**RUnoff Curve NUMBER METHOD: Examination**

**of the Initial Abstraction Ratio**

Richard H. Hawkins, Professor, University of Arizona, Tucson 85721\*;
Ruiyun Jiang, Research Associate, University of Arizona. Tucson 85721;
Donald E. Woodward, National Hydrologist (retired), USDA, NRCS,
Washington DC 20013; Allen T. Hjelmfelt, Jr., USDA, ARS, Columbia,
Missouri, 65203; J.E.VanMullem, USDA, NRCS, (retired) Bozeman, Montana
59715.

**[Abstract:]{.underline}** The Initial Abstraction ratio (Ia/S, or ) in
the Curve Number (CN) method was assumed in its original development to
have a value of 0.20. Using event rainfall-runoff data from several
hundred plots this assumption is investigated, and values determined by
two different methods. Results indicate a value of about 0.05 gives a
better fit to the data and would be more appropriate for use in runoff
calculations. The effects of this change are shown in terms of
calculated runoff depth and hydrograph peaks, CN definition, and in soil
moisture accounting. The effect of using =0.05 in place of the customary
0.20 is felt mainly in calculations that involve either lower rainfall
depths or lower CNs.

# **Introduction**

Originally developed by the Soil Conservation Service (SCS, now Natural
Resources Conservation Service or NRCS) in the 1950s for internal use,
the Curve Number method for estimating direct runoff from rainstorms is
now widely used in engineering design, post- event appraisals, and
environmental impact estimation. Background for this is found in the
NRCS document National Engineering a Handbook, Section 4, "Hydrology",
or "NEH-4" (SCS, 1985). The general runoff equation is

Q = (P-Ia)^2^/(P-Ia+S) for P Ia (1a)

Q = 0 for P Ia (1b)

Where Q is the direct runoff depth, P is the event rainfall depth, Ia is
an "initial abstraction" or event rainfall required for the initiation
of runoff, and S is a site storage index defined as the maximum possible
difference between P and Q as P. P- Ia is also called "effective
rainfall", or Pe.

All have units of length, and the equation is dimensionally homogeneous.
The storage index S, which has the limiting values of 0 and , is
transformed to the more intuitive "Curve Number" by the equation
CN=1000/(10+S), where S is in inches. CN, which is dimensionless, may
take values from 0 to 100, is an index of the land condition as
indicated by soils, cover, land use, and (perhaps) prior rainfall.

Though the developmental history and documentation is obscure, the
relationship between Ia and S was fixed at Ia = 0.2S. Inserting that
value into equation 1 gives

\*School of Renewable Natural Resources, University of Arizona. Tucson
AZ. 85721.

Telephone: 520-621-7273, email: rhawkins@ag.arizona.edu.

Q = (P-0.2S)^2^/(P+0.8S) P0.2S (2a)

Q = 0 P0.2S (2b)

The goal here is to examine the data-supported values of the Ia/S ratio,
called ("lambda"), and suggest accommodations for updating its role.

# **Methods**

Two techniques, Event Analysis and Model Fitting, were used for
determining Ia/S from field data sets. These are described in the
following:

![](media/image1.wmf){width="5.770833333333333in"
height="3.4368055555555554in"}**[Event Analysis]{.underline}**. Here,
concurrent synchronized break-point records of both rainfall and runoff
depth are required. The event rainfall depth recorded when the direct
runoff hydrograph begins is taken as Ia. Knowing the total event
rainfall P and the direct runoff Q, equation 1a is solved for S, and the
ratio simply taken Ia/S=. Here each event gives a separate value of ,
and the median for a large number of events is taken as the
representative watershed value. This procedure is portrayed in the
Figure 1.

Figure 1 Event Analysis Method

[General Model Fitting]{.underline}: Here the value of is simply
determined by iterative least squares procedure fitting for both and S
of the general equation.

# 

Q = (P-S)^2^/(P+(1-)S) for PS (3a)

#  Q = 0 for PS (3b)

The objective of the fitting is to find the values of and S such that

{Q - \[(P-S)^2^/(P+(1-)S)\]**}**^2^ (4)

![](media/image2.wmf){width="5.645833333333333in"
height="3.6458333333333335in"}is a minimum. Here each P:Q data set gives
only one value of . An illustration of such fitting is given in Figure
2.

Figure2: Model Fitting by least squares for WS26030 located at
Coshocton, OH with a drainage area of 303 acres. For the natural data
(squares): S = 4.0974 inches, CN = 70.8, = 0.0179, R^2^ = 50.50%, and SE
= 0.32 inch. For the ordered data (triangles): S = 2.0943 inches, CN =
82.6, = 0.1364, R^2^ = 99.17%, and SE = 0.0372 inches.

In each of the above two methods, only "larger" storms were used. This
was done to avoid the biasing effects of small storms towards high Curve
Numbers. With Event Analysis, only events with Pe = P-S 1 inch were
used. With Model Fitting, only events with P 1 inch were used. As shall
be seen, found values of Ia were often quite small, so that this
difference between the two techniques was slight. For statistical
analysis, only watersheds with more than 20 events with P 1 inch or Pe 1
inch were used.

In addition, for the model fitting determinations, both "natural" and
"ordered" data sets were used. Natural data pairs the P and Q as they
naturally occurred in time, and thus displays considerable variety in
runoff with rainfall. Ordered data matches (usually) unnatural
rank-ordered P and Q values, so that each has approximately the same
return period. This is in keeping with a major application of the
method, which in design work at least - matches the frequency of the
rainfall with the frequency of the runoff. For example, the 100-year
rainfall is assumed to produce the 100-year runoff.

### Data sets

Rainfall-runoff data from 307 watersheds or plots were used, originating
from USDA-Agricultural Research Service, US Forest Service, US
Geological Survey, and New Mexico State University. It covered 23
states, mainly in the east, midwest, and south of the United State.
There was no data from the northwestern 1/3 of the country, from roughly
California to Minnesota. A total of 28,301 events were available that
met the rainfall depth (P and Pe) criteria. For event analysis, only ARS
data was applicable, insofar as it alone contained the needed detailed
in-storm break-point information. All others were only rainfall and
runoff depths P and Q. This is summarized in Table 1.

Table 1. Data sets and sources

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

Data source \# Watersheds (w) Method used

or plots (p)

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

ARS 134 (w) Event Analysis, Model Fitting

USLE (ARS) 137 (p) Model Fitting

USFS 26 (w) Model Fitting

Jornada (NMSU) 6 (p) Model Fitting

USGS 4 (w) Model Fitting

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

These 307 watersheds all had 20 or more events which met the storm size
criteria. The ARS data is available from Web site"
<ftp://hydrolab.arsusda.gov/pub/arswater/>. The "USLE" plot data had
been used in the development of the Universal Soil Loss Equation, and
was downloaded from the web site:
[http://topsoil.nserl.purdue.edu/usle/.](http://topsoil.nserl.purdue.edu/usle/)
Forest Service data was in large part supplied in reduced form to the
author (RHH) by Dr J.D. Hewlett of the University of Georgia, who used
it in an earlier paper (Hewlett, et al., 1977; Hewlett and Fortson,
1984). The Jornada plot data, from site north of Las Cruces NM, was
supplied by Dr T.J. Ward, now at the University of New Mexico. It is
described in Hawkins and Ward (1998). The USGS data was supplied from
local sources for a number of urban and urbanizing watersheds in the
Tucson area.

# **Results**

In general, the results showed that is not a constant from storm to
storm, or watershed to watershed, and that the assumption of =0.20 is
unusually high.

**[Event Analysis]{.underline}**: Found Ia/S ratios varied greatly
between storms within watersheds, and also between the 134 watersheds.
An example of the array of found for events in a single watershed is
given in Figure 3. For each watershed the median was used to describe .
These median values also varied, and their distribution is shown in
Figure 4. The general findings are included in Table 2. Values of the
found (median) varied

![](media/image3.wmf){width="5.620833333333334in"
height="3.2708333333333335in"}from 0.0005 to 0.4910, with a median of
0.0476. There was a distinct negative skew, or a crowd of smaller
values. Over 90% were less than 0.2.

Figure 3: An example of the array of found by event analysis for
watershed 26030

# 

# ![](media/image4.wmf){width="5.620833333333334in" height="3.8208333333333333in"}

Figure 4 Cumulative frequency distribution of values from Event Analysis
method

**Table 2. Summary results of value for ARS watersheds (n = 134)**

  ------------ --------------- --------------------- ---------------------
               Event Analysis  Model                 Model Fiting(ordered)
                               Fitting(natural)      

  Min          0.0005          0.0000                0.000

  Median       0.0476          0.0001                0.0736

  Mean         0.0701          0.0555                0.1491

  Max          0.4910          0.5766                0.9682

  STDV         0.0812          0.0983                0.2001

  Skewness     2.5899          2.8364                1.8725

  \% 0.20      93.7            93.3                  72
  ------------ --------------- --------------------- ---------------------

# **Model Fitting**. As described previously, both natural and ordered data sets were fitted to the general runoff equation by least squares to determine Ia and S. Results were more varied than with Event Analysis, although this may be explained with the much larger sample size (N=307). For natural data, the range was from 0 to 0.996, with a median of 0, and for ordered data, the range was form 0 to 0.9793 with a median of 0.0618. A summary of these results is given in Table 3. Figure 5 shows the distribution of findings.

Table 3. Summary Results of values from model fitting

<table>
<colgroup>
<col style="width: 9%" />
<col style="width: 8%" />
<col style="width: 9%" />
<col style="width: 8%" />
<col style="width: 10%" />
<col style="width: 10%" />
<col style="width: 7%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 10%" />
<col style="width: 7%" />
</colgroup>
<tbody>
<tr class="odd">
<td></td>
<td colspan="2"></td>
<td colspan="4">Natural Data</td>
<td colspan="4">Ordered Data</td>
</tr>
<tr class="even">
<td></td>
<td>N</td>
<td><p>Total</p>
<p>Event</p></td>
<td>Max</td>
<td>Mean</td>
<td>Median</td>
<td>Min</td>
<td>Max</td>
<td>Mean</td>
<td>Median</td>
<td>Min</td>
</tr>
<tr class="odd">
<td>ARS</td>
<td>134</td>
<td>12499</td>
<td>0.5766</td>
<td>0.0555</td>
<td>0.0001</td>
<td>0</td>
<td>0.9682</td>
<td>0.1491</td>
<td>0.0736</td>
<td>0</td>
</tr>
<tr class="even">
<td>USLE</td>
<td>137</td>
<td>11140</td>
<td>0.996</td>
<td>0.0997</td>
<td>0</td>
<td>0</td>
<td>0.9266</td>
<td>0.1581</td>
<td>0.061</td>
<td>0</td>
</tr>
<tr class="odd">
<td>Others</td>
<td>36</td>
<td>4392</td>
<td>0.4727</td>
<td>0.04</td>
<td>0</td>
<td>0</td>
<td>0.9793</td>
<td>0.0992</td>
<td>0.0044</td>
<td>0</td>
</tr>
<tr class="even">
<td>Total</td>
<td>307</td>
<td>28031</td>
<td>0.996</td>
<td>0.0734</td>
<td>0</td>
<td>0</td>
<td>0.9793</td>
<td>0.1472</td>
<td>0.0618</td>
<td>0</td>
</tr>
</tbody>
</table>

# 

# **Applications**

# From the above results, it is obvious that a more appropriate "rounded" value of Ia/S would be in the vicinity of 0.05. Using this value the runoff equation is adjusted, for which a new set of CNs based on =0.05 must be determined

#  

**[Runoff equation]{.underline}**: Using Ia/S=0.05 the runoff equation
becomes

Q = (P-0.05S)^2^/(P+0.95S) P0.05S (5a)

Q = 0 P0.05S (5b)

![](media/image5.wmf){width="6.0152777777777775in"
height="4.291666666666667in"}However, the S values in the above equation
are not the same as previously used assuming Ia/S=0.20. They are defined
on a system of Ia/S==0.05.

Figure 5 The distribution of values from different data sets

**[Equivalent CN]{.underline}:** Based on the above experiences with the
data was fitted by least squares to the CN equation for each case: that
is, for = 0.05 and for the traditional value of = 0.20. The latter is
the basis for existing CN Tables. In 252 of the 307 cases (approximately
5 out of 6) the 0.05 fitting produced a higher r^2^ and lower SE.

The relationships found between the values of S~0.05~ and S~0.20~ were,
for natural and ordered data respectively:

S~0.05~=1.344S ~0.20~ ^1.149^ r^2^=99.38% (6)

S~0.05~=1.316S ~0.20~ ^1.164^ r^2^=99.44% (7)

# 

# Rounded consensus values of these two almost-identical findings condense to 

S~0.05~=1.33S~0.20~^1.15^ (8)

Preserving the basic definition of CN = 1000/(10+S), the above
relationship permits conversion from the 0.20-based CNs to 0.05-based
CNs. Making the substitutions and simplifying gives

100

CN~0.05~ =
\--**\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--**
(9)

1.879\[100/CN~0.20~ --1\]^1.15^ + 1

Consideration of equation 8 shows that S~0.05~=S~0.20~ at 0.148 inch, or
CN~0.20~98.5. They are also equal at S=0, or CN=100. At these
seldom-encountered levels we suggest they be considered equal. Table 4
gives the CN~0.05~ corresponding to current values of CN~0.20,~ taken
from equation 9, or "Conjugate" Curve Numbers

Table 4. Conjugate Curve Numbers and P~crit~

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

CN~0.20~ S~0.20~(in) CN~0.05~ S~0.05~(in) P~crit~ (in)

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

100.00 0.000 100.00 0.000 \-\-\-\--

95.00 0.526 94.02 0.636 2.44

90.00 1.111 86.95 1.501 1.72

85.00 1.765 79.64 2.556 1.95

80.00 2.500 72.39 3.815 2.27

75.00 3.333 65.31 5.311 2.63

70.00 4.286 58.51 7.091 3.05

65.00 5.385 52.03 9.219 4.51

60.00 6.667 45.90 11.785 4.04

55.00 8.182 40.14 14.915 4.64

50.00 10.000 34.74 18.787 5.35

45.00 12.222 29.71 23.663 6.15

40.00 15.000 25.03 29.947 7.13

35.00 18.571 20.71 38.285 8.35

30.00 23.333 16.73 49.777 9.92

25.00 30.000 13.08 66.457 12.02

20.00 40.000 9.75 92.517 15.04

15.00 56.667 6.75 138.095 19.81

10.00 90.000 4.08 235.088 \-\-\-\--

5.00 190.000 1.77 555.160 \-\-\-\--

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

[**Comparisons:**]{.underline} How does the modification of Ia/S affect
calculated values of runoff? First, by equating the runoff equations
using 0.05 and 0.20 and making the transformation of CNs using equations
8 or 9, the rainfall depth corresponding to equal runoffs for conjugate
CNs can be determined. This rainfall is shown as P~crit~ in Table 4.
There is no closed solution: P~crit~ was determined numerically. For P
greater than P~crit~ use of Ia/S=0.2 will result in higher calculated
runoffs. For lesser P values, a higher runoff will be found using
Ia/S=0.05.

Second: Figure 6 shows rainfall-runoff plots over a range of rainfalls
and conjugate CN values. P~crit~ is the locus of the crossing points. As
can be seen, the conjugate curves are similar at a wide range of
frequently-encountered values of P and CN. However, as shown in Figure 7
relative differences are substantially different in cases of lower
rainfalls and lower CNs.

![](media/image6.wmf){width="5.520833333333333in" height="3.98125in"}
Figure 6. Rainfall and runoff for selected conjugate CNs over a range of
rainfalls.

![](media/image7.wmf){width="5.420833333333333in"
height="3.9208333333333334in"}

Figure 7. Relative differences in runoff using =0.05, taken from Figure
6.

Third, in modeling hydrographs similar differences are seen. An example
is shown in Figures 8 and 9. Similar hydrographs, peaks, and timing
result for the higher CNs, but distinct differences are shown with the
lower CN. Figure 9 shows highlights the differences for the CN~0.20~=50
example. Using =0.05 calculates a smaller Ia, giving direct runoff
earlier in the event. Here this
![](media/image8.wmf){width="5.720833333333333in"
height="4.2965277777777775in"}leads to a peak about 60% increase in peak
flow.

Figure 8. Comparative hydrographs for conjugate CNs .

This low CN-small storm scenario will be important in small storms and
lower CNs. This situation characterizes forested watersheds, especially
area of modest return period storms depths (e.g., large portions of the
interior west). This condition would also prevail in continuous modeling
scheme, which consider all rainfalls -- usually on a daily basis --
regardless of size. Differences also exist at higher rainfall extremes,
but have not been evaluated here. The major effect seems to be at lower
CNs at lower rainfalls, or in general at low P/S situations.

![](media/image9.wmf){width="6.0152777777777775in"
height="4.121527777777778in"}Figure 9. Comparative hydrographs from
Figure 7 for the case of CN=50.

**Conclusions**

**[Ia/S]{.underline}**: As determined by two separate methods, Ia/S (or
) of 0.05 fits observed rainfall-runoff data much better than does the
handbook value of 0.20.

**[Runoff Equation]{.underline}**: With =0.05, the runoff equation
becomes

Q = (P-0.05S~0.05~)2/(P+0.95S~0.05~) P 0.05S~0.05~

Q = 0 P 0.05S~0.05~

**[Change of CN]{.underline}**: Altering requires a change of handbook
CNs. That is, if is changed,

a different CN must be used. The relationship is

S~0.05~ = 1.33S~0.20~1.15

(where S is in inches) relates the two conditions. It was determined
from data from 307 watersheds. New handbook CN tables for =Ia/S=0.05
might be constructed using this relationship.

[**Differences**:]{.underline} The most obvious differences in runoff
modeling are at lower CNs and lower rainfalls, or in general at low P/S
situations. This would prevail for low CN watersheds, more frequently
occurring rainstorm depths, and for climates where the more modest
design return period rainfalls are found.

**Acknowledgements**

This wok has been supported by the USDA, Natural Resources Conservation
Service, Water and Climate Center, and by the Arizona Agricultural
Experiment Station Much of the work shown here is from an MS Thesis by
Ruiyun Jiang. A presentation of similar title and content was give at
the NRCS Hydraulic Engineers workshop in Tucson AZ in November 2001, and
a paper on it is to be included in its Proceedings.

**References**

Hawkins, R. H., and T. J. Ward. 1998. Site and cover effects on event
runoff, Jornada

Experimental Range, New Mexico. Proceedings from American Water Resource
Association

Conference on Rangeland Management and Water Resources. Reno, NV,
361-370

Hawkins, R.H., R. Jiang, and D.E. Woodward. 2001. Investigation of the
Initial Abstraction

Ratio in the Curve Number Method. Presentation at American Institute of
Hydrology Annual

Meeting, Minneapolis MN, October 2001. \[PowerPoint\]

Hawkins, R.H., R. Jiang, and D.E. Woodward. 2001. Application of Curve
Number Method in

Watershed Hydrology. Presentation at American Water Resources
Association Annual

Meeting, Albuquerque NM, November 2002. \[PowerPoint\]

Hawkins, R.H., and A.V. Khojeini 2000. Initial Abstraction and Loss in
the Curve Number

Method. Arizona Hydrological Society Proceedings.

Hewlett, J.D., G.B.Cunningham and C.A.Troendle. 1977. Predicting
stormflow and peakflow

from small basins in humid areas by the R-index method. Water Resources
Bulletin.

13(2):231-253.

Hewlett, J.D. and, J.C. Fortson.1984. Additional tests on the effect of
rainfall intensity on storm

flow and peak flow from wild-land basins. Water Resources Research.
20(7):985-989.

Jiang R. 2001. Investigation of Runoff Curve Number Initial Abstraction
Ratio. MS thesis,

Watershed Management, University of Arizona, 120 pp.

USDA/USLE data web site[:
[http://topsoil.nserl.purdue.edu/usle/.](http://topsoil.nserl.purdue.edu/usle/)]{.underline}

USDA, Agricultural Research Service:
<ftp://hydrolab.arsusda.gov/pub/arswater/>

USDA, Soil Conservation Service. 1985. National Engineering Handbook,
Section 4

" Hydrology".

Cnpap3k.doc 10apr02