Runoff Curve Number Method: Beyond the Handbook

**Joseph A. Van Mullem, Hydraulic Engineer, USDA, NRCS, (retired)
Bozeman, MT**

**Donald E. Woodward, National Hydrologist (retired), USDA, NRCS, Wash.,
DC**

**Richard H. Hawkins, Professor, University of Arizona, Tucson,
Arizona**

**Allen T. Hjelmfelt Jr., Hydraulic Engineer, USDA, ARS, Columbia,
Missouri**

25 Annette Park Dr., Bozeman, MT 59715, <jvanmullem@in-tch.com> (406)
586-0701

[**Abstract**:]{.underline} The behavior of rainfall-runoff systems in
terms of the Curve Number method is described, including findings that
have become apparent and/or appreciated since the original development
in the mid-1950s; not all of which are shown in current versions of the
primary reference, NEH-4. These include the following: different modes
of application of the method; basic general patterns of rainfall-runoff
at variance with the CN equation; sensitivity information; the basic
intractability of the method with infiltration-based systems;
limitations in the role of prior rainfall; difficulties in the
hydrologic descriptions of soils; CN variation with season and land use;
and wide spread application to continuous models as a soil moisture
budgeting device.

# 

# Introduction

The Natural Resources Conservation Service (NRCS) and the Agricultural
Research Service (ARS), both agencies of the U.S. Department of
Agriculture formed a joint work group in 1990 to assess the state of the
Curve Number method and to chart it's future development. The need for
the work group resulted from questions and concerns about the method,
its unique role in applied hydrology, and how it was being used and the
recognition of the need to provide guidance for reanalysis and
clarification of the procedure (Woodward 1991). A number of goals and
objectives were developed including: reevaluating the procedure,
improving the documentation, updating the primary reference document
(NEH-4), developing a data base, standardizing the procedure for
developing curve numbers from data, etc.

It became apparent that much of the difficulty surrounding the procedure
was attributable to the presentation in the National Engineering
Handbook (NEH-4). Thus an early task was to rewrite those portions of
the NEH-4 pertaining to the procedure. Problems identified ranged from
incorrect and misleading statements to incomplete documentation. For
example it was incorrectly stated that S includes I~a~, whereas it can
be easily shown mathematically that S does not include I~a~. Fortunately
this is only significant for continuous simulation. Another example was
Table 4-1 that related 5-day antecedent rainfall to antecedent moisture
condition (AMC) for a local situation. This was not intended to have
nationwide application, though it was widely treated as such. Several
"Folklore" beliefs concerning Curve Numbers could also be attributed to
problems with documentation. An example of such is that the Curve Number
Runoff Equation is an infiltration equation.

In rewriting the Curve Number portions of NEH-4, the work group agreed
that they must agree with concepts expressed, references will be
included and the result must be technically defensible. The results of
the updating include such items as:

1)  Reference to Antecedent Moisture Condition (AMC) was removed and the
    terminology changed to Antecedent Runoff Condition (ARC).
    Variability of the curve numbers and event runoff is incorporated by
    considering the curve number as a random variable and the ARC-I and
    ARC-III conditions as bounds on the distribution.

2)  Reiteration of desirability of locally determined curve numbers was
    made. Although this was part of the original documentation in NEH-4,
    local calibration has been seldom pursued.

3)  Explicit expression of Curve Number runoff equation was made as a
    transformation of rainfall frequency distribution to runoff
    frequency distribution. This too was demonstrated in the original
    documentation.

4)  Expression of ARC-I and ARC-III as measures of dispersion about the
    central tendency (ARC II). This is a corollary of treating the CN as
    a random variable.

5)  Mathematical proof and demonstration that S does not include I~a~.

6)  The chapters are posted on the World Wide Web as approved.
    http://www.wcc.nrcs.usda.gov/water/quality/wst.html

As the work progressed several other areas of need became evident. These
were 1) to reconsider the hydrologic soils classifications recognizing
the vastly expanded database available today and the capabilities of
modern computers, 2) to reevaluate the use of I~a~ = 0.2S, and 3) to
reconsider the tables of curve numbers in terms of the expanded
rainfall-runoff database available (Agricultural Research Service 1995).
There was also considerable concern about the possibility of regional
and seasonal variation of curve numbers (Hjelmfelt et al 2001).

# Ordered Pairs and Asymptotes

One of the goals of the work group was to standardize the procedure for
calculating a curve number for a watershed from rainfall-runoff data.
The accepted handbook method was to plot the annual series of
rainfall-runoff on a scatter diagram and select the curve number that
best fit the data. This method doesn't make use of the many storms that
were not the largest annual events.

The runoff curve number equation is often used to transform a rainfall
frequency distribution into a runoff frequency distribution. That is,
for example, the 100-year rainfall is used to determine the 100-year
runoff, etc. This practice, called frequency matching leads to the idea
of ordered pairs (Hjelmfelt 1980). Hawkins (1993) followed this idea by
sorting the rainfall and runoff depths separately and re-aligning them
on a rank order basis creating new sets of rainfall-runoff pairs. These
might be thought of as having equal return periods with the individual
runoffs not necessarily associated with the original causative rainfall.

Using all the storms in the data set, Hawkins then calculated the curve
number for each of the ordered pairs and plotted them against rainfall.
Curve number was found to vary with storm depth, but - for most cases --
approaches a constant value at higher rainfalls. The curves that result
from these plots were then fitted with asymptotic equations to approach
this constant CN. The limiting CN that is approached as rainfall
approaches infinity is taken as the best fit curve number for the
watershed. The procedure is best illustrated by the figures in the
following section.

This method of determining a watershed curve number has the advantage of
being mathematical and therefore programmable. The results are mostly
influenced by the largest events, which is in keeping with the usual
intended applications, but all of the data is used. Additionally, the
decrease in scatter results in illustrative graphics which are
descriptive of the watershed behavior and therefore a valuable tool for
analysis. This method appears to give results consistent with the
present procedures and CN tables in NEH-4. The curve number work group
has adopted this procedure to determine curve numbers from local data in
the future.

# Behavior of Rainfall-Runoff Systems

The plots described above show several types of CN "behavior". These
have been defined by Hawkins (1993) as Complacent, Standard and Violent.
The Standard behavior fits the curve number model best; the Violent less
so, and the Complacent not at all.

Standard behavior is the most common scenario. The observed CN declines
with increasing storm size and then approaches a near constant value
with increasingly larger storms. An example of this behavior is shown in
Figure 1. About 70% of all watersheds evaluated showed this pattern
(Hawkins, 1993). The process of computing CN for small events biases the
CN toward high values. This is because CN can be computed only if direct
runoff occurs. If the event CN is low, and the event initial abstraction
(I~a~) is high, no runoff occurs and CN cannot be computed. Only high
CNs can be detected with low rainfalls. The plot of CN vs. rainfall
displays this bias and the storm magnitude at which the bias becomes
insignificant.

![](media/image1.wmf)

Figure 1. Standard behavior. The asymptotic CN for this example is about
68.

Complacent behavior is defined when the observed CN declines with
increasing rainfall and there is no appreciable tendency to achieve a
stable value. Curve numbers cannot be safely determined from data that
exhibit this pattern. The behavior indicates a partial source area
situation where the source area fraction may be quite small. An example
of this behavior is shown in Figure 2. There is no easy way of telling
if and at what rainfall that the source area or the curve number may
change or violent behavior may occur.

![](media/image2.wmf)

Figure 2. Complacent Behavior**.** No constant CN is determined for this
data.

![](media/image3.wmf)

Figure 3. Violent Behavior. The asymptotic CN for this example is about
93**.**

Violent behavior is seen where the observed CN rises suddenly and
asymptotically approaches an apparent constant value. This is often
accompanied by complacent behavior at lower rainfalls. From a source
process standpoint, this could be a threshold process at some critical
rainfall depth value. At rainfall above the threshold a high fraction of
the rainfall becomes runoff. An example of Violent behavior is shown in
Figure 3**.** For this example the threshold rainfall is about 0.80
inches**.**

# Modes of Application of the CN Method

The work group recognized three distinctly different modes of
application for curve numbers: 1) Determination of runoff volume of a
given return period, given total event rainfall for that return period.
2) Determine the direct runoff for individual events. This acknowledges
the variation between events and is the basis for the development. 3)
Process models, an inferred application as an infiltration model, or a
soil moisture-CN relationship, or as a source area distribution.

The first application, already discussed**,** is the most widely used in
engineering and uses the curve number to transform the rainfall
frequency distribution into a runoff frequency distribution. It was the
reason for the development of the model. The runoff volume that is
computed is often overlooked and the peak discharge, which is more
frequently the desired value, calculated with a unit hydrograph model
and used directly.

The second application, runoff from individual rainfall events, is the
basis for the original development, the P vs. Q plots which led to the
curve number concept. There is a wide variation of runoff from rainfalls
of the same magnitude which forces us to acknowledge that CN varies
between storms for a side range of reasons. The original handbook --
developed in large part for conditions in the humid east, south, and
mid-west - designated "Antecedent Moisture Condition" or AMC as the most
significant variable in explaining this. Average condition moisture was
called AMC II and applied to the curve number when flooding occurs. Dry
conditions (AMC I) applied to the low curve number, and wet conditions
(AMC III) applied to the high curve number. This condition is most often
determined by prior rainfall since soil moisture conditions are not
frequently monitored.

The work group studied the effect of soil moisture on curve number by
looking at infiltrometer studies (Van Mullem 1992). Four studies with
162 data pairs on 86 different soils from across the U.S. were used.
Although average CN increased from 9% to 40% between the studies from
the initial (dry to average condition) test to the wet condition test,
no significant relationship was found between soil moisture and CN. The
study indicated that the difference in CN that might be related to soil
moisture is much less than the variation between ARC I and ARC III.
Similarly, Hawkins and Cate (1998) showed for 25 agricultural watersheds
that 5-day prior rainfall (AMC5) was the only consistent factor in
explaining deviations from the central trend of runoff, but in only in
11 of the 25 cases studied, and at levels far below handbook
expectations.

Because prior rainfall explains only part of the variation of CN the
terminology has been changed to "Antecedent Runoff Condition" or ARC.
More importantly, the ARC I and ARC III conditions have been shown to be
the bounds on the distribution of CN. Figure 4 shows the ARC I, II, and
III curve numbers plotted on a rainfall-runoff scatter diagram. Figure 5
shows that the ARC I is the 10% exceedance and ARC III is the 90%
exceedance for a number of agricultural watersheds (Hjelmfelt et al
1981).

![](media/image4.wmf)Figure 4. Rainfall-Runoff scatter diagram showing
ARC I, II, and III CN.

![](media/image5.png){width="4.625in" height="2.972916666666667in"}

Figure 5. The 10% and 90% exceedance values compared to the ARC I and
ARC III curve numbers.

It might be inferred that the curve number model is an infiltration
model because in its application it is used to determine runoff
incrementally over the duration of the storm for input into a unit
hydrograph model. With this use it becomes a surrogate for an
infiltration model and this has created much confusion. The model
doesn't behave like most infiltration models/equations because the CN
losses don't always decline with time or with prior rainfall but may
increase when the rain intensity increases (Hawkins 1980). The CN model
behavior in this regard is the same as from a partial area saturation
model. Also the ultimate steady state rate is zero, it does not approach
a steady-state non-zero infiltration rate with time, as do (say) the
Horton or Green-Ampt Equations. There doesn't seem to be any consensus
as to whether this is the way watershed losses occur. However, the model
performs as originally intended, i.e., as an integrator of all the
losses from all processes over the watershed. It should be emphasized
that the CN model is not a point infiltration model and the difference
between rainfall and runoff is better defined as watershed "losses."
Watershed indices -- such as the curve number -- are lumped expressions
of net basin performance.

It is also sometimes inferred that the parameter S, defined as the
potential maximum retention, is a physical property of the site like a
soil moisture storage parameter, and the water in it can be accounted
for. This is has not been shown with any certainty. The parameter S (or
CN) is a model variable and is only a constant for a particular storm.
Although it is related to soil and cover characteristics, it is not an
identifiable physical property.

# Hydrologic Soil Groups

There has been a vast increase in basic soils property data since
Musgrave first proposed the concept of hydrologic soil groups in
Handbook of Agriculture (USDA 1955)**.** The data is now available in an
electronic database. Modern tools of data mining were explored for
analysis of this mass of data. Both neural networks and fuzzy sets were
tried with fuzzy sets being adopted. Soil Hydrologic Groups are assigned
to soil series and phase of series by soil scientists based upon their
interpretation of the published criteria. The soil scientist's
interpretation of the published criteria has varied across time and
between states or regions. Thus, the hydrologic group criteria are not
applied consistently across the United States. This is most evident in
the comparison of soils with similar soil hydrologic and physical
properties and dissimilar hydrologic group placement.

The Hydrologic Soil Groups are A, B, C, D, and dual groups A/D, B/D and
C/D. Soils in hydrologic group **A** have low runoff potential. Soils
that have a moderate rate of infiltration when thoroughly wet are in
hydrologic group B. Hydrologic group **C** soils that have a slow rate
of infiltration rate when thoroughly wet. Soils in hydrologic group
**D** have a high runoff potential. Dual Hydrologic Soil Groups (A/D,
B/D, and C/D) are given for certain wet soils that could be adequately
drained. The first letter applies to the drained and the second to the
undrained condition. Soils are assigned to dual groups if the shallow
depth to a permanent water table is the sole criteria for assigning a
soil to hydrologic group D.

A model or rule based automated system that provides for objective
placement of soils into Hydrologic Soil Groups was developed. The fuzzy
system model for assigning soils to hydrologic soil groups is based on
the published hydrologic group assumptions and criteria. The soil
surface is taken to be bare and the soil is not permanently frozen. The
soil physical and hydrologic characteristic which make up the hydrologic
grouping criteria are the depth to permanent water, depth to a
restrictive layer, minimum saturated hydraulic conductivity in the
soil's upper 100 cm, and the soil's texture.

The model was applied to 1828 unique soils using data from Kansas, South
Dakota, Missouri, Iowa, Wyoming, and Colorado and the correlation
between these soils' assigned and modeled hydrologic grouping was
analyzed. Table1 shows a detailed comparison by Hydrologic Soil Group
between the currently assigned HSG and the modeled HSG. The correlation
between the assigned and modeled HSG A and HSG D soils is higher than
the correlation between the assigned and modeled HSG B and HSG C soils.
There are several reasons for the poorer correlation between the
assigned and modeled groups B and C. The first is that of the boundary
condition that occurs when a soil has properties that do not fit
entirely into a single hydrologic group. In this case, the soil
scientist may have placed the soil into one HSG while the model placed
the soil into an adjacent group. Groups B and C are the most prone to
this error because

they are bounded by two groups whereas HSG A and D are only bounded by
one group. Another source of correlation inconsistency is that the
assigned HSG may be relatively correct but the data in the database may
not support the corresponding HSG determination by the model. Finally,
correlation inconsistencies can be attributed to the fuzzy modeling of
the subjective Hydrologic Soil Group criteria (Hjelmfelt et al 2001).

![](media/image6.wmf){width="5.999305555555556in"
height="1.4347222222222222in"}

TABLE 1: Correlation frequency between assigned and fuzzy modeled
Hydrologic Soil Group

# Variation of CN with Season and Land Use

Curve Numbers were derived from rainfall-runoff data for 15 distinct
land uses on 177 small watersheds in the United States (Rietz and
Hawkins 2000). Curve Numbers for each land use on each watershed were
calculated using the asymptotic method and evaluated at the local,
regional and national scale. Significant differences at the 5% level
were found between the CNs of almost all of the different land uses
tested. Significant differences in CN were also found on grazed and
ungrazed paired watersheds, and on watersheds that had undergone land
use conversions.

The general magnitudes and rank order of the average land use CNs were
in general agreement with expected handbook values. Meadows almost
always produced the lowest CN at both the local and regional level.
Forestland produced the lowest overall average CN at the national level,
but also displayed the largest variability. No significant differences
could be determined between curve numbers for pasture and rangeland at
the regional scale or between row crops and small grain at any scale.
Where comparable, pastures usually had higher CNs than meadows. None of
these comparisons considered hydrologic soil group or any other soil
parameter.

Seasonal variation of CN has also been noted. It is seen more readily in
the more humid settings, and is rare in arid and semi-arid watersheds.
Where evident it follows a pattern with higher CNs in the dormant season
when the ground has less cover and is likely to be wetter, and lower CNs
during the summer when the ground is dryer and vegetation is in a high
growth stage. Also, the seasonal variation in forest curve numbers may
be associated with the leafing stages (Price, 1998).

# Application to Continuous Models

The curve number is used in hydrologic models to divide rainfall into
storm runoff and associated losses. The losses are then added to the
soil moisture budget in the model. Rainfall for most of these models is
on a daily time-step. Poor results are often obtained because most
storms are small, and the CN model does not work well for small events,
forcing the model user to increase CN or to devise some other scheme to
increase runoff at certain times of the year. Most of these models use
the soil moisture budget as an index to vary the CN between ARC I and
III. As previously stated the soil moisture should be used to explain
only part of this variation.

Most models assume that ARC I is equivalent to wilting point and ARC III
is equivalent to field capacity of the soil. This allows the CN for ARC
II to be converted to appropriate daily CN. There generally is some
procedure for frozen ground and snow melt in each model. However the
basic CN technology was not developed for these conditions.

## Sensitivity

Over a wide range of rainfall depths, the direct runoff Q calculated by
the equation is more sensitive to Curve Number than to rainfall depth P.
(Hawkins, 1975). This is especially true close to the threshold of
runoff. Similarly, hydrograph peaks modeled using CN-derived sequential
pulses of rainfall excess are also usually more sensitive to CN than
rainfall depth. (Bondelid et al, 1982) The supporting data situation
contrasts this order of importance: rainfall is widely measured,
studied, analyzed, and reported, but ground truth for Curve Numbers is
rare.

# References

Agricultural Research Service Water Data Center, 1995. ARS Water Data:
ARS/Access CD. USDA-ARS Hydrology Lab, Beltsville, Maryland.
Agricultural Research Service Water Data Center, 1999.
<http://www.hydrolab.arsusda.gov/arswater.html>.

Bondelid, T.R., R.H. McCuen, and T.J. Jackson. 1982. "Sensitivity of SCS
Models to Curve Number Variation." *Water Resources Bulletin*, 18(1),
111-116. (February 1982.)

Hawkins, R.H. 1975. "The Importance of Accurate Curve Numbers in the
Estimation of Storm Runoff". *Water Resources Bulletin,* 11(5), 887-891.
(October 1975.)

Hawkins, R.H. 1993. Asymptotic determination of runoff curve numbers
from data. Journal of Irrigation and Drainage Engineering. Amer Soc Civ
Eng. 119(2): 334-3 45.

Hawkins, R.H, and A**.** Cate. 1998. Secondary Influences in Curve
Number Rainfall-Runoff. Presentation to Amer Soc Civ Engineers, Memphis
TN.

Hjelmfelt, A.T. 1980. Empirical investigation of curve number technique.
Journal of the Hydraulics Division. Amer Soc Civ Eng 106 (HY9):
1471-1476.

Hjelmfelt, A.T., D.A. Woodward, G. Conaway, Q.D. Quan, J. Van Mullem,
and R.H. Hawkins. 2001. Curve Numbers, Recent Developments, XXIX IAHR
Congress Proceedings**,** Beijing, China

Price, M. 1998. Seasonal Variation in Runoff Curve Numbers. MS Thesis,
Watershed Management, University of Arizona. 189pp.

Rietz, P.D. and R. H. Hawkins. 2000. Effects of land use on runoff curve
numbers. Watershed Management 2000, Am. Soc. Civil Engineers.
Proceedings Watershed Management Symposium, Fort Collins CO (CD ROM)

U.S. Department of Agriculture, Soil Conservation Service. 1993.
National Engineering Handbook, Section 4, Hydrology (NEH-4).

USDA, 1955 Yearbook of Agriculture. Washington, DC

Van Mullem, J.A. 1992. Soil Moisture and Runoff---Another Look. ASCE
Water Forum '92, Proceedings of the Irrigation and Drainage Session,
Baltimore, MD

Woodward, D.E. 1991. Progress Report ARS/SCS Runoff Curve Number Work
Group. ASAE Paper 912607. Chicago IL