WATER RESOURCES RESEARCH, VOL. 21, NO. 8, PAGES 1185-1194, AUGUST 1985
Moments
of Catchment
Storm
Area
PETER $. EAGLESON
Departmentof Civil En•]ineerin•t,MassachusettsInstitute of Technolo•]y,Cambrid•]e
WANG QINLIANG
YangtzeValley PlanningOffice,Ministry of Water Resourcesand Electric Power, Wuhan,People'sRepublicof China
The portion of a catchmentcoveredby a stationaryrainstormis modeledby the commonarea of two
overlappingcircles.Given that rain occurswithin the catchmentand conditionedby fixed storm and
catchmentsizes,the first two momentsof the distributionof the commonarea are derivedfrom purely
geometricalconsiderations.The varianceof the wetted fraction is shownto peak when the catchmentsize
is equal to the sizeof the predominantstorm.The conditioningon storm sizeis removedby assuminga
probabilitydistributionbasedupon the observedfractal behaviorof cloud and rainstormareas.
INTRODUCTION
World War II on what was called the "coverage"problem. As
formulated, this work required estimatingthe moments of the
When estimatingthe dispositionof storm rainfall on mesooverlappingarea of two geometricalplanar figuresof specified
scale land surfaces,whether it be for conventional catchment
shape and size. The "target" was a stationary figure, reprehydrology or for estimating grid square moisture fluxes in
sentedusually by a square,rectangle,or circle, on which anatmosphericgeneralcirculationmodels(GCMs), it is of critiother "covering" figure was randomly dropped to represent
cal importanceto accountfor the spatialvariability of precipithe damage area of an individual bomb. The center of the
tation [Milly and Eagleson,1982]. Various investigators,becovering figure was given an assumeddistribution, which was
ginningperhapswith Lecam[1961] and led more recentlyby usually circularly normal, Poisson, or uniform, about the
Waymire and Gupta [1981], have modeled the space-time
centerof the target(the "aiming"point).
structure of rainfall fields,but none has explicitly considered
Of particular importance is the work of Robbins [1944],
the variable of first-orderimportancein theseproblems,which
who formulates(but doesnot solve)the problem in a general
is the fraction of the catchmentor grid squarecoveredby the
way that includes geometrical figures other than rectangles,
rainstorm.
squares,and circles and deals with caseswhere the number,
Theories of flood frequency[e.g., Eagleson,1972] yield disposition, and orientation of the covering figures may all be
tributions conditioned upon the fraction of the catchment
probabilisticallyvariable. General analytical solutionsto this
being wetted by the precipitationand therebybypassingwhat
problem are apparently not available,however.
is surely a major contributor to streamflow uncertainty in
Garwood[1947] describesan unpublishedindustrialreport
many catchments.Algorithms to estimatethe short-term heat
(G. W. Morgenthaler, some target coverage problems, Rep.
andmoisturefluxesfromthelargegridelement(104-105km2)
P-59-69. The Martin Company, Denver, Colorado, 1959) conof atmosphericGCM's make similarone-dimensional
assumptaining tables of the overlappingareas of squaresand rectantions [e.g., Hansenet al., 1983].
gles obtained using analog-integratingdevices.Guenther and
In responseto thesedeficiencies
we estimatehere the uncerTerra•tno [1964] refer to a similar report for circular areas
tainty of the catchment area upon which rain actually falls.
(The Rand Corporation, offsetcircle probabilities,Rand Rep.
Perhaps the simplestsituation is that of the stationary rainR-234, Computer SciencesDepartment, Santa Monica, Calistorm. Eagleson[1984] has addressedthis particular problem
fornia, 1952].
using the geometricallysimple conceptualizationshown in
Garwood [1947] carried out laborious numerical compuFigure 1. In Figure 1 the catchmentarea Ac is representedby
tations using five combinations of circles, rectangles, and
a circleof radiusrc and the overallstormarea Asby a circleof
squaresfor three relative geometries:Ac/.As = 5, 1.0, and 0.56.
radius rs. We are interestedhere only in the casein which it
He used covering areas of fixed size and with Poissonspatial
rains on the catchment and will assume that this occurs whendistribution,and he calculatedthe varianceof the overlapping
ever the two circles overlap to give a nonzero catchment
(i.e., "covered")area. His results,excerptedhere in Table 1,
storm area Asc.Such an occurrencewill define a "storm" in
indicatean insensitivity(of the secondmoment) to the particuthis work. Owing to patchinessin the storm structure,Ascmay
lar geometry of the overlapping figures. This supports our
actually contain dry areas, and this is the subject of earlier
presentidealization of both storm and catchmentby the more
work by Eagleson[1984]. Here we are interestedin the probanalyticallymanageablecircle.
ability distributionof the overall area Asc.
PROBLEM FORMULATION
BACKGROUND
Problems of estimating bomb damage led to a flurry of
applied probabilisticeffort during and immediatelyfollowing
Copyright 1985by the AmericanGeophysicalUnion.
Paper number 5W0395.
0043-1397/85/005W-0395505.00
Given the storm radius, we will assume that its center is
distributed uniformly over the disc of radius rc + rs centered
at the catchment center. The conditional probability Pc[rlrc,
rs] that a storm as definedherein is centeredwithin r of the
catchmentcentergivenrc and rsis therefore[Eagleson,1984]
Pc[rlrc,rs]= Pc= r2/(rs+ rc)•
1185
(1)
1186
EAGLESON
ANDWANG' MOMENTS
OFCATCHMENT
STORMAREA
catchmentand storm circlesare 0c and Os,respectively.The
following definitionswill prove convenient'
/rs
(7)
and
X = {1 - [(1 + cz)P
c + 1 -•z-]2/(4Pc)}
1/2
////
(8)
wherePcis the probabilitydefinedin (1).
From the geometry[Eshbach,1936]
Asc= «(rs20s
+ rc20c)- rb/2
\\\\•
(9)
//•7 and
Fig. 1. Definition sketchfor catchmentstorm area [Eagleson,1984].
Os= 2 sin-• (b/2rs)= 2 sin-1 X
(10)
Oc= 2 sin-• (b/2rc)=2 sin-• (X/•)
(11)
The key to the analyticaldevelopment
of this geometryis the
normalization
o• --
From the geometryof overlappingcircleswe can find
As = g•(r, rc, rs)
(2)
r = g2(Asc,re, rs)
(3)
or
Asc
Os+ •20c bPc•/2
=
•(rs+ re)2 2n(1+ 002 (rs+ re)
(12)
We considerfirst the conditionillustrated,rc > rs(i.e.,0•> 1).
Case 1
The conditional probability that a storm has an area within
the catchmentequal to or greaterthan Ascgiven rs and rc is
obtainedfrom (1) and (3) as
Pc[Asclrc,
rs]= Pc= [g2(Asc,
re,rs)/(rc+ rs)]2
(13)
(re2 -- rs2)1/2< r <_rc+ rs
or, using (1),
•
< Pc < 1
(14)
(4)
If we have the probability density function (pdf) of the storm
radiusf(rs), we can remove the dependenceupon rs to obtain
the more usefulconditionalprobability
and
0 < Os< n
(15)
which yield
+ •)
Pc[Asc
Irc]
-; [g2(Asc,
re,
rs)/(rc
+rs)]2f(rs)
drs(5)• = {[sin-• X + •2 sin-• (X/•)]/(1
- XPc•/2}/[n(1+ •)]
Becausein practice we are concernedwith a particular catchment, rc is known, and we may alternativelywrite (5) as the
marginal probability
(16)
Case 2
r = (re2 - rs2)'/2
(17)
P[Asc(rc)]
=;[g2(Asc,
re,
rs)/(rc
+rs)]2f(rs)
drs(6)
or
Pc -
CATCHMENT/STORM
SCALERELATIONSHIPS
It is clearthat the criticalindependentvariablein this problem is the catchment/stormsizeratio, and it is well to have in
mind the important range of this ratio. This is shown in
Figure 2 where the relevant atmosphericphenomena have
beensizedaccordingto the scaleproposedby Orlanski[1975].
In this diagram the extreme catchment scale is that of the
GeM grid [Hansen et al., 19833.The remainder of this scale,
10 < Ac< 10'• km2, is somewhatarbitrarily chosenas the
TABLE 1. Variance of A•c/Ac for Poisson-DistributedStorms
[from Garwood,1947]
E[A•/A,]
Ac/As
5
rangeof interestof most catchmenthydrology.
GEOMETRY OF OVERLAPPING
CIRCLES
We begin by consideringthe geometry of overlappingcircles using notation shown in Figure 3. The desired area of
overlap Ascis composedof circlesand circular sectorsaccording to the four distinct casesillustrated.The common chord of
thesesectorsis b, while the sectoranglessubtendedby b in the
(18)
•+1
1.0
0.56
Geometry
0.25
0.50
0.75
circle on square
0.0254
0.0303
0.0186
circle on circle
0.0260
0.0310
0.0190
squareon square*
circle on square
0.0252
0.0789
0.0300
0.0964
0.0182
0.0608
circle on circle
0.0810
0.0983
0.0620
squareon square*
circle on square
0.0781
0.1026
0.0945
0.!262
0.0593
0.0816
circle on circle
0.1040
0.1281
0.0829
squareon square*
0.1002
0.1230
0.0790
*Sides remain parallel and centerlinescongruent.
EAGLESON
ANDWANG'MOMENTS
OFCATCHMENT
STORM
AREA
MESO-7
SCALE
MESO-/5
SCALE
105I T I
I
GCM
MESO-a
SCALE
I
GRID
SCALE/
102
1
1187
= -- =
1
10-1
10-2
Ac
(km2)
103
102
FRONTS
-'
SQUALLLINES
••
THUNDERSTORMS
• .•
AND
•
;.
HURRICANES
4 10
4
102
4
103
4
104
4
105
4
106
4
As (km2)
Fig 2. Storma•d catchmerescale•clatio•ships.
and
Case 1
Os• •
(19)
which yield
0 < r _<2rc
(29)
0 < Pc-< 1
(30)
or
co= [(;z/2)+ o•
2 sin-• (1/00- (0•2 - 1)•/2]/[•:(1
+ •z)2]
(20)
Case 3
which yields
re -- rs< r < (rc2 -- rs2)TM
(21)
co= [sin-• X- XPc•/e]/(2n)
(31)
Case 2
or
<Pc <•
•z+l
(22)
r -- 0
(32)
Pc= 0
(33)
co= 1/4
(34)
or
and
• < Os< 2•
(23)
which yields
which yield
co= {[n- sin-• X + 0•2 sin-• (X/•z)]/(1+ •z)
-- XPc'/2}/[n(1+ cz)]
(24)
Case 4
0 < r < rc -- rs
(25)
or
For thefinalcondition
rc< rs(i.e.,• < 1)weusethesymmetryinherent
in theproblem.
That is, we replace
0•in the
aboveequationsby
l?= 1/• = rs/rc
(35)
Theconditional
probability
Pcis plottedasa functionof co
in Figure4 for various
constant
valuesof • or /1> 1. The
curvesshowthat for large• or/1 thereis a maximumvalue
0 < Pc -<
+
(26)
and
entirelywithinthecatchment
or for /1= rs/rc to thecasein
Os= 2n
(27)
whichthestormcompletely
coversthe½atchment.
CONDITIONAL
MOMENTS
OFCATCHMENT
STORMAREA
which yield
co= 1/(1+ o02
Weseefrom(16),(24),and(31)alongwith(8)thatit willnot
(28) bepossible
to solve
explicitly
fortheconditional
probability
For the conditionrc= rs (i.e., 0•= 1) we recognizetwo
cases.
of coabovewhichthereis zeroexceedante
probability.
This
corresponds
for • = re/rsto the casein which the storm lies
Pc as wasanticipated
in (4) and (5). However,we canusethe
implicit relationsco(Pcl•)derivedabove to find the con-
ditional
moments
of Asc.
Fromthedefinition
of co(equation
1188
EAGLESON
ANDWANG.'MOMENTS
OFCATCHMENT
STORM
A•,EA
Defininga new variable
X • -- XPc•/2
CASE
(39)
andsolving(8) and(39)explicitly
for Pcasfunctions
of X and
X•, respectively,
wemaychange
variables
in (38)to obtain
1
VA•c
17cc
....(• - 1)
2
sin
-• X
•2(• + 1) =•2
+•
CASE 2
sin- • (X/oOd
q
L(=+ 1)•J
'
(• + 1)z
,1+•z,
fo•/"+"
{--2[.2
- ,.+l,:XlZ]
'/a
}
X• d,
•:
(40)
(• + 1)z
which yields
•21
=1/(•+1) z
•1
(41)
Returning
to dimensional
variables
[ora moment,
(41)is
CASE 3
E[A=Ir., r.] =nr. 2/(rc/r.+ 1)2
r. k r.
(42)
Asarguedbefore,if wehadreversed
notationat thestart,we
would have obtained
EEA=I r•, r.3=nr. 2Ars/r•
+ 1)2
r. k r•
(43)
which normMizesto be identicMto (41). The final resultfor
the conditionalexpectationfor all • is thus
I-A•c
•]=1/(•x
+1)2
• = rc/r•
CASE 4
(44)
Equation(44)is plottedin Figure5.
l/ariance
To utilize (37),we firstcalculate
Fig. 3. Geometryof overlapping
circles,rs< rc.
(12))we canwrite
(45)
=
w(Pc
dP½
(36)
1.0 -
and
0.9
(37)
0.8
0.7
Expectation
0.6
For • > 1 we use(16),(24),and (28)in (36)to write
0.5
0.4
f(•1)/(•+
1)In - sin- • X
•21 •[(•1)/(•+
1)]•
0.3
+•
0.2
+ • sin- • (X/•)
0.1
-(1 + •)XP• xn] dP•
i
0.0
0.025
+•
•2
0.050
0.075
O. lOO
o.125
o.15o
o.175
0.200
0.225
0.250
[sin- • X + • sin- • (X/•)
- 1)/(•
+1)
-(1 + •)XP• •n] dP•
Fig. 4. Conditional½xcecdanc½
probabilityof a dimensionless
(38)
catchment
storm area.
EAGLESONAND WANG: MOMENTS OF CATCHMENT STORM AREA
1189
1.0
•
' [ [ ] '' '']
-
•.
[ [ [ [ [[['I
LAclJ
•
[ [ [ [ , ],,[.2o
-.18
.16
-
-
.14
.12
.10
.08
.06
.04
.02
0
10-2
10-1
1
0
c•= (Ac/As)l/2
Fig. 5. Conditional momentsof the relativecatchmentstorm area.
For 0•> 1 we use(16),(24),and (28)in (45) to write
The integral in (47) can be evaluatedin seriesform as
=--8 <z+ oo
•-,)/(•+,)]2sin-'ZdP,
PcY
•2 +
.o
•,=o
f/,'
(11)
y'.
•'
.(or2
• 1)2'+'
(2n;
1)(2n- 2k1+ 1)2 ,50)
A2
- ø•
dP•
1½ff•(•')/(•
+')]2
Using (47), (50), and (44) in (37) givesthe conditional second
moment
1 d[(•
I(•-')/(•+
')[=- sin-' X + •2 sin-' (X/oO
"[7•'•4
- 1)/(•
+1)]2
+ 1)'*
Var[A•c•
cot> 1] •2(•
2•z+l-Y(•z)
•z>l
(51)
where
+•
[sin-• X + • sin-• (X/•)
_•)/(a
+•)
-- (1 + •)XP••/2]: dP•
(46)
which becomes
3/t2(Z3(Ot
+ 1)
+ ((Z2-•
4 1)2
• +1
y(oO
=(or2
+3)(3ot2
+21)
-2•2•
1)(o•in(l•-ll)
- •2•2(•2
+2
1) .=• n•0(
•2 •
+ 1)2nl
8(•1)
.(2n;
1) 1
,52,
(2n- 2k + 1)2
To complete the variance for • < 1, we reverse the subscriptsof r in the dimensionalform of (51) and (52) to get,
upon renormalization,
1
(or
2 + 3)(3ot
2 + 1)
ot2(ot
+ 1)2
3/t20t3(Ot
+ 1)2
(or
2+1)(o•
- 1)2
In(
(or1)
vc
2ota(ot
+2
1)•<'
,,_,)/(•+,)]•
VarkA,
• <1]=(l+•)•-
0•+1
sin- ' z
•
dpc
(47)
y = [-(o• + 1)2Pc
2 + 2(o•
2 + 1)Pc- (o•- 1)23'/2
(48)
P•y
where
(53)
Equations(51), (52), and (53) are plotted in Figure 5 along
with the expectation,and the resultsare very interestinghydrologically.Note in Figure 5 how important the relative scale
A,/A s is in determiningthe averagefraction of the catchment
involvedin hydrologicresponseto a singlesto•. For frontal
disturbances
in a GCM grid, for example,A,/As = 10-•
(Figure 2), and E[As,/A,] = 0.56. The sameconditionsobtain
for squall lines over a moderate-sized
catchment(say 103
km2).
and
z = [(ix+ 1)2p•-(a• + 1)]/(2ix)
(49)
Looking at the Vat [A•/A,] in Figure 5, we find a critical
1190
EAGLESONAND WANG: MOMENTSOF CATCHMENTSTORMAREA
scaleratio Ac/As= 10-x (i.e.,g = 0.32)at whichthe variance
is amplified.That this peak must occurfor someintermediate
Ac/As followsfrom limiting considerations.
The varianceof
Asc/Acmust approachzero when the catchmentis very much
larger than the storm, for then the wetted area is always a
negligiblefraction. Similarly, the variance must vanish when
the storm is very much larger than the catchment because
then (remembera storm meansAsc> 0) essentiallythe entire
catchmentis alwayswetted.
The "resonant"peak in variancefor intermediatevaluesof
Ac/As has particular significancefor the theory of extreme
floods. It implies that there is a critical catchmentsize in a
givenclimate(i.e.,predominantstorm size)for which the variance in catchmentstorm area contributesmost heavily to the
variancein runoff.The coefficientof variationof Asc/Acis also
plotted in Figure 5 and peaksat near unity for As -- Ac.
Comparisonof the circle-on-circlevaluesfrom Table 1 with
the variance curve of Figure 5 showsa larger variance when
the storm centersare Poisson-distributed(Table 1) than when
they are uniformly distributed (Figure 5). Circle-on-square
variancesare slightly smaller than circle-on-circlevariances
(Table 1).
(1+/1)
2
Ac>_8
(59)
wherefi = l/s, as definedin (35).
h=
_7.3
A (i(ED1
Ac _<Asm
(60)
I4=
(Asm/Ac)112
•1--D Ac > Asm
(61)
c/Asm)112
and
This gives
E[Asc/Ac] = D(Ac/8)-D/2Ii
E[As½/A½]
= D(Ac/8)-D/2112
+ 13]
E[Asc/Ac]= a(Ac/8)-a/2I½
Ac <8
8 _<A c <_Asm
A c > Asm
in which the integralsare evaluatedthrough binomial expansionas
ll = .=• x(--1)
"-x(n- 1n+ D)
PROBABILITYDENSITY FUNCTION OF STORM AREA
Lovejoy[1982] foundthat tropicalrain andcloudareasare
fractalswith dimensionD- 1.35 over the range 1 km2<
As< 106 km2. Kordak[1938] foundempirically
that theareas
of islandsobeya hyperbolicprobabilitydistribution
P0[A0> z] = Bz-c
(62)
(54)
and Mandelbrot[1977,p. 69] showedthat fractalislandshave
the same distributionwith C- D/2. Such "fat-tailed"distributionsare characterized
by infinitemeanand variance.
For analyticalconvenience
we havechosento represent
the
stormsby circleswhichare two dimensionaland hencenot
(63)
12= Y',(-1) "-x (n + 1n- D) 1--
(64)
n
13= •(-1) "-x (n- 1 + D)
- (n - 1 + D)/2
fractals. Nevertheless,.
we wish our storm areas to have the
-1
probabilitydistributionobservedin nature.Accordingly,
we
constrainthe rangeof sizesin order to assurefinite moments
and adapt the aboveresultsfor circlesto obtain [Eagleson,
n
I½= • (-- 1)
1984]
f (rs)= D(8/;r)a/2rs
- •- •
(8/•z)
x/2<_rs_<(Asm/•)
1/2
(55)
ß
where8 and Asmare the minimumand maximumstormareas,
respectively.
We will now use the above pdf of storm radius to remove
theconditioning
of themoments
of As½
uponthevalueof rs.
Expectation
The marginalexpectation
is calculated
by theweighting
EEAsc
Itc,rs]f(rs)drs
For analyticalevaluationof (57) we breakthe integralinto
=
(A,r/•)t/2
1
(i•zD-q-'
Ac<
d(A½lA,m) tl 2
(66).
The parameterAsm/8of Figure6 represents
the spreadof
size in the populationof stormsin the given climate.For
Asm/8
= 1 the pdf of stormsizereduces
to a unit impulseat
As= Asm
= 8,andthestormsizeisnota randomvariable.
The
curvefor Asm/8
= 1 is thusidenticalwith the conditionalexpectationpresented
earlierin Figure5. As we let Asm> 8, we
assumed
hyperbolicpdf of As.Initially (i.e.,Asm/8
= 10),for
largestormsand smallcatchments
(i.e.,Ac/8<<1) a broadening of the stormspectrum
will introducesomesmallerstorms
and thus will lower the expectationof Ascfrom its limiting
valueof unity.Againinitially(i.e.,Asm/8
= 10),but for small
stormsand largecatchments
(i.e. Ac/8>>1) a broadening
of
the storm spectrumwill introducesomelarger stormsand
thuswill raisethe expectation
of Ascfrom its limitingvalueof
zero.As Asm/8getsverylarge,however,the marginalexpectation of Ascbecomes
independent
of Asm/8and is largerthan
(rc
+rs)
2drs (57)
four pieces
(66)
(56) seethe effectof weightingthe conditionalexpectationby the
Using (44), this becomes
EFAscq
=D
- 1
Equation(62)is plottedin Figure6 for D- 1.35using(63)-
MOMENTS OF CATCHMENT STORM AREA
E[Asc
Irc]-- EEAsc
]= !
(n + 1 - D)
(58) the conditional value at all Ac/8.
EAGLESONAND WANG: MOMENTS OF CATCHMENT STORM AREA
1191
1.0
D = 1.35
.9
.8
.7
.6
EAI-•--c
] .5
.4
.3
-
103
-
10
102
.2
.1
0
10-1
10-2
1
Ac/C
Fig. 6.
Expected value of the relative catchmcnt storm area.
and
Variance
The marginalvarianceis calculatedby the weighting
VarLAc•zi -• bo-Jr'E7bi• i+4
Var IAscI rc] -- Var IAsc]
1_<•<10
where
Var [AscJrc, rs]f (rs)drs
• I(Asm/•)l/2
(67)
-
Using (54) this becomes
=
•D
Var
•
d•
(68)
Breakingthe integrand into four parts
ao = -0.0010154478
b0 = 0.0000004385
a•-
b• = 1.3567479700
1.4397069216
a2 -- -- 5.8425507545
b2 = -4.9889183044
a3 = 13.1218589544
b3 = 9.4151695967
a4 = -19.4853985310
b4 = -11.1284917593
as = 18.5074794292
bs = 8.2469664812
a6 -- -- 10.0027712584
b6 = -3.4935194552
(76)
I•= Fc,,
•-• Var
LAc
• <1 d• Ac<s
(69)
I6 =
(70) a7= 2.3152309954b7 = 0.6444504783
d(Ac/Asm)•/2
fi-{• +D)Var
I, =
(75)
i=1
L Ac
•z> 1 dfi
Ac > •
•z
ø-'VarIAsc
<1]d•zAc_<
L Ac •z
(71)
The approximations(equations(74) and (75))are compared
with the exact equations(52) and (50), respectively,by the
plottedpointsin Figure 5 using(76).The agreementis within
0.2%overtherange10-'• _<Ac/As_<102.
and
The integralscan now be evaluatedto obtain
18=
(72)
Is= • (i+a,
D)(_._•)(,+
a)/2
[1- (?)-(,+a)/2]
fi-{•
+D)
Var
LAc
(77)
i=0
Asm
Ac
10-'* <--<
This gives
D-•(Ac/œ)
D/2Var [Asc/Ac]= I s
D-X(Ac/s)
D/2Var [Asc/Ac]= 16+ 17
D-•(Ac/8)D/2Var [Asc/Ac]= I8
Ac < s
• _<Ac _<Asm (73)
Var
LAc•z-•• ai•
i 10-2
• • <1
i=0
-1 ]+ •1(i+4-bo[(•)D/2
biD)
16--'•
Ac> Asm
i=
1-
The conditionalvariancesof (69)-(72) are givenby (50) and
(52), and their exact integration is cumbersomeat best. We
will settlefor their approximation by the more easily integrable polynomials
1
1 _<-œ < 102
(78)
17
• (i+
D)
:,=o
a,
[l_(?)-('+a'/2(•)('+a'/a
1
(74)
(79)
•o-• <(zd•c•/(•sm•
\•-//\•-/ < •
1192
EAGLESONAND WANG' MOMENTS OF CATCHMENT STORM AREA
.20
'
' ''
[["l
'
' [ ] '"[i
'
' ' '''"1
[
[ ' '']['
D = 1.35
.18
_
.16
.14
.12
VA
RIA'•-'•-I
.10
.08
102
105
.06
Asm/• = 1
_
10
.04
-
_
.02
-
_
•
0
i
] ] i,],l
10-2
i
,
t , ,,I,[
10-1
1
10
Ac/•
Fig. 7. Varianceof the relativecatchmentstorm area.
and
Equation (73) is plotted in Figure 7 for D = 1.35 using(77)(80). In Figure 7 we see that the effect of weightingthe conditional variance(i.e., A_sm/8
= 1) by a storm spectrumof increasingwidth is to shift the peak to somewhatlarger values
of Ac/8. Once again, a limiting variance is approachedfor
large A_sm/t•
at which the marginal varianceof Ascbecomes
independentof Asc.
The marginal coefficientof variation of Asc/Acis shownin
Figure 8 again for D - 1.35. In Figure 8 we seethat the resonant value of Ac/e increaseswith As,n/e,as doesthe resonant
+ • (i+4-D)
i=
(80)
peakin CV[Asc/Ac].
To reachthepeakfor A_sm/8
= 103,it was
necessaryto exceedthe range of (75). This was done using a
third polynomial approximation
VarLAc
•z_•• cil?
i+• 10_<•<103
(81)
i=l
2.2
I I I llll I
I
I I I I I l, I
i
I [ I [ I Ill
D= 1.35
2.0
1.8
103
1.6
1.4
cvI-•l
1.2
1.o
• I Ill•,l
o
lO-1
,
1
, , ,,,,,I
,
lO
I , I,,ttl
,
lO 2
Ac/•E
Fig. 8. Coefficientof variationof the relativecatchmentstormarea.
i
i
i
i
iii
103
EAGLESON AND WANG.' MOSmNTS OF CATCHMENT STORM A•y.•
where
when the catchment size and predominant storm size are
"tuned" at A`:/As= 1, indicating the important contribution
-0.0000003718
that
C2 '-- 0.0000464831
c3 = -0.0027903169
c,• = 1.5035021156
(82)
cs - -6.9985353947
variable
c7 = -42.5362448692
Let the integrandof (72) be I o and rememberthat the conditional variance included therein is given by (75). We will
now introduce Io* to representthe same integrand with the
variance given instead by (81). We then define the additional
integrals
I o* dfl
10%_>Ac > 102Asa
(Asm/Ac)l/2
Io* dlt
A`: areaof catchment,
km2.
Ao areaof fractalisland,km2.
As areaof storm,km2.
102t•_<Ac '< Asm
(84)
As`: area of storm within catchment("catchment
Io* dlt
102t•_<Ac < Asm
stormarea"),km2.
Asa maximumstormarea,km2.
(85)
ai coefficientsof varianceexpansions.
(Asm/Ac)l/2
Io* dlt
102e_<A½_<102Asm (86)
B
coefficient.
b
common chord of storm and catchment sectors,
dO.1
123=
to the total
NOTATION
(83)
/At)l/2
I•2 =
area can make
Assigningthe storm areasAs a pdf which is consistentwith
observations,the marginal expectation,variance, and coefficientof variationof As are derived.This showshow a large
catchmentin a climate having a wide range of storm sizeswill
experiencea great amplificationin the coefficientof variation
of storm runoff for purely geometricalreasons.
.1
122=
storm
uncertainty in storm runoff under certain scalerelationships.
Said another way, there appearsto be a critical size of catchment in a given climate for which the variance of streamflow
will be amplified due to the uncertainty in the wetted area.
These resultsmay be usefulin removingthe one-dimensional
assumptionfrom many practicesof catchmentscalehydrolo-
•}(e/A½)I/2
12o=
catchment
gY.
c6= 22.0687894821
19=
1193
Io* di•
km.
bi coefficientsof variance expansions.
C parameter of distribution.
c• coefficientsof variance expansion.
102e_<Ac _<102Asm (87)
/At)l/2
whereuponthe marginalvariancemay be written
D
fractal dimension.
For A`:/Asmg 1
I•
i
k
n
integrals.
integer counting variable.
integer counting variable.
integer counting variable.
D-2(Ac/tOD/2
Var [As/A`:] = I s
D-2(A`:/e)
D/2Var [As`:/A`:]
= 16+ 17
10-½< A`:/e< 1
1 < Ac/e< 102 (88)
r
D-X(A`:/e)
m2Var [As`:/A`:]
= 17+ I•o + 122
102_<Ac/e
center, km.
r`: radius of catchment,km.
rs radius of storm, km.
X probability function.
For 102 _>A`:/Asa> 1
D-•(Ac/e)•>/2Var [As`:/A`:]
= 18
D-2(Ac/tO
•)/2Var [As`:/A`:]
= I•2 + I23
1 _<Ac/e< 102
102--<Ac/t•
(89)
For 106> A`:/Asm
> 102
D-2(A`:/e)z'/2Var [As/A`:] = I9
102_<A`:/e
(90)
In practicalterms,Figure 8 showshow a large catchmentin
a climate having a broad range of storm sizescan produce
extreme amplification of the coefficientof variation of storm
runoff for purely geometrical reasons.We believe that this
phenomenonis describedhere quantitativelyfor the first time
and shouldbe helpfulin improvingtheoreticaldescriptionof
the frequencyof extremefloods.
distance of storm center from catchment
CV[
E[
f(
0(
The catchmentstorm area As,:has been representedby the
P[
common area of two overlapping circles,and the first two
moments of the area have been calculatedfrom geometrical
considerationsconditional upon valuesof the catchmentand
storm size.For catchment/stormscaleratios of practicalimportance the wetted catchmentfraction As`:/A`:has a mean
value much lessthan one, emphasizingthe dangerof invoking
one dimensionalityin hydrology. The variance of As`:peaks Var [
Y
variance
y
z
•
fl
e
Oc
0s
ß
probability function.
value of fractalislandarea, km2.
catchment/stormscaleratio, r`:/rs.
storm/catchmentscaleratio, rs/r`:.
minimumstormarea, km2.
catchmentsectorangle,rad.
storm sectorangle,rad.
probability function.
ro
dimensionless
]
]
)
)
]
function.
catchment
storm area.
coefficientof variation of[ ].
expectationof[ ].
probability densityfunction(i.e.,pdt)
of().
function of().
marginal exceedanceprobability of catchment
storm
area.
]
conditional exceedanceprobability of
]
exceedanceprobability of fractal island
]
variance of [
catchment
storm
area.
].
area.
1194
EAGLESONAND WANG: MOMENTSOF CATCHMENTSTORMAREA
Lecam, L., A stochasticdescriptionof precipitation,in Proceedingsof
the Fourth Berkeley Symposiumon Mathematical Statistics and
Probability, vol. 3, edited by J. Neyman, pp. 165-186, University of
NAG 5-388. We are indebted to Randal Koster for his assistance with
California Press,Berkeley,1961.
problem formulation.Supportfor W.Q. while at the Massachusetts Lovejoy, S., Area-perimeter relation for rain and cloud areas, Science,
216, 185-187, 1982.
Institute of Technologywas providedby the Yangtze Valley Planning
Office of the Ministry of Water Resourcesand Electric Power, Mandelbrot, B. B., Fractals: Form, Chance, and Dimension,W. H.
Freeman, San Francisco, Calif., 1977.
People'sRepublic of China. We wish to expressour appreciationfor
Milly, P. C. D., and P.S. Eagleson,Infiltration and evaporation at
this support.
inhomogeneousland surfaces,Rep. 278, 180 pp., R. M. Parsons
Lab., Dep. of Civ. Eng.,Mass.Inst. of Technol.,Cambridge,1982.
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Acknowledgments.This material is basedupon work supportedby
the National ScienceFoundationunder grant ATM-8114723 and by
the National Aeronautics and Space Administration under grant