A revisit of NRCS-CN inspired models coupled with RS and GIS for runoff estimation

ABSTRACT

The Natural Resource Conservation Service – Curve Number (NRCS-CN) methodology is a widely used tool for estimating surface runoff, which is of prime importance in hydrological engineering, agricultural planning and management, environmental impact assessment, flood forecasting, and others fields. This article reviews the methodology and associated hydrological models used for runoff estimation along with their advantages and limitations. Furthermore, discussion focuses on the potential applications of Remote Sensing (RS) and Geographical Information System (GIS) techniques for estimating hydrological variables, such as rainfall, soil moisture and CN required for the NRCS-CN methodology, as well as future research and opportunities for improved runoff estimation at the macro scale.

EDITOR D. Koutsoyiannis

ASSOCIATE EDITOR A. Efstratiadis

KEYWORDS:

• curve number
• surface runoff
• hydrological model
• remote sensing
• geographical information system

Abbreviations

ADI	=	Antecedent Discharge Index
AGNPS	=	Agricultural Non-Point Source Pollution
AMA	=	Antecedent Moisture Amount
AMC	=	Antecedent Moisture Condition
AMC I	=	AMC for dry
AMC II	=	AMC for average
AMC III	=	AMC for wet
AMSR-E	=	Advanced Microwave Scanning Radiometer – Earth observing system
AMSU	=	Advanced Microwave Sounding Unit
ANN	=	Artificial Neural Network
APEX	=	Agricultural Policy/Environmental Extender
API	=	Antecedent Precipitation Index
ARS	=	Agriculture Research Service (of the USDA)
ASCAT	=	Advanced Scatterometer
ASM	=	Antecedent Soil Moisture
ASTER	=	Advanced Spaceborne Thermal Emission and Reflection
AWC	=	Antecedent Wetness Condition
CELTHYM	=	Cell-based Long-term Hydrological Model
CMORPH	=	Climate Prediction Centre Morphing
CN	=	Curve Number
CN1	=	Curve Number I

CN2	=	Curve Number II
CN3	=	Curve Number III
CN4GA	=	Curve Number for Green-Ampt
CREAMS	=	Chemicals, Runoff and Erosion from Agricultural Management Systems
DEM	=	Digital Elevation Model
DMSP	=	Defense Meteorological Satellite Program
EPIC	=	Erosion Productivity Impact Calculator
ERS	=	European Remote Sensing
ESA	=	European Space Agency
ESRI	=	Environmental Systems Research Institute
ET	=	Evapotranspiration
EVI	=	Enhanced Vegetation Index
GEWEX	=	Global Water and Energy Cycle Experiment
GIS	=	Geographical Information System
GLCF	=	Global Land Cover Facility
GLEAMS	=	Groundwater Loading Effects of Agriculture Management System
GMS	=	Geostationary Meteorological Satellite
GOES	=	Geostationary Operational Environmental Satellite
GPM	=	Global Precipitation Mission
GR4J	=	Génie Rural à 4 paramètres Journalier
GRACE	=	Gravity Recovery and Climate Experiment
HEC-HMS	=	Hydrologic Engineering Centre’s Hydraulic Modeling System

HELP	=	Hydrologic Evaluation of Landfill Performance
HSG	=	Hydrological Soil Group
IRS	=	Indian Remote Sensing
L-THIA	=	Long-term Hydrological Impact Assessment
LTHS ASMA	=	Long-term Hydrological Simulation Advanced Soil Moisture Accounting Model
LTHS-MICHEL	=	Long-term Hydrological Simulation – Michel Model
LULC	=	Land use/Land cover
MADRAS	=	Microwave Analysis and Detection of Rain and Atmospheric Structures
MODIS	=	Moderate Resolution Imaging Spectroradiometer
MW	=	Microwave
NASA	=	National Aeronautics and Space Administration (USA)
NASDA	=	National Space Development Agency (USA)
NATSGO	=	National Soils Geographic Database (USA)
NEH	=	National Engineering Handbook (USA)
NLEAP	=	Nitrogen Leaching and Economic Analysis Package
NOAA	=	National Oceanic and Atmospheric Administration (USA)
NRCS-CN	=	Natural Resource Conservation Service – Curve Number
PERSIANN	=	Precipitation Estimation from Remotely Sensed Information using Artificial Neural Networks
PRZM	=	Pesticide Root Zone Model
RS	=	Remote Sensing
SAR	=	Synthetic Aperture Radar
SCS-CN	=	Soil Conservation Service Curve Number
SM	=	Soil Moisture
SMA	=	Soil Moisture Accounting
SMAP	=	Soil Moisture Active Passive Mission
SMOS	=	Soil Moisture Ocean Salinity
SPUR	=	Simulation of Production and Utilization of Rangelands
SRTM	=	Shuttle Radar Topography Mission
SSM/I	=	Special Sensor Microwave Imager
SSURGO	=	Soil Survey Geographic Database
STATSGO	=	State Soil Geographic Database
SWAT	=	Soil and Water Assessment Tool
SWI	=	Soil Wetness Index
SWIM	=	Soil and Water Integrated Model

SWOT	=	Surface Water and Ocean Topography
SWRRB	=	Simulator for Water Resources in Rural Basins
TDR	=	Time Domain Reflectometry
TRMM	=	Tropical Rainfall Measuring Mission
USDA	=	United States Department of Agriculture
VIS and IR	=	Visible and Infrared
VSA	=	Variable Source Area
WSR 88-D	=	Weather Surveillance Radar 1988 Doppler

1 Introduction

Estimation of the amount of rainfall that is transformed into surface runoff is of prime importance in hydrological engineering and watershed management. Among the various methods for runoff estimation developed over time, the NRCS-CN methodology (SCS 1993, USDA 1986), previously known as SCS-CN, is widely accepted and popular, as indicated from its frequent use in the literature. Since its inception, the method has been the focus of extended discussion and debate, although the actual intent of the procedure has often been ignored. In particular, it has attracted intensive and extensive exploration into its formation, rationality, applicability and extendibility, aspects of its pros and cons, and physical significance (e.g. Ponce and Hawkins 1996, Mishra and Singh 1999, Mishra et al. 2005, 2008b). The advantages of the NRCS-CN method are its simplicity, predictability, stability, applicability for ungauged watersheds, reliance on only two parameters – namely the CN that specifies the soil storage capacity (S) and initial abstraction (I_a), which is expressed as a percentage of S – its convenience of use and capability of incorporating easily accessible watershed characteristics, such as soil type, land use/treatment, hydrological conditions and AMC (Mishra et al. 2004, 2005, Sahu et al. 2007, 2012b, Jiao et al. 2015). Due to these many advantages, the method is still commonly used today among water resources practitioners (Singh et al. 2015) and enjoys fruitful applications across the globe.

Initially, the method was developed for computation of direct surface runoff from storm rainfall events in small agricultural watersheds in the USA, but very soon it was adopted for several regions, land uses and climatic conditions (Romero et al. 2007, Soulis and Valiantzas 2013). In the past two decades, runoff estimations and predictions have been strongly improved, after coupling of RS and GIS techniques within the NRCS-CN approach. Due to recent developments in sensor functionality at both temporal and spatial resolution, RS techniques can monitor large regions and provide images at short time intervals, on a repetitive basis. These advantages can be used in two different ways: (a) in terms of model input data and (b) within model parameter estimation. Remote sensing data can be acquired from various satellites (e.g. Landsat, IRS, NOAA, Meteosat, Megha Tropiques) and global missions, and can be used to provide reliable estimations of input data, such as rainfall, soil moisture and CN values. In addition, multi-temporal imagery offers unique opportunities to collect datasets on, for example, soil type, soil texture, LULC and its changes, and other properties that are required for rainfall–runoff modelling of large areas (Schultz 1988). Furthermore, visible, microwave and thermal spectral measurements have also proven to have major benefits for hydrological applications. Furthermore, the GIS technique has powerful capability to process spatial and non-spatial data and provide new platforms for data analysis, management and visualization. It also makes it feasible to simulate large areas in many widely used hydrological models.

In this perspective, the purpose of this article is to (a) provide a critical review of the NRCS-CN concept and associated models, its advantages and limitations; (b) discuss recent improvements in the CN concept; (c) discuss the applications of RS and GIS techniques for estimating model inputs and parameters; and (d) summarize the outcomes of this review for future courses of action in research.

2 NRCS-CN method: origins, derivation and limitations

2.1 Historical background

The NRCS-CN method was pioneered and developed by the USDA in 1954, for estimation of surface runoff from a given storm event and construction of design hydrographs in US agricultural watersheds of less than 250 km². Since its first documentation (Section 3, NEH-4), it has been revised several times (SCS 1956, 1964, 1971, 1972, 1985, 1993, 2004). Its origin was based on Sherman’s proposal (1942, 1949) for plotting direct runoff vs storm rainfall. The subsequent work of Mockus (1949) focused on estimating surface runoff in ungauged watersheds using information on soil, land use, antecedent rainfall, storm duration and average annual temperature. Andrews (1954) also developed a graphical procedure for estimating runoff from rainfall, for several combinations of soil texture and type, vegetation cover and conservation practices, which are referred to as the soil-cover complex. Finally, the empirical rainfall–runoff relationship of Mockus (1949) and the soil-cover complex of Andrews (1954) led to the foundation of the NRCS-CN methodology (USDA 1986).

2.2 Derivation of the traditional NRCS-CN method

The traditional NRCS-CN method is an event-based, lumped rainfall–runoff model. It is based on the combination of the water balance equation:

(1)

and the following two proportional equality hypotheses, expressed as:

(2)

and

(3)

where P is total rainfall, I_a is initial abstraction, F is cumulative infiltration excluding I_a, Q is direct surface runoff, S is potential maximum retention and λ is the initial abstraction coefficient. A combination of Equations (1) and (2) yields the popular form of the NRCS-CN method:

(4)

Parameter S is mapped to the CN using:

(5)

where S is expressed in mm and CN is a dimensionless runoff index varying in the range 0–100 and also varying with AMC, generally described using a 5-day antecedent rainfall.

This method assumes a long-term average soil moisture condition prior to the start of runoff, and a ratio of I_a to S of 0.2. The first assumption is not always the case for the period of interest in a given area and can result in unreasonable sudden jumps in estimated runoff (Michel et al. 2005, Sahu et al. 2010), while the second assumption has no scientific basis, is very ambiguous and has raised inconclusive arguments (Sahu et al. 2007, Shi et al. 2009, Wang et al. 2010). Later, Steenhuis et al. (1995) stated that runoff generation from rainfall is not due only to infiltration excess, which, in fact, is consistent with both infiltration excess and saturation excess or VSA hydrology concepts. They derived the fraction of the watershed (A_f) contributing runoff by differentiating Equation (4) as:

(6)

where P_e is the effective rainfall (cm) and P – I_a is the amount of water required to initiate runoff.

2.3 Limitations of the traditional NRCS-CN method

The NRCS-CN method has a number of limitations and misinterpretations due to the basic empirical structure, as described below:

1. It was derived from approximately 10 years of rainfall–runoff data collected exclusively from agricultural and rangeland. Therefore, it works best on agricultural sites, fairly on range sites and poorly on humid, semi-arid climatic and forested regions (Hawkins 1984, Mishra and Singh 2003b).

2. It was developed simply to estimate the daily runoff that could result from daily rain storms, without taking care of the effect of the antecedent moisture in its basic formulation.

3. It does not contain any expression for time and, as a result, ignores the impact of rain intensity and duration. In addition, there is no explicit provision for the spatial variability of rainfall. Therefore, in principle, it may not be appropriate for the sub-daily time resolution. Consequently, the use of this procedure in small catchments may be erroneous and prone to relatively large errors as it requires hourly or sub-hourly temporal resolution of the net rainstorm hyetograph (Garen and Moore 2005, Eli and Lamont 2010).

4. It is an event-based model, and not a continuous simulation model. Its advancements have significantly altered CN notions and vocabulary well beyond its original definition and validity (Woodward et al. 2010).

5. It works well at the watershed scale, not at the point, plot or field scale (Garen and Moore 2005). Thus, it is not specific to a streamflow generation process. In principle, it is incorrect to imply that the runoff is generated entirely from overland flow and not only by infiltration excess from the whole land surface.

6. It implicitly assumes that infiltration excess is the primary runoff mechanism, but it is also composed of the saturation excess mechanism.

7. It is not suitable for computation of infiltration during a storm event (Eli and Lamont 2010).

8. Its development took place before computers and GIS were in wide use; these now provide extensive spatial data on terrain, soils and vegetation.

3 Improvements in the NRCS-CN methodology

Despite several limitations and misinterpretations, as mentioned above, and even questionable credibility reported from time to time, the NRCS-CN method has been in continuous use due to its satisfactory performance at the field level. It has also gone through several phases of critical reviews (Rallison 1980, Chen 1982, Ponce and Hawkins 1996, Mishra and Singh 2003b, 2006, Garen and Moore 2005, Michel et al. 2005, Walter and Stephen 2005, Mishra et al. 2007, Singh et al. 2010).

In 1990, NRCS and ARS, both agencies of the USDA, formed a joint working group to assess the state of the CN procedure and to chart its future development. The primary task of the committee was to rewrite those portions of the NEH-4 where confusion and problems arose from incorrect and misleading statements or incomplete documentation. The group concluded that the added portion must agree with concepts expressed, references should be included, and the result must be technically defensible. As the work progressed, the vastly expanded database of hydrological soils classification and the capability of modern computers were considered. The use of initial abstraction, I_a, equal to 0.2S was also considered during re-evaluation. Finally, there was significant concern about the possibility of regional and seasonal variations of CNs (Hjelmfelt et al. 2001).

The joint working group recognized three distinct different modes of application for the CN method:

1. determination of runoff volume of a given return period and a given total event rainfall for that return period;

2. determination of direct runoff for individual events, explaining the variability from event to event, as used in continuous simulation models; and

3. determination of infiltration rates for short time intervals, as used within unit hydrograph development of flood hydrographs.

In addition to the above agencies, several researchers have also proposed improvements to overcome the limitations of the traditional NRCS-CN method and its modified version with respect to SMA (e.g. Williams and LaSeur 1976, Ponce and Hawkins 1996, Michel et al. 2005, Sahu et al. 2007, Wang et al. 2012b, Singh et al. 2015, Ajmal et al. 2015b, 2016a), infiltration (e.g. Gabellani et al. 2008, Grimaldi et al. 2013), as well as potential retention (e.g. Smith and Williams 1980, Wang et al. 2008, Sahu et al. 2010, Babu and Mishra 2012) and baseflow (e.g. Coustau et al. 2012) processes. Some modified versions of the NRCS-CN method that are applicable for continuous hydrological simulation have also been suggested (e.g. Huber et al. 1976, Williams and LaSeur 1976, Knisel 1980, Woodward and Gbuerek 1992, Mishra and Singh 2004a, Geetha et al. 2007, Kannan et al. 2008, Williams et al. 2012, Durbude et al. 2011, Jain et al. 2012), with less interest for ungauged watersheds, because several new parameters were included and, as a consequence, “convenience” and “simplicity” were reduced.

The implementation of the NRCS-CN method in a continuous simulation model within hybrid modelling schemes was also developed in order to represent the variability of rainfall and AMC (e.g. Camici et al. 2011, Grimaldi et al. 2012a, 2012b). In particular, Svoboda (1991) used the CN concept to calculate the soil water content for deriving the rainfall contribution to direct runoff and groundwater. Steenhuis et al. (1995) modified the NRCS-CN equation using the partial-area hydrology concept by simply assuming that runoff occurs from areas that are saturated and that the remaining watershed does not contribute any runoff, giving the expression of A_f (Equation (6)). Furthermore, Lyon et al. (2004) developed a modified approach called the CN-VSA method. They combined the traditional method with a distributed topographic index to estimate the saturation probability of a specific area of watershed for predicting distributed runoff volume. Thus, these modifications took into account saturation excess or VSA hydrology.

To overcome limitation 2 (Section 2.3), several improvements in the NRCS-CN method have been attempted and used with different degrees of success over many years, explicitly on the SMA procedure. In particular, Mishra and Singh (2002a) incorporated a new parameter, antecedent moisture, M_c, which can be estimated from prior 5-day accumulated rainfall. Furthermore, Mishra et al. (2006a) proposed a relationship for estimating M_c () to obviate sudden jumps in CN variations with AMC. Michel et al. (2005) reviewed the SMA process and pointed out several structural inconsistencies in the traditional method. They found that these inconsistencies arise partly from the confusion between intrinsic parameter and initial condition, and partly from an incorrect use of the underlying SMA procedure. They also proposed two new parameters: soil moisture store level at the beginning of the event, V₀, and the threshold moisture for runoff generation to compute the direct runoff, S_a. Later, Sahu et al. (2007) amended the Michel et al. (2005) model, suggesting that V₀ be equal to the sum of pre-antecedent moisture level (V₀₀) 5 days before the onset of rainfall and a fraction of the part of rainfall that is not transformed into runoff. Furthermore, the Sahu-Mishra-Eldho model (Sahu et al. 2012b) reformulated the Mishra and Singh (2006) model and hypothesized that the ratio of Q to (P – I_a) is equal to that of the sum of cumulative infiltration after runoff starts and the antecedent soil moisture store level to the maximum retention S₀, and that I_a is a fraction of the difference between S₀ and antecedent soil moisture store level. The model assumes that the antecedent soil moisture store level can be estimated as a function of P₅ and that a watershed of interest is always dry 5 days before the onset of a rainfall event. Geetha et al. (2007) adjusted event CN values for AMCs by modifying the existing NRCS-CN method in two approaches: AMC and AMA. However, Woodward et al. (2010) highlighted major abuses related to the use of CN values in continuous models proposed by Geetha et al. (2007) that are only remotely linked to the NRCS-CN method. Williams et al. (2012) revised the soil moisture index proposed by Williams and LaSeur (1976) for a continuous method, instead of using 5-day antecedent rainfall. They also tested the ability of the soil moisture index to predict runoff over a range of soil properties and conditions with two different runoff models. Recently, Ajmal et al. (2015b, 2016a) also proposed an advanced rainfall–runoff relationship for obviating sudden jumps based on both P and P₅.

To overcome limitation 3 (Section 2.3), Grimaldi et al. (2013) proposed a CN4GA semi-empirical model from the modified Green-Ampt (Green and Ampt 1911) infiltration equation and tested it on 100 observed rainfall–runoff events. This model is also applicable to runoff prediction in ungauged basins. In contrast, Kabiri et al. (2013) found the runoff values computed from the SCS-CN method did not differ from those of the Green-Ampt method. Mockus, the developer of the original method, noted in his later years that the saturation- and infiltration-excess mechanisms are part of runoff generation. This means that landscape factors should also be accounted for in addition to land use, soil class and land management conditions when assigning the CN values. Recently, Bartlett et al. (2016) proposed a theoretical framework (ProStor) to create an extended NRCS-CNx model. The framework describes a spatially lumped, event-based rainfall–runoff modelling to overcome spatial variability of runoff, one of the limitations of the traditional method. It may also be applied with different rainfall–runoff responses at a point and different spatial probability density functions for the watershed variables that are representative of watershed heterogeneities. Furthermore, the spatial distribution of runoff may be mapped to the watershed using the topographic index (e.g. Lyon et al. 2004).

To move beyond the limitations of the NRCS-CN method in the context of using advance GIS techniques: Patil et al. (2008) developed a GIS interface to estimate runoff by adopting the typical NRCS-CN method and three modifications: (a) modified CN-I, considering I_a = 0; (b) modified CN-II, considering I_a = λS and λ = αt_p, where t_p is time of ponding and α is the Horton constant; and (c) modified CN-III by expressing cumulative infiltration, F, as a sum of static (F_c) and dynamic (F_d) components, and substituting them against appropriate parameters of the traditional NRCS-CN method. They tested the above modifications using recorded data for the Banha watershed in Jharkhand, India. Their results showed that modified CN-I performed best, followed by modified CN-III, while modified CN-II performed worst for all AMCs. Moreover, under AMC-based conditions, modified CN-I also performed best for AMC II, while modified CN-II performed worst under all AMCs.

Thus, the above-mentioned recent contributions have provided significant improvements in the understanding of the NRCS-CN method and, consequently, its potential for a wide range of applications. Although primarily developed for event-based rainfall–runoff modelling and flood design on ungauged watersheds, it has been successfully applied in the broader domain of hydrology, watershed management and environmental engineering. The method has also been successfully applied to sediment yield modelling (Mishra et al. 2006b, Singh et al. 2008, Tyagi et al. 2008, Bhunya et al. 2009), subsurface flow estimation (Yuan et al. 2001, Mishra and Singh 2004b, Jain et al. 2012), urban hydrology (Kibbler 1982, Pandit and Gopalakrishnan 1996), water quality (Ojha 2012, Narain et al. 2015), rainwater harvesting (Kadam et al. 2012, Singh et al. 2013, Zende and Atal 2015) and estimation of ET (Mishra et al. 2014b). It is also used to determine soil erosion, and nutrients and pesticides transported from fields and into streams (Garen and Moore 2005). This vast applicability clearly proves the prominence of the NRCS-CN technique among prevailing hydrological models.

The widespread application of the NRCS-CN method has led to its inclusion in continuous hydrological simulation models for runoff estimation, such as CREAMS (Knisel 1980), AGNPS (Young et al. 1989), HEC-I and HEC-HMS (US Army Corps of Engineers), SPUR (Wight and Skiles 1987), GLEAMS (Leonard et al. 1987), EPIC (Sharpley and Williams 1990), SWRRB (Arnold et al. 1990), HELP (Schroeder et al. 1994), L-THIA (Harbor 1994), PRZM (Carsel et al. 1997), NLEAP (Shaffer et al. 1991), SWIM (Krysanova et al. 2000), APEX (Williams et al. 2000), CELTHYM (Choi et al. 2002) and SWAT (Neitsch et al. 2002, Gassman et al. 2007). In addition, the capabilities of RS and GIS have encouraged and improved the expanded use of these models worldwide. High-resolution satellite data products from various sensors appear poised to capture the spatial variations in the hydro-meteorological variables and high temporal resolution sufficient to represent the dynamics of the hydrological processes, while GIS tools have proved to be highly useful for the generation of precise and updated information for characterizing drainage basin parameters (Grohmann 2004, Korkalainen et al. 2007, Hlaing et al. 2008, Javed et al. 2009, Pankaj and Kumar 2009, Singh et al. 2014). The published literature highlights the successful application of the NRCS-CN method in distributed watershed modelling (White 1988, Moglen 2000). Hong et al. (2007) demonstrated the potential for using this simple method when computing runoff values from satellite rainfall data for the globe and for medium to large river basins.

4 Recent improvements in the CN concept

The CN is a dimensionless watershed index. Originally, it was determined using only three inputs, namely soil type, land cover and land management conditions, which are published in SCS NEH-4 tables. The NEH-4 tables provide three CN values: those for the upper and lower bounds and the median, for various typical land–soil complexes. However, the tables do not provide guidance for selecting other CN values throughout the expected range and even its origin seems to be lost; the lack of scientific testing of the results raises some doubts when very accurate results are needed (Hawkins 1984).

The CN values determined from look-up tables are usually treated as fixed parameters, although they are random variables that vary across different sources of spatio-temporal variability, such as the effects of storm intensity or of antecedent rainfall and associated soil moisture (Ponce and Hawkins (1996). In addition, CN values also change with regional rainfall–runoff behaviour, quality of the measured data, hydrological soil groups, and seasonal and land-use condition. The CN value theoretically varies from 0 to 100, but, in practice, as validated by experience, it lies between 40 and 98 (Van Mullem 1991) when considering I_a = 0.2S. Owens et al. (2003) reported that CN ranged from 97 under bare conditions to 57 under full cover on a hard-setting Sodosol with low surface infiltration rates under bare conditions. For a heavy clay soil with varying levels of cover (25–90%), Foerster and Milne-Home (1995) similarly found optimized CN ranging between 56 and 96 for runoff plots under wheat for different tillage systems in Gunnedah, Australia. Later, Efstratiadis et al. (2014) found that basins such as karst with negligible I_a showed overestimated CN values, in fact much lower, of about 30–40. Therefore, the CN represented a much wider range than values for similar land use and HSGs in NEH tables, thus resulting in significant uncertainty in the determination of CN values.

However, several studies have already indicated that many parameters in addition to soil type and LULC, such as 5-day antecedent rainfall, stage of vegetation growth, soil moisture and interception, explain the variation in individual CN for a watershed with storms. Therefore, it seems necessary to account for these additional parameters to reduce the uncertainty of modelling results and promote more common use of adjusted CN in practice. Soulis et al. (2009) and Steenhuis et al. (1995) noted that variation of CN values according to AMC category alone cannot justify the observed variability of CN values in every case. Yuan et al. (2014) reported that runoff estimation can be very sensitive to initial abstraction coefficient, λ, especially for relatively low rainfall amount and for semi-arid watersheds.

In the following sections, we explain how these parameters affect CN.

4.1 Factors affecting CN

The CN is primarily affected by major runoff-associated watershed characteristics, such as soil type, type of vegetation, land use and treatment, hydrological condition, AMC and climatic regime (Singh 1992, Mishra and Singh 2003b, Sahu et al. 2007). These characteristics mainly affect the infiltration/retention capacity of a watershed. This is elaborated below.

4.1.1 Soil type

Musgrave (1955) described a hydrological classification of soils depending on their infiltration rate. He grouped all soils into four basic groups (A–D) depending on the minimum infiltration capacity. Soils in groups A, B, C and D exhibit high, moderate, low and very low rates of infiltration, respectively. Hence, the CN increases as the soil type changes from HSG A to HSG D. This hydrological classification system was a major stepping stone for the NRCS-CN method (Woodward et al. 2002).

4.1.2 Land use/land cover

The SCS defined three major classes of LULC as urban, cultivated, and woods and forests. These classes were further categorized into various subclasses on the basis of land treatment practices, such as contoured, terraced, straight row, bare, etc.

4.1.3 Hydrological conditions

The SCS defines the hydrological condition of a watershed in terms of the percentage area of vegetation cover. A watershed is in good hydrological condition if it is lightly grazed and more than 75% of the area is covered with vegetation. If a watershed is not heavily grazed and 50–75% of the area is covered with vegetation, it is referred to as being in “fair” hydrological condition, whereas “poor” hydrological condition of a watershed is considered if it is heavily grazed and there is vegetation cover on less than 50% of the area. Watersheds in “good” hydrological condition have high CNs and vice versa.

4.1.4 Antecedent soil moisture conditions

The AMC refers to the soil moisture present in the soil profile before the start of an event. On the basis of P₅, the NEH categorized AMC into three levels: AMC I, AMC II and AMC III, which refer, respectively, to dry, average or wet conditions and correspond statistically to 90, 50 and 10% cumulative probability of exceedence of runoff depth for a given rainfall event (Hjelmfelt et al. 1982). The higher the soil moisture, the higher the CN and vice versa.

4.1.5 Climate and initial abstraction

The water held by initial abstraction in the form of surface detention, infiltration and interception finally goes back to the atmosphere through evaporation and transpiration. These processes are mainly governed by climatic factors, such as humidity, radiation, temperature and wind. The NRCS-CN accounts for the climatic effect in terms of I_a. Moreover, temperature also affects the viscosity of water, which may slightly affect the infiltration rate (Lin et al. 2003).

4.1.6 Rainfall duration and intensity

For a given rainfall amount, the storm duration and rainfall intensity are inversely related. The greater the rainfall intensity, the shorter will be the storm duration and vice versa. A storm of shorter duration allows less time for rainwater to stay on the land surface, which leads to smaller infiltration and consequently greater surface runoff. The reverse also holds true.

4.1.7 Slope

The CN₂ values mentioned in the NEH-4 tables are appropriate for slopes of around 5% (Sharpley and Williams 1990), because these reference values were identified from small agricultural watersheds in Texas, where slope does not play a major role. However, this relationship was never thoroughly tested. According to Hawkins et al. (2009), some researchers suggest employing negative slope adjustments for steeper slopes, which could be related to the erosion of potential surface seals before they can form.

4.2 Determination of representative CN

As mentioned above, the CN can vary within a significant range for a given watershed (Hjelmfelt 1991, McCuen 2002, Tedela et al. 2012). Hawkins (1993) reported that the runoff calculated from the NRCS-CN method was much more sensitive to the CN chosen than to rainfall depth. Thus, several approaches are available to estimate representative CN for a watershed (Hawkins et al. 2009), and these can be categorized into two broad groups: (a) hydrological soil-cover complex number procedure, and (b) derivation of CN from field data.

4.2.1 Hydrological soil-cover complex number procedure

This procedure consists of two main approaches: (i) a composite (area-weighted) CN approach, and (ii) a weighted-Q approach.

4.2.1.1 Composite CN approach

Standard CN tables of NEH-4 (SCS 1956, 1964, 1971, 1972, 1985, 1993, 2004) are used to estimate CN at fine spatial scales (e.g. grid-cell scale), on the basis of land-use type and HSG combinations. The CN values of the respective hydrological soil-cover complex are multiplied by the respective percentage areal coverage of the complex. The area-weighted CN is computed using:

(7)

where CN_n and A_n are CN value and area of the nth watershed, respectively, with n the total number of segments in the watershed and A the total area of the watershed.

Narayana (1993) recommended the use of the composite CN technique in India, with I_a = 0.1S for black clay soil and 0.3S for the remaining soils. In contrast, Grove et al. (1998), Soulis and Valiantzas (2012) and Jena et al. (2012) found that significant errors were produced in runoff estimates using single composite CN values instead of weighted Q. This may be due to the nonlinear form of the NRCS-CN formula. Lantz and Hawkins (2001) also discussed possible errors caused by the use of a single composite CN value.

In the past decade, this approach has been employed with the aid of RS and GIS techniques in distributed watershed modelling. Pandey et al. (2003), Tejaswini et al. (2011), Dhawale (2013), Joycee and Santhi (2015) and many others used RS and GIS to estimate representative CN for a watershed. Randusová et al. (2015) compared different approaches of CN determination using natural and rank ordered data and highlighted the uncertainties of tabulated CN values.

4.2.1.2 Weighted-Q approach

In this approach, runoff (Q) is computed for each sub-area of a watershed using Equations (1) and (5), utilizing the CN value derived for each respective hydrological soil-cover complex of the sub-area from standard CN tables of NEH-4. Finally, the area-weighted Q is computed. Obviously, the weighted-Q approach is superior to the weighted-CN method, as the former is more rational than the latter for water balance reasons. However, the weighted-CN is easier to work with if the watershed has many complexes or for a series of storms. Mishra and Singh (2003b) pointed out that the runoff computed using the hydrological soil-cover complex CN approach would deviate significantly for a wide range of CN values for various complexes in a watershed.

4.2.2 CN from field data

Several researchers have highlighted the difficulty in selecting accurate CN and suggested that CN values could be determined from recorded (P–Q) data. This involves transformation of S, which varies in the range 0 ≤ S ≤ ∞. A value of S can be calculated as follows (Equation (8)), if λ = 0.2 (Hawkins 1993) and CN is later computed via Equation (5):

(8)

Furthermore, the above equation is transformed (Equation (9)) to determine S, characterizing the watershed, and CN is later computed via Equation (5):

(9)

However, CN values calculated using Equations (8) and (9) vary systematically with rainfall depth. In this case, a single asymptotic CN value observed for very high rainfall depths is preferred. In addition, different approaches are also available in the literature to determine CN values from field data, as given below.

4.2.2.1 Asymptotic (rank order) approach

Hjelmfelt (1980) suggested a rank-ordered approach, in which P–Q data are sorted and rearranged so as to have equal return periods. This technique is also referred to as “frequency matching”. The CNs derived from a rank-ordered P–Q dataset give a secondary systematic correlation that always emerges in the watershed as three different behaviours: complacent, standard and violent. The standard and violent cases lead to a constant CN with increasing storm size, whereas complacent behaviour does not lead to a stable determination of CN (Hawkins 1993). The standard and violent relationships can be expressed by Equations (10) and (11), respectively:

(10)

(11)

where is constant as and k is a fitting constant.

Several researchers have used this approach (e.g. Hawkins and Ward 1998, Hjelmfelt et al. 2001, Van Mullem et al. 2002). Hawkins (1993) suggested that a single asymptotic CN value can be safely determined from measured data for a standard watershed, whereas the CN value cannot be determined for a complacent watershed. A similar suggestion was also made by Hjelmfelt et al. (2001). Bonta (1997) proposed an improvement to the Hawkins (1993) method for the asymptotic determination of CN from measured data in violent and standard watersheds using derived distributions. Rietz and Hawkins (2000) applied this approach for estimating CN for each land use in a watershed at three scales, i.e. local, regional and national. Schneider and McCuen (2005) developed a new lognormal frequency distribution method to estimate CN from measured P–Q data.

However, all the above methodologies for estimating a single asymptotic CN are only applicable for high rainfall depths and they also fail to determine a final CN value in complacent watersheds. More recently, Soulis and Valiantzas (2012) observed a correlation between calculated CN values and rainfall depths in standard and complacent watersheds by considering the spatial variability of soils and land cover; they introduced a two-CN heterogeneous system for calculating watershed runoff and found that, for any watershed characterized by heterogeneous land use, runoff determined with this approach was very similar to the recorded runoff. They also proposed two CN system approaches for runoff calculation, for high rainfall (P > 3000 mm) and low rainfall, and found them more accurate for runoff evaluation as compared to a single asymptotic CN value approach.

4.2.2.2 Median CN approach

Originally, CNs were developed using daily P–Q data corresponding to maximum annual flows from gauged watersheds, for which information on soil-cover complex was available (SCS 1972, 1985). The P–Q data were plotted on arithmetic paper having a grid of plotted CN (Fig. 1). The value corresponding to the curve that parted one half of the plotted data from the other half was taken as the median CN for the watershed, and assigned to AMC II (CN₂). The upper and lower enveloping curves were taken as representative of AMC III (CN₃) and AMC I (CN₁), respectively. Schneider and McCuen (2005) stated that the mean or median of the storm-event CNs is the best estimate of CN for the watershed. The median is more suitable for small samples, since it reduces the influence of outliers (Schneider and McCuen 2005) and it is more reliable in an operational setting (Hjelmfelt 1991).

Figure 1. Determination of CN for AMC I, AMC II and AMC III using the original SCS-CN method. Source: Mishra and Singh (2003b).

4.2.2.3 Geometric mean approach

This approach has been used to determine a representative CN value if individual values calculated using measured P–Q data from storm events are lognormally distributed. Yet, no one seems to have established the lognormality of CN distributions. The major strength of this approach is quantification of uncertainty with the standard deviation and confidence intervals. The geometric mean is obtained by calculating the average of the series logS, and then estimating the geometric mean maximum potential retention . Then, the geometric mean CN is computed via:

(12)

where α = 254 mm. Tedela et al. (2012) and Gundalia and Dholakia (2014) compared this approach of CN estimation with other approaches in their studies.

4.3 CN variation with AMC

The runoff depth estimated using the NRCS-CN method was found to be critically influenced by three main factors: total rainfall, rainfall intensity and AMC (Roberts and Crane 1997, Menziani et al. 2001, Pfister et al. 2003, Weiler et al. 2003, Brocca et al. 2008). Although AMC is usually considered the most significant factor (Mishra et al. 2004), several researchers have suggested that the catchment’s initial conditions are not critical, particularly in the case of large events (Merz and Plate 1997, Castillo et al. 2003, Brocca et al. 2009b, Gamage et al. 2015). For any change in AMC (say from AMC II to AMC I or AMC III) in a given catchment, a sudden jump in CN value (i.e. from CN₂ to CN₁ or CN₃) occurs, thus giving the impression of a discrete nature of the CN–AMC relationship.

The CN₂, which corresponds to AMC II, depends on the cumulative 5-day precipitation (P₅) and is converted to CN₁ and CN₃ using analytical formulae (SCS 1972, McCuen 1989, Ponce 1989, Singh 1992, Mishra and Singh 2003b). Table 1 shows conversion expressions developed by Sobhani (1975), Hawkins et al. (1985), Chow et al. (1988), Neitsch et al. (2002) and Mishra et al. (2008a), and directly from the NEH-4 table. The formula by Sobhani (1975) was found suitable for CN₁ conversion and that by Hawkins (1985) for CN₃ conversion. Mishra et al. (2008a) developed AMC-dependent conversion formulae that have been compared with the existing ones, using field data taken from the USDA-ARS database. The results showed that their formulae provided the best performance and those of Neitsch et al. (2002) formulae the poorest.

Table 1. AMC-dependent CN conversion formulae.

Reference	AMC I	AMC III
Sobhani (1975)
Hawkins et al. (1985)
Chow et al. (1988)
Neitsch et al. (2002)
Mishra et al. (2008a)

Source: Mishra et al. (2008a).

4.4 Slope adjustment to CN

Table 2 provides some recently proposed formulae for adjusting slope to CN. Sharpley and Williams (1990) first presented a slope-corrected CN equation. In another attempt, Williams and Izaurralde (2005) developed a relationship for adjusting CN₂ for slopes of more than 5%. However, the initial approach led by Sharpley and Williams (1990) has not been intensively tested in the field. Hence, Huang et al. (2006) adopted a simplified approach and developed a relationship for slope-corrected CN₂ for slope gradients ranging from 0.14 to 1.4. Recently, Ajmal et al. (2016b) modified their approach for slope adjustment accompanied by a lower λ value and validated its efficacy using a dataset of rainfall–runoff events for South Korean watersheds.

Table 2. Correction formulae for CN₂ against slope.

Reference	Equation for slope-adjusted CN2
Sharpley and Williams (1990)	where CN2 and CN3 correspond to AMC II and AMC III, respectively
Williams and Izaurralde (2005)	where S2α is a retention parameter associated with slope-adjusted CN2 and α is the average slope of the watershed (%)
Huang et al. (2006)	where CN2α is the slope-adjusted CN2, and α is the slope
Ajmal et al. (2016b)	where CN2α is the slope-adjusted CN2, and α is the slope

Chaudhary et al. (2013) and Jha et al. (2014) experimentally investigated the effect of watershed slope on runoff generation and resulting CN for a given soil (HSG-C) and land use of sugarcane. They found that a 5% slope yielded the largest runoff and the derived CN values were fairly close to NEH-4 values. Shrestha et al. (2013) found an increment in CN values with an increase in slope and vice versa. Mishra et al. (2014a) and Lal et al. (2015) also reported that slope-corrected CN values provided significantly improved runoff.

4.5 Impacts of rainfall characteristics

As highlighted by Smith (1997) and Yu (1998), among others, the NRCS-CN method does not consider the impact of rainfall intensity or storm duration. Several studies (Hawkins 1978, Smith 1978) proposed a CN-based infiltration relationship and found that CN decreases as storm duration increases. Rallison (1980) attributed the runoff variation to varying infiltration rates at the soil surface strongly affected by rainfall intensity. Smith (1997) proposed accounting for not only rainfall intensity but also its pattern, since in NRCS-CN methodology a particular storm can produce different runoff hydrographs, depending on rainfall intensity. Jain et al. (2006) and Sahu et al. (2012a) incorporated the storm duration and presented enhanced versions of the traditional NRCS-CN model. Mishra et al. (2008b) developed a rain-duration-dependent procedure to estimate CN for the computation of direct surface runoff from long-duration rains. Mishra and Singh (2002b) presented a NRCS-CN-based time-distributed runoff model. Finally, Mishra and Singh (2004b) established the criterion for the applicability of the NRCS-CN method and extended the concept to derive a time-distributed runoff model and infiltration model.

4.6 Advances in the I_a–S relationship

The I_a–S relationship has been a topic of wide research for a variety of practical applications. Plummer and Woodward (2002) stated that I_a was not accounted for in the original version of the NRCS-CN method during its initial formulation, but, with the continuation of developmental phases, it was assumed to be in a constant ratio of 0.2Ia to S (Equation (3)) for all watersheds. It was described as pre-empting a regionalization based on the geological and climate setting. The validity of the constant 0.2 is also matter of debate and discussion (Woodward et al. 2003).

Since its introduction in 1954, this linear relationship has been used to avoid an independent estimation of I_a (NRCS 2001). Nevertheless, more representative values have been derived that seem to have less uncertainty for some watersheds. For example, the Central Unit for Soil Conservation (1972) recommended a λ value of 0.3 for all regions of India, except for the black cotton region, for which λ = 0.1 is applied under AMC II and III conditions. Aron et al. (1977) suggested λ ≤ 0.1 and Golding (1979) provided values for urban watersheds depending on CN with λ ≤ 0.075 for CN ≤ 70, λ ≤ 0.10 for CN ≤ 80 and λ ≤ 0.15 for CN ≤ 90. Springer et al. (1980) found that the most commonly used value of λ = 0.2 is not appropriate for either humid or arid watersheds and cautioned against its use for other watersheds.

Several studies indicated that the typical value of λ = 0.2 is strongly overestimated. For instance, Jacobs and Srinivasan (2005) obtained higher correlation coefficients with λ = 0.05 and 0.1 than with 0.2. Lim et al. (2006) computed Q utilizing λ = 0.05 in urbanized areas with HSG-D and also obtained better agreement than those with λ = 0.2 when compared with the observed ones. In a later study, Baltas et al. (2007) determined λ of a small watershed in Attica, Greece. They found that the average ratio of the entire watershed was equal to 0.014, whereas the corresponding ratio at a sub-watershed was 0.037. They attributed this difference to the different spatial distribution of land uses and geological formations at the watershed extent. Beck et al. (2009) computed observed S in 186 Australian watersheds by means of P–Q pairs with a proposed λ = 0.05. Shi et al. (2009) also reported a similar study using the event rainfall–runoff data from the Three Gorges area of China. Their results indicated that λ ranged from 0.010 to 0.154, with mean and median values of 0.053 and 0.048, respectively. They recommended a modified λ value equal to 0.05, for better prediction of runoff from large storm events. Yuan et al. (2014) reported that runoff estimation can be very sensitive to λ, especially for relatively low rainfall amount and for semi-arid watersheds. They found a variation of the I_a/S ratio of between 0.01 and 0.53 for different Walnut Gulch (USA) catchments. A similar study carried out in India by Lal et al. (2015) reported that median and mean λ values were, respectively, 0 and 0.034for original P–Q data, and 0.033 and 0.108 for ordered P–Q data.

Hjelmfelt (1991) revealed that several factors related to storm and landscape interact to explain the initial abstraction. In this context, numerous studies have been carried out across the globe (SCS 1956, 1964, 1971, 1972, 1985, 1993, 2004, Springer et al. 1980, Ramasastri and Seth 1985, Bosznay 1989) and values with a variation of λ within the range (0, 0.3) have been reported. However, as the initial abstraction concept accounts for short-term losses, such as interception, depression storage and infiltration before runoff commences, λ can theoretically take any non-negative value, even higher than 1 (Mishra and Singh 1999). Ponce and Hawkins (1996) advocated the treatment of λ as a regional parameter to enhance the model responsiveness to a diversity of geological and climatic settings. Mishra and Singh (1999, 2003a) also suggested λ to be implicitly related to both S and P. In this context, Jain et al. (2006) proposed a nonlinear I_a–S relationship, incorporating P. Mishra et al. (2006a) modified this relationship by including antecedent moisture, M, since I_a largely depends on it. These relationships were tested using a large dataset of the USDA ARS water database. The results revealed enhanced performance of the nonlinear relationship as compared to the existing ones. Finally, Babu and Mishra (2012) highlighted the dependencies of I_a on AMC and S that vary from event to event, and they related I_a with potential maximum retention (S_abs) and M linearly, which gave improved performance.

4.7 Impact of vegetation on CN

The spatial and temporal variability of vegetation is important because it affects hydrological processes (Ishidaira et al. 2008, Temimi et al. 2010, Gonzalez et al. 2015). Processes such as infiltration, evapotranspiration and water balance are nonlinearly related to soil moisture. Therefore, it is essential to accurately monitor the spatial and temporal variation of vegetation to improve the performance of the NRCS-CN method. Ponce and Hawkins (1996) suggested the necessity of adjusting CN to account for the changing canopy over time. Hawkins and Ward (1998) examined the behaviour of CN and vegetation cover for several bush and grassland sites in New Mexico (USA). Elhakeem and Papanicolaou (2009) estimated a range of CN values in agricultural fields by means of runoff simulators, and proposed a modification to the original CN equations for autumn and summer seasons. Gonzalez et al. (2015) proposed adjusted CNs using the greenness fraction (GA) as proxy for vegetation density to improve the runoff estimation by means of the NRCS-CN method.

4.8 Overparameterization

The scientific community always prefers the simplest hydrological models with a small number of parameters required and superior performance (Merritt et al. 2003). This is because overparameterization in hydrological modelling increases the number of processes considered and associated parameters run the risk of having a high degree of uncertainty associated with the method inputs that are translated through to the method outputs. The ultimate factor determining a method’s value is its simplicity relative to its explanatory power (Steefel and Van Cappellen 1998, Skaugen et al. 2015). Researchers have also demonstrated that overparameterization prevents the possibility of locating a better calibrated value for unidentifiable parameters (Jakeman and Hornberger 1993, Grimaldi et al. 2013).

In this context, some researchers (e.g. Michel et al. 2005, Sahu et al. 2007, Babu and Mishra 2012, Ajmal et al. 2015b) also simplified their improved models to a single-parameter model as the traditional NRCS-CN method for field applications (Table 3). However, all simplifications achieved by optimization of the selected variables might make these models less appealing for runoff prediction in ungauged watersheds in other climatic settings (Ajmal et al. 2015b).

Table 3. Equations used in NRCS-CN-based hydrological simulation models.

Model name	Empirical equation	Accounting of antecedent moisture
Williams and LaSeur (1976)		(13) where Sabs is absolute potential maximum retention equal to 20 (14) where t is time, bc is the depletion coefficient, M is the soil moisture index at the beginning of the first storm, M(t) is the soil moisture index at any time t, E(t) is the average monthly lake evaporation for day t and T is the number of days between storms (15) where CN2 is the CN corresponding to AMC II and subscripts µ and G correspond to ungauged and gauged watersheds, respectively
Hawkins (1978)	(16)	(17) where (t, t+∆t) is the duration ∆t between intervals t and (t+∆t) (18)
Smith and Williams (1980)		(19) (20) (21) where Wi is a weighting factor, SWi is the initial soil moisture content in layer i, SATi is the upper limit of the soil moisture storage in layer i, Di is the depth to the bottom of storage i, RD is the root zone depth, Smax is the maximum retention and CN1 is the CN corresponding to AMC I
Mishra et al. (1998)	for P ≥ 0.2S, Q = 0 otherwise (24) where DRO is the direct runoff at the outlet and d1, d2, d3 are regression coefficients	(22) (23)
Mishra-Singh (Mishra and Singh 2002a)	(25)	(26) where P5 is the amount of 5-day antecedent rainfall
Choi et al. (2002)	where qdr is the direct runoff of a grid, (27, 28) where Qsub is the direct runoff produced by the sub-watershed and Qdr is the total direct runoff of the watershed	(29) where CNadj is the adjusted CN For ASM < 50% of ASMmax: (30) with (31) For ASM ≤ 50% of ASMmax: (32) with (33) where a is the ratio of soil moisture of above/below 50% of ASMmax and WP is the wilting point
Mishra and Singh (2003b)	(34) where Fc is the field capacity	(35)
Mishra and Singh (2004a)	(36) (37) (38)	(39) (40) where E(t+Δt) is pan evaporation during the period Δt
Michel et al. (2005)	(42)	(41) where V is the soil moisture store level at the time when the accumulated rainfall is equal to P, and V0 is the soil moisture content prior to rainfall occurrence
	(43)
	(44) where Sa = (Ia + V0) is an intrinsic parameter independent of the initial conditions
Jain et al. (2006)	(45) (46) with (47) where Iad is adjusted initial abstraction, Pc is adjusted rainfall, Po is observed rainfall, tp is the storm duration, is the mean storm duration, and α and β are coefficients	(48) where γ is a coefficient and P5 is the antecedent 5-day precipitation amount
Mishra et al. (2006a)	(49)	(50) where Mc is antecedent moisture
Sahu et al. (2007)	(52) where β is a model parameter ranging from 0 to 1	(51) where V00 is the old moisture level available on day 5 before the onset of rainfall, and γ is a parameter ranging from 0 to 1
	(53)
	(54)
Geetha et al. (2007)	(55) (56) (57)	(58) where AM is antecedent moisture and ANTRF is antecedent rainfall (59)
Kim and Lee (2008)		(60) where CNW is the temporally weighted average CN value and α is a temporal weight varying from 0 to 1
Wang et al. (2008)		(61) where θsat is the saturation soil moisture, ψ is the wilting point, Se = (θ – ψ)/(θsat – ψ) is the relative saturation and a and b are coefficients
Sahu-Mishra-Eldho (Sahu et al. 2010)	(62) (63)	(64)
Durbude et al. (2011)	(65)	(68) (69) where β and γ are model parameters and V00 is the old moisture level available 5 days before the onset of rainfall (70) (71) where AM is antecedent moisture and SM is soil moisture
	If , (66)
	If , (67)
Babu and Mishra (2012)	for Pc > Ia, Q = 0 otherwise (72)	(73) (74) where Sabs is the constant quantity for a watershed irrespective of storms and γ is a proportionality coefficient
Modified Sahu-Mishra-Eldho (Sahu et al. 2012a)	(75) (76) (77) where fc is the minimum infiltration rate and T is rainfall duration	(78) (79) where S0 is the absolute potential maximum retention, and CN0 is the CN corresponding to a completely dry watershed
Wang et al. (2012b)	(80) (81) where Csat is the runoff coefficient for the saturation condition, Se is relative saturation, P is throughfall (mm) and α and λ are coefficients
Singh et al. (2015)	(82)	(85) (86) (87) where Sa is the threshold soil moisture, Sb is the absolute potential maximum retention, and α and β are coefficients
	(83)
	(84)
Jiao et al. (2015)	(88) (89)
Single-parameter simplified model as traditional NRCS-CN model
Pandit and Gopalakrishnan (1996)	for P ≥ 0.2S, Q = 0 otherwise	(90) (91)
Mishra-Singh (Mishra and Singh 2002a)	(92) (93)
Michel et al. (2005)	For AMC I: (94)
	For AMC II: (95)
	For AMC III: (96)
Mishra et al. (2006a)	(97) (98)	(99) where Mc is the antecedent moisture
Kannan et al. (2007)	(100) where S(t – 1) is retention parameter at the previous time step, PET is potential evapotranspiration at the present time step, B is the depletion coefficient that varies from 0 to 2, P is the rainfall depth at the previous time step, Q is the runoff depth at the previous time step and Smax is the maximum value of S	If depletion occurs at a faster rate under saturation, (101) If depletion occurs at a much slower rate as S approaches Smax, (102)
Sahu et al. (2007)	(103)	(106) where V0 = 0.4P5 if P5 ≤ 0.1S
	(104)
	(105)
Ajmal et al. (2015a)	(107)
Ajmal et al. (2015b)	(108)
Ajmal et al. (2016a)	(109)

5 Modified NRCS-CN-based hydrological model

A short review on long-term hydrological models based on the NRCS-CN method was presented by Mishra and Singh (2004a). This section provides an updated list of the empirical models and discusses their assumptions, limitations and advantages. The governing equations of these models are given in Table 3 and their advantages and limitations are discussed in Table 4.

Table 4. Advantages and limitations of reviewed models.

Model name	Advantages	Limitations
Williams and LaSeur (1976)	It eliminates sudden quantum jumps in CN values when changing from one AMC level to another. It can be applied to nearby ungauged watersheds by adjusting the curve number for the ungauged watershed in proportion to the ratio of the CN2 to the average predicted curve number for the calibrated watershed.	It utilizes an arbitrarily assigned value of 20 inches for absolute potential maximum retention, and assumes physically unrealizable decay of soil moisture with lake evaporation, which is a false assumption. It begins simulations 1 year before the actual calibration period because of a prior determination of initial soil moisture conditions. The actual soil moisture conditions are represented at the end of the first year, which leads to loss of 1 year of data.
Hawkins (1978)	It avoids sudden quantum jumps in CN values when changing from one AMC level to another.	It fails to distinguish dynamic infiltration from static infiltration and also couples the interim drainage with ET, which is quite conflicting because the water drained down to the water table may not be available for ET. The model formulation shows the combined participation of Ia and S in a dynamic infiltration process rather than S alone, which is contradictory because initially abstracted water is not available for transpiration due to high capillary suction. It allows an additional storage space of 20% of S to be available for water retention at every time step.
Smith and Williams (1980)	The model is more physically based.	It underestimates runoff (Shirmohammadi et al. 1997). It ignores the impact of rainfall intensity or rainfall duration.
Pandit and Gopalakrishnan (1996)	It is useful for small watersheds because it ignores routing, which is minimal in daily runoff computation.	It determines the actual AMC based on the previous 5-day rainfall and modifies CN such that CN does not exceed 98, which contradicts the traditional NRCS-CN concept where CN values can vary between 1 and 100. It allows sudden jumps in CN values and ignores evapotranspiration, drainage contribution and watershed routing.
Mishra et al. (1998)	It allows for transformation of rainfall excess to direct runoff, accounting for baseflow and making it applicable to large basins.	It allows sudden jumps in CN values while changing from one AMC to another. It does not distinguish between dynamic infiltration and static infiltration. The use of a linear regression equation raises the problem of mass balance. It stated that the baseflow is a fraction of F, which is not rational because the water retained in the soil pores may not be available for baseflow; rather the water that percolates down to the water table may appear at the outlet as baseflow.
Mishra-Singh (Mishra and Singh 2002a)	It obviates sudden jumps in CN values and, hence, in computing runoff through incorporation of the expression of M replacing the three AMCs.	Ia relies on interception, surface storage and infiltration, and all these factors depend greatly on the antecedent moisture, but explicit dependency of Ia on M is not represented.
Choi et al. (2002)	The approach of continuous linear variation of CN avoids sudden jumps in the calculation of CN values to a certain extent.	It ignores the number of storms and rainfall distribution within a day; CN estimation methods based on soil moisture at the end of the previous day cannot entirely account for the decrease in the retention parameter and the increase in CN through storms, which may lead to underestimates of runoff.
Mishra and Singh (2003b)	It is able to obviate the problem of a sudden jump. It is more reliable as it accounts the effect of storm duration.	It requires a priori knowledge of minimum infiltration rate.
Mishra and Singh (2004a)	It is more suitable for high runoff producing watersheds than forest watersheds, where the most sensitive model parameter Fc plays a significant role in the rainfall–runoff generation process.	It allows sudden jumps in CN values when changing from one AMC level to another. It is not capable of simulating high and low flow satisfactorily.
Michel et al. (2005)	It is more consistent with the SMA point of view by introducing the term V0 (initial soil moisture store level).	It does not provide any mathematical expression for V0, which plays vital role in the SMA procedure. S is assumed to be constant for a storm and variation in runoff is largely attributed to variation in P and V0.
Jain et al. (2006)	It performs better than the previously existing models.	It describes S as potential maximum retention and consider only one S (or CN) value for a watershed, whereas the actual NRCS-CN equations ((1)–(3) and (6)) estimate different S (or CN) for each P–Q data item (i.e. array of S or CN) on the same watershed.
Mishra et al. (2006a)	The model is more close to reality as it links the initial abstraction with the initial moisture as well as retention parameter S.	It ignores the impact of rainfall intensity or rainfall duration on the runoff.
Sahu et al. (2007)	It provides an expression for V0, which incorporates the concept of P5. The model is applicable to a continuous hydrological simulation.	It does not provide any expression for moisture threshold (Sa); therefore, it needs to be modified for recasting the SMA procedure by incorporating the variation of daily CN with respect to antecedent rainfall and moisture availability prior to the rainfall.
Geetha et al. (2007)	It obviates the sudden jumps in the CN values with change in antecedent moisture condition. It is able to identify the dominant runoff generation mechanism.	It does not perform well for the watersheds with low runoff factor.
Kannan et al. (2007)	It is found more suitable for shallow soil conditions.	It uses the rainfall amount of the previous day and does not take into account the effects of the amount of rainfall on a given day.
Kim and Lee (2008)	It is most suited to the estimation of high peak flows in a watershed having rapidly varying daily rainfall–runoff characteristics.	It usually overestimates very low peak flows.
Sahu et al. (2010)	It shows an explicit dependency of Ia on M, which leads to better performance of the model.	It ignores the impact of rainfall intensity or rainfall duration on the runoff.
Durbude et al. (2011)	It provides an expression for V0 based on AMA, thus leading to the advanced SMA procedure. This model performs better in high runoff producing watersheds than in low runoff producing watersheds.	The problem of simulating the peaks still remains in these models. The model is difficult to apply in the field as it needs many parameters that are generally not available for ungauged watersheds.
Babu and Mishra (2012)	It shows an explicit dependency of Ia on M, which leads to better performance of the model.	It ignores the impact of rainfall intensity or duration on runoff.
Modified Sahu-Mishra-Eldho (Sahu et al. 2012a)	The model is able to exclude the static infiltration component as it is identical to initial abstraction. It shows an explicit dependency of Ia on M, which leads to better performance of the model.	The model is difficult to apply in ungauged watersheds, as it needs the minimum infiltration rate, which is generally not available.
Singh et al. (2015)	It is able to obviate the sudden jump in CN or S. It is structurally more reliable from the SMA viewpoint.	It ignores the impacts of rainfall duration or intensity on runoff.
Jiao et al. (2015)	The model is structurally more stable as it deals with the dynamic depletion of antecedent daily effective rainfall induced by evapotranspiration and seepage.	It may not perform well if applied in watersheds with low evapotranspiration or seepage
Ajmal et al. (2015a)	It is simple and easy to use as it relies on a single parameter and retains all the simplicity of traditional models. It is well suited for mountainous steep slope watersheds.	It may not be suitable for mild or gentle slope watersheds. It ignores the effect of the initial wetness condition of the watershed.
Ajmal et al. (2015b)	It is able to obviate the sudden jumps and maintains all the simplicity of traditional models.	It ignores the impact of rainfall duration or intensity on runoff. Its applicability should be assessed in the watersheds of other biomes.
Ajmal et al. (2016a)	It is able to obviate sudden jumps and maintains all the simplicity of traditional models. It performed well in the forested watersheds of South Korea.	It may not perform well in the watershed of other agro-climatic conditions as well as low storm rainfall events (P < 25.4 mm).

5.1 Advanced model

5.1.1 Williams and LaSeur (1976) model

Williams and LaSeur (1976) developed a continuous simulation model by relating the retention parameter, S, of the traditional NRCS-CN model to soil moisture, M, for the computation of direct surface runoff. They were the first to incorporate the antecedent moisture, M, in the traditional methodology (Equation (13)), as it is depleted continuously between storms by ET and deep storage. The parameter M was assumed to vary with lake evaporation, as given by Equation (14). For applying the model in ungauged watersheds, they calibrated the average CN value of the watershed using Equation (15).

5.1.2 Hawkins (1978) model

Hawkins (1978) formulated a continuous SMA procedure by linking ET and CN for use in continuous hydrological simulation modelling. This model uses the volumetric concept to account for the site moisture on a continuous basis. Daily simulations are implemented by using Equation (16), which shows that, as P → ∞, the maximum possible water loss is equal to S_t (total space available for water storage), which is equal to 1.2S. The maximum water loss at a higher time level, t + ∆t, for storm duration equal to ∆t, can be derived from the moisture balance, given in Equation (17), by coupling these water loss relationships, and through substitution of Equation (6) into the resulting expression, as given in Equation (18) (Mishra and Singh 2004a). For known values of ET, P and Q within ∆t, and the computed value of CN_t, the next value of CN_(t+∆t) is calculated using Equation (18).

5.1.3 Smith and Williams (1980) model

Smith and Williams (1980) improved the NRCS-CN method by modifying the retention parameter, S. As shown in Equation (19), they introduced the weighting factor, W_f (Equation (20)), to account for the initial moisture in the soil profile and related it to the retention parameter. They reduced CN₂, corresponding to AMC II, to CN₁ (i.e. CN for AMC I) to identify the maximum retention, S_max (Equation (21)), and then adjusted it to the moisture distribution in the soil profile using the weighting technique.

5.1.4 Mishra et al. (1998) model

Mishra et al. (1998) suggested a NRCS-CN-based model for long-term hydrological simulations. They defined AMC based on the NEH-4 table and varied CN with time, as given in Equations (22) and (23). They used a linear regression approach similar to the unit hydrograph scheme to convert rainfall excess into direct runoff, which is given in Equation (24).

5.1.5 Mishra and Singh (2002a) model

Mishra and Singh (2002a) assumed that C = S_r, where C = Q/(P – I_a) is the runoff coefficient, and S is the degree of saturation. Using this concept, they modified the equation of direct runoff by incorporating the antecedent moisture, M (Equation (25)). They also formulated Equation (26), for estimating M.

5.1.6 Choi et al. (2002) model

Choi et al. (2002) proposed a cell-based long-term hydrological model, CELTHYM, for estimating direct runoff from small agricultural watersheds at daily time steps. The direct runoff at the grid scale (qdr) is computed using the traditional NRCS-CN method. Grid runoff is further averaged together to get the sub-watershed direct runoff, Q_sub (Equation (27)). Finally, the total direct runoff from the main watershed, Qdr, is obtained by the area-weighted average of the sub-watershed direct runoff, using Equation (28). Choi et al. (2002) calculated the retention parameter using adjusted CN, CN_adj, as given in Equation (29). The CN is assumed to vary linearly from CN₁ at AMC I to CN₂ at AMC II and CN₃ at AMC III (Equations (30)–(33), respectively), to allow for the representation of varying soil moisture conditions. In this model, AMC II is assumed at 50% of maximum available soil moisture, i.e. 0.5ASM_max, AMC I is considered the soil moisture at the wilting point, and AMC III is the soil moisture at the field capacity, because the soil moisture varies from wilting point to field capacity after drainage of excess soil water by deep percolation processes.

5.1.7 Mishra and Singh (2003b) model

Both Mishra and Singh (2002a) and Mishra and Singh (2003b) modified the traditional NRCS-CN formula by accounting for the static portion of infiltration, F_c, and antecedent moisture, M. The modified method assumed that, once the soil reaches saturation, a constant rate of infiltration will continue, which accounts for the seepage and percolation losses from the soil; this final constant rate of infiltration is referred to as F_c. At saturation, S = 0, so Q is the amount of rainfall after deducting F_c. The resulting modified expression for Q proposed by Mishra and Singh (2003b) is given in Equation (34).

5.1.8 Mishra and Singh (2004a) model

Mishra and Singh (2004a) proposed a versatile NRCS-CN model for long-term hydrological simulations. They described the water balance equations of Mishra and Singh (2002a) by discretizing them for time t, using Equation (36), and modified the runoff equation of Mishra and Singh (2002a) by means of Equation (37). They computed a dynamic infiltration, F_d, during ∆t using Equation (38), which represents an increase in amount of soil moisture, M, in the soil profile during ∆t. The soil moisture budget is expressed by Equation (39), which yields AMA for the next storm to modify S(t) for the next storm. The sum S_abs = S + M represents the absolute potential maximum retention, which corresponds to the completely dry condition of the soil and represents the maximum possible available void space in the soil profile ranging from zero to the upper bound of S. They also expressed ET(t) in terms of S(t) and S_abs, as shown in Equation (40).

5.1.9 Michel et al. (2005) model

Michel et al. (2005) developed a sounder methodology, hypothesizing that the NRCS-CN model is valid not only at the end of the storm but also at any instant during a storm. Their findings were based on an analysis of the continuous SMA implied by the NRCS-CN equation. They used the SMA process, which was based on the notion that high moisture storage in the soil will lead to higher rainfall conversion into runoff. Finally, they developed a procedure that is more consistent from the SMA viewpoint, by introducing two new variables: V₀, representing the initial soil moisture store level (Equation (41)) and S_a (= I_a + V₀); here I_a can be either negative or positive and is not a fraction of S, whereas the traditional NRCS-CN method is valid for I_a > 0 only. Based on the above hypothesis and Equation (42), Michel et al. (2005) showed several limitations in the traditional NRCS-CN procedure, due in part to confusion between intrinsic parameters and initial conditions, and partly from an incorrect use of the underlying SMA procedure. The generalized model, with new insight into the SMA process, is expressed through Equations (42)–(44).

5.1.10 Jain et al. (2006) model

Jain et al. (2006) proposed a new model formulation to enhance the NRCS-CN methodology, by incorporating the storm duration and a nonlinear I_a–S relationship, expressed by Equation (45). The modified I_a–S relationship relies on the adjusted rainfall P_c and S (Equation (46)). Jain et al. (2006) also developed a relationship to estimate antecedent moisture content using P₅ (48).

5.1.11 Mishra et al. (2006) model

Mishra et al. (2006a) experienced the high dependency of initial abstraction, I_a, on antecedent soil moisture, M, which is very close to reality because the larger the antecedent moisture, the smaller will be the initial abstraction, and vice versa. In this respect, they modified the I_a–S relationship, as given in Equation (49), and used it in the model of Mishra and Singh (2002a) for computing runoff.

5.1.12 Sahu et al. (2007) model

Sahu et al. (2007) proposed an expression for V₀ of the Michel et al. (2005) model, primarily based on the assumption that V₀ depends not only on P₅, but also on S. Considering S_a = αS, consistent with Michel et al. (2005), and using other suitable hydrological assumptions (Equation (51)), they derived Equations (52)–(54) for the three AMC conditions.

5.1.13 Geetha et al. (2007) model

Geetha et al. (2007) modified the existing NRCS-CN method by attributing CN to AMA in place of AMC, to eliminate the sudden jumps in CN values. They considered the current space available for retention, S_t, which is inversely proportional to CN_t, assuming CN_t = CN₀ or S_t = S₀ for the first 5 days (Equations (55–57)). For following days, S_t is calculated by Equation (58) and antecedent rainfall is computed using Equation (58). The AMA is a fraction of the antecedent 5-day rainfall, which is expressed through Equation (59).

5.1.14 Kim and Lee (2008) model

Kim and Lee (2008) developed a temporally weighted average CN method for daily runoff simulation, considering daily rainfall and ASM conditions. The model is based on the notion that existing approaches use daily time steps to provide estimations of streamflow, and a rainfall event is defined as the total sum of all rainfall that occurs during the calendar day, ignoring the number of storms and distribution of rainfall within a day. They developed Equation (60) for the computation of the weighted average CN on the current rainy day, by introducing a temporal weight that represents the degree of the effect of increase in soil moisture during the current rainy day.

5.1.15 Wang et al. (2008) model

Wang et al. (2008) proposed a modified curve number (MoCN) model which overcomes limiting the accuracy of Equation (4) by computing S as the difference between S₁ and a continuous variable M that represents a decrease in retention due to increase in soil moisture. While I_a is computed using Equation (46) (Jain et al. 2006), the other parameter M is represented as a power function of soil moisture, θ (Equation (61)).

5.1.16 Sahu-Mishra-Eldho (2010) model

Sahu et al. (2010) analysed the limitations of the Mishra and Singh (2002a) model and presented a hydrologically more sound procedure, by defining S = S₀ – M (here S₀ is the absolute maximum retention capacity). They modified Q and I_a estimations through Equations (62) and (63), respectively, and also introduced Equation (64) to calculate M. It may be noted that Equation (64) assumes I_a to be explicitly dependent on M, which is close to reality, given that the higher the antecedent moisture, the lower the initial abstraction, and vice versa.

5.1.17 Durbude et al. (2011) model

Durbude et al. (2011) modified the Michel et al. (2005) model to avoid the unrealistic sudden jumps in V₀ by incorporating a conceptual SMA procedure and using daily CN variation based on AMA rather than AMC. Between these two modified models, the long-term hydrological simulation LTHS MICHEL model used V₀ for AMC, as proposed by Michel et al. (2005), whereas the LTHS ASMA (advanced SMA) model re-conceptualized the Michel et al. (2005) SMA procedure to avoid the unrealistic sudden jumps in CN, and further in V₀, by incorporating an expression for V₀ based on AMA. They modified Equations (42)–(44) as Equations (65)–(67), respectively, to compute runoff by using expressions for V₀, V₀₀, AMA and SM (Equations (68)–(71)).

5.1.18 Babu and Mishra (2012) model

Babu and Mishra (2012) modified the Jain et al. model (2006) by including a new parameter, S_abs, to overcome most of the limitations existing in the NRCS-CN method and developed a new model consisting of five parameters, α, β, γ, λ and S_abs, as given in Equations (72)–(74). They assumed S_abs to be the sum of S and M (here S is a variable quantity from storm to storm and inversely varies with M). Furthermore, to make the model more realistic, M was modified as a function of P₅ as well as S_abs, which is represented as Equation (74). The model was developed with the same existing I_a–S relationship, but considering S_abs in place of S in its expression (Equation (72)).

5.1.19 Modified Sahu-Mishra-Eldho (2012a) model

Sahu et al. (2012a) modified the earlier Sahu-Mishra-Eldho (Sahu et al. 2010) model by considering the effect of cumulative infiltration, F_c, on runoff. They separated it from total infiltration, F, and excluded it as it was similar to I_a. They developed an improved equation for computing surface runoff (Equations (75) and (76)) and incorporated the storm duration for computing the static infiltration, F_c, as given in Equation (77).

5.1.20 Wang et al. (2012b) model

Wang et al. (2012b) proposed a more robust modified rational equation (MoRE) by coupling the NRCS-CN model and the rational equation (Q_p = CiA, where Q_p is peak discharge in m³ s⁻¹, C is a runoff coefficient, i is the average rainfall intensity in mm h⁻¹ for the time of concentration, and A is the area of the watershed in ha). The method includes four intrinsic parameters: CN; permanent wilting point, ψ; field capacity, θ; and saturated soil moisture, S_sat, but excludes I_a. It was found to be an improved version of the traditional NRCS-CN method and the methods of Michel et al. (2005), Sahu et al. (2007, 2012b) and Wang et al. (2008). The proposed formula for runoff computation is given in Equations (80)–(81).

5.1.21 Singh et al. (2015) model

Singh et al. (2015) used the C = S_r concept given by Mishra and Singh (2002a) and a modified basic hypothesis of the NRCS-CN methodology by incorporating V₀ in it. They formulated a set of equations, as given in Equations (82)–(84), and developed an enhanced version of the model. For the computation of initial soil moisture (V₀) and threshold soil moisture (S_a) required for surface runoff generation, they used well-tested functions, as given in Equations (85) and (88), respectively.

5.1.22 Jiao et al. (2015) model

Jiao et al. (2015) introduced the potential initial abstraction (I_d), which determines the maximum rainfall storage before runoff and the threshold of daily effective rainfall, and a decay constant, K, which describes the dynamic depletion of antecedent daily effective rainfall induced by ET and seepage. They employed an exponential function to estimate the water depletion and included it for computing the initial abstraction (Equation (89)).

5.2 Single-parameter approaches

5.2.1 Pandit and Gopalakrishnan (1996) model

Pandit and Gopalakrishnan (1996) developed a continuous simulation model for estimating the annual runoff and extended it for determining the annual pollutant loads. This model is useful for urban areas characterized by a given percentage imperviousness, which shows application of the NRCS-CN method in urban hydrology. It involves the following steps (Mishra and Singh 2004a):

1. Determine the pervious CN for AMC II.

2. Determine the directly connected impervious area of the urban watershed.

3. Estimate the daily runoff depth for both pervious and impervious areas separately, using Equation (5).

4. Determine the actual AMC based on P₅ and modify CN (using Equations (90) and (91)).

5.2.2 Mishra-Singh (2002) model

Mishra and Singh (2002a) replaced the λ value with 0.2, because it was highly accepted among researchers, and proposed a simplified version of their model for field application. The formulations are given in Equations (92)–(93).

5.2.3 Michel et al. (2005) model

Michel et al. (2005) simplified Equations (42)–(44) by letting S_a be a set fraction of S, i.e. S_a = 0.33S. Additionally, the parameter V₀ was replaced by a fraction of S corresponding to lower, median and upper deciles as representative values for AMC I, AMC II and AMC III, respectively. They finally suggested single-parameter models to compute the runoff for these three AMCs, as given in Equations (94)–(96), which retain the original definition of the SCS-CN method.

5.2.4 Mishra et al. (2006a) model

Mishra et al. (2006a) applied their model to 84 small agricultural watersheds in the USA. On the basis of their results, they replaced the parameter α with their mean value (0.72) and parameter λ with their median value (0.08) and proposed a single-parameter model for field application (Equation (97)).

5.2.5 Kannan et al. (2007) model

Kannan et al. (2007) used the expression given by Williams et al. (2000) for ET estimation and developed a simple one-parameter continuous SMA procedure to use within the NRCS-CN methodology for continuous hydrological simulation. The retention parameter at the present time step and an initial value of S are estimated using Equation (98) and the existing value of CN_2. Furthermore, they converted Equation (100) for two different conditions in order to estimate retention (Equations (99) and (100)). Equation (101) can be used when soil moisture depletes at a faster rate, whereas Equation (102) can be used when soil moisture depletes at a much slower rate.

5.2.6 Sahu et al. (2007) model

Sahu et al. (2007) applied their advanced models (Equations (52)–(54)) on 82 watersheds, which resulted in a number of optimized values of α and β coefficients. The median values among them were taken as 0.1 and 0.4 for α and β, respectively, and these simplifications yielded a single-parameter model, as described by the set of Equations (103)–(105), with V₀ defined in Equation (106).

5.2.7 Ajmal et al. (2015a) model

Ajmal et al. (2015a) considered the high dependency of runoff on the rainfall event and developed a new model for runoff computation. They assumed that I_a is 2% of the rainfall amount, which resulted in the runoff equation described in Equation (107).

5.2.8 Ajmal et al. (2015b) model

Ajmal et al. (2015b) developed a single-parameter model combining the concept of the traditional NRCS-CN model and the SMA procedure from Michel et al. (2005), which was used in the formulation of an event-based empirical GR4J model. They also applied the typical assumption I_a = 0.2S and V₀ values suggested by Michel et al. (2005) for the three different AMCs (i.e. V₀ = 0.33S, 0.61S and 0.87S for AMC I, AMC II and AMC III, respectively). The continuous variation of S was adjusted on the basis of both P and P₅ and replaced using the expression in place of S, in order to avoid sudden jumps in runoff prediction. The runoff equation is given in Equation (108).

5.2.9 Ajmal et al. (2016a) model

Considering continuous I_a after ensembling the traditional NRCS-CN model and the GR4J model, Ajmal et al. (2016a) proposed a single-parameter lumped model by replacing M_c of Equation (44) (Mishra et al. 2006a) with V₀ (Michel et al. 2005) and taking I_a for the three AMCs (i.e. I_a = 0.15S, 0.124S and 0.107S for AMC I, AMC II and AMC III, respectively) to avoid the problem of sudden jumps in runoff prediction. Furthermore, different transformed values of S were used, as in the Ajmal et al. (2015b) model. The proposed runoff equation is given in Equation (109).

6 Use of RS and GIS techniques

The RS technology is now being widely used by the hydrological community, because it considers two important characteristics of hydrological processes, i.e. variability over time and variability in space. Temporal variation has always been accounted for in hydrological measurements, but spatial variation has not been treated adequately with traditional techniques. The spatial distribution of variables is a unique feature of RS, which simplifies their incorporation into GIS. This potential makes several hydrological models more transparent and enables the fast communication of their operations and results to a large group of users. The growing literature on the integration of GIS within hydrological modelling confirms the recognition of such coupled benefits (e.g. DeVantier and Feldman 1993, Maidment 1996, Kherde and Sawant 2013). Numerous studies have also reported the use of GIS in hydrological modelling (e.g. Zhang et al. 1990, Bhaskar et al. 1992, Srinivasan and Engel 1994, Nourani et al. 2009, Fry et al. 2013).

Furthermore, with the advancement in satellite imagery and digital data, topography, LULC and HSG data are now available in digital format. Given the accessibility of digital databases, NRCS-CN application to distributed hydrological modelling has become very popular in the past two decades. Before describing the extended application of NRCS-CN methodology coupled with RS and GIS, it is appropriate to provide a brief account of the availability of digital databases for predicting CN to distributed hydrological modelling.

6.1 Availability of digital data

6.1.1 Topography

Spatially distributed topographical details are available in the form of DEMs. These models have been used widely to derive hydrological features which serve as inputs to various hydrological models. Currently, elevation data are available from several sources and at different spatial resolutions. Globally, DEMs can be downloaded from either the Shuttle Radar Topography Mission (SRTM) at http://srtm.usgs.gov/(90-m resolution, except for USA with 30-m resolution), or ASTER (30-m resolution). Using a GIS, the process of determining any of the required quantities from a DEM is largely an automated process.

6.1.2 Land use

Land use is an important characteristic of the runoff process that affects infiltration, erosion and evapotranspiration. In digital format, land use is often described by a series of polygons over a region, with each polygon indicating a region of assumed homogeneous land use. There are several sources available to describe land use at different resolutions. Landsat Thematic Mapper (TM) (30 m resolution) images represent valuable and continuous records of the Earth’s surface over the past three decades. Moreover, the entire Landsat archive is now available freely to the scientific community, which helps in identifying and monitoring changes in manmade and physical environments. The satellite data covering a study area can be downloaded from the Global Land Cover Facility (GLCF; http://glcfapp.glcf.umd.edu:8080/esdil) and the USGS Earth Explorer site (http://earthexplorer.usgs.gov/).

6.1.3 Soil data

Soil hydrodynamics information is one of the crucial inputs needed to assess the impacts of existing and alternative land-use management practices on runoff. Thus, it is important to evaluate the effects of the spatial scale at which the soil database is developed. The NRCS maintains spatially different soil databases in three GIS formats: (a) STATSGO (mapping scale: 1:250 000); (b) SSURGO (mapping scale: from 1:12 000 to 1:31 680); and (c) NATSGO, which can be used to derive soil water parameters such as saturation soil moisture, wilting point and field capacity needed for runoff estimation. The STATSGO database contains data as polygon coverage and is available throughout the USA. SSURGO is a more detailed soils database; however, not all areas are covered in digital format. However, NATSGO data describe variations in soil type from a multi-county to state scale (USA). Mednick (2010) studied the effects of spatial resolution (STATSGO vs SSURGO) on runoff across Wisconsin, USA, using a synthetic rainfall dataset and concluded that STATSGO-based estimates under-predicted the impact of rainfall intensity.

6.2 Coupling with NRCS-CN methodology

Integration of GIS and RS in hydrological modelling can enhance the quality of a hydrological model. Brooner et al. (1987) described many hydrologically significant parameters that can be obtained through RS, including land cover, vegetation properties, near surface soil moisture, surface roughness, ET, thermal and moisture indices and imperviousness.

Event-based hydrological simulations can be implemented through modification of the CN based on ASM, which can be retrieved using RS data. The most practiced approach for determining AMC is through API, by means of 5-day precipitation (SCS 1971), which can be retrieved using RS data. Blanchard et al. (1981) and Rosenthal et al. (1982) also related API to remotely sensed surface soil-moisture content. In contrast, long-term hydrological simulation models have a comprehensive structure, which requires a number of components, for example, infiltration, ET, percolation and subsurface flow. Among these, ET and surface runoff can also be estimated using remotely sensed data (Karimi and Bastiaanssen 2015).

GIS utilities have been exploited for the pre- and post-processing of spatially distributed hydrological data. Mancini and Rosso (1989) examined the spatial variability of the SCS-CN using a raster GIS and a distributed hydrological model. Choi et al. (2002) developed the CELTHYM model which can be integrated with a GIS to predict continuous streamflow from small agricultural watersheds on a daily time step. They used the NRCS-CN method considering soil moisture on a daily basis for surface runoff estimation (Savvidou et al. 2016). Therefore, the GIS is found to be an acceptable alternative, greatly simplifying data preparation and processing.

There are many articles that discuss the role of RS techniques for any particular application, namely, estimation of rainfall (Petty and Krajewski 1996, Kidd and Levizzani 2011), land surface evaporation (Kalma et al. 2008), water quality (Ritchie et al. 2003) and runoff (Schultz 1996). Several studies are also available evaluating the multidimensional applications of RS in hydrology and water resources (Kite and Pietroniro 1996, Kumar and Reshmidev 2013). In this article, we emphasize the role of RS and GIS in estimating and representing input data (e.g. rainfall, soil moisture) and watershed properties (e.g. LULC), as well as model parameters (e.g. CN) that are used within NRCS-CN methodology.

6.2.1 Rainfall

Rainfall exhibits high space–time variability and requires frequent observations to ensure adequate representation. Such observations are not possible through surface-based measurements over large areas, particularly in inaccessible regions (Huffman et al. 2007). Thus, observations through high-resolution satellite sensors have massive potential for use in hydrological modelling, particularly in developing countries, as an alternative to conventional raingauges, which are typically sparse. Satellite-based rainfall estimation techniques can be grouped broadly into three categories: (a) VIS and IR; (b) active and passive MW; and (c) a combination of both. The VIS and IR sensors offer data at very high temporal resolution but with crude estimation of rainfall, whereas MW sensors (e.g. AMSU, MADRAS) provide more accurate estimation of rainfall but at coarse temporal resolution. Therefore, the third category is the best for rainfall estimation, as recognized by numerous researchers (Adler et al. 1993, Jobard and Desbois 1994, Vicente and Anderson 1994, Mishra et al. 2009, Prakash et al. 2009, Behrangi et al. 2011). Furthermore, soft computing techniques, such as fuzzy logic and ANN, make use of IR and MW at various degrees of complexity and targeting different rainfall regimes (Sorooshian et al. 2000, Todd et al. 2001).

Over the past four decades, satellite rainfall products have been used widely for runoff estimation (Hong et al. 2006, Su et al. 2008, Ashenafi and Hailu 2014). Yilmaz et al. (2005) investigated the use of a satellite product called PERSIANN in streamflow forecasting, using a lumped hydrological model over several medium-size basins in the southeastern USA. Their analysis showed the improved performance of remotely sensed rainfall data in hydrological modelling when the model was calibrated with satellite data. Hossain and Lettenmaier (2006) described a number of issues that must be addressed prior to the adoption of global satellite rainfall datasets in hydrological models. Artan et al. (2007) demonstrated the enhanced performance of remotely sensed rainfall data in hydrological modelling by using satellite data instead of gauged rainfall to recalibrate the GeoFSM hydrological model for four sub-basins of the Mekong and Nile rivers. Milewski et al. (2009) used TRMM (3B42.v6) rainfall data for estimating runoff and observed that the regionalization approach is not a substitute for traditional methodologies that rely on extensive field datasets from raingauge and streamflow networks. Yet, they claimed that these can provide first estimates for rainfall and runoff over large sectors of the arid world lacking adequate coverage with spatial and temporal precipitation and field data. Sagintayev et al. (2011) also reported similar results and suggested the use of extensive field datasets. Their methodology was found useful for less-studied watersheds of the world, especially in areas where field data are inadequate or accessibility is limited.

In recent years, several applications using satellite-derived rainfall within CN-based hydrological models have also been reported. For instance, Jacobs and Srinivasan (2005) evaluated several versions of the NRCS-CN method for estimating near real-time runoff for naturalized flow using high-resolution radar (WSR-88D) rainfall data for watersheds in various agro-climatic regions of Texas, USA. They found that the use of the CN₁ value and a reduced initial abstraction ratio (equal to 0.05) produced the most statistically significant comparison between observed and estimated runoff. Hong et al. (2007) presented a global runoff model applying the NRCS-CN method. They used three RS global datasets: (a) rainfall data (TRMM 3B42), (b) LULC data (MODIS) and (c) soil data (TERRASTAT database). Their model calculated pixel-based CN values and runoff depths using the standard approach that has been practiced in hydrology for several decades. Their results showed that calculated runoff volume was larger than observed runoff for the flat and larger watersheds. Yilmaz et al. (2010) evaluated the global runoff model developed by Hong et al. (2007) and found that the model was able to simulate the onset of flood events produced by heavy precipitation; however, the simulation performance declined in the later stages, which pointed out the need for an improved routing component. Global-scale analysis showed that the model is able to detect 38% of the observed floods; however, it suffers from region-dependent bias. Curtis et al. (2007) also used satellite-derived rainfall and gauged runoff data to estimate CN for basins in eastern North Carolina, USA. Harris and Hossain (2008) found that the NRCS-CN method is more robust than more complex schemes in terms of handling uncertainty that exist in satellite rainfall data products. Collischonn et al. (2008) noticed that TRMM-driven daily simulations performed well for representing low flows, but peak flows tended to be overestimated. Finally, Coustau et al. (2012) developed a distributed rainfall–runoff model, coupled with the NRCS-CN method, and assessed its performance using radar rainfall data. The calibration results were found satisfactory by using hourly time steps, and the radar rainfall inputs significantly improved the simulations of the initial conditions for five events at the beginning of autumn (mostly in September and October), but were less efficient for the events at the end of autumn.

6.2.2 Soil moisture

The RS techniques, by virtue of their large area coverage and frequent revisit capability enabling repeated estimates on regular basis, are highly suitable for soil moisture estimation. Runoff estimations have been strongly improved after the addition of remotely sensed AWC data (Aubert et al. 2003, Jacobs et al. 2004, Crow and Ryu 2009, Brocca et al. 2010). These data can be retrieved from different RS methods, which can be classified into three main groups: (a) active, (b) passive and (c) a combination of both. The combination of high spatial resolution information from active sensors with high temporal resolution information from passive sensors on an extended area provides more accurate AWC. Microwave combined algorithms, statistical analysis techniques and neural network applications with both active and passive data are also highlighted in the literature.

Numerous researchers have used remotely sensed soil moisture products for hydrological modelling and found promising results (Goodrich et al. 1994, Wagner et al. 1999, Tischler et al. 2007, Wang et al. 2011, Tramblay et al. 2012, Brocca et al. 2013, Massari et al. 2014, 2015). In particular, in the context of CN-based applications, Hong et al. (2007) used API as a proxy of AMC to estimate time-varying CN values determined by 5-day normalized API. Their results demonstrated the potential of satellite rainfall data for AMC estimations, to then be used for runoff estimation. Beck et al. (2009) showed that CN-based runoff estimates at 186 Australian catchments were improved after inclusion of proxy soil moisture estimations based on remotely sensed data (AMSR-E and TRMM). In particular, the largest improvement was detected in drier catchments with low EVI and topographical slope. Brocca et al. (2009a) investigated the potential of the ERS scatterometer for assessment of wetness conditions in three sub-catchments of central Italy. They derived SWI to estimate the retention parameter of the NRCS-CN method and compared it with AWC derived from selected flood events. The results showed the great reliability of SWI for estimation of wetness conditions at the catchment scale and it was highly correlated with observed maximum potential retention. The computed runoff depth estimation provided good performance with an error of less than 25% for 80% of the investigated rainfall–runoff events. In another study, Brocca et al. (2009b) examined the relationship between soil moisture measurements of in situ and satellite sensors and soil potential maximum retention parameter. An inverse linear relationship was found, with the coefficient of determination decreasing with catchment area, but significant for the largest catchment. Tramblay et al. (2010) used TDR observations of soil moisture to estimate the potential retention of a small headwater catchment in France, and they found good correlation between TDR predictor and calibrated values of S. Tramblay et al. (2012) evaluated the ability of RS data (ASCAT and AMSR-E) to produce AWC for use in the NRCS-CN method. They used three AWC estimators (ADI, API and SMA) based on rainfall and ET. The results showed that both satellite products were able to reproduce daily soil moisture dynamics simulated by the SMA model with satisfactory accuracy. Finally, Meier et al. (2011) developed a forecasting model, based solely on RS data, providing soil moisture and rainfall estimates for prediction of discharge and applied it to three different sub-basins of the Zambezi River basin (Africa). The application of this model in a real-time framework yielded good results in watersheds where soil storage is an important factor.

6.2.3 Curve number

The CN depends on soil and land-use characteristics, the spatial variability of which is better represented through RS (Ragan and Jackson 1980). The use of RS also helps in updating land-use changes, particularly in areas where the land-use pattern is highly dynamic, causing significant variation in the hydrological processes. Then, an RS map can be integrated with HSGs (A-D) of the watershed using GIS, which gives more reliable CN values.

White (1988) applied the NRCS-CN method for runoff estimation by combining watershed information with daily rainfall data, using GIS. Stuebe and Johnston (1990) used GIS to predict the runoff volume for six basins. Tauer and Humborg (1992) used RS data and GIS to determine potential sites for water harvesting. Gumbo et al. (2002) found that the CN method worked well in the GIS environment because of its relatively simple formulation. Akhondi (2001) pointed out that increasing watershed area decreases correlation between observed and estimated discharge using the CN method. Jasrotia et al. (2002) determined the rainfall–runoff relationship for the Tons watershed (India), by deriving CN through RS and GIS techniques. Sandholt et al. (2003) also investigated the use of RS data in distributed hydrological modelling.

Zhan and Huang (2004) developed the ArcCN runoff tool, an extension of ESRI ArcGIS software, which can be applied to determine CNs and calculate runoff for a storm event within a watershed. They also suggested that the implementation of a precipitation time series and consideration of factors such as dry and wet AMCs would improve the runoff predictions. Kamuju (2015) compared runoff estimated by the ArcCN runoff tool and ArcSWAT and found that the first gives 90% of rainfall in the form of runoff, while 75% of rainfall becomes surface runoff from ArcSWAT.

Nayak and Jaiswal (2003) found a good correlation between measured and estimated runoff depth using GIS with CN, and concluded that GIS is an efficient tool for preparation of most of the input data required for the NRCS-CN method. Many researchers applied this technology to calculate runoff curve numbers using Landsat, IRS-1A and IRS-1B data. Ebrahihimian et al. (2012), Fan et al. (2013) and many others have estimated runoff using the CN of a basin and RS data in a GIS environment. Ebrahimian et al. (2012) and Garg et al. (2013) tested the effect of slope on runoff using the NRCS-CN method and RS and GIS. Many others used NRCS-CN coupled with RS and GIS technologies for runoff estimation (e.g. Sharma and Singh 1992, Chakraborti 1993, Tiwari et al. 1997, Chatterjee et al. 2001, Tripathi et al. 2002, Ashish et al. 2003, Bhanumurty et al. 2003, Gupta and Panigrahy 2008, Ramakrishnan et al. 2009, Gajbhiye and Mishra 2012, Dhawale et al. 2013, Nagarajan and Poongothai 2012, Nayak et al. 2012, Sindhu et al. 2013, Thakuriah and Ranjan 2014, Zende et al. 2014, Topno et al. 2015). Finally, Hernández-Guzmán and Ruiz-Luna (2013) developed a vector-based interface to estimate CN and runoff volume for hydrological evaluations. The interface allows users to select different AMC and λ to fit the local conditions.

6.2.4 Land use/land cover (LULC)

Pandey and Sahu (2002) pointed out that LULC, which is used for the determination of CN, is an important input of the NRCS-CN method. Ragan and Jackson (1980) showed that urban land uses, as well as various categories of agriculture and forest cover, can be determined accurately through RS, and can then be used to estimate CN values. Still and Shih (1984, 1985, 1991) used Landsat data to develop a basin-wide runoff index and successfully demonstrated how RS data can be used to track the changes in runoff that occur in a basin due to land-use change. Sharma et al. (2001) digitally analysed multi-date satellite imagery for improving accuracy in mapping LULC in a multi-crop cultivated watershed.

7 Advantages and limitations of RS and GIS-based approaches

The rising accessibility of high-resolution satellite data products is providing hydrologists and water resource practitioners with ample opportunities to improve their simulations, especially in developing countries and inaccessible regions where in situ measurement networks are sparse (Anagnostou 2004, Hong et al. 2006, Hossain and Lettenmaier 2006, Ebert et al. 2007, Gourley et al. 2010, Yong et al. 2010, Bitew and Gebremichael 2011, Wang et al. 2012a, Xue et al. 2013, Awadallah et al. 2016). As indicated so far, the success of NRCS-CN methodology is often governed by the precision of CN and I_a parameters, which are very sensitive. Typically, these are assumed constant over space and time. However, even for the same location, these values are highly variable due to changes in land use, crop cover, crop growth, land treatment etc. Therefore, RS and GIS techniques provide inputs more efficiently, as well as changing parameters (i.e. CN) to NRCS-CN methodology, as discussed in previous sections.

Despite the multiple advantages offered by RS and GIS techniques in hydrological modelling, they have some limitations. For example, in most satellite systems the revisit time can be a critical problem in studies involving rapidly changing conditions, such as surface soil moisture, and, especially, if a constant viewing angle is important, the revisit time can be much longer. Thus, optimizing the time and frequency of coverage is a critical problem for measuring soil moisture. Microwave remote sensing offers unique advantages over other spectral regions. However, major difficulties exist because the MW signal is influenced by various terrestrial factors (e.g. surface roughness, vegetation) and the resolution in space (passive MW) is too coarse for many hydrological tasks. AMSR may be the optimal option for mapping soil moisture in regions of low vegetation cover (<1 kg m⁻² vegetation water content), because it operates at high frequency, i.e. 6.9 GHz (Njoku et al. 2003).

In the case of precipitation, there are many RS possibilities, including ground-based radar, VIS and thermal IR satellite imagery and MW satellite data. Radar systems provide high-resolution precipitation data over space and time. However, their spatial extent is inadequate to detect global precipitation and to evaluate hydrological models at global and continental scales. This limitation highlights the importance of satellite-based global precipitation data for hydrological and other studies. Over the past decade, the use of precipitation data observed by satellites in hydrology has advanced greatly with the launch of various satellite series (i.e. GOES, DMSP) and research platforms and with the development of more sophisticated retrieval algorithms (e.g. TRMM 3B43, CMORPH, PERSIANN). Despite the significant developments in satellite observation of precipitation, the coarse spatial and temporal resolution of the data are still inappropriate for most hydrological and climate studies. In this respect, there exist a number of challenges, such as: (a) the development of hybrid solutions to combine the benefits of satellites and ground observations (e.g. ground radars and gauge networks); (b) the use of advanced multi-spectral algorithms to improve the accuracy and spatial resolution of satellite precipitation; (c) improving computational efficiency; (d) enhancing the ability to detect liquid and solid precipitation, (e) the development of IR-based algorithms for capturing extreme precipitation more reliably; and (f) understanding and communicating uncertainties of precipitation estimates and forecasts.

8 Future research and opportunities

In general, future research and opportunities are related to present and upcoming information requirements over large areas, and the expected evolution of the associated data and methodology. In all these areas, the NRCS-CN methodology has received a strong boost. The methodology still has issues requiring further improvement, which can be achieved by incorporating the effects of spatial and temporal variations of rainfall, soil moisture and LULC parameters, treating λ as a regional and climatic parameter, estimating CN using RS and GIS techniques, transferring model parameters to regions with similar hydro-climatic and physiographic characteristics (e.g. slope), and refining the I_a–S relationship for different regions and different soil, land-use and climate conditions.

In addition to the above suggestions, another major inconsistency of the method is that it implicitly seems based on infiltration excess and similarity of the Horton-based approach. Saturation excess, however, is dependent on landscape factors that are rarely or never used to defined CN values. Thus, there is a need to develop methods using saturation excess as an important hydrological process.

Recent development in RS techniques and in current satellites and sensors provide many of the necessary data to supplement conventional data, to expand the NRCS-CN methodology in new directions. Ongoing development and future satellite missions, as well as innovative GIS techniques, will offer entirely new data types and forms that will improve the functioning of the method, which will also help hydrologists tackle previously unsolved problems. In addition, combinations of RS data from different sources, as well as combinations with spatially distributed in situ measurements (e.g. digital maps, digital terrain models), provide very promising approaches for the purpose of CN-based runoff computations in the future.

Current satellites/sensors and missions also offer new opportunities for rainfall and soil moisture estimation, in general, and hopefully for MW applications, in particular. Rainfall can be estimated from geostationary satellites. It is also hoped that MW data (passive or active) may be used for the determination of soil moisture in the upper layer (i.e. few centimetres), as well as in deeper layers (i.e. up to several decimetres). If such information is available, a breakthrough in hydrological modelling could be expected.

Future research on data assimilation techniques, error properties in different climate regions, storm regimes and seasons, sources of error (i.e. satellite error) are also necessary to evaluate the quality and reliability of satellite data estimates across different regions and climate conditions. This task will require improved retrieval algorithms, uncertainty models and bias adjustment techniques.

For high resolution and reliable precipitation or rainfall data, future research should also focus on: (a) evaluation of extremes using ground reference measurements (e.g. ground radars and gauges) to understand to what extent one can observe extremes reliably; (b) multi-spectral and multi-satellite retrieval techniques; (c) improvements in GPM datasets by combining downscaling and artificial intelligence techniques; and (d) developing data assimilation techniques in order to integrate assimilation as a part of the modelling and algorithm development approach. For soil moisture estimation, future research is highly desired in integrating the space-borne measurements from multiple sensors, especially high-resolution SAR images, with radiometric data (e.g. SMAP mission). In the context of global climate modelling and regional weather forecasting, large-scale hydrological modelling is of increasing relevance. Here, runoff computations are of particular importance for calibrating such combined atmospheric and hydrological models, and the use of RS data for these purposes at large scales is gaining relevance (Schulz 1988). The GPM mission as an international network of satellites, which is available globally, is perhaps an extensive use of RS data for large-scale hydrological modelling. Furthermore, ground validation ensures that the satellite estimates are statistically accurate.

9 Conclusions

This article presents a critical review of NRCS-CN-based approaches for runoff estimation, coupled with RS and GIS applications. In view of the above discussion, it can be realized that in recent years the methodology has been improved by using SMA procedures, by incorporating AMA, storm duration and slope adjustment, as well as linear and nonlinear I_a–S relationships, by linking the dependency of initial abstraction on initial moisture amount, and by accounting for absolute retention, static and dynamic infiltration, ET, baseflow computation, etc. Despite several modifications, there is still need for further refinement in the critical components of the methodology, such as the SMA procedure for water balance study, simulation of peak of hydrograph, use of spatially distributed data, and GIS interfaces for initial data processing.

Thus, further research and investigation are needed in the context of model development and validation. Both RS and GIS techniques and the data derived using RS methods can and should play a pivotal role in future research. The capability of high spatial resolution satellites/sensors and missions for global data has been found to be advantageous in many hydrological applications, including the NRCS-CN methodology. Ongoing research for deriving high-accuracy parameter and input data, based on RS and GIS, can be expected to ensure more accurate runoff estimations. In particular, by 2020, there will be an unprecedented stream of concurrent observations from the GPM, SMAP, GRACE and GRACE-FO missions, and eventually the SWOT and ECO stress missions. As we look ahead, these satellites will soon provide routine global monitoring of all stocks and fluxes for continental- to regional-scale hydrology.

Acknowledgements

The authors are thankful to Associate Editor, Dr Andreas Efstratiadis, and the anonymous reviewers for their thoughtful comments, which helped to improve the article.

Disclosure statement

No potential conflict of interest was reported by the authors.