VOL.
14, NO. 5
WATER
e
RESOURCES
RESEARCH
OCTOBER
1978
Climate, Soil, and Vegetation
The Distribution of Annual PrecipitationDerived
From ObservedStorm Sequences
PETER S. EAGLESON
Department of Civil Engineering,MassachusettsInstitute of Technology
Cambridge,Massachusetts02139
Point precipitation is representedby Poissonarrivals of rectangularintensitypulsesthat have random
depthand duration. By assumingthe stormdepthsto be independentand identicallygammadistributed,
the cumulative distribution function for normalized annual precipitation is derived in terms of two
parameters of the storm sequence,the mean number of storms per year and the order of the gamma
distribution. In comparison with long-term observationsin a subhumid and an arid climate it is
demonstratedthat when working with only 5 yearsof storm observationsthis method tendsto improve
the estimate of the variance of the distribution of the normalized annual values over that obtained by
conventionalhydrologic methodswhich utilize only the observedannual totals.
INTRODUCTION
by consideringthe processes
to be stationaryrather than periodic.
Point precipitation is a random time seriesof discretestorm
In this work we will continuethe simplifyingassumptionof
eventswhich under certain restrictionsmay be assumedto be
stationarity
but will replace the exponential distribution of
mutually independentand in which the averagetime between
events is large with respect to the average duration of the individual storm depths by the more general two-parameter
events themselves.
gammadistribution,as was done earlier.by Todorovicand
For reasonsof physicalvalidity an engineeringmodel of this
processmust retain thosenatural featureswhich are important
to the problem at hand, while for economy of computation
and for clarity of behavior it should omit those features
deemed unessential.Maximum understandingof hydrologic
variability would be achievedwith a model of the precipitation
processwhich incorporatesthe geophysicaldynamicsthrough
which atmosphericdisturbancesare generatedand propagated
and produce local precipitation. Unfortunately, the base of
scientificknowledge doesnot yet exist for this model. Instead
we will take the phenomenologicalapproach,which represents
the essentialobservedfeaturesof the precipitation time series
by a stationary stochasticgenerating processthat is analytically tractable.
The essentialfeaturesof precipitationin terms of a physically oriented model of water balance processesare (1) the
time betweenstorms,sincethis is the 'window' for evapotranspiration, (2) the duration of the storms, since this starts and
stopsthe infiltration process,and (3) the storm depths,since
thesedetermine the water availability for infiltration.
Various models have been used for this process(see the
reviewsby Grace and Eagleson[1966] and Gupta [1973]), but
the Poissonprocess[Benjaminand Cornell, 1970,pp. 236-249]
seemsto provide the bestcompromisebetweenthe conflicting
demandsof simplicity and generality.
Perhapsthe most general developmentof the Poissonprecipitation models is that of Todorovic [1968], in which he
considersthe individual storm depths to be distributed exponentiallywith a mean depth which is annuallyperiodic.The
Yevjevich[1967] and more recentlyby Ison et al. [1971]. We
will extend these developmentsto the case of a randomly
variable length of the rainy seasonand will demonstratethe
value of the techniqueto the derivation of the frequencyof
annual rainfall from only a few years of storm observations.
Poisson arrival rate of these events is also considered to be
annually periodic. In a subsequentsummary of this work,
Todorovicand Yevjevich[1969] apply it with some successto
the descriptionof the probability densityfunctionsof seasonal
precipitation.
Benjaminand Cornell [1970, p. 310] simplify this approach
Copyright ¸
The classical derivation
PROCESS
of the Poisson distribution
of storm
arrivals at a point assumessuccessiveeventsto be independent. The probability of one arrival in time interval/xt is p and
that of more than one arrival in this interval is negligible.This
leads to the binomial distribution for the probability po(v) of
having 0 = v storms in total time t = m/Xt:
po(v) =
which
! - p
(1)
has mean
E[O] = mp
(2)
Var [0] = mp(1 - p)
(3)
and variance
It should be noted that the independenceassumption is
strictlyapplicableonly for instantaneous
arrivals--in thiscase,
storms of zero duration. A storm of finite duration occurring
in intervalAtj mayoverlapinto the subsequent
intervalAtj+•,
thereby decreasingthe effectivelength of that interval and
hence the probability of obtaining another storm arrival
therein. A measureof the validity of this assumptionin application to an idealized storm seriessuchas is shown in Figure 1
will be the 'smallness'of the ratio of the averagestorm dura-
tion to the averagetime betweenstorms,mrs/into.
If we let At
-• 0 without otherwisechangingthis stationaryprocess,p -• 0
and m -• •o, thus giving
mp = oot
(4)
whereo0is the averagearrival rate of the stormeventsand t is
1978 by the American GeophysicalUnion.
Paper number 8W0185.
0043-1397/78/058W-0185501.00
THE POISSON ARRIVAL
713
714
EAGLESON:
CLIMATE,SOIL, ANDVEGETATION
RAINFALL
INTENSITY
where h is the total precipitation(depth) from a singlestorm.
We definefv{.)(Y)as the probabilitydensityfunctionof P(v).
The probability densityof the total precipitationP in time t
is thengivenby summingthe probabilitydensitiesf•.{•)(y)for
each of the (mutually exclusiveand collectivelyexhaustive)
numberof stormsv whichcoulddelivery precipitationin this
time, each weightedby the discreteprobability Po•t(v)that
exactlythat number of stormswill occur. Analytically, this
gives a compound distribution composedof the continuous
'T
o
o
(o)
ACTUAL
density function
RAINFALL
INTENSITY
2r+-
CLIMATIC
s -•i
GENE
RATING
fe<t)(Y)
= • fe<•,)(y)Po,t(v)
y>0
.... ,
.,.,
(14a)
plus the impulse
Po•t(O)= e-'"t
DEPTH h = i. tr
PROCESS
y = 0
(14b)
To complete this derivation, we need the density function
fvl•(Y), whereP(v) is the sumof randomvariableshj.
We will assumetheseh• to be independent,and, althoughin
many natural cases,seasonaldifferencesare present,we will
Fig. 1. Model of precipitationevent series.
assumethe h• to be identically distributed. This assumption
may be removed (at considerablecomputational cost, howthe timeperiodof interest.Equation(1) becomes
the Poisson ever) should the circumstancesso dictate.
distribution
Having made the independenceassumption,we will selecta
distributionfor the storm depthswhich meetsthe following
v!
v = 0,1,2,'"
(5) criteria: (1) it provides a good fit with observationsover a
range of climatic types and yet (2) it is simpleenoughto be
for the probability of obtainingexactlyv eventsin time t. analytically tractable. Both of theseconditionsare met by the
From this distribution we can find the mean
gamma distribution,
(b)
MODEL
Po,t(v)
= (øøt)"e-øøt
E[OIt] -- m. = oot
(6)
Var [01t] -- a.: = wt
(7)
fu(h)
= G(K,
3,)=,X(Xh)K-•e-Xn
F(t•)
(15)
and the variance
INTERARRIVAL
which has mean and variance given by
TIMES
If T is a randomvariablerepresentingthe time until the first
mu = •/X
(16)
a.' = t•/(X)'
(17)
and
storm arrives,
Prob [T > t,,] = 1 - Fr(t.)= po,t.(O)
(8)
Defining the mean storm depth alternatively as
By using(5) this becomes
mu = r/-'
Fr(ta) = 1 - e-'"t.
(18)
(9)
we have
which is the probabilitythat the first stormwill arrive after
3, = r/t•
(19)
elapsedtime ta. Sincethe Poissonprocessis assumedto be
stationary,the origin of the time interval ta is arbitrary and
Another advantage of the gamma distribution is that it is
may be thoughtof as coincidingwith one of the instantaneous 'self-preserving,' or 'regenerative.' This means in our case
occurrencesshown in Figure lb. Fr(G) then becomesthe cumulative distribution function (cdf) of the continuousrandom variable: t. is the interarrival time. Differentiating (9)
givesthe exponentialdistributionfor interarrivaltimesof the
Poissonprocess:
fr(t.) = ooe-'"t.
X. = 2,
(11)
(12)
(21)
Thus considering (13) and assuming independenceand
identicaldistributionfor the hi, we have from (15),
TOTAL PRECIPITATION IN A GIVEN TIME
fv,.)(y
) = G(w,
2,)=
The total precipitatioladeliveredby v eventsis
P(v)= • hj
(20)
1
and
and
0'%:= •-:
t•.= • t•=.t•
( 1O)
which has mean and variance
mr. = w-'
[Benjaminand Cornell, 1970, p. 292] that the sum of v independentG(., 3,) gammavariableswith common3,and Kis also
gamma distributed, with
F(vK)
(22)
which is the density function of total precipitation from v
(13) storms.
The mean and variance of this distribution
are
EAGLESON:CLIMATE, SOIL, AND VEGETATION
me(v)= vK/h
(23)
715
where'y[a,x] is the incompletegammafunction.Equation(32)
is now
and
av(,)•' = vK/h•'
(24)
FvA(Y)=
Prob [Pa < y] = e-øømr
'Substituting(5) and (22) in (14) and using(19), we havethe
(compound) distribution of cumulative point precipitation:
fvr(Y)•.2r/•(r/•Y)v•-•e-n•Y
(wr)Ve-øør
y) 0
Po•.(O)
= e-'o"
y= 0
ß 1+
v! P[v•,rl•y]
=
(34)
where P[vK,•y] is Pearson'sincompletegamma function; that
(25)
is,
P[a, x] -= 'y[a, x]/_I'(a)
where r is the length of the rainy season.
Equation(25) assumesa constantvaluefor the lengthof the
rainy season,which is reasonablefor humid regionswherer is
often 365 days. In arid regions, however, r will be a random
variable, and (25) gives the (conditional) distribution of annual rainfall given the seasonlength. That is,
(35)
It will be convenientto have (34) in dimensionlessform. The
natural choice for a normalizing parameter is the climatic
variablemvA.By using(18) and(27) andlettingz equaly/mvA,
PA
fV(r)O')-• fV.v(YI r)
ß 1+ =
We find the desired marginal distribution of annual rainfall
•i
P[v•,wm•z]
(36)
depthre(Y) by integratingthe joint distributionof this depth This gives the cdf for annual point precipitation under the
and the seasonlength over all values of the latter variable:
following simplifyingassumptions:(1) The point stormprecipitation is a (stationary) Poissonarrival process,(2) individual
storm depthsare identicall:?distributedaccordingto G(•, X),
and (3) the coefficientof variation of the seasonlength is small.
Although this integration is readily performed for simple Note that the normalized distribution is specifiedin terms of
fr(r) (such as the uniform distribution), we will omit this only two parameters,the mean number of independentstorms
refinementhere, assumingthe coefficientof variation of r to be per year com•and the order K of the gamma distribution of
small.We will thusreplacethe random variabler in (25) by its storm depthsß
We will now verify (36), usingstorm observationsfrom both
mean value m•.
moist and dry climates.
Taking the expectedvalue of (13) gives
fv(y) -
fv,r(y, r) dr -
•0
fv.r(yI r)fr(r) dr
•0
E[Pa]=
mv,
=E[• h,
•=E[Omn]
=m•mn
(26)
OBSERVED DISTRIBUTIONS OF STORM PROPERTIES
j=l
If we use(6) and (11), this givesthe mean annual precipitation
as
mv,= E[wr]mn = wm•mn
The variance
(27)
can be written
Var [P,] • av,•= E[P, •] - •[P,]
(28)
which can be reduced,in the manner of (26), to
(29)
By using(6), (7), (16), (17), and (19) the varianceof the mean
annual precipitationbecomes
avff = (mvAZ/comr)(1
+ •-•)
(30)
The cumulative distribution function of the annual point
precipitationis of primary interesthere. This is definedby
EvA(Y)
=
Prob [Pa(r) < y] =
fvA(x)dx
(31)
To assembleobserveddistributionsof storm properties,we
first need a criterion for separatingthe time seriesinto independentevents.Graceand Eagleson[ 1966]usedthe rank correlation coefficientto test for linear dependenceamong successive 10-min rainfall depths. They found for two locations in
New England that linear dependencewas insignificant(at the
5% level) betweentwo rainy 10-min periods separatedby 2 h
or more. Although this test does not assureindependencein
any but the linear sense,a dry interval of to- 2 h was adopted
here as the criterion for distinguishing independent consecutive events at Boston.
From an analysisof 5 years of hourly precipitation data at
Boston, Massachusetts,Grayman and Eagleson [1969] found
the following to be true.
1. Storm duration is distributed exponentially at Boston
according to
frr(tr) = be-btr tr >--0
(37)
•J=mtr-•
(38)
where
By using(25), lettingz equal r/Kx,and replacingr by its mean
The observedand fitted distributions are shown in Figure 2.
2. The time betweenstorms to is distributed exponentially
at Boston accordingto
value mr,
:
r(v•)v•
•0
fro(to)= l•e-t•(to-to) to->to
The integral is [Gradshteynand Ryzhik, 1965, p. 317]
Since•St0<< 1, we will use the approximation
•Ky
e-Zz
w-•
dz='y[v•,
r/•y]
(33)
fro(to)= •Se
-ato
to > 0
(39)
716
EAGLESON:CLIMATE, SOIL, AND VEGETATION
0.12
0.60
OBSERVED
O. 50
RELATIVE
Fr(ta):
f f fTr,To(tr,
to)dR
=fotadto
fota-tø
t•t•e-fitø-•tr
dr,(46)
O. lO
2
Integrating
FREQUENCY
o
z
SAMPLE
uJ 0.30
--
0.20
rr
0.10
fi e-i•t
r.(t.): 1+ ••• e-fit. •-b
.
o.o8 >:
0.40
__
•
-
SIZE
0.06
r-,
and differentiating,we get the densityfunction,
546
EVENTS
5
•
FITTED
YEAR
S
-O 13tr
15 20
t
25
= STORM
30
35
40
45
(48)
from which
o
0.02
n-
I
0
- e-•t,]
f.(t.)= • fi•
_ • [e-•t,
0.04 •
EXPONENTIAL
•DISTRI
BUTION
5
(47)
z
m.=
•
=w-,
(49)
For Boston we have found that
50
DURATION (HOURS)
•/•
<< •
Fig.2. Distribution
of stormdurations
at Boston,
Massachusetts.
thus for large t. the distribution of interarrival times approachesthe exponential,as is requiredfor a Poissonprocess.
This exponentialapproximationis givenby (10), w beingde-
where
fi = mtb-'
(40)
The observedand fitted distributionsare shown in Figure 3.
3. Storm intensityi is distributedexponentiallyat Boston
accordingto
f/(i) = ae-'"'
i _>0
(41)
fined by (49).
5. The storm depth h is related functionallyto the storm
duration and average intensity by
h = itr
(50)
and its calf is thus given by
F•(h)
=f fn•,f(i,
tr)didtr
where
(51)
(42)
The observedand fitted distributionsare shown in Figure 4.
4. The storm interarrival time may be written (seeFigure
lb)
t. = tr Jr-to
What is thejoint distributionf(i, tr)9.Are i and tr independent?Assumingfor the momentthat they are independent,we
can write
f(i, tr) = f(i)f(tr) : ott}e-at-&r
(43)
Assumingthat tr and to are independent(this has been found
to be true by Grace and Eagleson [1966]), we can write their
joint distribution,
ft,,ro(tr,to) = ft,(tr)frb(to)
(44)
which, using(37) and (39), is
fTr,To(tr,
to)= t•t}e
-gto-•tr
whereupon(51) can be integratedto yield
(53)
F.(h) = 1 - 2(abh)'/2K,[2(abh)'/21
whereK•[ ] is the modifiedBesselfunctionof the first order.
Equation(53) is comparedwith Bostonobservations
in Figure
5. In spiteof its consistency
with the chosenmarginaldistribu-
(45)
0.30
We can now derive the cdf of storm interarrival times t.
accordingto
I
I
I
I
I
I
I
I
30
I
FREQUENCY
SAMPLE
SIZE
5
YEARS
_
0.20
o.i 41-- IJ OBSERVEDRELATIVE
'
__•
OBSERVED
RELATIVE
0.25
1.6
>-
(52)
--
25'-'
_
20
•r- / FREQUENCY
o0,12•
[•
•
z
z
• 0.10•- •
ø
0.15
FITTEDEXPONENTIAL
I-I I•X
-
DISTRIBUTION
u_ 0.08
•- 0.10
_o.o1111
o-o1111111
O•
0
•
iii iiiii
i
20
•
I
40
I
I
60
I
I
80
*-------'---
FITTED
N
DISTRI
BUTION
--
N•.
15
-
EXPONENTIAL
I0
o
5 n
0.05
I
I
I00
I
I
120
I
I
140
I
I
160
I
I---1
180
I---I
200
lb: TIME BETWEEN STORMS(IN HOURS)
Fig. 3. Distribution of time betweenstormsat Boston, Massachusetts.
m
fI(i):30e-30i
0
0
i
2
3
4
5
6
7
8
9
10
I
i=POINTSTORM
INTENSITY
xlO2 (INCHES/HOUR)
Fig. 4. Distribution of point storm intensitiesat Boston,Massachusetts.
EAGLESON:CLIMATE, SOIL, AND VEGETATION
717
%
I-FH(h)
O.001
0.01
0.1
I
99
99.99
9
7
THAN
70
50
20
5
I
0.1
0.01
I I
\
B-
6
L ESS
90
0 GROUPED
OBSERVATIONS
I OCT.
- $OAPR 78 EVENTS
__
?-
5
-
,, ..... -'•h.'"'"'•
• \,
'• =50HOUR
S/ INCH
-
F:i(h,,):
i••H
•n•='r/e
.._,•h
' f|....•-•",,,.
. _
X,,•\ • =0.
I5HOURS-'
•0
5 -
_.
X(•%
-xh
••
•DODISTRICT
6-
k( kh)K-le'kh
q.-
•
'•
5-
_
0 fH(h)
=•--•-•
--'
•,:0.186
INCHES
-I
•k
--
•'T[:0.25
K, Xh]
•\
_
>_-
0 GROUPED
OBSERVATIONS
1956, 1957,
-
1962, 1964, 1965, 546
0
EVENTS
I -
I
I
I
I0
3
0.01
-I
I0
%
GREATER
I
I00
O. I
I
THAN
5
20
50
% GREATER
Fig. 5. Point storm precipitationat Boston, Massachusetts.
70
90
99
99.99
THAN
Fig. 7. Point storm precipitationat Santa Paula Canyon, California
(at Ferndale Ranch).
tions for intensityand duration, (53) is unsatisfactoryfor our It is believedthat this is due to southernCalifornia precipipurposes.Its density function is not self-preserving,as we tation being composedof two distinct storm typesrather than
required in deriving (22). We thus will fit the observedstorm a singlehomogeneouspopulation, as was assumed.
depthswith the gamma distributionof (15) by the methodof
OBSERVED NUMBER OF STORMS PER YEAR
moments, acknowledgingthat this distribution is inconsistent
with (37) and (41).
The monthly averageprecipitation for Boston, MassachuIn Figure 5 the grouped observationsare also compared
with (15) and with the specialK = 1 caseof (15), which gives setts,and for Santa Paula, California, is shown in Figure 8 for
the exponentialdistributionhaving3, = rt. Of the three alterna- 5 years of observation.From this it is seenthat the average
tives presented,it seemsclear that the gamma function pro- length of the Boston rainy seasonis mr = 365 d. For Santa
vides the best fit to the observations.
Paula it is not so clear, but from thesemonthly totals we can
A similar 5-yr analysis of storm data was made for Pasa- pick the nominal value mr = 212 d. Using this as a guide, we
dena, California, and for Santa Paula, California (in Ventura can count the number of storms per seasonand averagethis
County about 75 mi northwest of Pasadena).In both of these number to obtain for Santa Paula,
cases,only daily rainfall recordswere available, and the criterhv = Wmr = 15.7
rion for independenceof successive
storm eventswas taken to
be their separation by at least 1 day that has no recorded If we usemr = 212 d, the minimal storm arrival rate is defined
rainfall. The fitting of (15) to the observedstorm depths is to be
shown in Figures 6 and 7. The values found for mn and K, as
5.7/212 = 0.074 d -x
well as for the other stormpropertiesdetermined,are givenin
Table
1.
Notice in Figures6 and 7 that for large storm depthsthe
chosengamma distribution does not fit the observationswell.
%
99.99
LESS
90
99
70
THAN
50
20
5
I
which will differ somewhatfrom that calculatedby using(49),
as is shown in Table 1. This problem can be eliminated by
using an actual (i.e., first storm to last storm) seasonlength
rather than a nominal (i.e., calendar) one when computingw.
It does not arise, of course, when the seasonis a full year.
0.1 0.01
8
"•0
0 GROUPED
OBSERVATIONS
I OCT. - 30 APR
•o
_
1968-1972
}
86
TABLE 1. Five-Year Average Properties of Storm Series
EVENTS
Properties
(• u.s.
NATIONAL
WEATHER
SERVICE
-
5
•t•, d
0.32
mto,d
kh]O(•
0.01
0.1
I
5
20
% GREATER
50
Pasadena
1.43
10.42
rhu, cm
0.86
3.41
2.37
--
•
0.50
0.25
0.33
;: o.•
_
rhr, d
365
my
rh,, cm/d
109
2.03
7'[Kkh]= p[K,
øøø
•
2.98
Santa Paula
,• =0.353 INCHES
-•
•)Q•,, 1"
(K)
•
Boston
70
0
90
99
_
--
99.99
15.7
m, '" rhu/rht•,
cm/d
2.69
2.38
co,d-'*
0.30
0.084
co= rhv/rfir,
d-•
0.30
0.074
212
17.2
0.081
meA,cm•'
102.6
54.4
51.5
meA,cm
94.1
53.6
40.8
THAN
Fig. 6. Point storm precipitationat Pasadena,California.
212
* From (49).
•'Full station record.
718
laJ
EAGLESON:CLIMATE,SOIL, AND VEGETATION
totals for the period 1886-1974 at Boston.(Earlier observations
were discardedwhen a cumulativeprecipitationcurveshowed
an apparent abrupt changein slopein about 1886.) We may
OCTI NOVIDECI JANI FEBI MARlAPR
I MAYIjU N I JULI AUG• SEP
BOSTON,MA.-'1956,'57, '62
--
^
-- mr BOSTON
=3•
'64,
'65-
consider these observations to define the true statistics of PA,
SANTA PAULA, CA: 1970-1974
'
-
r3•,Y.q
which we are attemptingto estimatefrom a short 5-yr period
of observation.
- r•rSANTA
PAULA=?,:
>DAYSJ
_
•
The solidcurvein Figure9 represents
the cdf of P,•/m•,Aas
-i.... L___
derived by (36) by using valuesof the parametersmy =
and K estimated from the same 5 years of individual storm
observations.Notice the closeagreementbetweenthis predic-
F-
tionandthe;population'
cdf(givenby thecircles).
Useof
the commonlyneglected(althoughnot alwaysavailable)storm
propertiesappearsto provide, through the Poissonmodel, an
estimateof the variance of the annual precipitation,which is
considerablymore accuratethan that obtained through useof
the annual totals alone. One might intuitively expectthis kind
of 'leverage'due to the larger samplesizefor the parametersto
be estimatedprovided, of course,that the Poissonmodel is a
true representationof the process.
Fig. 8. Seasonalityof annualprecipitation.
ANNUAL PRECIPITATION
Humid climates. In Figure 9 the total annual point precipitation at Bostonfor each of the 5 yearsfor which the individual stormswere studiedis presentedin the form of the cumulative distribution function. The probability wascalculatedfrom
an ordered ranking of the five observationsaccordingto the
Thomas [1948] relation,
Probm•
PA<
_ N+
Clinton, Massachusetts,is in the Nashua River basin about
50 mi northwest
of Boston. It seems safe to assume that the
stormpropertieswill be homogeneous
over this shortdistance.
By usingthe stormparametersmyand K determinedfrom the 5
years of Boston record the cdf of normalized annual point
precipitation
wascalcuhited
for Clinton(sincetheparameters
are identical, the cdf are identical) and is presentedas the
10. Here it is compared with the total
availablerecord of annual point totals at Clinton, as is shown
by the circlesplotted accordingto (54). Once again the agree-
(54) solid line in Figure
1
where m, is the rank order of observationof magnitudez and
N is the number of yearsof record. The five valuesare shown ment is remarkable.
A rid climates. In arid climatesit is generallyaccepted[Linby the solid symbolsin Figure 9.
In a humid climate such as that of Boston it is generally sley et al., 1975,p. 356] that annual precipitationis distributed
accepted[Linsleyet al., 1975,p. 356] that annual precipitation log normally. In Figures 11 and 12 we presentthe cdf of
is distributednormally. It is conventionalthereforeto fit such annual precipitationfor Pasadenaand Santa Paula, Califoran a priori distribution to small data sets.The dashedline of nia. The comparisonsare similar to those for Boston and
Figure 9 representsthe normal distribution fitted by the Clinton, Massachusetts,except that log probability paper is
used.Here the divergenceof the observedand the calculated
method of momentsto the five annual precipitationtotals.
Shown as circles on Figure 9 are the annual precipitation distributionsin both tails may be due to the apparentdual
%
99
99.99
I I
I I
I
I
I
LESS
70
90
I
I
THAN
50
I
I
20
I
I
5
I
I
I
I
I
I
01
I
I I
WITH
t •' =0.50
ARS
OF
0.0
I
-STORM
DATA
0.8--
NORMAL
DISTRIBUTION,
5 YEARS
mX
ANNUAL
TOTALS
1956,'57,
g2,'64,'65 ß '-•
0.6--
•
ANNUAL
TOTALS
1886-1974
_
-
0.4--
0.2--
I
I
I I
I
I.OI
1.2 5
RECURRENCE
2
INTERVAL
5
IO
-
20
50
I00
YEARS
Fig. 9. Annual point precipitationat Boston,Massachusetts.
IOOO
EAGLESON:
CLIMATE,SOIL,ANDVEGETATION
%
90
99
9999
I
I
LESS
70
I
I
THAN
50
I
719
I
20
I
I
I
i
5
I
I
o.i
O.Ol
I
1.4
•
1.3
PROB
[• zJ=e
t'•Z•i
•'i'ß'-•z•'•-•"%•
1.2
--
•
•
• •= •m r =109
WITH •
L; =0.50
o
•
BOSTON STORM DATA
1.0
0.9
0.8
0.7
•
0
OBSERVATIONS
USGS
WSP
1904
0•,
- 1933
846
•.0•
L25
2
RECURRENCE
5
INTERVAL
-
•0
20
50
•00
•000
YEARS
Fig. 10. Annual precipitationin Nashua River basinat Clinton, Massachusetts.
This distribution allows the generation of the frequency
curve of annual precipitation,givenshort-periodobservations
SUMMARY
of storm characteristics,and is shown through comparisons
By usinga Poissondistributionof point rainstormarrivals with observationsin both arid and subhumidclimatesto proand a gamma distribution of point rainstorm depthsthe cu- vide a better estimate of the variance of the annual precipimulativedistribution
functionof annualpointprecipitation
is tation than do conventionalhydrologictechniqueswhich utilize the observedannual totals.
derived.
storm populationmentionedearlier or to the assumptionof
constant
•-.
%
99
99.99
I0
I I
I
90
I
I
I
LESS
70
I
I
THAN
50
I
20
I
I
I
5
I
I
I
I
I
I
0.1
I
O. Ol
I I I
8--
4
--
•
( I•=tom.[-:
17.2
WITH /• 0.33
ROM
5 YEARS
OF
xx
ß
6
m
4
---
S
_TORM
DATA
0
O0
ß ANNUAL
TOTALS1968-1972
LOG-
0
NORMAL
ANNUAL TOTALS
DISTRIBUTION,
5 YEARS
0
'•
1883-1972
I I
0,1
1.01
1.25
RECURRENCE
2
5
I0
20
50 I00
INTERVAL-YEARS
Fig. 11. Seasonalpoint precipitationat Pasadena,California.
IOOO
720
EAGLESON:
CLIMATE,SOIL, AND VEGETATION
%
99
99.99
io
!
4
I
90
I I
I
I
LESS
70
I
I
THAN
50
I
I
20
I
I
I
--
•.•
I
I
I
I
I
I
0.1
I
0.01
I II
--
FROM
2
5
I
•'""•
WITH
•_
--
o
ß
5 YEARS
Of
STORM
DATA
r r.,,--,.,..
=•5.?
^
-
L•<
=0.25
_
.{
0.8-0.6M
LOG
-
ß
0.4--
•
ANNUAL
TOTALS
1970
-1974
o
% ",,•
__
ANNUAL
TOTALS
1957-1975 %
0.2
m
I I
0.1
i
1.01
1.25
RECURRENCE
2
5
INTERVAL
I0
-
20
50
I00
I000
YEARS
Fig. 12. Seasonalpoint precipitationat Santa Paula Canyon, California (at FerndaleRanch).
NOTATION
h
i
number
t
fl
number of storms in wet season.
parameterof gammadistributionof stormdepth.
parameterof gammadistributionof stormdepths,
of time intervals.
mean storm depth, centimeters.
mean storm intensity,centimetersper second.
mPA averageannual precipitation, centimeters.
rnta mean storm interarrival time, days.
mean time between storms, days.
mt r mean storm duration, days.
rank order of observationof magnitudez.
my
mean number of stormsper year.
mr
mean length of rainy season,days.
N
number of years of record.
P precipitation, centimeters.
PA annual precipitation,centimeters.
p probability value.
T time to arrival of first storm, days.
ta
to
tr
to
a
reciprocalof mean storm depth, equal to mu-2,
centimeters -2.
storm depth, centimeters.
precipitationrate, centimetersper second.
counting variable.
time, seconds.
storm interarrival time, days.
time between storms, days.
storm duration, days.
storm spacingfor independence,days.
reciprocalof averagerainstorm intensity,equal to
rnt-2, secondsper centimeter.
reciprocalof averagetime betweenstorms,equal
to mto-2,days-2.
At arbitrary small time interval, seconds.
b reciprocal of average storm duration, equal to
rnt•-2, days-2.
p
O'x
O'x
2
T
O)
œ[ ]
r[ ]
f(
)
G(, )
equal to K/mu, centimeters-2.
counting variable for number of storms.
standard deviation
of x.
lengthof rainy season,days.
averagearrival rate of storms,days-2.
expectedvalue of [ ].
cumulative distribution
function.
probability densityfunction of (
gamma distribution.
Bessel function
P[, ]
Var[ ]
r( )
?[a,x]
( )
of x.
variance
).
of order 1.
Pearson'sincompletegamma function.
varianceof [ ].
gamma function.
incomplete gamma function.
estimateof ( ).
Acknowledgments.This work wasperformedwhilethe authorwas
a visiting associatein the EnvironmentalQuality Laboratory at the
California Institute of Technology on sabbatical leave from MIT.
Publication has been supported by a grant from the Sloan Basic
Research Fund at MIT.
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Grace, R. A., and P.S. Eagleson,The synthesisof short-time-in-
EAGLESON:
CLIMATE,SOIL,ANDVEGETATION
crement rainfall sequences,Hydrodynam. Lab. Rep. 91, Dep. of
Civil Eng., Mass. Inst. of Technol., Cambridge,Mass., 1966.
Gradshteyn, I. S., and I. M. Ryzhik, Table of Integrals, Series and
Products, Academic, New York, 1965.
Grayman, W. M., and P.S. Eagleson,Streamflowrecord length for
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Gupta, V. K., A stochasticapproach to space-timemodelling of
rainfall, Techn.Rep. Natur. Resour.$yst. 18, Univ. of Ariz., Tucson,
1973.
Thomas, H. A., Frequencyof minor floods,J. BostonSoc. Civil Eng.,
35, 425-442, 1948.'
Todorovic, P., A mathematicalstudy of precipitation phenomena,
Rep. CER 67-68PT65, Eng. Res. Center, Colo. State Univ., Fort
Collins, 1968.
Todorovic, P., and V. Yevjevich, A particular stochasticprocessas
applied to hydrology, p.aper presentedat International Hyd.rology
Symposium,Colo. State Unig., Fort Collins, 1967.
Todorovic, P., and V. Yevjevich, Stochasticprocesses
of precipitation,
Colo. State Unit). Hydrol. P•tp.35, 1-61, 1969.
Ison, N. T., A.M. Feyerherm,and L. D. Bark, Wet period precipitation and the gamma distribution,J. Appl. Meteorol., 10(4), 658•665, 197t.
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Hydrologyfor
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721
(Received August 19, 1977;
revised December 27, 1977;
acceptedFebruary 16, 1978.)