Introduction

A rainfall–runoff model forms the basis for several engineering applications such as hydraulic structure design, flood peak discharge computation, irrigation scheduling, reservoir operation, minimizing downstream flood hazards and water balance studies. One of the most globally used methods to estimate runoff is the Soil Conservation Service Curve Number (SCS-CN) method applicable for ungauged watersheds. It was developed by US Department of Agriculture (USDA)–SCS (SCS 1956/1972). This method is renamed as Natural Resource Conservation Service Curve Number (NRCS-CN) method (Hawkins et al. 2010).

The method is simple, easy to understand and apply and accounts for major runoff producing watershed characteristics (Garen and Moore 2005; Mishra et al. 2006; Sahu et al. 2012). It relies on only one parameter ‘CN’ which depends on watershed climatic and geographic factors (Mishra and Singh 2003). It is of common experience that a watershed can have a set of CNs which may be attributed to spatial and temporal variations of rainfall and watershed properties, quality of measured rainfall–runoff data, variability of antecedent rainfall and associated soil moisture amount (Hjelmfelt 1991; Hawkins 1993; Soulis and Valiantzas 2012). The usually identified source of variability is antecedent moisture condition (AMC) (Steenhuis et al. 1995; Soulis et al. 2009) which can be overcome by incorporation of initial soil moisture (Mishra and Singh 2002; Jain et al. 2006; Sahu et al. 2007). It plays a vital role in restructuring of the SCS-CN method and permits smooth variation of CN and thus avoids sudden jump in runoff estimation. It invoked the concept of Soil Moisture Accounting (SMA) procedure for development of enhanced SCS-CN-based models. Mishra and Singh (2002) incorporated the effect of antecedent moisture amount and developed an improved version of the SCS-CN inspired model. Several researchers (Mishra and Singh 2002, 2004, 2005; Jain et al. 2006; Babu and Mishra 2012) have developed different expressions of antecedent moisture on the basis of 5-day antecedent rainfall amount. Singh et al. (2010) presented an updated hydrologic review on the latest developments in SCS-CN methodology and discussed its physical and mathematical significance in hydrologic applications. More recently, Ajmal et al. (2015) examined this method and its inspired versions using the data of 15 watersheds of South Korea (total 658 large storm events). However, there are only a few studies providing an insight into the structural soundness of SMA procedure of the existing SCS-CN methodology (Verma et al. 2017).

Michel et al. (2005) diagnosed the structural foundation of the SCS-CN method and revealed the underlying inconsistencies arising partially from the misperception between intrinsic parameters and initial conditions and partially from an improper use of the underlying SMA procedure. They emphasized the incorporation of initial soil moisture (V 0) rather than an impractical intrinsic parameter in the form of initial abstractions (I a ). With the changed parameterization of threshold soil moisture (S a ) required for runoff generation in place of I a and underlying SMA procedure, they proposed an advanced SCS-CN inspired model. This model does not include any expression for computing the input model parameters (V 0) and (S a ). Expressions were, however, provided for the simplified Michel et al. (2005) model. Singh et al. (2015) improved the SMA procedure proposed by Michel et al. (2005) by providing expression for initial soil moisture.

Looking into the versatility of the SCS-CN method and associated inconsistencies and changed parameterization, this study aims at to (a) propose an enhanced version of the SCS-CN inspired model based on SMA procedure for estimating runoff depth and suggest simple expressions for V 0 and S a estimation to avoid sudden jumps in V 0 and AMC and (b) compare the performance of the proposed model with the original model, Mishra and Singh (2002) model, Michel et al. (2005) model and Singh et al. (2015) model using a large (= 38,169) storm events derived from 152 USDA watersheds varying in area from 0.3 to 12,990 ha.

Methodology

Original SCS-CN method

The SCS-CN (original) method employs the water balance equation:

$$P = I_{a} + F + Q$$
(1)

and the following two proportional equality hypotheses expressed, respectively, as:

$$\frac{Q}{{(P - I_{a} )}} = \frac{F}{S}$$
(2)
$$I_{a} = \lambda S$$
(3)

where P is the rainfall depth, I a is the initial abstraction, F is the cumulative infiltration excluding I a , Q is the runoff depth, S is the potential maximum retention and λ is the initial abstraction coefficient, which is taken as 0.2 for usual applications. Hawkins (2009) found that the variation of λ between 0 and 0.05 are more realistic. A combination of Eqs. (1) and (2) yields the popular form of the SCS-CN method:

$$\begin{aligned} & Q = \frac{{(P - I_{a} )^{2} }}{{P - I_{a} + S}} {\text{for}}\;\;P \ge I_{a} \\ & Q = \, 0 \, \;\;\;\;\;\;{\text{for }}P \le I_{a} \\ \end{aligned}$$
(4)

Parameter S is mapped onto CN using Eq. (5) as:

$$S = 25.4\left( {\frac{1000}{\text{CN}} - 10} \right)$$
(5)

where CN is a dimensionless quantity varying from 0 to 100 and S is expressed in mm. CN varies with watershed characteristics such as soil type, land use, hydrologic condition and AMC (Chow et al. 1988; Mishra and Singh 2003).

Mishra and Singh (2002) model

Mishra and Singh (2002) used the C = S r concept, where C = Q/(P−I a ) is the runoff coefficient and Sr = (F/S) is the degree of saturation. Using this concept, they modified the equation of surface runoff by incorporating antecedent moisture equal to V 0 (Fig. 1) is given in Eq. (6):

$$\begin{aligned} & Q = \frac{{\left( {P - I_{a} } \right)\left( {P - I_{a} + V_{0} } \right)}}{{\left( {P - I_{a} + V_{0} + S} \right)}},\;\;{\text{if}}\;\;P > I_{a} \\ & Q = 0,\;\;{\text{otherwise}} \\ \end{aligned}$$
(6)

They also formulated an equation for the computation of V 0 as:

$$\begin{aligned} & V_{0} = 0.5\left[ { - 1.2S + \sqrt {0.64S^{2} + 4P_{5} S} } \right],\;\;{\text{if}}\;\;P_{5} \ge 0.2S, \\ & V_{0} = 0,\;\;{\text{otherwise}} \\ \end{aligned}$$
(7)

where P 5 is the 5-day antecedent rainfall.

Fig. 1
figure 1

Explanatory diagram showing soil moisture store in the SCS-CN inspired models

Full size image

Michel et al. (2005) model

Michel et al. (2005) reviewed SMA procedure of the SCS-CN method and unveiled major inconsistencies in treatment of AMC and proposed a renewed sounder methodology. They assumed that the SCS-CN model is valid not only at the end of the storm but at any instant along a rainfall event. Their findings are based on an analysis of the continuous SMA procedure based on the concept that higher the moisture store level, higher the portion of rainfall that is transformed into surface runoff. They developed a procedure which is more stable from SMA viewpoint, by introducing the term V 0, i.e., initial soil moisture store level.

Based on their hypothesis and water balance theory, Michel et al.(2005) model shows several variations in the original procedure of SCS-CN method due to confusion between intrinsic parameters and the initial condition and partially from an inappropriate use of the basic SMA procedure. The model eliminated initial abstraction I a and introduced a new parameter ‘S a ’ threshold moisture for generating surface runoff (S a  = I a  + V 0) to compute the surface runoff (Fig. 1).The generalized Michel et al. (2005) model with new insight into SMA procedure is given as follows:

$${\text{If}}\;\;V_{0} \le \, S_{a} {-}P,\;\;{\text{then}}\;\;Q = 0$$
(8)
$${\text{If}}\;\;S_{a} {-}P < V_{0} < S_{a} ,\;\;{\text{then}},\;\;Q = \frac{{\left( {P + V_{0} - S_{a} } \right)^{2} }}{{\left( {P + V_{0} - S_{a} + S} \right)}}$$
(9)
$${\text{If}}\;\;S_{a} \le V_{0} \le S_{a} + S,\;\;{\text{then}},\;\;Q = P\left[ {1 - \frac{{\left( {S + S_{a} - V_{0} } \right)^{2} }}{{S^{2} + \left( {S + S_{a} - V_{0} } \right)P}}} \right]$$
(10)

With the introduction of initial soil moisture (V 0) and replacement of parameter I a by S a , the procedure became more reliable from SMA viewpoint.

Singh et al. (2015) model

Singh et al. (2015) used the concept of C = S r and modified the proportionality hypothesis (Eq. 2) as:

$$\frac{Q}{P - Ia} = \frac{{F + V_{0} + Ia}}{{S + V_{0} + Ia}}$$
(11)

They incorporated the concept of S a  = I a  + V 0, and proposed a set of equations for runoff estimation as:

$${\text{If}}\;\;V_{0} \le \, S_{a} {-} \, P,\;\;{\text{then}}\;\;Q = 0$$
(12)
$${\text{If}}\;\;S_{a} - P < V_{0} < S_{a} ,\;\;{\text{then}}\;\;Q = \frac{{\left( {P + V_{0} - S_{a} } \right)\left( {P + V_{0} } \right)}}{{P + S + V_{0} }}$$
(13)
$${\text{If}}\;\;S_{a} \le V_{0} \le S_{b} ,\;\;{\text{then}}\;\;Q = P\left[ {1 - \frac{{\left( {S_{b} - V_{0} } \right)^{2} }}{{SS_{b} + \left( {S_{b} - V_{0} } \right)P}}} \right]$$
(14)

where S a is the threshold soil moisture, S b  = S+S a is the absolute potential retention (Fig. 1), V 0 is the initial soil moisture. For computing V 0 and S a , they used the existing equations as:

$$V_{0} = \alpha \sqrt {P_{5} S}$$
(15)
$$S_{a} = \beta S$$
(16)

Need of modification in existing concept

Equation 7 (Mishra and Singh 2002) yields unrealistic negative values of V 0 when P 5 < 0.2S, leading to occurrence of inconsistent sudden jumps in SMA computation. Later, Michel et al. (2005) model suggested Eqs. (8–10) to account for these jumps. Since this model relies on the original SCS-CN model (Eq. 4), it does not incorporate V 0 in its expressions. Recently, Singh et al. (2015) further modified Mishra and Singh (2002) model employing the same C = Sr concept and including initial abstraction in proportionality hypothesis (Eq. 11). This, however, does not appear to be rational considering the initial abstraction to be a component of precipitation loss contributing neither to infiltration nor to runoff. It invokes the development of the proposed model.

Proposed model

For the proposed version, the modification began from Mishra and Singh (2002) model which was developed on C = S r concept employing antecedent moisture (V 0). To make this model suitable within a continuous watershed model, the model should be valid not only at the end of the storm but at any instant along a storm. In this perspective, rainfall (P) and runoff (Q) are assumed as functions of time t and it holds at the completion of the storm too when the continuous model overlaps the concept of Mishra and Singh (2002) model.

Let us assume the rainfall rate (p) and the runoff rate (q) are equal to dP/dt and dQ/dt, respectively. Also, consider soil moisture volume store which collects that portion of rainfall trapped in soil profile and not converted into runoff. Assume V 0 is the level of soil moisture at the beginning of the storm and V is the level of soil moisture at any instant of time t, when the collected rainfall depth equals P.

The water balance Eq. (1) can be written as:

$$V = V_{0} + P_{{}} - Q$$
(17)

Replacing Q by Eq. (6) (Mishra and Singh 2002) to derive the following:

$$V = V_{0} + P - \left[ {\frac{{\left( {P - I_{a} } \right)\left( {P - I_{a} + V_{0} } \right)}}{{\left( {P - I_{a} + V_{0} + S} \right)}}} \right]$$
(18)

Differentiating Eq. (6) with time (t) yields

$$q = \frac{{\left( {P - I_{a} + V_{0} } \right)\left( {P - I_{a} + V_{0} + S} \right) + S\left( {P - I_{a} } \right)}}{{\left( {P - I_{a} + V_{0} + S} \right)^{2} }}p, \, {\rm if}\,P > I_{a}$$
(19)

where q = dQ/dt

Substituting P from Eqs. (18) into (19) yields

$$q = \frac{{2S\left( {V - I_{a} - V_{0} } \right) - \left( {V - I_{a} - V_{0} } \right)^{2} + SV_{0} }}{{S\left( {S + V_{0} } \right)}}p\;\;{\text{if}},\;\;V > V_{0} + I_{a} ,\;\quad q = 0,\;\;{\text{otherwise}}$$
(20)

Replacing (I a  + V 0) with the intrinsic parameter (S a ) gives

$$\begin{aligned} & q = \frac{{\left( {V - S_{a} } \right)\left( {2S + S_{a} - V} \right) + SV_{0} }}{{S\left( {S + V_{0} } \right)}}p,\;\;{\text{if}}\;\;V > S_{a} \\ & q \, = 0,\;\;{\text{otherwise}} \\ \end{aligned}$$
(21)

Thus, hydrologically, the soil moisture store initially collects all the rainfall until it reaches the threshold value (S a ). Since the soil moisture store has a fixed volume, the extreme capacity of soil moisture store can be reached to V when the soil is fully saturated and all rainfall converts to runoff, i.e., q = p. At this point of time, there is no space in the soil moisture store for rain to enter; according to the derivative of Eq. (17) which suggests dV/dt = 0. For maximum capacity of V, i.e., when V = S + S a , Eq. (21) yields

$$q \, = \, p.$$
(22)

The continuity equation can be obtained by differentiating Eq. (17) as:

$$\frac{{{\text{d}}V}}{\text{dt}} = p - q$$
(23)

A coupling of Eqs. (21) and (23) results into complete model of soil moisture store.Equation (6) can be rewritten using intrinsic parameter S a instead of I a as:

$$\begin{aligned} & Q = \frac{{\left( {P - S_{a} + V_{0} } \right)\left( {P - S_{a} + 2V_{0} } \right)}}{{\left( {P - S_{a} + 2V_{0} + S} \right)}}{\text{if}}\;\;P + V_{0} > S_{a} , \\ & Q \, = \, 0\;\;{\text{otherwise}} \\ \end{aligned}$$
(24)

Here, if V 0 = S + S a , Q should be equal to P because the soil moisture store is completely filled and there is no space left in the store where the excess rainfall can enter. Q derived from Eq. (24) is larger than P, as below:

$$Q = P + \frac{{S\left( {2S + S_{a} } \right)}}{{\left( {P + S_{a} + 3S} \right)}}$$
(25)

which is not possible in reality and it is a drawback of Mishra and Singh (2002) model. Thus, Eq. (24) is flawed.

The exact formulation can be obtained by recalculating the formula for the total amount of P and Q by integrating Eq. (23) and using the value of q from Eq. (21) as:

$$\frac{{{\text{d}}V}}{\text{dt}} = \frac{{\left( {V - S_{a} - S} \right)^{2} }}{{S\left( {S + V_{0} } \right)}}p$$
(26)

It can be rewritten as:

$$\frac{{{\text{d}}V}}{{\left( {V - S_{a} - S} \right)^{2} }} = \frac{{p{\text{dt}}}}{{S\left( {S + V_{0} } \right)}}$$
(27a)

Since the soil moisture varies from V 0 to V, after integration, one gets

$$\int\limits_{{V_{0} }}^{V} {\frac{{{\text{d}}V}}{{\left( {V - S_{a} - S} \right)^{2} }}} = \frac{1}{{S\left( {S + V_{0} } \right)}}\int\limits_{0}^{t} {p{\text{dt}}}$$
(27b)
$$\left[ {\frac{ - 1}{{\left( {V - S_{a} - S} \right)}}} \right]_{{V_{0} }}^{V} = \frac{P}{{S\left( {S + V_{0} } \right)}}$$
(27c)
$$\frac{1}{{S_{a} + S - V}} - \frac{1}{{S_{a} + S - V_{0} }} = \frac{P}{{S\left( {S + V_{0} } \right)}}$$
(27d)

Now, replacing V from Eq. (17), one obtains

$$Q = P\left[ {1 - \frac{{\left( {S_{a} + S - V_{0} } \right)^{2} }}{{\left( {P\left( {S_{a} + S - V_{0} } \right) + S\left( {S + V_{0} } \right)} \right)}}} \right]$$
(28)

The computations reveal that the derived model, i.e., Eqs. (24) and (28) are two different parts of the complete proposed model. Three cases arise from the presence of S a as below:

Case I When P is not enough to overcome the initial moisture deficit of the soil moisture store (less than threshold value, S a ), i.e., V 0 + P < S a , then Q = 0, Thus,

$${\text{If}}\;\;V_{0} \le S_{a} - P,\;\;{\text{then}}\;\;Q = 0$$
(29)

Case II When V 0 < S a but P is large enough to overcome the initial moisture deficit and generate surface runoff, an initial part of P will be used to overcome the initial moisture deficit without generating runoff and the remaining part (P + V 0−S a ) will generate runoff.

$${\text{If}}\;\;S_{a} {-}P < V_{0} < S_{a} ,\;\;{\text{then}}\;\;Q = \frac{{\left( {P - S_{a} + V_{0} } \right)\left( {P - S_{a} + 2V_{0} } \right)}}{{\left( {P - S_{a} + 2V_{0} + S} \right)}}$$
(30)

Case III When V 0 > S a , an amount (= V 0−S a ) of water is taken out from the soil moisture store and added to P to further increase runoff.

$${\text{If}}\;\;S_{a} \le V_{0} \le S_{a} + S,\;\;{\text{then}}\;\;Q = P\left[ {1 - \frac{{\left( {S_{a} + S - V_{0} } \right)^{2} }}{{\left( {P\left( {S_{a} + S - V_{0} } \right) + S\left( {S + V_{0} } \right)} \right)}}} \right]$$
(31)

Equations (29)–(31) represent the proposed SCS-CN model as it is more logical and physically more stable compared to the other models under study for surface runoff estimation. For computation of V 0 and S a , the formulations used by Singh et al. (2015) (Eqs. 15 and 16) is used here. The advantage of this expression is that it physically relates V 0 to P 5 and S, in the sense that a higher P 5 or S will give a higher V 0. Moreover, it obviates the sudden jump of V 0 with S or CN.

For convenience, the original SCS-CN model, Mishra and Singh (2002) model, Michel et al. (2005) model, Singh et al. (2015) and the proposed model are referred as M1, M2, M3, M4 and M5, respectively, in the forthcoming text. Table 1 summarizes the formulation of all these models.

Table 1 Model formulations
Full size table

Application

Study watersheds and data

The proposed model is evaluated and compared with existing models (M1, M2, M3 and M4) using the data derived from 152 USDA agricultural watersheds varying in area from 0.3 to 12,990 ha (Fig. 2). The rainfall–runoff data for 3–35 years are available at http://www.ars.usda.gov/arsdb.html as well as at http://hydrolab.arsusda.gov/arswater.html. The number of rainfall–runoff events varied from 8 to 979 (total = 38,169) for different watersheds.

Fig. 2
figure 2

Location of ARS experimental watersheds

Full size image

Measures of model performance

The models’ performance is evaluated using four widely accepted statistical indices: (1) root-mean-square error (RMSE) (Deshmukh et al. 2013) (2) coefficient of determination (R 2), (3) percent bias (PBIAS) (Moriasi et al. 2007) and (4) Nash–Sutcliffe efficiency (NSE) (Gupta et al. 1999; Nash and Sutcliffe 1970).These are expressed as:

$${\text{RMSE}} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {Q_{\text{obs}} - Q_{\text{comp}} } \right)_{i}^{2} } }$$
(32)
$$R^{2} = \frac{{\left( {\sum\nolimits_{i = 1}^{N} {\left( {Q_{\text{obs}} - \bar{Q}_{\text{obs}} } \right)_{i} \left( {Q_{\text{comp}} - \bar{Q}_{\text{comp}} } \right)_{i} } } \right)^{2} }}{{\sum\nolimits_{i = 1}^{N} {\left( {Q_{\text{obs}} - \bar{Q}_{\text{obs}} } \right)_{i}^{2} } \sum\nolimits_{i = 1}^{N} {\left( {Q_{\text{comp}} - \bar{Q}_{\text{comp}} } \right)_{i}^{2} } }}$$
(33)
$${\text{PBIAS}} = \left[ {\frac{{\sum\nolimits_{i = 1}^{N} {\left( {Q_{\text{obs}} - Q_{\text{comp}} } \right)_{i}^{{}} } }}{{\sum\nolimits_{i = 1}^{N} {\left( {Q_{\text{obs}} } \right)_{i}^{{}} } }}} \right] \times 100$$
(34)
$${\text{NSE}} = \left[ {1 - \frac{{\sum\nolimits_{i = 1}^{N} {\left( {Q_{\text{obs}} - Q_{\text{comp}} } \right)_{i}^{2} } }}{{\sum\nolimits_{i = 1}^{N} {\left( {Q_{\text{obs}} - \bar{Q}_{\text{obs}} } \right)_{i}^{2} } }}} \right] \times 100$$
(35)

where Q obs is the observed storm runoff, Q comp is the computed runoff, \(\bar{Q}_{\text{obs}}\) is the mean of observed runoff values in a watershed, N is the total number of rainfall runoff events and i is an integer varying from 1 to N. The lower value of RMSE indicates better performance and vice versa. It means RMSE = 0 shows a perfect agreement between the observed and the predicted values. R 2 describes the proportion of the total variance explained by the model in the observed data. It is a meaningful indicator of the accuracy of predictions. It ranges from 0 to 1 with higher values indicating better agreement with the observed data. Published literature (Aitken 1973; Mishra and Singh 1999; Santhi et al. 2001; Garen and Moore 2005) indicates acceptable model performance if R 2 > 0.6. The PBIAS measures the model’s average tendency to predict higher or lower values than the observed data. The ideal value is zero; however, negative value indicates over prediction, whereas a positive value indicates under prediction. According to Moriasi et al. (2007), the model performance for flow simulation can be interpreted as ‘unsatisfactory’ if PBIAS > ± 25%; ‘satisfactory’ if ± 15% ≤ PBIAS ≤ ± 25%; ‘good’ if ± 10% ≤ PBIAS < ± 15%; and ‘very good’ if PBIAS ≤ ± 10%. The NSE (%) = 100 indicates perfect agreement between observed and computed values, whereas poorer agreement is indicated by decreasing value (Fentie et al. 2002). NSE (%) = 0 indicates that the model predictions are as accurate as the mean of the observed data, implying that the model predictions are equal to the average of the observed data (Coffey et al. 2004). The negative value of NSE indicates that the average observed value is a better estimate than the model predicted value (Fentie et al. 2002; EI-Sadek et al. 2001). Deshmukh et al. (2013), Singh et al. (2015) among many others used this criterion for models comparison.

Parameter estimation

Model parameters were estimated using Marquardt (1963) algorithm of constrained least squares. In M1 and M2 applications, the initial estimate of CN is taken as 50 and it was allowed to vary in the range (0, 100). For models M3, M4 and M5, the initial estimate of parameter S was taken as 125 mm and was assumed to vary in the range (0, 2500) mm. For M3, V 0 and S a parameters were allowed to vary in the range (0, 500) mm with its initial estimate of 100 mm. For the M4 and M5, α and β were allowed to vary in the range of (0, 1) with an initial estimate of 0.01. The statistics of estimated values of model parameters are given in Table 2.

Table 2 Range of parameters obtained from model application in 152 watersheds
Full size table

Results and discussion

The values of NSE (%), RMSE, R 2 and PBIAS resulting from model (M1–M5) applications to 152 USDA watersheds are summarized in Appendix I and II. A model can be ranked superior if it shows higher NSE (%) or R 2 values and vice versa. Conversely, lower values or RMSE and PBIAS (either positive or negative) also indicate better performance of models. Figure 3 shows the models’ performance based on RMSE which shows that the proposed model (M5) has lowest RMSE for most of the watersheds. When the models’ performance was evaluated on the basis of NSE (%), it is found that the model M1, M2, M3, M4 and M5 shows NSE (%) > 75 for 14, 40, 36, 59 and 62 watersheds, respectively. Figure 4 shows the models NSE (%) for all 152 watersheds. It is clear here that the model M5 shows significant improvement over M1, M2 and M3 and marginal improvement over model M4. Similar performance is observed from Fig. 5 depicting the models’ performance on the basis of PBIAS. It shows that model M5 shows lowest PBIAS (either positive or negative) as compared to rest of the models. The models M1, M2 and M3 underestimate the runoff by showing positive value of PBIAS. According to the evaluation criteria set by Moriasi et al. (2007), M1, M2, M3, M4 and M5 models shows very good performance in 53, 29, 117, 129 and 134 watersheds, good in 17, 30, 13, 10 and 6, fair in 33, 41, 11, 9 and 10 and unsatisfactory in 49, 52, 11, 4 and 2 watersheds, respectively.

Fig. 3
figure 3

Models’ performance on the basis of RMSE for 152 watersheds

Full size image
Fig. 4
figure 4

Models’ performance on the basis of NSE (%) for 152 watersheds

Full size image
Fig. 5
figure 5

Models’ performance on the basis of PBIAS for 152 watersheds

Full size image

The models’ performance based on R 2 indicates that model M1, M2, M3, M4 and M5 shows acceptable performance (R 2 > 0.6) for 86, 107, 100, 116 and 117 watersheds, respectively. Figure 6 shows the R 2 of all models indicating the significant improvement as compared to M1, M2 and M3 and marginal improvement over M4.

Fig. 6
figure 6

Models’ performance on the basis of R 2 for 152 watersheds

Full size image

The models’ performances are also evaluated using scatter plots (Fig. 7a, b for WS-ID 9004 and 17003, respectively), which compare the predicted runoff with the observed runoff for all the five models.

Fig. 7
figure 7

Scatter plot of models (M1–M5) under study

Full size image

The models’ performance were further assessed using RGS based on efficiency suggested by Mishra and Singh (1999) and Singh et al. (2015), by assigning ranks (I)–(V) to the above five models in the order of their merit in applications to the dataset of a watershed. The rank (I) shows the highest NSE and rank (V) the lowest NSE. The ranks of models in each application and their overall ranks (I–V) from the overall score of each model are shown in Table 3. This table shows that M5 scores the highest (= 691) marks with overall rank I followed by M4 with 642 marks and overall rank II, M3 with 376 marks with overall rank III, M2 with 362 marks with overall rank IV and M1 with 209 marks, and overall rank V out of the maximum 2280 marks. Therefore, based on the overall results obtained here, it can be clearly deduced that M5 is rated as the best model followed by M4, M3, M2 and M1 models.

Table 3 Evaluation of models rank based on NSE (%) and RGS
Full size table

Conclusions

An enhanced SMA-based SCS-CN-inspired model was proposed and tested for its suitability using the large dataset of 152 US watersheds. Based on RMSE, R 2, PBIAS and NSE (%), the proposed model (M5) was found to perform marginally better than Singh et al. (M4) model and significantly better than both Mishra and Singh (2002) (M2) and Michel et al. (2005) (M3) models as well as the original method (M1). It was also supported by ranking and grading systems, which show M5 to have scored the highest marks/rank.