The curve number (CN) method was developed more than half a century ago and
is still used in many watershed and water-quality models to estimate direct
runoff from a rainfall event. Despite its popularity, the method is plagued
by a conceptual problem where CN is assumed to be constant for a given set of
watershed conditions, but many field observations show that CN decreases with
event rainfall (

The estimation of runoff from a rainfall event is of primary importance in applied hydrology. It is necessary in the engineering design of small structures, post-event appraisals, environmental impact work, and other applications (Hawkins, 1993). One of the most popular techniques used for this purpose is the curve number (CN) method, which has been in use for more than half a century (D'Asaro and Grillone, 2012; Hawkins et al., 2008; Kent, 1968; Ponce and Hawkins, 1996; Rallison and Miller, 1982; Soil Conservation Service, 1956, 1972). The method uses a parameter called curve number, which is assumed to depend mainly on land cover, soil types, and antecedent conditions within a watershed.

Curve number varies spatially due to watershed heterogeneity and temporally due to changes in soil moisture, land cover, temperature, and other processes (Hawkins et al., 2008; Ponce and Hawkins, 1996; Rallison and Miller, 1982). CN also varies with the magnitude (D'Asaro and Grillone, 2012; Hawkins, 1993; Hjelmfelt Jr. et al., 2001) and spatiotemporal distribution of rainfall (Hawkins et al., 2008; Van Mullem, 1997). When heterogeneity is known at sufficient detail, CN variation can be accounted for by using a distributed parameter model, e.g., SWAT (Gassman et al., 2007). Otherwise this approach can introduce more parameters than can be reliably estimated from the available data (Soulis and Valiantzas, 2013) and can potentially cause large uncertainties in the predicted runoff. There are several ways to account for temporal variation of CN, each with its own advantages and shortcomings (Santikari, 2017). CN variation with the distribution of rainfall is usually ignored (Hawkins et al., 2008). The CN method is most commonly applied as an event-scale lumped-parameter model, which is simple but also limited in its ability to account for the variations of CN. This diminishes the accuracy of its runoff predictions (e.g., Soulis and Valianzas, 2012).

The objective of this work is to improve the event-scale lumped-parameter application of the CN method by describing an approach for incorporating the spatiotemporal variations of CN. The investigation is described in two papers, which build on Santikari (2017). In this paper, effects of spatial variation of CN (heterogeneity) at the watershed scale are analyzed. Insights gained from this analysis are used to create modified models that account for heterogeneity. The modified models are evaluated using the runoff generated by a distributed parameter model applied to a hypothetical heterogeneous watershed. In a companion paper (Santikari and Murdoch, 2018) and in Santikari (2017), the modified models are refined by including an approach that accounts for the temporal variation of CN using antecedent moisture. The refined models, which account for spatial and temporal variability, are then evaluated using data from real watersheds.

The CN method assumes that a rainfall event produces runoff (

The conceptual basis that defines the curve number method comes from the
following assumption (Hawkins et al., 2008; NRCS, 2003; Ponce and Hawkins,
1996; Rallison and Miller, 1982; Woodward et al., 2002):

Presumed variation of the ratios in Eq. (2) with event rainfall (

To eliminate the need for an independent estimation of

For convenience (Hawkins et al., 2008; Ponce and Hawkins, 1996),

Variation of CN (

The curve number method is appealing because it is based on an intuitive concept (Eq. 2), relies on only one parameter, has a large body of literature (Hawkins et al., 2008), and has a comprehensive database (NRCS, 2003; USDA, 1986). It has been included in many watershed and water-quality models such as SWAT (Soil and Water Assessment Tool) (Neitsch et al., 2005), CREAMS (Chemicals, Runoff and Erosion from Agricultural Management Systems), GLEAMS (Groundwater Loading Effects of Agricultural Management Systems) (Knisel and Douglas-Mankin, 2012), AnnAGNPS (Annualized Agricultural Non-point Source Pollution Model) (Bingner et al., 2011), EPIC (Environmental Policy Integrated Climate), APEX (Agricultural Policy/Environmental Extender) (Wang et al., 2012), and HydroCAD (HydroCAD, 2015). A physically based modeling framework, such as the diffusive-wave approximation for overland flow coupled with the Richard's equation for unsaturated subsurface flow, e.g., Panday and Huyakorn (2004), may improve accuracy and resolution of model predictions compared to the CN method, when the necessary input data, expertise, and computing resources are available. However, the CN method will likely remain popular for many applications in runoff modeling because of its ease of use, wide knowledge base, and less demand on computational resources than many physically based models.

Curve number is assumed to be a watershed property that depends on the
current conditions, but it also varies with

In watersheds showing a standard behavior, CN was treated as an
asymptotic function of

A standard behavior of CN was also observed in two watersheds (BC5 and BC1) near Greenville,
South Carolina, USA (Fig. 2a and b). In these watersheds, CN
(calculated using

The approach used in Fig. 2a and b avoids the commonly used frequency
matching (e.g., Hawkins, 1993). Each CN value in the plot was calculated
using the

The hypotheses given by Hawkins (1993) are valid, but insufficient to
explain the standard and complacent behaviors. It may be true that small rainfalls produce runoff only
under wet (large CN) conditions and therefore only the large CN values are
recorded. However, if one has a large enough sample of storms, some of the
larger storms also must have occurred during wet conditions. For the larger
storms, therefore, one would expect to see the whole spectrum of CN values
ranging from the largest to the smallest. However, this is not the case. As

Soulis and Valiantzas (2012) hypothesized that the observed variation of CN
with

In a later paper, Soulis and Valiantzas (2013) suggested using spatial
information on land cover and soils to delineate the areal extent of
subareas and constrain their respective CNs. This approach would reduce the
number of calibrated parameters by half because it only requires the
calibration of the CNs for the subareas. In essence, the multiple-subarea
approach is similar to a distributed modeling approach that calculates the
watershed runoff as the area-weighted average of the runoffs from the
subareas, e.g., SWAT (Gassman et al., 2007). The approach used by Soulis and
Valiantzas (2013) attempts to match the observed and simulated values of CN,
whereas that used by SWAT attempts to match the observed and simulated
values of

Using a single value of CN independent of

The quantities CN,

To evaluate the link between heterogeneity in

By the theoretical definition of

However, it is difficult to detect the exact moment of generation of runoff
and determine the corresponding value of

It may be difficult to measure

Strictly adhering to the definition of

The inconsistencies in the usage of

Spatial distribution of

Consider a watershed with four HRUs mainly characterized by their land use
types, viz. open waterbody (

Consider a general case of a heterogeneous watershed with

Then what is the effective initial abstraction of the watershed for a given
rainfall event? Consider an event where the rainfall falls within the range:

Variation of

To investigate the variation of

The analysis presented so far represents a discrete case where each HRU is
homogeneous and has a finite area. The values of

Representing areal distribution of

For the case of a continuous distribution of

Variation of

Let us hypothesize that

Differentiating Eq. (10) and using Eq. (19) gives

CN

The shape of the CN

The functional form of

For each distribution, the corresponding functional form of

For the purpose of comparison, symmetrical versions of the distributions
were considered such that all of them have the same minimum, mean, and
maximum values of

The kurtosis (peakedness) of

Skewness of

The analysis also shows that a watershed cannot be characterized or compared
with other watersheds using a single value of CN (such as CN

Similar to the case of

Effect of skewness, mean, and range of

Writing an expression for

To illustrate the effect of heterogeneity on

The similarities between

Functional forms of

The analysis from previous sections shows that

Variation of

The distributed parameter modeling approach, Eq. (21) with the application
of mass balance (Eq. 1) at watershed and HRU scales, shows that

The sigmoid-shaped function of

Here the emphasis is placed on treating

Lumped-parameter application of the CN method was modified by treating

Conventional CN models are defined by Eqs. (1) through (5) and by the
assumption that

VIMs are defined by Eqs. (1), (2), (4), (5), and (23), and they have three
free parameters. If

Lumped-parameter models described in the previous section were evaluated in
their ability to predict runoff and account for watershed heterogeneity.
Accounting for heterogeneity means that the model accurately predicts

Evaluation of lumped-parameter models requires the data for

In a distributed parameter model, Eqs. (1) through (5) are applicable at the
HRU scale, with the assumption that

The distributed parameter model was applied to an idealized synthetic
watershed with the storage distribution shown in Table 2, for the cases of

The reason for using a synthetic watershed here is that the heterogeneity can be precisely defined and used to evaluate the predictions of heterogeneity by the lumped-parameter models. In real watersheds, the heterogeneity has to be determined by calibration, and there can be non-uniqueness when multiple HRUs are present. Application of these modified models to data from real watersheds is discussed by Santikari (2017) and Santikari and Murdoch (2018).

Storage distribution in a hypothetical heterogeneous watershed used to
illustrate the variation of

Each lumped-parameter model was calibrated by minimizing the sum of the
squared residuals between its predicted runoff (

NSE values were calculated for the model predictions of runoff (NSE

NSE

The results show that using variable initial abstraction improved the
accuracy of model predictions of runoff and heterogeneity (Table 3). Based
on their overall performance, the models can be arranged from the best to
the worst as VIM

Variable

Variable

Variable

The performance of lumped-parameter CN models that were calibrated to
the runoff data generated using a distributed CN model for two cases of a
synthetic watershed with the storage distribution shown in Table 2.
SEE,

In the models where

The storage in a watershed is distributed between

Storage transfer is evident when the values of

A transfer of storage from

Storage transfer from

The models were also evaluated using rainfall-runoff observations from 9
real watersheds located in different parts of the world (Santikari, 2017;
Santikari and Murdoch, 2018). The models' ability to predict the observed runoff
was assessed using NSE

The degree of heterogeneity, defined as the sharpness of change in CN,

Comparing the results (Tables 3 and 4) shows that the performance of VIMs
remained nearly the same, whereas the performance of CM0.2 decreased and
that of CM

The results from real watersheds (Santikari, 2017; Santikari and Murdoch,
2018) also show that the performance of CM0.2 was poor, NSE

Performance of the models for the cases of

One of the main objectives of this study was to improve the predictive
ability of the CN method while maintaining its simplicity. Using the number
of calibrated parameters as an indicator, the models can be arranged in the
order of increasing complexity as CM0.2 (one)

CM

Variable

When the watershed heterogeneity is known in great detail such that the
number of calibrated parameters

A strength of the models proposed in this paper is that they provide a
compact way to account for the spatial variation of CN,

Another limitation of VIMs is that the CN values calculated using Eqs. (5)
or (10) are incompatible with the standard CN values (NRCS, 2003; USDA,
1986) derived using CM0.2. However, this limitation is not unique to VIMs
because any method, including CM

Watershed heterogeneity causes calculated values of

This paper uses synthetic data, which can be generated using Table 2 and the procedure described in Sect. 5.1.

VS conceived the idea and performed the analysis. LM supervised the analysis and influenced the overall direction and content of the work. Both VS and LM wrote the paper.

The authors declare that they have no conflict of interest.

Primary funding for this study was provided by the USDA Natural Resources Conservation Service (NRCS-69-4639-1-0010) through the Changing Land Use and Environment (CLUE) project at Clemson University. Additional support was provided by the USDA Cooperative State Research, Education, and Extension Service under project number SC-1700278. Edited by: Thomas Kjeldsen Reviewed by: two anonymous referees