JOURNAL OF Grorrivs•cA/.
RrS•ARCH
VOL. 69, NO. 8
What
Is Watershed
ArRn.
15, 1964
Runoff?
ROGER P. BETSON
TennesseeValley Authority, Knoxville, Tennessee
Abstract. A nonlinear mathematical model, starting with the integral of an infiltration
capacity function, is developed to analytically equate the differencebetween rainfall and runoff to hydrologic variables. Only the three independent variables--storm rainfall, duration,
and soil moisture--are used, and an equation is evolved in which the identity of the coefficients is kept intact and unusually good statistical control is maintained. The coefficientsof
the equation appear to be stable over a range of watershed sizesand conditions.The equation
strongly indicates that runoff usually originates from a small, but relatively consistent,part
of the watershed. The function can be manipulated to show a 'function of apparent watershed
infiltration capacity.' This function characterizesthe infiltration capacity of that portion contributing to runoff, on the average, and should prove to be a useful infiltration capacity index
with which watersheds can be compared. The equation itself provides insight into why in situ
measurementsof infiltration capacity seldom agree with the capacity determined from rainfallrunoff data. It also indicates why storm runoff frequently is not linear with respect to causative
factors.
INTRODUCTION
Infiltration,by definition,is the movementof
water throughthe soil surfaceinto the ground.
The conceptof infiltration is easilyunderstood.
Measuringand understandingthe physicalfactors controlling the infiltration process,however, is exceedinglydifdcult.
The rate of infiltration of precipitation, in
general,equalsthe rate of precipitationas long
as the infiltration capacity of the soil is not
exceeded.When this capacity is exceeded,the
excessprecipitationis either storedtemporarily
on the soil or beginsto run off. Sincean understandingof the surfacerunoff processis basic
to many problemsin engineering,agriculture,
municipalplanning,etc., and sincesurfacerunoff is closelyrelated to infiltration capacities,
there has been a considerableeffort expended
in trying to evaluate measuresof infiltration
capacity.
Measurements of infiltration capacity have
generallyfollowedtwo directions.Actual measurements of infiltration capacity are made in
situ using artificial sprinkling devicesor small
surface runoff plots. Indirect determinationsof
infiltration capacity are made from measurements of storm precipitation and watershedrunoff. These indirect determinations
often result
in more than an index of the infiltration c•pac-
ity, sinceresultsfrequentlyvary from storm to
1541
storm and seldom agree with in situ measurements. It is important to note, however, that
the infiltration capacity obtained from these
indirect determinations represents an average
for the watershed,whereasthe in situ or plot
measurementsapproacha point, or local, measure of infiltration capacity. Point measurescan
vary considerablyfrom the average.
Indirect measurementsof infiltration capacity
have generallybeen hamperedby the fact that
this capacity is not a constant value. It is a
complex function, principally of soil moisture
but also of many other factors. It may change
considerablyduring the courseof a singlestorm.
Computationsof either mathematicalinfiltration
capacity functions or indices have generally
been limited to individual storms. An analytic
approach to the computation of an infiltration
capacity function has been lacking because
available techniquesfor this type of problem
solution usually required linear or linear-transformableequations.
With the availability of electroniccomputers,
the limitations of simplicity and linearity have
been removed as requirementsof analytic solutions. Computer techniques are now available
that will evaluate almost any equationthat can
be devised for data adjustment. This paper
showsthe developmentof a complexfunction of
watershed infiltration capacity using analytic
techniques. This development starts with an
1542
ROGER
P. BETSON
available empirical model and continueswith
improvements based on the results obtained in
fitting the model to actual data.
THE INFILTRATION PROCESS
The role of infiltration in the hydrologiccycle
was probably first recognizedby Horton [1933].
This infiltration pertained to the passage of
water through the soil surface. Once the water
is in the soil its further
movement
is defined as
percolation. The maximum rate at which water
can enter the soil is dependenton many factors.
]:[orton recognizeda maximum capacity and a
minimum capacity.The maximum capacityfor
any given rain occursat the beginningof the
rain. This rate decreasesrapidly at first because
of changesin the structure of the surface soil
and increasesin surfacesoil moisture,and then
gradually approachesa somewhat stable minimum. The decay in the capacity results from
such processesas changein surface soil moisture, rain packing of the soil, closingof sun
watershed conditions(i.e., cover, soil, state of
tilth, etc.), the integral of this function over
time would determine the maximum
amount of
precipitationthat couldbe infiltratedin a given
time. If the storm rainfall is intenseenoughto
exceedthe infiltration capacityduringthe entire
storm, the differencebetweenthe storm rainfall
and this volume resultingfrom the integration
would be approximatelyequal to the surface
runoff volume. Thus it should be possibleto
determinethe coefficients
in this equationfrom
rainfall-runoff data. Evaluating this equation
from data, however,would necessitateusing
nonlinearfitting techniquessincethe function
cannotbe easilylinearized.
Earlier work in the TennesseeValley Author-
ity (TVA) hasshownthe feasibilityof usingthe
multivariate technique of componentanalysis
in the solution of complex equationsby the
method of nonlinear least squares [Snyder,
1962]. The technique for solutionby nonlinear
least squaresmay be found under the name of
checks,swellingof crumbstructure,in-washing, 'method of differential corrections'[Nielsen,
and breaking down of the crumb structure. The
1957]. Whenever the compositetechnique of
minimum infiltration capacity approachesthe nonlinearleast squaresand componentanalysis
evaluationof
percolationrate of the soil profile. Under con- has been tried in TVA, successful
ditionsof saturationthe percolationrate and,
mathematical models has been achieved. All the
solutionsof the variousformsof the developed
by the permeabilityof someleastpervioussoil equationwere obtainedfrom an availablespehorizon.
cial-purposeIBM 704 program [TVA, 1963].
Horton definedthe infiltrationcapacity,/(p),
in turn, the infiltration capacitycan be limited
as the maximum rate at which a given soil can
absorbprecipitation as it falls. The actual infiltration during a stormis equal to the capacity
only when the rainfall intensity equalsor exceeds/(p). Horton proposedthat the relation
between /(p) and the rainfall duration can be
expressedby
l(p) =
+ (Io - IOe-'
where
f(p) = instantaneous
infiltrationcapacityin
inchesper hour at time T.
/c
/0
= minimuminfiltrationcapacity.
= maximuminfiltrationcapacityat time
T
= time in hours.
zero.
e
= NapierJanbase.
k is a positiveconstant.
DEVELOPMENT OF AN INFILTRATION
CAPACITY FUNCTION
Basic data. The data usedin the development of the watershedinfiltrationcapacityfunction were collected at the Western North Caro-
lina CooperativeResearchProject locatednear
Waynesvilleand Asheville,North Carolina.This
proiect has been describedin variousprogress
reports as well as in special reports on the
project [TVA, 1960a]. The purposeof this
smallwatershedresearchprojectis to determine
water-land relationshipsfor agriculturaldevel-
opmentof someof the principalsoilsand crops
in westernNorth Carolina.In accomplishing
the project objectives,informationis being
obtainedon the effectsof singleagricultural
practices
on (a) runoff,(b) soilmoisture,and
(c) groundwaterlevels.The programinvolves
basicresearch
in the development
of hydrologic
If this infiltration capacity function can be relationships.
considered
constantfor a given set of physical
The four experimental watershedsare located
WATERSHED
in the Blue Ridge subregion on the southern
Appalachian soils region. The soils are all acid
and low in available phosphorusand they vary
in their content of potassium. Subsoils are
usually permeable and well drained. Average
slopesin the watershedsrange between 22 and
38 per cent, some slopesbeing as steep as 60
per cent. The watershedsare generally bowlshaped,varying in size from 3.7 to 5.6 acres.A
metal cutoffwall is installedto a predominantly
metamorphic bedrock at the lower end of each
watershed. Continuousdischargemeasurements
are madewith either a 2-foot H-type flumeor a
l«-foot San Dimas flume. Surfacerunoffmeasurements are also made on 0.05-acre subplots
RUNOFF
1543
tions that follow, has been found to agreewell
with near-surface measurements of soil moisture.
The initial trial to developa watershedinfiltration capacityfunctionwasbasedon Horton's
infiltration capacity equation.The area under
the infiltration capacity curve for time T is a
volumetric loss. If storm runoff can be considered an excess that results when the infiltration
capacity is exceeded,this lossshouldequal the
difference between rainfall
and runoff.
L=R-RO
=fo
z•+
(2)
where
within some of the watersheds.
The developmentof the infiltration capacity
function
was based on data obtained when the
L
= loss in inches.
R
= storm rainfall in inches.
RO = storm runoff in inches.
watersheds were in either a moderately or
D -- storm duration in hours.
heavily grazed pasture cover. These two covers
c, b, and n are coefficients
to be determined.
were used so that changesin the structure of
They correspond
to )•c,0•0-- )•c),andk,
the soil surfaceresultingfrom either the impact
respectively,in (1).
of high-energyrainfall or tillage would be minimized in the initial stages of the development.
Initial assumptions. Various investigators
Integrationof the equationin this form,howhave shown that the initial infiltration capacity ever,startsat the maximuminfiltrationcapac-
is a function of soil moisture. As the soil mois-
ity, or initiallydry condition,
whereas
theinitial
ture at the beginningof a storm becomeshigher, infiltration capacityfor a given storm might
the initial infiltration capacity becomeslower, be considerably
lessthan the over-allmaximum.
approachingthe percolationrate as a condition In fact, whenthe groundis saturated,the maxiof saturation is reached. If the infiltration camum infiltrationcapacityat the beginningof a
pacityfunctionof a watershedcanbe considered storm may not be much different from the
relatively constant under a given watershed minimumcapacity(c). The equation,therefore,
regime,it shouldbe possibleto usesoilmoisture needsto be modifiedto incorporatesoilmoisture
to predict the initial infiltration capacityor, in in such a manner that it indicates the timeother words, the beginningpoint of the func- beginningpoint of the equationor, in other
tion. If, in addition,storm runoff can be con- words,the lowerlimit of integration.This can
sidered to be the flow that results when rainfall
be done easilybecausethe exponentialfunction
exceedsthe infiltration capacity, it should be
possibleto evaluate a watershedinfiltration
capacity function from conventionalhydrologic
measurements.The extent to which that portion
of flow known as subsurfaceflow, or interflow,
may invalidate this assumption will be discussed later.
The soil moisture used in this study is an
index computedon a special-purpose
IBM 704
program[TVA, 1963].The basicconceptof this
index is a bookkeepingsystem of daily values
of soil moisturebalance.An upper- and a lowerlevel soil moisture index are computed. The
upper-levelindex, which was usedin the equa-
has the characteristicof being piecewisecontinuous in the interval 0 =• t --• T; therefore,
the exponentialfunction can be divided into a
finite number of subintervals in each of which
F(t) is continuousand possesses
finite left- and
right-hand limits at every point 0 =• t --• T.
If a time-soil moisture functional relationship
can be establishedto determinethe time equivalence of the computedsoil moisture index, this
soil moisture function can be added directly in
order to determine the beginning time in the
infiltration capacity equation,thus allowingthe
equation to vary from storm to storm. With
this provision, the equation would be
ROGER
1544
P. BETSON
shedhydrologicdata. The only stormsincluded
in the data were continuous,with relatively
high rainfall intensities throughout. It was
hoped, at this stage, that if any storms were
(½+
D+•(•m)
=
where [(sm) -- function of soil moisture.
Integrating, we get
L = cT -- (b/n)e-nT I,•+,m)
included
(4)
The functional scaling relationship between
t•e
and the soil moisture
index is u•no•.
The function must, however,be able to relate
the index, which v•ries from zero for dry con•tions to 1.0 for wet conditions,to an equivalent t•e in hours, and probably should allow
for a nonlinearrelation.Fu•her, the t•e eq•v•lence of the soilmoistureparmeter shouldbe
zero for sto•s that occurwhen the soil is d•
(sm index equals zero). A pol•omial relation
can satisfy these two conditionsso that
that
contained
some rainfall
at
in-
tensitiesbelow the infiltration capacity the prediction error resulting from the fitting could
be used to further modify the storm list. The
resultsobtainedin fitting this equationto watershed 1 pasture data are shown in equation 1,
Table 1. Theoretically,the value of the polynomial
soil moisture-time
function
at
a soil
moisture value of 1.0 should be approximately
equal to the length of time it takes the exponential function to become insignificant. In
other words, the exponentialfunction ideally
shoulddrop out of the equationfor soilmoisture
values of 1.0. Equation 1, Table 1, however,
indicates that at a soil moisture value of 1.0
f(sm) -- mS -]- gS2
(5)
where$ -- soilmoistureindexand m and g are
coeffcients.
Substituting the functional form for i(sm)
given in (5) for the limits of integration in (4)
yields
L = cD •- (b/n){exp [--n(mS •- gS•)]
- exp[--n(D + mS + g$•')]} (6)
Figure 1 graphicallyshowsthe relationshipof
the variables.
The equationin this form was fitted to water-
the time equivalentof the soilmoistureis negative; this doesnot have a physicalinterpretation. In view of the low degreeof adjustment
achievedin fitting the equation,as indicatedby
a multiple correlationcoefficientr ---- 0.61, it
appears that the additional degreeof freedom
allowed by the polynomialtime-soil moisture
functionwasusedto adjustfor someof the large
residualerror.Sincethe polynomialrelationship
cannotbe evaluated,a linear time-soilmoisture
relationship
shouldbe adequate.
Modification for interception losses. It was
apparent from the results of fitting the first
o
'"
Loss-f•D
+f
(smJ
(.c+
be-nT)dt
(sin)
•.
z
_
StormDurati9.n,
C
_
•'•-•.-•
_
o0 '
TIME: IN HOURS
Fig. 1. Graphical representationof infiltration capacity function.
WATERSHED
RUNOFF
1545
TABLE 1. Preliminary Equation Solutions
Equation 1 (watershed1)
L -- 0.138D+ (0.0002/1.69)
lexp[-1.69(7.905- 12.18S2)]
- exp[-1.69(D + 7.90S- 12.18S2)]]
Standard
deviation
Se -- 0.36
Multiple correlationcoefficientr -- 0.61
Equation 2 (watershed1)
L - 0.378- 0.0767Sq- 0.0901Dq- (1.358/4.322)[exp
[-(4.322 X 1.189S)]-exp [-4.322(D q- 1.189S)]]
Se r
-
0.34
0.66
Equation 3 (watershed5)
L - 0.266 - 0.068Sq- 0.026Dq- (3.87/2.82)[exp[-(2.82 X 0.518S)]-exp [-2.82(D q- 0.518S)]]
$e -- 0.39
r
- 0.65
Equation 4 (watershed1)
L - -0.117 q- 0.099S q- 0.0016D q- (0.853/5.433)
ß[exp[-(5.433 X 0.296S)]-exp [-5.433(D q- 0.296S)]]q- 0.948R
Se -- 0.0178
r
-- 0.99925
where
L
D
S
R
-- loss in inches.
•- storm duration in hours.
-- soil moisture index.
-- storm rainfall in inches.
e ----NapierJanbase.
equationto data that somemajor factor in-
In this form the equationwas againfitted to
volved in the infiltration capacity relationship watershed1 and, in addition,to watershed5
had been ignored.This factor couldbe inter- data. The resultsof thesetwo fittings are shown
2 and 3, respectively.
The
ception.Interceptionlossesrepresenta reduc- in Table 1, equations
tion in the effective rainfall, or throughfall. coe•cients of both of these equationsare reaStudies by Wollny [Baver, 1938] in 1880 sonable.The indicatedinterception(a -- gS) at
showed that rainfall interception by plants
could amount to a substantial percentage of
the rainfall. Since interception lossesare not
dependenton the duration of the storm, the
equationcannotin its presentform adjust for
this type of loss.Adjustmentcouldbe madeby
the addition of a constantto the equation.It is
probable, however, that this interception loss
is not a constant; rather it is high when conditions are dry and lesswhen conditionsare wet.
To approximate this condition it was assumed
that the interception loss varied with the soil
a soil moisture value of zero is 0.38 and 0.27
inch, respectively,
on watersheds1 and 5. For
wet conditions where soil moisture equals 1.0,
the interceptionvaluesdecreaseslightly to 0.30
and 0.20 inch,respectively.
The minimuminfiltration capacity(c) is 0.09 in./hr on watershed
1 and 0.026 in./hr on watershed5, and the
maximum infiltration capacity for these two
watersheds(b -]- c) is 1.45 and 3.9 in./hr, re-
spectively. These values are in substantial
agreementwith other measurementsmade on
thesewatershedsand also agreewith published
data.
moisture as follows:
Despitethe apparentreasonable
magnitudes
•(i) -- a- gS
(7) of the coe•cients,the degreeof adjustmentof
where•(i) -- functionof intereeption,$ -- soil this equationis low for bothsetsof data.Figure
moisture,and a and g are coefScients.
The as- 2, a residualerror plot of watershed1 data,
sembledequation is now
L--- a--
gS-•-cD
--n(D+m•)
d- (b/n)(e
-nms-- e
)
(8)
showsthat the degreeof adjustmentis low becausea predictionbias existsthat is related to
the dependentvariableloss.
Modification •or partial area runoff. The
1546
ROGER P. BETSON
1.0
0.5--
ß
ß
ß
ßß ße
0
ß
ae
ß
ee
ß
ßß ß
ß
ß
ß
...
ß
ß
ß ß
ße
ß
--I.0
0
Fig. 2.
0.5
1.0
i.$
2.0
2,5
Residual error of prediction versus observed loss for Waynesville watershed 1 (equation 2, Table 1).
residual error bias shown in Figure 2 in itself
providesno insight as to the causeof the bias.
Large residual error (R, ---- observed-- predicted) results when either the observed data
runoff and therefore more proportionalto the
loss.The equationwith the linear rainfall term
R added (the interceptat zero rainfall will be
absorbedinto the constanta of the equation) is
are in error or the model does not fit the data.
This could mean that either the runoff does not
occuraccordingto the infiltration capacityconcept or the model itself is incorrect. Since the
L=
a--
gSnt-cD
q- (b/n)(e-"r•s-- e-n(D+m$))
-]- hR
(9)
observedhydrologicdata are considered
reasonThe resultsof fitting this equation to waterably correct,the model must be revised.
shed1 pasturedataare shownin Table 1, equaThe consistencyof the bias shown in Figure tion 4. A high degree of adjustment was
2 suggests
that a linear relationshipcan be fitted achievedby the equation,as indicatedby a
betweenthe residualerror and loss,the depend- multiple correlationcoefficientof 0.00925.The
ent variable of the equation. If this equation very low standarderror, 0.02 inch,is within an
were linear, the bias could not have occurred, acceptablerangefor hydrologicdata. The sign
since the addition
of such a linear loss function
to the equation,in effect, amountsto a simple
rescalingof each of the coefficientsof the independent variables,a solutionwhich would have
beenfoundin the fitting process.
However,since
the equation is nonlinear a simple rescalingof
all the independentvariables is not possible.
This same effect can be achievedif the dependent variable,loss,is rescaledby includingeither
storm rainfall or storm runoff as an independent
variable in the equation.This fact is developed
further in a following section. Actually, storm
rainfall wasselectedas the rescalingindependent
variable becauseit is considerablylarger than
and values of all of the coefficientsexcept the q
term are within reasonable limits. The reversed
signof the coefficient
g, however,doesnot have
physicalsignificance.
It indicatesthat the interceptionlossincreases
with soilmoistureandthat
the equationis probably overdetermined.
This
term can therefore be omitted from the equa-
tion. The physicalsignificance
of the term h will
be developedin the followingsection.
The final form of the infiltration capacity
equation is
L = a q-cDq- (b/n)(e
-"r•s-- e-"(z•+r•s))
q- hR
WATERSHED
TABLE
2.
RUNOFF
1547
Equation Solutions
Coefficients
Watershed
Cover
Condition
Season
a
c
b
Statistics
n
m
h
r
Se
Watershed
1
Pasture
All
--0.032
0.391
8.61
0.035
0.954
0.9995
Watershed
5
Pasture
All
--0.139
--0.045
1.025
1.52
0.61
0.765
0.975
0.113
Summer
Summer
Dry summer
Wet summer
Dry summer
Wet summer
Winter
-0.016
--0.017
0.001
0
--0.006
--0.022
--0.176
0.0045
0.006
0.011
0.002
0.011
--0.128
0.005
0.394
0.287
0.351
0.258
0.133
0.307
1.923
3.31
2.86
3.55
3.91
5.59
.133
2.56
0.21
0.47
0.53
0.37
0.10
2.45
0.41
0.782
0.870
0. 772
0.862
0.921
0.739
0.559
0.995
0.993
0.996
0.996
0.9997
0.969
0.950
0.040
0.052
0.048
0.018
0.015
0.084
0.108
Summer
Winter
--0.059
0.022
0.0006
0.009
0.785
0.427
4.58
2.01
0.12
0
0.856
0.612
0.985
0.976
0.104
0.143
All
--0.0002
0.0009
0.487
4.07
0.43
0.849
0.9925
0.060
0.0031
0.606
4.31
0.29
0.793
0.9905
0.066
0.043
1.13
2.73
0.21
0.142
0.888
0.122
Parker
Parker
Parker
Parker
Parker
Parker
Parker
Branch
Branch
Bzanch
Branch
Branch
Branch
Branch
Complex
Complex
Complex
Complex
Complex
Complex
Complex
Middle Creek
Middle Creek
Complex
Complex
Watershed 2
Pasture
Watershed 2
Pasture
Copper
36 %
Basin
Calibration
Evaluation
Calibration
Calib:ation
Evaluation
Evaluation
Evaluation
Moderately
grazed
Heavily
grazed
All
All
0.0044
--0.021
0.0014
0.015
1 W
In this form the equationhas been fitted to
several sets of data from the single-practice
western North Carolina watershedsas well as
several other watersheds.
INTERPRETATION
OFRESULTS
century. Watershed 1W, a 5-acre area, was
gaged from 1943 through 1951 as part of a
project to determine the effect of an erosion
control program. Cover on this watershed,
which was left untreated, never exceeded36
per cent of the area. These areasare shownon
Table2 shows
theresults
of fittingthefinal thelocation
map,Figure
3.
equation
to datafromvarious
watersheds.
The Thecoef•cients
oftheequation
forthevarious
Parker
Branch
watershed,
described
in varioussolutions
shown
in Table2, witha fewexcepprogress
reports
andspecial
reportson the tionswhichwillbeexplained
later,show
surproject[TVA, 1960b],is a 1.51-mi
• researchprising
consistency.
Thisconsistency
generally
arealocated
in thesame
general
region
asthe remains
despite
thefactthatthedrainage
areas
western
NorthCarolina
project
watersheds.
This forthewatersheds
varyfrom3.7acres
to 32.7
complex-practice
agricultural
watershed
was mi•, arelocated
fromtheBlueRidgephysiocalibrated
hydrologically
from1953to 1955. graphic
region
to theGreatValley,andvaryin
Starting in 1955 the farmers were given an intensive farm development program designed
to achieveoptimum economicwell-beingof the
people.Evaluationof the effectsof this program
was begun in 1958 and ended in 1962. The
Middle Creek watershed is a 32.7-mi ' area
within ttiwassee River watershed. This area is
in the Great Valley of east Tennessee. This
agricultural complex-practice watershed was
gaged from 1944 through 1961 to demonstrate
the effectof TVA's applicableregionaldevelopment integratedon a singlewatershed.The Copper Ba•sin area in the southeastern corner of
Tennesseewas denuded primarily as a result
of open-air copper ore roastingin the late 19th
KNOXVILLE
T E N N.
/-.-/
..,-'.J/
~/
IdlDDLE CREEK
•-•--.-
/
PARKER
BRANCH
,."'1
ß
WAYNESVILLE-5
e
COPPERBASIN-IW
__ *.[........
• I
_..J •'"'-_'-......•'-""
/
Fig. 3. Location map.
ASHEVILLE
I
1548
ROGER
P. BETSON
coversfrom single-practicepasture, to denudation, to complexpractice. The storms from the
relationwouldoccuraccording
to the infiltration
capacityconcept.Runoff, unlike rainfall, can
Middle
Creek watershed have been divided
be rescaledmathematically.The observedvolinto summer and winter seasons and in addiume of watershedrunoffin cubicfeet per section to that division, the Parker Branch data ond times one day (ds/) is convertedto waterhave been subdividedinto stormswith signifi- shedinchesby dividingby the drainagearea
cant rainfall within the three days before the
(DA) and an appropriateconstant(K) as in
storm, i.e. initial wetnessand dryness.
The indicatedminimum infiltration capacity
for most data setsis of the order of 0.01 in./hr
inches
= DA_
XK
or less; the maximum rate is approximately
% in./hr. This consistency
alsogenerallyfollows
If both sidesof (13) are divided by the term
through the coefficientn which determinesthe (1 -- h) this term becomesmore meaningful.
shape of the exponentialdecay function and
through the coefficientm which determinesthe
inches
ds/
=
time equivalent of soil moisture. In most cases
the multiple correlationcoefffcientis unusually
Storm runoff can be scaled upward in conhigh, whereasthe standarderror is very low
verting
to the watershedinch unit if the effecrelative to the mean dependentvariable, the
tive drainagearea is reduced.The term (1 -- h)
loss,and even the mean runoff.
Despitethe consistency
of the coet•cientsand is actually a measure of this effective runoffthe high degreeof adjustmentachievedby the producingarea of a watershed.It has been the
equation,most of the indicated infiltration ca- conversion to the watershed inch runoff unit
pacitiesare low in comparisonwith published with its implication of total watershed area
data. The reason for this can be better under-
contribution that has been complicatingthese
stoodby studyingthe coefficient
h. Rearranging
the equationwill showthe rescalingnature of
studies.
this term
runoff-producingarea or the runoff is scaled
upward to the watershed inch unit over a re-
Loss = R--
hR -- RO = a + cD
q- (b/n)(e
-'""s-- e-"(z)+"'s')
= /(ip) (11)
or R(1 -- h) -- RO =
where/(ip) -- functionof infiltrationcapacity.
If the reductionin rainfall indicatedby the
If
either
the
total
storm
rainfall
is
reduced to an amount that fell over the effective
duced drainage area, the resulting rainfallrunoff relationdoesoccuraccordingto the infiltration capacity concept. The effective runoffproducingarea of a watershed,then, is not the
same as that delineatedby the topographical
divide.
That the entire watershedmay not contribute
term (1 -- h) is real, there is an implication to runoff during moderate-size storms is not
that not all of the storm rainfall is effective in
surprising. Amorocho and Orlob [1961] and
producingrunoff,if runoffis to occuraccording Moldenhauer et al. [1960] had indicationsof
to the infiltration capacity concept.It is not this possibility, and the SoutheasternForest
obviousat this point why this is so.
Experiment Station [1961] estimated that at
The dependentvariable could also have been the CoweetaHydrologicLaboratory in western
rescaledby usingrunoff as an independentvari- North Carolina no more than about 40 per cent
able and an equally good fit of the equation of the drainage area of their watershedsconwould have been achieved. This can be shown
tributes to storm runoff even in a relatively
if the equation is rearranged
heavy rainfall area. It is the low percentageof
the total area that doescontributeand the apRO
parent consistencyof the size of this area on
R-- (1-- H)= (1-- h)/(ip) (12) some
watershedsthat are surprising.For the
This rearrangement indicates that if it were
possibleto rescalestorm runoff by a relatively
constant factor, the resulting rainfall-runoff
equationsolutionsshownin Table 2, the term
(1 -- h) indicatesan averagecontributionthat
varied f,roma low of only 4.6 per cent of water-
WATERSHED
RUNOFF
1549
shed1 to a maximumof 85.8 per cent of Copper
from only 4.6 per cent of the watershed, or,
conversely,the stormswithin the rangeof these
Verificationof the magnitudeof the computed data did not exceedthe infiltration capacity of
effectiverunoff-producing
area of a watershedis about 95 per cent of this watershed.On this
difficult on the Parker Branch or Middle Creek
watershed,installation of a sheet-metal cutoff
watershedsbecauseof the complexpatterns of wall has resulted in a permanently swampy
land use. Ideally, to verify that the term area above the flume. The swampy area covers
(1 -- h) is a measureof the effectiverunoff- about i to 2 per cent of the watershed. It is
producingarea of a watershed,the equation evident,by the quick responseof the streamflow
should be fitted to data obtained from waterdischargeto changesin rainfall intensity and
shedswith known runoff-producingareas. One the unusually low yield of storm rainfall, that
way to do this would be to use relatively im- most of the storm runoff that was measured
pervious watershedssuch as those in urban from this watershedunder pasture cover came
areas.For this type of watershedthe coefficient from this swampy area. The equationindicates
that most of the storm runoff results from an
h should have small values, so that (1 -- h)
approachesunity. Lacking this type of data, area about twice the sizeof the swampyportion
some verification of this interpretation of the of the watershed.
term (1 -- h) can be madeusingdata from the
If the conceptthat, on the average, runoff
western North Carolina project watershedsand usually resultsfrom a relatively fixed percentage
from Copper Basin 1W watershed.These two of the watershedarea is valid, further verificaprojects represent extremesin land-use prac- tion shouldbe possibleon watershed2. On this
tices. Figure 4 showsa verification plot of the watershed three subplots are maintained to
residual error versus the loss for watershed 1.
measure surface runoff. These 1/20-acre subThe biashasbeen removedfrom the error, leav- plots are located around the watershed about
ing a more normal pattern, and the errorsthem- •/3 to % of the distanceup to the divide. Some
selves are reduced to acceptable values. The subplot surface runoff was measuredfrom about
•/3 of the storms included in this data set. This
more normal error pattern indicates that with
the dependent variable rescaledthe equation seemsto contradictthe result of the equation
does adjust to these data. The relatively small that indicates that average runoff from the
remaining error indicatesthat the percentage watershedresultsfrom only 15 per cent of the
of watershedcontributingto runoff is relatively area. The amount of storm runoff recorded,
constant over the range of storms included in however,supportsthe interpretation.Only durthe data set. The storm precipitationwithin this ing one of the storms included in the set was a
data set includesstorm rainfall amountsvary- substantial volume of runoff recorded from all
ing from 0.16 in. to 2.28 in. and intensities three plots.In general,surfacerunoff from these
up to 2.6 in./hr. The coefficient
h indicatesthat, plots amountedto less than 0.01 inch, and
on this watershedfor storms in the light-to- seldomwasrunoffrecorded
from all threeplots
moderaterange,runoff occurson the average during a given storm. The only time substantial
Basin 1W.
.050
,..,
o
,-,-
.025
0
ß
•- -025
-o5o
,
I
o.5
t
•.o
J
ß
•.5
OBSERVED LOSS IN INCHES
Fig. 4. Residual error of prediction versusobserved loss,Waynesville Watershed 1.
550
ROGER P.
runoff was measuredat the plots, the storm was
very intense.
BETSON
cient for the minimum infiltration capacity
resulted. The storms that
were used for the
Parker Branch evaluationperiodstudy, labeled
sonablenessof the term h that results from 'summer data with wet antecedentconditions,'
fittingthe equationto CopperBasin1W water- on the other hand, containedmany eventswith
shed data shouldbe possible.This watershed, high runoff that resultedbecausethe moisture
with 64 per cent of the area completelyde- storagecapacity was reducedwhen the upper
Someintuitive judgmentconcerning
the rea-
nuded,is subjectto relatively heavy, rapid soil horizonswere near saturation.The equation
runoff through an advancedsystemof gullies resultingfrom these data is an almost random
This resultmust occurwhen
that completelydissectthe area. Streamflow array of coefficients.
is intermittent and most of the runoff can be
considered surface runoff. Under these condi-
large variations in the runoff occur becauseof
conditionsexternal to the equation.This result
tions,runoffshouldoccurfrom a largepercent- did not occurwith the CopperBasin 1W fitting
ageof the watershed,
andthe indicatedinfiltra- becausethe data were relatively 'homogeneous.'
tion capacitiesshouldbe relatively low since Further modificationsof the equationwill have
crusting of the soil surface resultsfrom fre- to be made to accountfor, at least,the effectof
quent,intensestorms.The coeffcienth for the changesin the volume of moisture stored within
CopperBasin1W data (Table 2) indicatesthat
the soil on runoff.
the average watershed area contributing to
If the infiltration capacity function that has
runoff was 86 per cent. The indicated infil- been developedis to measure the watershed
tration capacities,however,are not Iow. They infiltration capacity,stormsurfacerunoffshould
are actually among the highest of the various have been used rather than total storm runoff.
sets of data used. In fact, these are the only Present methods of hydrograph analysis, howindicated infiltration capacities, of all those ever, do not permit reliable separationof the
determined
by the equation,that are reasonable. hydrograph into surface and subsurfaceflow.
This reasonableness
suggests
that the indicated Becausetotal storm runoff has been used, the
infiltration capacityvalues are related to the lossespredicted by using the function will be
amount of the watershedcontributingto runoff. somewhatlower than they would have been if
The residual error distribution that results surfacerunoff only had beenused.However, the
from fitting this equationto any of the data degreeto which this equationadjustsfor a fairly
setsgenerallyfollowsa seasonal
pattern.High wide range of hydrologicaleventsindicatesthat
runoff (negativepredictions)occursduringthe these two componentsof storm runoff, comwinter seasonand low runoff occursduring the bined to evaluate the equation,are not incomsummer.This separationsuggeststhat during patible as data. If this is true, the function
the winter,whenthe amountof moisturestored should be a relatively stable index of the infilin the soilprofileis high,runoffprobablyoccurs tration capacity of a watershed.
from a somewhatlarger portion of the waterAN APPARENT INFILTRATION
shed than during the summer when this conCAPACITY FUNCTION
tributing area is smaller.The fitting technique
sets the equationat what might be considered It would be helpful if the coefficientsthat
an averagecondition.This interpretationhelps result from fitting the equationhad more reato explainsomeof the exceptionsto the con- sonable values. It was noted above that the
sistencyof the coefficients
foundin Table 2. In- magnitude of the coefficientsis related to the
consistencies
were found in the fittings when the sizeof the contributingarea. Equation 12 shows
might be rescaledto remove
data containeda disproportionatelyhigh num- how the coefficients
ber of winter storms with 'apparently' high the effectof the sizeof the contributingarea on
runoff, or when the actual infiltration capacities their magnitude.By dividingthroughthe equaof the watershedwere low, so that occasional tion by the term (1 -- h), the coefficients
a, c,
high runoffresultedwhenmore than the average and b are scaledupward. This has the effect of
watershed area contributed during a very in- scalingthe coefficients
of the infiltration capacity
tense storm. Both these conditions occur at
function upward to the equivalent value they
watershed5. A negative, uninterpretablecoeffi- would have had if runoff had occurred from the
WATERSHED
TABLE
3.
1551
RUNOFF
Apparent WatershedInfiltration Capacities
Coefficient
Watershed
Watershed
Cover
1
Condition
Complex
Complex
Complex
Complex
Complex
Complex
Middle Creek
Middle Creek
Complex
Complex
Watershed
2
Pasture
Watershed
2
Pasture
Calibration
Evaluation
Calibration
Calibration
Evaluation
Evaluation
b
n
m
-0.70
0.03
8.5
8.61
0.035
Summer
-0
07
Summer
-0
13
0.02
0.05
0.05
0.014
0.14
0.009
1.8
2.2
1.5
1.9
1.7
3.4
3.31
2.86
3.55
3.91
5.59
2.56
0.21
0.47
0.53
0.37
0.10
0.41
Dry summer
0 004
Wet summer
0 0
Dry summer -0
08
Winter
-0
31
Summer
Winter
-0.41
0.06
O. 004
0. 023
5.45
1.1
4.58
2.01
O. 12
0
Moderately
grazed
All
- 0.001
0. 006
3.2
4.07
0.43
Heavily
grazed
All
0.015
2.9
4.31
0.29
0.050
1.32
2.73
0.21
All
36 %
c
a
All
Pasture
Parker Branch
Parker Branch
Parker Branch
Parker Branch
Parker Branch
Parker Branch
Copper Basin
Season
0.021
-0.025
1W
entire watershed in a manner similar to that
tions,therangeof coefficients
obtained
in fitting
consistent.
This
from the contributingarea. The shapecoeffi- this equationis surprisingly
cient n and the time-soil moisture coefficient m
consistency
seemsto holdoverwatersheds
rangremainunchanged.
Table 3 showsthe computed ing in sizefrom 3.7 acresto 32.7 mi2 and for
with decidedlydifferenttopography,
coefficients
of the apparent infiltration capacity watersheds
of
for the equationsolutionsincludedin Table 2. agriculturalpractices,and subclassifications
The solutions for watershed 5 and Parker Branch
that broke down have not been included for rea-
sonspreviouslyexplained.
The value of the minimum and maximum ap-
parent infiltrationcapacities(c and c q- b, respectively) as shown in Table 3 are not too
different across the set of solutions. The mini-
stormswithin a data set. It would appear, then,
that the apparent watershedinfiltration capacity function is a relatively stable measure
with which the runoff, or, more directly,the
loss,from a watershed
may be characterized.
CONCLUSIONS
The stepwisedevelopment
of a hydrologic
mum capacityrangedfrom a low of 0.006in./hr
to evaluateapparentwatershed
infilon the moderatelygrazedpastureof watershed equation
2 to 0.14 in./hr for all summer storms during
the calibrationperiod at Parker Branch. The
maximum rate rangedfrom 8.53 in./hr on the
watershed1 pastureto 1.12 in./hr for winter
storms on Middle Creek. The coefficients of the
trationcapacityhasbeenpresented.
This equation can be usefulboth as a hydrologictool and
as a meansof developing
a better understanding of the storm runoff process.
The resultsobtainedfrom fitting the equation
equationfitted to Copper Basin 1W data are
well within the range of values obtained from
other watershedswhere considerablylessof the
to various sets of data indicate that storm run-
watershed contributed
of the watershed area. This was found to be
true on small test watershedsin pasture cover,
to runoff. This consist-
ency indicatesthat thesecoefficients
shouldbe
a measure of the infiltration capacity of that
area of the watershed that on the average con-
tributes to runoff.Consideringthat the equation
containsonly the three independentvariables-stormduration,rainfall, and a soil moistureindex-and that no restrictions are imposed on
the magnitudesof the coefficientsof the equa-
off, at leastin the geographic
regionof the
study,frequently
occurs
fromonlya smallpart
and it appearsto be true alsofor largerwatershedswith complexland-usepatterns. These
results also seem to indicate that the size of the
runoff-contributingarea .may not vary much
under normal conditions.Logically,the size of
the area must changeas the moisture storage
capacityof the soilchanges,
andin addition,the
1552
ROGER P. BETSON
size of this area is also related to rainfall
in-
tensity. Itowever, it appears that these two
factors do not causelarge variations in the size
of the runoff-producingarea of many watershedsexcept under unusual conditions.If this
is true, the conceptof partial area storm runoff
helps to explain the nonlinear nature of many
hydrologic relations. Mathematical models relating storm rainfall and modifyingvariablesto
runoff,peak discharge,
or other dependentvariablesfrequently underpredictlarge events,particularly if a wide range of storms is used. A
typical storm list contains a large number of
small-to-moderate
events where the runoff-con-
tributing part of the watershedis a relatively
small area.
Because of the number
of these
The equationin its presentform has a somewhat limited application.On small test areasit
will work well only with data that do not contain too many stormswith unusuallyhigh runoff. On larger areasit appearsto fit consistently,
probably becauseof the integrating effect of
many factors. To be useful, the model requires
further improvements,and these improvements
are currently under development.From a practical point of view, and judging from results
obtainedthus far, the model shouldadjust for
increased runoff resulting from reductionsin
soil moisture storage, changesin agricultural
cover, and extreme rainfall intensities.If these
factors can be incorporatedin the model successfully,a far better understandingof water-
smaller events, the model is forced to fit this shed runoff should result.
typical event. When during an extreme storm
i•EFEREI•CE S
runoff does occur from a much larger area
within the watershed,the small increasein the Amorocho, J., and G. T. Orlob, An evaluation of
inflow-runoff relations in hydrologic studies,
measuredindependentvariablesseldomjustifies
Water Resources Center Contrib. 4;1,p. 59, Unithe large increasein the dependentvariable.
versity of California, Berkeley, 1961.
Actually,it appearsthat differentrelationships Bayer, L. D., Ewald Wollny, A pioneer in soil and
can be justifiedfor largeand smallevents.
water conservation research, Soil Sci. Soc. Am.
The resultsfrom this study also help to exProc., $, 330-333, 1938.
plain why infiltration studiesbased on rainfall- I-Iorton, R. E., The role of infiltration in the hydrologic cycle, Trans. Am. Geophys. Union, 14,
runoff data seldomagree with in situ measure446-460, 1933.
ments.If storm runoffusuallyoccursfrom only M oldenhauer, W. C., W. C. Barrows, and D.
a small part of a watershed,the infiltration
Swartzendruber, Influence of rainstorm characteristics on infiltration measurements, Trans.
capacity of a greater part of this watershedis
Intern. Congr. Soil Sci., 7th, Madison, Wis., 1,
seldomexceededduring normal storms. It ap426-432, 1960.
pears, then, that in situ measurementsof in- Nielsen, K. J., Methods in Numerical Analysis,
filtration capacity will always equal or exceed
The Macmillan Company, New York, 1957.
measurements obtained from rainfall-runoff
data.
The two techniques will yield similar results
only when either the in situ measurementsare
made on the runoff-contributingarea or the
function of apparent infiltration capacity is
evaluated for relatively imperviousor saturated
areas.
The function that was developed should be
useful as a means of estimating storm runoff.
Becausethe equation was constructedlogically,
most of the coefficientshave a physical interpretation and thus are relatively stable and reproducible.This stability is an important factor
if the function is ever to be usedfor predicting
hydrologiceventson an ungagedwatershed.Although only three independent variables are
used,the standarderrorsthat resultfrom fitting
are generally lessthan 10 per cent of the mean
of the dependentvariable.
Snyder, W. M., Some possibilities for multivariate
analysis in hydrologic studies, J. Geophys. Res.,
67(2), 721-729, 1962.
Southeastern Forest Experiment Station, Report
yor 1961, pp. 62-63, Asheville, North Carolina,
1961.
Tennessee Valley Authority, I-Iydrology of small
watersheds
in relation
to various
covers and soil
characteristics(a pictorial brochure), in cooperation with North Carolina State College of Agriculture and Engineering, Knoxville, Tenn.,
1960a.
Tennessee Valley Authority, An analysis of the
Parker Branch watershed project 1953 through
1959 (a progress report), in cooperation with
North Carolina State College of Agriculture
and Engineering, Knoxville, Tenn., 1960b.
TennesseeValley Authority, TVA computer programs for hydrologic analyses, Res. Paper 3,
Knoxville, Tenn., 1963.
(Manuscript received June 15, 1963;
revised December 17, 1963.)