Rc) and CNs for the groups of different land uses, HSGs and slopes
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Fig. 4
figure 4

Relationship of a mean runoff (Q m, mm) and b curve number (CN), antecedent moisture conditions II (AMC II), with infiltration capacity (fc, mm/hr) of soil for all 27 agricultural plots

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P–Q data based CN determination

First, event-wise CN values were derived for individual plots using Eqs. (3) and (4) (with λ = 0.20). The derived CNs from all 27 plots were comparable. The Kolmogorov-Smirnov test revealed event-wise CNs for all individual plots to be normally distributed. The Student t-test for equality of means revealed a statistically significant difference between event CNs of all 27 study plots.

The effect of land use, infiltration capacity (fc), and slope on event-wise CNs was also studied using similar analysis (or tests) as discussed previously for Rc. As seen from Table 5, land uses did not show any significant difference in CNs except sugarcane, which produced significantly (p < 0.05) higher CNs than blackgram and fallow land uses. Furthermore, slope also did not show any effect on CNs as all three slope groups (i.e. 5, 3, and 1 %) were statistically insignificant. In the present study, CNs are seen to be influenced by the infiltration capacity (fc) of soil because all three groups of soil (i.e. A, B and C) exhibited significantly different CNs.

CNs were also estimated for 27 plots for both natural and ordered datasets using optimization (Eq. 5), and the results are presented in Table 2. As shown, CNs for study plots ranged widely from 64.73 to 90.33 and 67.47 to 90.59 for natural and ordered datasets, respectively. All 27 CNs, when combined into one group, were found to be normally distributed when tested using the Kolmogorov-Smirnov test.

As already analyzed, fc is the main explanatory variable for runoff production in the study plots. An inverse relationship between CN and fc for all 27 study plots was also detected with significant correlation (R 2 = 0.461, p < 0.01; Fig. 4b). The results from this analysis (Fig. 4b) support the applicability of NEH-4 tables where CNs decline with fc (or HSG).

Comparison of CNHT, CNLSn, and CNLSo

The NEH-4 curve numbers (CNHT) are compared with those due to both natural and ordered P–Q datasets observed on 27 plots (Table 2). CNHT ranged from 58 (plots 19, 20, and 21) to 88 (plots 25, 26, and 27). The optimized values of CNLSn ranged respectively from 64.73 (plot 19) to 90.33 (plot 25), and CNLSo from 67.47 to 90.59 for ordered dataset. Both CNLSn and CNLSo values were higher than CNHT (Table 2). As seen in Fig. 5a, CNHT and CNLSn do not compare well, as 17 out of 27 CNLSn values are higher than those for CNHT; and both exhibited a greater difference for values lower than 75; however, the difference diminishes with increasing values. The group of CNHT lower than 75 shows a higher PBIAS (= −12.84 %) than the group of CNHT higher than 75 (= 1.03 %). Overall, pair-wise comparison showed a significant difference (p < 0.05) existing between CNHT and CNLSn means. Such an inference is consistent with the general notion that the existing SCS-CN method performs better for high P–Q (or CN) events.

Fig. 5
figure 5

CN comparison for a CNLSn vs CNHT, b CNLSo vs CNHT, and c CNLSn vs CNLSo

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From Fig. 5b, CNHT with CNLSo compare similar to that in Fig. 5a; however, PBIAS of the group of CNHT lower than 75 is = −14.87 % compared to 0.12 % for the group higher than 75. From Fig. 5c, CNLSo values are seen to be higher than those for CNLSn, which is consistent with that reported elsewhere (Ajmal et al. 2015a; D’Asaro and Grillone 2012; D’Asaro et al. 2014; Hawkins et al. 2009; Stewart et al. 2012). CN values derived for individual plots using ordered datasets differ from 0.15 to 3.22 CN compared with those derived from natural data (Table 2). The trend between CNLSo and CNLSn allows a conversion as follows:

$$ {\mathrm{CN}}_{\mathrm{LSo}}=0.005\ {\left({\mathrm{CN}}_{\mathrm{LSn}}\right)}^2+0.163\ {\mathrm{CN}}_{\mathrm{LSn}}+37.449;{R}^2=0.990;\ \mathrm{S}\mathrm{E}=0.552\ \mathrm{C}\mathrm{N} $$
(15)

Table S2 of the ESM shows the performance statistic used to test the accuracy of all three sets of CNs, with respect to CNHT, CNLSn, and CNLSo for the data of 24 plots (plots 1–24 of Table 2). Plot Nos. 25–27 were excluded from comparison due to unavailability of their corresponding P 5 data. Both NSE and R 2 show the estimated runoff based on all three CNs to be poorly matching (except for a few plots) the observed runoff. In general, CNLSo performed the best of all, and CNLSn better than CNHT. The reason for CNHT to have performed the most poorly is that these are the generalized values derived from small watersheds of the United States for high-magnitude P–Q events (or high CN values). As seen from Table S2 of the ESM, for 15 out of 24 plots, a simple mean of the observed runoff was a better estimate (due to negative NSE values) than that due to CNHT, the estimates of which reasonably correlated (NSE > 0.50) with observed runoff for only two plots. Similarly, the mean of the observed runoff series was a better estimate for 8 out of 24 plots than that due to CNLSn or CNLSo; while the runoff estimated by CNLSn and CNLSo was reasonably close (NSE > 0.50) to the observed runoff for 5 and 9 plots, respectively.

From Fig. 5a,b, Table 2 and Table S2 of the ESM, it is evident that the general agreement between CNHT and CNLSn or CNLSo is poor, which is consistent with that reported elsewhere (D’Asaro et al. 2014; Fennessey 2000; Feyereisen et al. 2008; Hawkins 1984; Hawkins and Ward 1998; Sartori et al. 2011; Stewart et al. 2012; Titmarsh et al. 1989, 1995, 1996; Tedela et al. 2012; Taguas et al. 2015). As an alternative to CNHT, the best CN-values based on the highest R 2, NSE (or lowest RMSE; from Table S2 of the ESM) are suggested for each of the 24 plots. As seen, CNLSo ranked first for 20 out of 24 plots, whereas each of CNHT and CNLSn ranked first on only 2 plots. Therefore, CNLSo is suggested as a preference over CNHT for use in areas with similar plot characteristics and climatic conditions.

Derivation of λ

From Table 2, the optimized λ-values derived for both natural (ranging from 0 to 0.208) and ordered (ranging from 0 to 0.659) P–Q datasets are seen to vary widely from plot to plot with 0 as the most frequent value. The cumulative frequency distribution of λ-values for both datasets shows that λ-values are larger for ordered data, the distribution is skewed, and most λ-values (26 for natural and 21 for ordered P–Q datasets out of total 27) are less than the standard λ = 0.20 value. The respective mean and median λ-values are 0.030 and 0 for natural, and 0.108 and 0 for ordered data, quite less than 0.20 but consistent with those reported elsewhere (Ajmal et al. 2015a; Baltas et al. 2007; D’Asaro and Grillone 2012; D’Asaro et al. 2014; Elhakeem and Papanicolaou 2009; Fu et al. 2011; Hawkins and Khojeini 2000; Hawkins et al. 2002; Menberu et al. 2015; Shi et al. 2009; Yuan et al. 2014; Zhou and Lei 2011). In addition, the existence of a I a–S relationship for different plots was also investigated using all the data of 27 plots. In contrast to the existing notion, I a when plotted against S (Fig. 6), exhibited no correlation for both natural and ordered datasets, which is consistent with the findings of Jiang (2001).

Fig. 6
figure 6

Relationship between I a and S for data from the 27 plots for a natural and b ordered data

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Performance evaluation of the proposed model

Table 6 shows the performance indices (R 2, NSE, RMSE, n t and PBIAS) for fitting of Eq. (2) with λ = 0.20 (existing SCS-CN method i.e. M1) and λ = 0.030 (proposed method i.e. M2). As seen from the table, the runoff estimates with λ = 0.030 (M2) provide larger NSE and lower RMSE for 26 out of 27 plots than those due to λ = 0.20 (M1). Based on NSE, performance of the existing SCS-CN method (M1) is seen to be unsatisfactory, satisfactory, good, and very good for the data of 12, 5, 3, and 7 plots out of 27 plots, respectively. On the other hand, the performance of the proposed method (M2) is unsatisfactory, satisfactory, good, and very good on 8, 5, 5 and 9 plots out of 27 plots, respectively. Based on the mean values of NSE, M2 performed satisfactorily (NSE = 0.565) compared to M1 (NSE = 0.392).

Table 6 Performance statistics for runoff estimation using Eq. (2) with λ = 0.20 (model M1) and λ = 0.030 (model M2; used all runoff producing events)
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The positive PBIAS values resulting for both the methods indicate that the existing SCS-CN method (i.e. M1) underestimated the average runoff; however, these values for M2 were much lower than those due to M1, indicating an improvement in model performance. M1 performance was unsatisfactory, satisfactory, good, and very good as regards the data of 6, 10, 1, and 9 plots out of 27 plots, respectively. On the other hand, M2 performance was unsatisfactory, satisfactory, good, and very good on 4, 4, 6 and 13 plots out of 27 plots, respectively; thus, based on the mean PBIAS values, M2 performance was good (= 10.78 %), whereas M1 performed satisfactorily (= 16.90 %). For further analysis based on n t, M1 exhibited satisfactory or good performance regarding 11 out of 27 plots. The performance improved for 16 plots when M2 was used. The improved M2 model performance is also supported by the higher r-value. As shown in Fig. 7, the significant improvement in NSE (or r) using the M2 model was observed in 26 out of the 27 study plots. In contrast, the runoff predictions by M2 model were debased (r ≤ 0) in only one plot. Overall, as seen from Table 6 and Fig. 7, M2 performed better than M1.

Fig. 7
figure 7

The cumulative frequency distribution of improvement using the r criterion

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Sensitivity of λ to CN and runoff

This sensitivity analysis was carried out using the data of only 5 plots. To this end, as shown in Fig. 8, for a plot dataset and a given λ-value, S (or CN) was optimised using Eq. (2). As seen, the rising trends are similar to each plot. In general, CN is seen to increase with λ, which is due to the fact that for the given P–Q data, an increase in λ would require an increase in CN (or decrease in S) to obtain the same Q-value for a given P. Furthermore, variation in CN narrows down with increasing λ-values.

Fig. 8
figure 8

Variation in CNs (AMC II) with λ for data from five plots

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To indicate the most appropriate λ-value, variation of NSE with λ was plotted (Fig. 9). In general, NSE showed a decreasing trend with λ for all five plots, consistent with the findings of Woodward et al. (2004) and Yuan et al. (2014), which implies that a low λ-value provides a better prediction of runoff, and vice versa. To show the sensitivity of λ to runoff (using Eq. 8), for a given CN = 78.92 and P = 30 mm, the estimated runoff increased by 165 % when λ decreased (by 90 %) from 0.2 to 0.02.

Fig. 9
figure 9

Variation in NSE with λ

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Conversion of CN0.20 to CN0.030

The existing NEH-4 CNs are based on a λ value equal to 0.20; therefore, a transformation of CNs from λ = 0.20 to λ = 0.030 is imperative before using λ = 0.030 in runoff modeling. To this end, an empirical conversion equation, based on direct least squares fitting of 27 plots with natural datasets for converting CNs associated with λ = 0.20 (CN0.20) to λ = 0.030 (CN0.030), is proposed as follows:

$$ {S}_{0.030}=0.614\ {\left({S}_{0.20}\right)}^{1.248};{R}^2=0.995;\ \mathrm{S}\mathrm{E}=0.035\ \mathrm{mm} $$
(16)

In Eq. (16), maximum potential retention (S) is in mm and S 0.030 = S 0.20 at 7.148 mm or CN0.20 = 97.268. The ratio of S 0.030 to S 0.20 (i.e. S 0.030/S 0.20) was seen to be inversely related to the mean ratio of Q to P (i.e. Rcm). The substitution of Eq. (16) into the definition of CN yields

$$ {\mathrm{CN}}_{0.030}=25400/\left[254+0.614{\left(25400/{\mathrm{CN}}_{0.20}-254\right)}^{1.248}\right] $$
(17)

The applicability of Eq. (16) (or Eq. 17) to prediction of runoff using the NEH-4 tables curve number (CNHT) is also investigated. To this end, the estimated NEH-4 CNs (or CNHT0.20) based on plot characteristics for 24 plots (1–24 plots of Table 2) were first converted to CNHT0.030, and then employed for runoff estimation as shown in Table S3 of the ESM, along with R 2, NSE, and RMSE. As seen, CNHT0.030 from Eq. (16) estimates the runoff more accurately than did CNHT0.20. Besides, the r-statistic (Fig. 7) also shows the use of Eq. (16) to have significantly improved NSE in 22 out of 24 study plots.

Limitations of the study

The results of this study are limited to the experimental boundaries such as plot size, slopes, soils, agricultural land uses, and climatic conditions. Replication of such a study for a wider range of physical and climatic settings is imperative for indicating its broader applicability. In this regard, an automation of measurement of data may further help refine the results of the present study which is based on manual data collection.

Conclusions

The following conclusions can be drawn from the study:

  1. 1.

    Compared to land use and slope, infiltration capacity (fc) is the main explanatory variable for runoff (or CN) production in the study plots.

  2. 2.

    CN is inversely related to infiltration capacity (fc), which supports the applicability of CNs from the NEH-4 tables declining with fc (or HSG).

  3. 3.

    P–Q derived CNs are higher than those from NEH-4 tables. However, these are closer for higher CN values, which is consistent with the general notion that the existing SCS-CN method performs better for high P–Q (or CN) events.

  4. 4.

    Mean and median λ-values are respectively 0.030 and 0 for natural P–Q data, and 0.108 and 0 for ordered P–Q data. λ was greater than 0.20 for only one natural plot data and six ordered plot data.

  5. 5.

    Runoff estimation improves as λ decreases, for 26 out of 27 plots by changing λ-value from 0.20 to 0.030.

  6. 6.

    There exists a relationship between CN0.20 (λ = 0.20) and CN0.030 (λ = 0.030), useful for CN conversion for field application.