HIDROLOGÍA con modelos estocásticos

1. VOL. 14, NO. 2 WATER RESOURCES RESEARCH APRIL 1978
Stochastic Models of Floods
P. TODOROVIC
Ecole
Polytechnique,
University
ofMontreal,
Montreal,
Quebec,
Canbda
There
areessentially
twodifferent
approaches
totheproblem
offlood
analysis.
Onecorresponds
tothe
streamflow
annual
floodseries
(AFS), andtheothertothestreamflow
partialduration
series
(PDS).Here
we discuss
characteristics
of theseapproaches.
Threestochastic
modelsof floodspresented
in thispaper
arebasedon thestreamflow
PDS. Eachmodeldepends
on certainassumptions
concerning
properties
of
exceedances
of a baselevelx0.The second
andthird modelsrepresent
improvement
vis-h-vis
thefirstone
in the sense
that they are basedon lessrestrictiveassumptions.
The time whenthe largestexceedance
occursis alsostudied,and the distribution of this time is determined.Each exceedance
is characterized,
roughlyspeaking,by its duration and its volume. Here a methodis proposedfor determiningthe
distributionof thisvolume.The distributionfunctionof thelargestvolumein anintervalof time [0, t] is
alsogiven.Goodagreement
between
theoretical
andobserved
distributions
shows
thattheassumptions
concerning
the exceedances
are not unduly restrictive.
INTRODUCTION
The conceptof partial duration serieshas proved to be
usefulin many facetsof flood analysis.Roughly speaking,
such a seriesis obtained by retaining only the hydrograph
peaks that exceed a certain base level. Since the number of
these peaks in an arbitrary but fixed interval of time is a
random
variable,
weobviously
have
to•lealhere
withabivar-
iate point process(the term 'marked' point processis also
used).Mathematicalmethodsfor determining
thedistribution
functionof variousfunctionals
of thisprocess
werepresented
in a previouspaper [Todorovic,1970].Theseresultswerethen
usedto developmodelsof thefloodphenomenon.
Thepresent
paper is concernedwith stochasticmodelsof floods basedon
the concept of partial duration series.Our aim here is to
presentsomenewresultsandto extendand refinetheprevious
ones.We shall also discuss
recentinvestigations
in this area
andtry to elucidate
certainquestions
relatedto thisdevelop-
ment.
To provide a perspectivefor this presentation, we shall
brieflydiscuss
the socialand economicimpactof floodsin our
time.Oddly enough,after somanycenturies
of experience
and
struggleto control thisphenomenon,it seems
that the losses
in
propertyand humanlivesandthe disruptionshaveneverbeen
greater, and chancesare that in the future they will increase
evenmore. How and why?Does it meanthat the rainfall and
runoffrelationships
havechangedor that hydrologicalfactors
responsible
for creationof floodshavemultipliedall of a sud-
den? There is evidence that this new situation is not due to a
drastic shift in natural balance;instead,the escalationof dam-
agesdue to floodsis a resultof interactionsof many factors
recentlyemergingin our society.We shall identify someof
them below.
In manyof the highlyindustrialized
and densely
populated
areasoftheworld a reductionof thenaturalretentionspace
of
the floodplainhastaken place.Because
of this fact the flood
wavehasincreased
in amplitudeand accelerated,
resultingin
more flood damage downstreamthan had ever been antici-
pated. In fact, there are reachesof someEuropeanriversin
whichthelastfewyearshaverepeatedly
broughtfloodswhich
exceeded
the 100-yeardesignflood, on whichthe designs
of
bridgesandflood protectionworkswerebased.In someother
parts of the world the increasein flood damageis due to the
rise in the price of and incomefrom agriculturalproduce
Copyright
¸ 1978by theAmerican
Geophysical
Union.
Paper number 7W0892.
0043-1397/78/027W-0892503.00
(particularly as a result of the green revolution), increased
croppedareas,adoption of a modern multiple croppingpat-
tern, investments
in fertilizersand pesticides,
higherstandard
of living, increasedpopulation, etc.
Flood control measures
are simple.The obviousway is to
build walls alonga streamsofloodswill be confined.Another
way is to widen, deepen,or just cleanup the streamsoit will
carry more water before it overflows. However, too many
wallsalongsidea river may causea changein the flow regime,
resultingin amplificationof the flood waveand more damage
elsewhere.
A remainingmethodreducesthe sizeof floodsthat
willarrive.Thisisdonebytemporary
detention
in upstream
reservoirs,which store high peaks of floods and releasethe
water later at low, controlled, safe rates.
Althoughthe heightof a leveeis usuallyusedastheprinci-
pal measure
of floodprotection,thestructuremaybedamaged
or destroyedby occasionalfloodsof varyingmagnitudes.
The
frequencywith which suchdamagemay occurmustbe taken
into accountin determiningthe sizeor strengthof the struc-
ture, its location, or the feasibilityof building it at all. Infor-
mation concerningflood frequencyis alsonecessary
in insur-
anceand floodzoning,an activitywhichis nowconsidered
on
a broad scale.
The distribution of the number of flood occurrences in a
specific
interval of time hasbeenconsidered
by many authors
[Borgman, 1963; Shaneand Lynn, 1964; Kirby, 1969]. For a
completedescriptionof the flood phenomenon,however,it is
necessary
to considersimultaneously
not onlythe frequencyof
flood events but also the magnitude of the corresponding
hydrographpeaks.This problem was recentlyconsideredin
several papers [Todorovicand Zelenhasic, 1970; Zelenhasic,
1970; Todorovicand Woolhiser,1972].
DEFINITIONS AND PRELIMINARIES
There are essentiallytwo differentapproachesto the prob-
lem of flood analysis.As we havealreadyseen,oneisbasedon
thestreamflow
partial durationseries
(PDS); theotherisbased
on thestreamflowannualfloodseries
(AFS). To clarifycertain
questionsconcerningthe definition of thesetwo series,con-
sidera streamflowhydrograph•'(s), s > 0. Sincethe surface
runoff flowsvary in a random mannerwith time,
•' =/•'(s); s > 0}
is a continuousparameter stochasticprocess.In Figure 1 a
samplefunction of this processis given.
345

2. 346 TODOROVIC:
STOCHASTIC
FLOOD
ANALYSIS
X•
i I
• I
• I
i i
i i
0 • • •n
Fig. 1. Samplefunctionof theprocess
•'(s).
Denote
bYM(t) themaximum
value
of•'inaninterval
of
time [0, t]; i.e.,
M(t)= sup •'(s) (1)
o_<s_<t
.
When the interval [0, t] is a water year, M(t) is calledan
• .
ahnualflood. If we have a recordof •' over an n-yearperiod
andforeac•year
wefindM(t),thesequence
ofnobserved
v•.lues
ofM(t) isthen
anAFS.
Let us selecta certain base level Xo (see Figure 1) and
consider
only
those
flows
thatexceed
this
level.
In this
paper
we will call 'exceedances'
the truncatedpart of the process
•'
above
thebase
level
Xo.
Denote
by
T1, T2, ''', Tk, '''
thetimesof localmaximaof/' whichexceed
thislevel.In other
,words,
r• for •½ry k = 1, 2, ... isthetimeof thekih
hydrograph
peakwhichexceeds
the baselevelXo.Let r/(t)
standfor the number of thesemaxima in [0, t]; i.e.,
r/(t) = supIk; rn < t} (2)
Then
it•'S
clear
that
rt(i)isaninteger-valued
process
such
that
for everyt > 0, r/(t) = 0, 1, "'.
Denote
Xo = 0 X• = •'(r•) - Xo k = l, 2,... (3)
andconsider
thesequence
of a random
number
of random
variablesin [0, t]:
X•, X•., "', X.(tl (4)
Theserie
so•observed
values
of (4) overann-year
period
is
called
th• PDs,correSp6nding
to thehydrograph
•'(s)[see
Langbein•
1949;
Dalrymple,
1960].
Whenan exceedance
isa
multiple-peaked
hYdrograph
[Chow,1964,p. 14],onlythe
largestpeakis takeninto account.An interesting
discussion
concerning
these
twoseries
andtherelationbetween
themhas
beengivenbyChow[1950,1951
].
Inthefollowing,
weshall
designate
byx(t) thelargest
ofall
X• in the interval [0, t]' i.e.,
x(t) = sup X• (5a)
r•<t
or equivalently
x(t) = sup Xn (5b)
o_</e_<n(t)
It isapparent
thatx(t) isastochastic
process
ofnondecreasing
step-sample
functions;
i.e.,x(t•) _<x(t•.)whent• < t•..Investi-
gationof thisprocess
is oneof the mainobjectives
of this
study.
SOMEREFLECTIONON AFS AND PDS
With everysamplefunctionof thestochastic
process
•' ob-
served
duringa wateryearperiodwe canalwaysassociate
a
finiteanduniquely
determined
maximum
value.Thusaperiod
ofn years
givesonlyn observed
valuesof theprocess
M(t). In
other
words,
thecorresponding
AFSconsists
ofonly
nobser-
vations,by meansof whichwe haveto studyproperties
of
M(t). Someof thest•
observed
maximumvaluesmay be so
smallthat theycannotevenqualifyasfloods.
Attempts
toconstruct
a 'feasible'
stochastic
modeloffloods
basedon the streamflow
AFS are hamperedby manydiffi-
culties.One of the main problemshere is 'analyticin-
sufficiency'
andinadequacies
inherentin the useof empirical
procedures.
For instance,
thecommon
approach
to theprob-
lemofdetermining
thedistribution
function
q•(x,t) = P{M(t) < x} (6)
is based on a criterion of 'best curve fit' to the observedvalues
of themaximum
M(t). A listof thedistribution
functions
that
are mostfrequentlyusedfor this purposeincludes
the log
normal,logPearson
type3,two-parameter
gamma,
andGum-
bel extreme value distributions.
The
best
curve
fitprocedure
seems
somewhat
'adhoc'
on
theoretical
andphysical
grounds.
However,
it hasa longtradi-
tion,andit wasused
longbeforesophisticated
statistical
meth-
odswereavailable.In addition,it hasstrongintuitiveappeal,
andit isverysimple
to apply.Ontheotherhand,thecomplex
natureof the streamflow
process
and the lack of adequate
statistical
development
maketheproblemof determining
the
distribution
function•(x, t) theoretically,
in a mathematically
tractable
form,extremely
difficult.
Not all engineers
haveaccepted
thisempirical
procedure
as
a reasonable
workingmethod[e.g.,Dalrymple,1970].How-
ever,
apartfromtheluckofthetheory
tosupport
themethod
proposed,
the following
arethe mostfrequent
objections.
First,themethoduses
onlyoneordinateof thewholesample
functionof thestochastic
process
•'(s),andthustheinforma-

3. TODOROVIC:
STOCHASTIC
FLOOr)
ANALYSIS 347
X•
Xo
M(t) = SupS(s)
O_<s_<t
----------X(t) + Xo= SupZv + Xo
vv _<t ,•
Fig.2. Sample
functions
of •'(s),M(t), andx(t) + x0.
tion providedby other ordinatesof this functionis lost. Sec-
ond, the method does not take into account the seasonal
variations.Third, the value M(t) in somecasescan be quite
small,sothat it doesnot qualifyasa floodat all. Fourth, there
is no way to calculatethe flood volumefrom M(t).
On the other hand, stochasticmodelsof floodsbasedon the
streamflow PDS have a solid theoretical base, and most of the
resultsare in a mathematicallytractable form. For instance,
the distributionfunction of the stochasticprocessX(t),
Ft(x) = Plx(t) _<x} (7)
has, for the most part, a simple analytical expressionand
depends
on two parameters
whichcanbeeasilyevaluated.In
addition, it is the resultof a theoreticalprocedurerather than
an empiricalone.
The number of exceedancesof the level x0 in an interval of
time [0, t] is a random variablerift), which dependson this
level and of course on this interval. The choice of the base level
depends,generallyspeaking,on the particular engineering
problemunderconsideration.
It isalsostipulatedby ourdesire
to make theseexceedances
mutually independentevents. It
seems
intuitively
clearthat if thetruncation
level)Co
is suf-
ficientlyhigh, the assumptionof stochastic
independence
be-
co.
mesphysically
plausible.In practice,the baseis usually
chosenin sucha way that on the averagenot morethan two or
three exceedances
are includedfor each year [seeLangbein,
1949;Dalrymple,1960].Thiscriterionmaylook somewhat
like
a 'rule of thumb" however, it has the most interestingcon-
sequences.
The methodfor flood analysisbasedon the PDS usesmore
information from a samplefunction of the process•' (all ex-
ceedancesof the base level x0 are used). Although it may
happenthat in someyearsthe number of exceedances
is zero
(no floods?)
on theaverage,in ann-yearperiodwemayexpect
3n of them. This method also takes into account seasonal
variationsand allows the possibilityof evaluatingthe time
when the largestexceedances
in [0, t] will occur.
It would be of interestto comparethe stochasticprocess
M(t) with x(t). In Figure 2, samplefunctionsof M(t) and of
x(t) + x0 are presented.It is evidentthat thesetwo processes
(i.e., M(t) and x(t) + x0) differ only in the interval [0, r•].
Outside this interval, i.e., in (r•, co), they (for all practical
purposes)overlap. This of courseis not a precisestatement;
however,we arenot goingto dwellanylongeron thisproblem,
because
certainof its aspects
arewell beyondthe scopeof this
paper.
By virtue of this it followsthat the distributionfunctions
ß (x, t) F•(x - Xo) x _>Xo
will (roughly speaking)overlap if t is sufficientlylarge (here
Ft(x - Xo)represents
thedistributionof theprocess
x(t) + x0).
In other words, if the interval of time [0, t], in which we
observethe flood phenomenon,is sufficientlylarge, the two
approaches
lead to the sameresult. If we bear in mind that
thesetwo distribution functionsare of central importancein
flood analysis,it is certainlyinterestingto know that (roughly
speaking)
ß (x, t) • F•(x - Xo) x _>Xo
if the time t is sufficientlylarge.
For the background material relevant to the secondap-
proach, seethe work by, among others, Todorovicand Rous-
selle [1971], Rousselle[1972], Kartweliswilly[1975], Guptaet
al. [1976], and Karr [1976].
EXCEEDANCES AND RELATED DISTRIBUTIONS
In the previoussectiona number of argumentswere pre-
sented in favor of a stochastic model of floods based on the
streamflow PDS. However, one of the main factors favoring
the PDS wasthat the truncation of thehydrographby the base
level x0 frequently providesus with an insight into the phe-
nomenon which enables us to attribute a certain stochastic
structureto the processes
{r•(t);t > 0} and {X,•}•© which in the
context of the streamflow AFS was neither evident nor rele-
vant. Here we refer to certaintypesof independence
assump-
tions (suchasthe hypothesisthat X•, X•., ... is a sequence
of
mutually independentrandom variablesindependent'ofr•(t)),
amongothers,that now becomephysicallyplausible.
One of the most remarkable resultsthat the hydrograph
truncation produces is the (empirically established)prop-
erty that the number of exceedances,r•(t), is a time-
nonhomogeneous
Poissonprocess;i.e., for every t > 0 and
n-- 0, 1, ...,
P{r•(t) = n} = e-'•(t){A(t)}'*/n! (8)
whereA(t) = E{r•(t)}. This important property of the random
variable r•(t) was noticedby a numberof authors [Borgman,
1963;ShaneandLynn, 1964].However,theyassumed
that rift)
is a time-homogeneous
Poissonprocess,i.e., that A(t) - X. t,
and ignoredthe effectof seasonal
variations.
This partic.ular
propertyof r•(t) may be explainedby the
light densityof eventsand by the Poissonnatureof precipi-
tation events[Todorovicand Yevjevich,1969]. Its theoretical
explanation,however,canbe foundin certainworksby Lead-
better[e.g.,CramerandLeadbetter,1967,p. 256].As hedem-
onstrated,if •'(s) is a Gaussianprocess,
then underrelatively

4. 348 TODOROVIC:
STOCHASTIC
FLOOr>
ANALYSIS
I'0
i t:20days
t:60days
t:I00
days t:140
days t:160
days
0.8
0.6
0.2
K _K K K
0.0 0 I 2_ 0 I 2_• 4 0 I 2_• 4 5'"-
0 I 2_• 4 5 6
0.8
f t:
180
days t:
200
days t:
2_20
days
o.e
0.4
o.o o
I'0
f
0.8
t: 240days t=365days --Observed
0.6
.,•..
_•.•w,w•
.... TheOretica!
(Poissonian)
0.4 K=Number
of exceedance$
0.2 "•"
- K K
O0 - , ---
ß 0 I
I 2 • 4 5 6 7 '"'- 0 I 2 $ 4 5 • 6 ?•"-
Fig.3. Observed
andcorresponding
theoretical
(Poisson)
distributions
ofthenumber
ofexceedances
forintervals
of20,
60,100,140,160,180,
200,220,240,and365days
fortheGreenbrier
RiveratAlderson,
West
Virginia.
mild regularityconditionsthe number of upcrossings
of the
baselevel x0 converges
to a Poissonprocessas x0 -• o•. Al-
thoughtheproofexistsonlyfor a Gaussianprocess,
thereisno
reason to assume that it cannot hold in some other cases.
To provide someevidenceto support assumption(8), we
shall analyze a 72-year record of the Greenbrier River at
Alderson, West Virginia. Flood data in the form of partial
duration series
covertheperiod 1896-1967.The baselevelwas
x0= 17,000
ft3[U.S.Geological
Survey,
1963-1968].
During
this period, 205 exceedances
occurred.In Figure 3, observed
and correspondingtheoretical(Poisson)distributionsof the
number of exceedances
for periodsof 20, 60, 100, 140, 160,
180, 200, 240, and 365 daysare presented.Very good agree-
ment betweentheoreticaland observeddistributionssupports
our hypothesis.
When the truncation level x0 of the process•'(s) is suf-
ficientlyhigh,theannualnumberof exceedances
of thislevelis
relatively small (an ad hoc analysisof severalrivers in the
United States showed that the average annual number of
exceedances
wasof the orderof 3). This indicates
that thetime
lag between two exceedances
must be rather large. Con-
sequently,physicalintuition is not violated by assuming
X•,
X•., ... to be mutually independentrandom variables.One of
themostsurprising
aspects
of thisanalysis
isthefindingthat in
many casesa PDS sequence
{Xn}•
© consists
not only of inde-
pendentbut alsoof identicallydistributedrandom variables,
the commondistributionbeing of the exponentialtype. In
other words,for everyn = 1, 2, ... we have
H(x) = PIXn< x} H(x) = I - e-xx (9)
whereE{X•} = 1/X. Even when{X•}x
© wasnot a sequence
of
identically
distribut6d
random
variables,
each
H•(x) wasstill
of the exponentialtype [Todorovic
andRousselle,
1971;Rous-
selle, 1972].
In Figure4 the observed
andcorresponding
theoreticaldis-
tributionsof the PDS for theGreenbrierRiverarepresented.
Good agreementbetweenthem supportsour assertion.
By
meansof a Fourier seriesfit procedureit was found that for
this river the parameterA(t) has the followingexpression
[Zelenhasic,
1970;Todorovic
and Woolhiser,
1972]:
A(t) • 0.24754- 0.1583t+ 0.5086cos[(2•rt/18) + 0.6841•r]
and
+ 0.0556cos[(2•'t/9) - 0,147&r]
+ 0.0154cOS
[(2•rt/6) + 0.7780,r]
+ 0.0142cos[(2•rt/3) + 0.6742•'] (1o)
8.821 X 10-• (ft3 s-X)-x
The proposedfitting procedureis not necessarily
the best
possible
methodfor evaluatingA(t) if we bear in mind the
rathercumbersome
analyticalexpression
for thisfunction.It is
quite possiblethat more elegantand simpleanalyticalex-
pressions
canbefound.In Figure5 we presented
graphically
theseasonal
occurrence
of exceedances
andtheparameter
A(t)
fortheGreenbrier
River(fora 1-year
period).
Thetopgraph
is
very instructive.It showsthat most of the exceedances
occur
duringthe winter season.
This explainsthe seemingly
con-
tradictory
findingthatthePDSsequence
{Xrt}l
©corresponding
to thisparticularriver consists
of identicallydistributedran-
dom variables,
whenour experience
showsthat the greatest
floods
occur
duringthewinterperiod.However,
thefactthat
sup X•< sup X•
l</•--<n I 1</• <n•
if n• < n• and the top graphin Figure5 may providean
explanation
for thiscontradiction.
In otherwords,thelargest
floods occur during the winter seasonnot becausethe ex-
ceedances
corresponding
to thisseason
arelargerbut because
theyare more frequent.
It shouldbeemphasized
that therandomvariables
r/(t), r•,
X•, and of courseX(t) dependon the truncationlevelXo.
However,
throughout
thispaper,Xowillbefixed,sothatthere
isno needfor Xoto figureexplicitly
in all thesefunctions.
DISTRIBUTION
FUNCTION
OFX (t)
As wehaveseen,
oneof themainproblems
in constructing
stochastic models of floods based on the streamflow AFS is
that,generally
speaking,
weknowverylittleabouttherandom
process•'(s) and its stochasticstructure. However, whatever
the propertiesof this processmay be, we have established
(roughlyspeaking)
thatthenumberofexceedances
of a certain
baselevelx0in an arbitraryintervalof time [0, t] is a time-

5. TODOROV!C:
STOCHASTIC
FLOOD
ANALYSIS 349
0.8
0.6
0.4
0.2
0.0
9000
O. 1533
........... Summer and Fall
-------- Spring
• Year
..... Winter
18000 27000 3,6000 45000 54000 6:5000
X (cfs)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
H(x)
Observed
.... Theoretical (Exponential)
0.2
x (cfs)
0.0
9000 18000 27000 36000 45000 54000 63000
Fig.4. Greenbrier
River
atAlderson,
West
Virginia.
(Top)Observed
distribution
functions
ofexceedances
forfour
different
periods.
(Bottom)
Theoretical
andobserved
(1.year
period)
distribution
function
ofexceedances.
nonhomogeneous
Poisson
process
if x0issufficiently
large.
In
addition,
thecorresponding
PDSsequence
{X,,t•
ø•consists
of
mutually
independent
random
variables
withthecommon
dis-
tributionfunction
(9). Below
weshallshow
howthese
propers-
tiescanbeused
to develop
stochastic
models
of floods
based
on the streamflowPDS. In what follows, we shall first deter-
minethedistributionfunctionFt(x) of theprocess
X(t).
Themostgeneral
formof thisdistribution
wasdetermined
in a previous
paper[Todorovic,
1970]
in terms
of thejoint
distributionof r/(t) and {X,}•ø•;i.e.,
Ft(x) = Plr/(t) = 0}
+••1P
{sup
x•
•<
xln(t)=
n}P{rl(t)=n}
= l<k<n
(11)
In ourcase
theprobability
P{r/(t)= n}isgivenby(8) forevery
n = 0, 1, ß... To determine
effectively
thedistribution
function
(11), we haveto evaluatetheconditionalprobability
P{
sup
Xn_<x[rl(t)=n}
(12)
•</•<n
for everyn = 1, 2, ....
Thesimplest
waytodothisistoassume
(asweshall
do)that
for everyn = 1, 2, ... the sequence
{Xn}•"is independent
of r, andr,+• (13)
Thenwhen.(9)is'keptin mind,it followsthat
P{
sup
Xn<xln(t)=n}=P{
sup
Xn<x}
l<k<n l</•<n
=•-IPtx•
_<
x}
=(•- e-X•)
• (•4)
Byvirtueof thisresult
and(8) wehavethatin ourcase,
Ft(x)= e-A(t>
1+ • (1-

6. 350 TODOROVlC:
STOCHASTIC
FLOOD
ANALYSIS
Winter
ø"øt J
I
t
:5.00t.A..(t)
=,/X(s)
ds
2.00 -
Observed
1.00 Fitted
,;,, ,,,. t
0.00 m'l•-I I ' ' • • ] • • • ] I •
Fig. 5. Greenbrier
Riverat Alderson,
WestVirginia.(Top) Sea-
sonal occurrenceof exceedances.
(Bottom) Observedfunction A(t)
and fitting function.
Hence
where
Ft(x) = exp[-A(t)e-xx] (15)
A(t) = Elr/(t)} E{X,•} = 1/X
Notice that the distributionfunctionFt(x) hasa discontinuity
of thefirstkind at thepointx = 0 equalto Ft(O)= e-'x(t),
and
thusthecorresponding
density
functiondoesnot exist.
It is apparentthat the parameters
A(t) and X completely
determinethe distributionfunction(15). Valuesof thesetwo
parameters
for theGreenbrier
Riveraregivenby(10).Com-
parison
of thetheoretical
andobserved
distribution
functions
for t = 140,t = 180,and t = 365 daysis givenin Figure6.
Good agreement
between
the theoretical
and observed
re-
sultsshows
that theassumptions
concerning
the distributions
of the randomvariables
r/(t) and {X•}•© and their mutual rela-
tionarenotundulyrestrictive.
It isfortunate
thatwewereable
to verifymostofthem(atleastpartially)byobservation.
Some
of themare difficultto verify,but theyarenot in violationof
ourphysical
intuition.The onlyexception
ishypothesis
(13).
The nextsectionisconcerned
with thisparticularassumption,
whichapparentlyrepresents
a restriction
on the model.We
shall see that the values of the correlation coefficients between
random variablesr• and X• and alsobetweenr2 andX2 for the
Greenbrier River are small. This information givesuscertain
comfortbut not theproofthat theassumption
holds.Finally,
whenhypothesis
(13) doesnothold(or cannotbeassumed
to
hold) anymore,newmethods
arerequiredfor evaluation
of
the conditionaldistributionfunction (12). In the sequelthis
problem
isalsoconsidered,
andasa result,
a newformofthe
distributionfunction Ft(x) is proposed.
•Ft(x)
/ t=14odays •,•'-
o.,I-
•' / - Theoretical
•}lo•I I I I I
0 20000 40000 60000
Fig.6. Theoretical
andobserved
distribution
functions
ofthelarg-
estexceedance
for intervalsof 140, 180,and 365 daysfor the Green-
brier River at Alderson, West Virginia.
REMARKSON THE INDEPENDENCEASSUMPTION
Mostoftheassumptions
in thelasttwosections
concerning
the PDS eitherwerenot in violationof our physicalintuition
orwereabletobesupported
(reasonably
well)byobservation.
In thisrespect,
assumption
(13), whichis oneof the most
important
in thispaper,
isanexception.
No evidence
of any
kindwaspresented
to support
thishypothesis.
In whatfol-
lows,weshallattemptto elucidate
certain
questions
related
to
thisproblem.
Because
ofitsextraordinary
importance
wehave
decided
to devotea separate
section
to itsanalysis.
To this end, let us remind ourselves
that the following
relationshold for everyn = 0, 1, 2, ...:
{r/(t)= n} = {r,•_<t _<r,•+•}= {r,• _<t}- {r,•+•< t}
Byvirtueof thiswecanwrite(9) asfollows:
P{
sup
X•_<x,r,•_<t<
r,•+•}
l_</•_<n
Plr/(t) = n}
l_<g_<n l_<g_<n
Plr/(t) = n}
From thisand assumption
(13), (14) follows.
Theassumption
thattherandomvector(X•, X•, ..., X,•)is
independent
of therandomvariable
r,•+•for every
n = 1,2,
ß.. seemsreasonable.On the other hand, after somereflection
on the seasonal
variationsthe independence
assumption
be-
tween
(X•, X•, .-., X,•)andr,•seems
less
acceptable.
However,
at least for the Greenbrier River the observeddata support
assumption
(13).In Table1wepresent
theobserved
frequency
distributionof therandomvariable(X•, r•). A relativelysmall
correlation coefficient between X• and r•,
rx•,r•= 0.088
seems
tosupport
theindependence
assumption.
Wefound
that
thecorrelationcoefficient
betweenX• andr• isalsoverysmall:
rx,,r, = 0.120
It followsfrom the presentanalysisand someprevious
analysis
thatassumption
(13)(atleast
fortheGreenbrier
River
and severalother rivers)is not undulyrestrictive.
This, of
course,
cannotbe expected
to holdin all cases
(i.e.,for all

7. TODOROVIC:
STOCHASTIC
FLOOD
ANALYSIS 351
PDS,
10a
fta/s
TABLE 1. ObservedBivariate Distribution of (X•, r•)
Time, days
1- 31- 61- 90- 120- 150- 180- 210- 240- 270-
30 60 90 120 150 180 210 240 270 365 fx,
0-5 1 4 7 7 2 4 1 1 I 0 28
5-10 1 1 2 5 3 1 0 0 0 0 13
10-15 2 2 0 0 1 3 1 0 0 0 9
15-20 1 1 2 1 1 1 0 1 0 0 8
20-25 0 0 1 0 1 0 0 0 0 0 2
25-30 0 0 0 2 0 0 0 0 1 0 3
30-35 0 0 0 0 2 0 0 0 0 0 2
35-40 0 1 0 0 0 1 0 0 0 0 2
fr, 5 9 12 15 10 10 2 2 2 0 67
rivers).Thusit isdesirableto extendour resultsto cases
where
condition(13) doesnot hold.To thisend,weshallreplace(13)
with the assumptionthat
{X•}•n is independentof (Xn+•, r,•+•) (16)
for everyn - 1, 2, .... This hypothesisseemsintuitively less
restrictivethan assumption(13), because{X•}•• is a sequence
of independent
random
variables
andit seems
reasonable
to
suppose
that {X/•}I
rtis independentof •'n+lfor everyn = 1, 2,
ß.. Oneshouldpointout that theindependence
{X•}i" of X•+•
and r•+l doesnot necessarily
imply the independence
{X•}inof
(X,+i, r,+l). However,assumption
(16) impliesthe independ-
ence{X/•}lrtof •'n+l (and of courseof Xr•+l).
Now that a newandlessrestrictiveregularitycondition(16)
hasbeenidentified,it isreasonable
to attemptto determinethe
form of the distribution function Ft(x) which correspondsto
this new assumption.As was true in the previouscase,the
main problemhereis to computethe conditionaldistribution
(12). The derivationof thisfunction,basedon condition(16),
isgivenin AppendixA. Then by virtue of (11), after a simple
transformation, we obtain
f0
Ft(x) = 1- exp[-A(u)e-XX]K[(x,•), u]dA(u) (17)
wherefor everyn = 1, 2, ...,
K([0, x], u) = PIX,, < x Ir,, = u} (18)
The distribution function (15) is a specialcaseof (17). To
seethis is enough to supposethat X,• and rn are mutually
independent
randomvariables.In thiscase(oneshouldpoint
out that in the courseof the derivation of distribution (17) we
assumed
that (8) and (9) hold), it followsfrom (18) that
K[(x, •), u] = e-xx
Thus after a simpleintegration,(17) becomes
(15) (we assume
of coursethat 3_(0) = 0). Consider
P{X• _<x} = P{X• _<x lr• = u}dP{r• _<u}
By virtueof (8), (9), and(18) thisbecomes
e
-xx=
1
- fo
•ø
K([0, x], u)e-At")dA(u)
LOCATION OF THE SUPREMUM
In this sectionwe are concernedwith the location problem
of the maximum x(t). In this context, two'aspects of this
problemareof interest.
Thefirstisthequestion
of order(i.e.,
of whichof the exceedances
hasthelargestlocalmaxima).The
second
question
isitslocation(i.e., thelocationof x(t) onthe
time scale). In the sequel,one of our main concernsis the
randomvariableN(t), t > O,definedsothat N(t) isthe random
index n such that
T
L.(B) =2
I'
I
,I
Xz X$
I I
I
Fig.7. Graphical
presentation
of thebivariate
pointprocess
L.
• $

8. 352 TODOROVIC: STOCHASTIC FLOOD ANALYSIS
- Xit = sup Xv
Tv--
• t
In what follows, we shall determinethe distribution function
and mathematical
expectation
of N(t). We shalldo thisfirst,
assumingthat condition (13) holdsß
For everyn = 1, 2, ..., denote
pit(t)= PIN(t)= n} po(t)= e-n(t) (19)
If {Xit}x
©isan arbitrarysequence
of randomvariables
andthe
distributionof r/(t) is givenby (8), we have
fI ))
pit(t)
=e
-*(t) Hj(x)dH,•(x
/A(t)}n(20)
Finally, whencondition(13) isreplacedwith thelessrestrictive
condition(16), onecanshowthat in thiscase(seeAppendixB)
we have that
po(t) = e-*(t)
and for n = 1, 2, ...,
pit(t)
= K([0,
x],
s)e
-*(s)
• 1- A(s)
Hv(x)
V--It V
ß
[A(s)]•-•
dA(s)
•hHj(x)
dHit(x)
(27)
(v- 1)*
ß j4n
If weassume
thatXit andrit areindependent
randomvariables
for everyn = 1, 2, ..., (27) becomes
(20).
whereH•(x) = P{X• < x} [seeTodorovic
andShen,1976].If H
= H• = H2 .... ,
pn(t)
=e-n(t)
n•{A(t)}n
__ k:•i (21)
By virtue of (21) we have
ß E{N(t)} = [A(t) + 1 - e-•(t']/2 (22)
Considernow an intervalof time [0, t]; amongrandom
timesr•, r2, ßßßobservedin thisinterval, oneisassociated
with
thelargestobservation
Xit.The instantr, whichcorresponds
to themaximum
observation
in [0,t], will bedenoted
by T(t).
In other words,T(t) is the time in [0, t] whenthe extremeis
achieved.By virtue of thisdefinition,it followsthat
T(t) = 'rN(t) 0 --<T(t) _<t (23)
It isapparent
thatthe{T(s)•s G [0,t]} isa stochastic
process
of nondecreasing
samplefunctions,sothat
PIT(t) = 0} = P/r/(t) = 0}
(whichmeans
thatthecorresponding
distribution
functionhas
a discontinuity
at the point s = 0). It canbe shownthat the
most generalform of its distributionfunctionis (on the as-
sumption,of course,that condition(13) holds)
P{T(t)_<u}
=P{r/(t)=
0}+ • • • P{r/(u)
n=l k=n v=l
= v, tl(t) - tl(u) = k - v}P{Nn= n} (24)
[seeTodorovic
andShen,1976].
Whenr/(t)isaPoisson
process,
(24) then becomes
PIT(t) -< u} = e-n(t)+ [A(u)/A(t)][1 - e-*(t)] (25)
[see Todorovicand Woolhiser,1972].It is remarkablethat the
distributionfunctionis completelydeterminedwith a single
parameterA(t) (which for the GreenbrierRiver is givenby
Thejointdistribution
function
oftherandom
variables
T(t)
and x(t) was determinedby Gupta et al. [1976] under the
followingconditions:
{Xit}•
©isa sequence
of independent
iden-
ticallydistributed
randomvariables
withH(x) = P{XIt< x},
and assumption(10) holds;then
P{x(t) < x, T(t) < u} = P{tl(t) = 0}
+• {H(x)_••_}
P{r•
<u,
n(t)
=n} (26)
= v=• n! --
REMARKS ON THE FLOOD VOLUMES
Inmany
flood
studies,
one
isconcerned
notonly
with
peak
flowsbut alsowiththeproblemof durationoffloodsandwith
flood volumes.For instance,one may wishto know for what
periodof timea highway
adjacent
to a stream
islikelyto be
flooded.In the problemof interiordrainageof a leveedarea
we need information on those flood volumes in which the main
riveristoo highto permitgra.
vity drainage.In thissection
we
shall
briefly
examine
several
aspects
ofthis
particular
problem.
To this end, somenew notation is required.
Denote
by a• thekth upcrossing
oi
• thelevelx0by the
process
•'(s)(seeFigure1). It isapparent
thata• isa random
variable
foreveryk= 1,2,... andthat0_<
a• < a•< .... We'
alsowrite T• for the duration of the kth exceedance.
Then the
volume V• of this exceedanceis
f•,•,
+
Tt
V•= ' •'(s)
ds- x0'T• k = 1,2, ... (28)
Of particularinterestis thelargestvolumeV(t) in an interval
of time [0, t]. To givea precise
definitionof the process
V(t),
denoteby No(t) the numberof upcrossings
of the levelx0by
the process•'(s) in [0, t]; i.e.,
No(t) = sup{k; a• _<t} (29)
It is apparentthat for everyt > 0,
•(t) < No(t) < •(t) + 1
Then V(t) is definedas (seeFigure 1)
V(t) = max sup V•, •'(s)ds- (t- a•v•t•)'Xo
•'ot
k_<
t N(t)
If •'(s) < x0for all s G (a•v(t),
t], (30) becomes
(30)
V(t) = sup Vn (31)
ot•_•t
To clarifythedefinitionof theprocess
V(t), consider
Figure
1 Thenaccording
to (30) forthisparticularcase,V(t) becomes
V(t) = max V•, ..., Vit_•, •'(s)ds- (t- ait)'Xo (32)
If thelastexceedance
in [0,t] issuchthat OtN(t)
-t-rN(t)--•t, then
(30) holds.
For everyk = 1, 2, ... we shalldefineV• as
V• = X•. Tn/2 (33)
Then,assuming
X• and T• to be mutuallyindependent,
we
have

9. TODOROVIC:
STOCHASTIC
FLOOD
ANALYSIS 353
PIV• < u} = P T• < 2 dHn(x) (34)
whereH•(x) = PIX• < x}.
Denote
q't(v) = P{V(t) < v} (35)
Thenby virtueof (30) wehave(V0 = 0)
q•t(v)= P •'(s)ds_<v + Xot,N(t) = 0
+ • P sup
V•
<v, •'(s)
ds
= l_<k_<n ot
n
-(t
-c.)x0,
N(t)
=n} (36)
The rather complex structure of distribution function (36)
makesitspracticalapplicationratherdifficult.To alleviatethis
difficulty,we shallreplace(36) with anotherdistributionfunc-
tion ß0(x, t), which has a much simplerstructureand repre-
sentsa good approximation of q't(x). The general form of
ß0(x, t) is given by (for more information seethe work by
Todorovic[1971])
q'o(V,t) = PIN(t) = 0}
+,,•
P{
sup
V•<v,N(t)=n}
= l<k_<n
(37)
We shallassumethat a regularityconditionsimilarto (13)
holds;i.e., we shallsuppose
that for everyn -- 1, 2, -..,
{V•}•n is independentof a, and a,+• (38)
We shallalsoassumethat {(X,, T,)}•© is a sequence
of mu-
tually independentrandom vectors,suchthat
H(x) = P{X,• < x} G(s) = PIT, < s} (39)
for n = 1, 2, .... Then it caneasilybe verifiedthat
ßo(•, t) = PIN(t) = 0}
+ •:• G 2-•-dH(x)
PIN(t)=
n} (40)
where it can safely be assumedthat the countingrandom
variableN(t) is a time-homogeneous
Poisson
process.
We will
not dwellon this problemany longerhere.However,it will be
the subjectof a forthcomingpaperin whichit will be consid-
eredin greatdetailand illustratedby several
examples.
SOME GENERALIZATIONS
In this sectionwe shall outline anothergeneralizationof a
resultcontained
in a previoussection.
Here we referto (15),
which is basedon assumptions
(8), (9), and (13). In what
follows,weshallgeneralize
result(15) by replacing
therather
restrictivehypotheses
(9) and (13) (however,we shallassume
that (8) holds) with weaker ones(i.e., with lessrestrictive
ones).This generalizationreflectsthe fact that the old result,
givenby (15), is a particular caseof a new one.
Below we supposethat the following conditionsare satis-
fied.
1. The randomvariablesX•, X2, "' are conditionallyin-
dependentin relation to {r.}• ©.
2. For everyn = 1, 2, ... and x > 0,
PIX, < xl(r,• ©}= PIX,< xlr,} (41)
Thefirst
condition
means
thatforevery
finite
subsequence
X,2,... , X,• of {X,,}•
©the following
holds:
P{X,,C B•, ... , X,• C ©}
k
= H P{X,,
GBjI(r,),"} (42)
j=l
[see
Chung,1974,p. 306],whereBj isa Borelsubset
ofthereal
line.
Denote
(0, x]) = ?tX _<xinu}
Then onecanshow[seeChung,1974,p. 307]that conditions1
and 2 aboveare equivalentto the following:
P{X• < x[(r.)• ©,(X•)•,,.
©} = K•(r., (0,x]) (43)
Intuitively,thisresultcanbeinterpretedasfollows.Giventime
r., thecorresponding
X. isindependent
of otherX• andtimes
# n.
Before we proceedwith further development,somenew
definitions
arerequired.A bivariatepointprocess
[seeBartlett,
1963;CoxandLewis,1972]frequently
calleda 'markedpoint
process'is obtained if a real-valued random variable X. is
associated
witheveryr. (seeFigure7). It iscompletely
charac-
terizedby the nonnegative
integer-valued
randommeasureL,
definedon the classof all Borel subsets
of R+= (the non-
negativepart of the real Euclideantwo-dimensional
space)
defined as
L( )= •] e•T,,x,,() (44)
tl=l
wheree<•,•istheDiracmeasure
onR+•, i.e.,forevery
G C R+•.
e(•,•,(G) = 1 (s, x) G G
= 0 (s,x) O
Thusby virtueof (44), for everyBorelsubset
B C R+•, L(B)
denotes
thenumberof points(r,, X,) in R: belonging
to the
setB (seeFigure7). Thisobviously
impliesthatL(B) = O,1,2,
ß.. for every Borel setB C R+:.
It is apparentthat for everyt • 0,
L[(0, t]x(0, •)] = ,(t)
where•(t) is definedby (2).
Onecanshowthatthefollowing
fundamental
property
ofL
holds.
Suppose
that• isatimedependent
Poisson
process
with
parameter
A; thenunderconditions
1and2thebivariate
point
process
L isalsoa Poisson
process
with theparameter
tg,(u,
(0,
x])A(du) (45)
In other words,
t)x(O,xl] = k} = exp - K•(u,(0,x])A(du)
ß K•(u, (0, x])A(du) •)-• (46)
(seeAppendix C).
Let us now considerthe process
x(t); since

10. 354 TODOROVIC:
STOCHASTIC
FLOOD
ANALYSIS
{x(t) < x} = {L[(0, t]x(x, m)] = 0} (47)
it followsfrom (46) that [seeKarr, 1976]
Ft(x) = PILl(0, t]x(x, •o)] = 0}
:exp{-fK[u,(x,o)lA(clu)}
(48)
The distribution function (15) is evidently a specialcaseof
(48), obtainedby putting
K[u, (x, o)1 = - H(x)
By allowingthe distributionof anexceedance
to dependonthe
timeatwhichit occurs,
model
(48){scapable
ofrepresenting
the seasonal variations.
mathematically
tractablethanmodel(15). Anothergeneral-
izationispresented
inthehstsection.
Hereassumption
(13)is
replaced
with thehypothesis
that 'giventimer,,' thevalueof
X, isindependent
(stochastic.
ally)ofallother
Xj andrj,j • n.
Underthishypothesis
andthe assumption
that r/(t) is a time
dependent
Poissonprocess,
it wasshownthat the numberof
points(X,•, r,) in any Borel subsetB C R+•' is a bivariate
Poisson
process,
irrespective
of the distribution
of X,•. By
virtue
ofthisresult
anew
model
offloods
(48)isobtained.
Finally,wealsoconsidered
theproblemofdurationofflood
and flood volumes. The distribution function of the volume
associated
with the kth exceedance
is givenby (34). Under
assumption
(38) thedistribution
of thelargest
volumein [0,t]
is thengivenby (40).
GENERAL CONCLUSIONS
The subjectmatter of this paper is stochasticmodelingof
the flood phenomenon.The formulation of a mathematical
modelin generalis alwayga compromise
betweenmathemati-
cal intractabilityand inadequatedescription
of the phenome-
non beingmodeled.Thereusuallyis a choiceof mathematical
models between these two extremes. For this reason, one usu-
ally talks about 'a' mathematicalmodel for the phenomenon
and not 'the' mathematical model.
Among the variousapproaches
to theproblemof stochastic
modelingof floods,one can distinguishtwo basicallydifferent
methods.One of the methodsisbasedon the streamflowAFS.
Here the main problemhasbeento determine(in mathemati-
cally tractableform) the distributionfunction of the largest
flow (i.e., the maximum value of the streamflowhydrograph)
in a certain period of time. Exceptin one or two particular
cases,
no solutionof [hisproblemfoundso far hasbeen
mathematicallytractableenoughto be usedin applications.
The other method is based on the streamflow PDS. The
principalproblemin thiscaseisalsoto determinethedistribu-
tion functionof thelargestlocalmaximumof •'(s)that exceeds
a certainbaselevelx0in a givenperiodof time. Bothextremes
(i.e., onecorresponding
to the AFS and anothercorrespond-
ing to the PDS) are stochasticprocesses
of nondecreasing
samplefunctions.For a comparisonof thesetwo processes,
see
Figure2. It turnsout that (exceptfor a certainrelativelyshort
period of time in the beginningof the observations)thesetwo
randomprocesses
overlap.
One of the main advantages
of the streamflowPDS method
is the fact that the truncation of the streamflowhydrograph
makessomeassumptions
(concerningthe stochastic
structure
of the process•'(s)) physicallyplausible.Here we refer to
certaintypesof independence
assumptions
concerning
the se-
quencesof random variables{Xr•}l
© and {Tn}l
© and alsoto the
assumption
concerningthe Poissonnatureof the numberof
upcrosgings
of the truncationlevel.Theseassumptions
in the
context of the total duration series are neither relevant nor
evident.The last propertyhasfor sometime beenknown to
hold for certainclasses
of stochastic
processes.
The previous two assumptionsare used to determinethe
distributionof the largestlocalmaximumof •'(s)that exceeds
thetruncationlevelx0andalsoto investigate
certainproblems
concerningthe location of this maximum on the time scale
(equations(15), (20), (25), and (26)).
Substitutingfor the independence
assumption(13) the less
restrictiveassumption
(16), we obtaindistribution(17), which
representsa new model of floods based on the streamflow
PDS. Although it is more general,model(17) is clearlyless
APPENDIX
A: DERIVATION
OFDISTRIBUTION
(17)
Bearing
in mindnotation(18), wehave
PIX, < x, r, _<t} = P{X, _<
x I r, = u}dPlr,•< u}
= K([0, x], u) dPlr,• < u}
By virtueof this,(9), andcondition(16),
P{
sup
X•_<x,r,_<
t}
l_<•_<n
= P{X1 •- x, '' ', X•_ 1• x}P{X, _<x, r,• <_t}
fo
t
= (1 - e-XX)
'•-1 K([0,x], u)dPlr,•< u}
On the otherhand,it is apparentthat
P{
sup
X•J,
Tn+l•
t}
=(1-e-X•)"P{r,+l•t
15•n
Sincefor everyv = 1, 2, ...,
Plr•
• t}= • Plr•
• t}=e
-a(•' {A(u)}a
•=• •=• kl
dPlr• • t} = e-a{a'lA(u)}•-•dA(u)/(v -
it followsby virtueof (49) and(50) that
lS•Sn
= e_a<•)
{A(u)(l
- e-X')"
_ -'}.K([0,
x],
u)
- e
-a<• )}dA(u)
[Pin(t)=
n}]
-•
n•
From thisand(11), (17) follows.
APPENDIX
B' DERIVATION
OFDISTRIBUTION
(27)
In the derivationof distribution(27),
PIN(t)= n}= • PIN(t)= n,,?(t)=v}
On the other hand,
(49)
(5O)
(51)
(52)

11. Tor>oRovlc:
STOCHASTIC
FLOOr>
ANALYSIS 355
=fohH•(x)dH,•(x)
K([0,
xl,
s)
dP{r,•
_<
s}
_ H•(x)
dH,•(x)P{r•+•
_<
t}
J•n J•n
= K([0, x], s) dP{r• _<s} - P{r•+• _<s}H•(x) H•(x) dH,•(x)
J•n
= K([0,
x],
s)e
-'•1
- A(s)
H,(x)
•••i•,•,,
p
when(51) and(52) arekeptin mind.Thisprovestheassertion.
APPENDIX
C' DERIVATION
OFDISTRIBUTION
(46)
When (48) is kept in mind, it is enoughto prove (47)
for k =0:
PIL[(0, tlx(0, xll = 0}
= • PIL[(0,
t]x(0,
xl]=0, r•(t)
=n} (53)
n=0
On the other hand, we have
P{L[(0, t]x(0, x]] = 0, n(t) = n}
=f{ PIL[(0,
tlx(0,
xll
=01(•)1
•}
dP
r/(t)=n}
=• PIX1
>x,...,
Xn
>xl(rs)l
©}dP
rt(t)=n}
=f•n
(t)=n}
=f•, hK•[r,,
(x,
c•)]
dP
(t)=n} j=l
The last two lines follow from conditions 1 and 2. On the other
hand,a fundamental
propertyof a Poisson
process
states
that
givenr/(t) = n, then timesr•, r=, "', r,•belonging
to (0, t] are
independent
identicallydistributed
on (0, t] according
to the
probabilitymeasure
A(du)/A(t). Thus
f. flKx[rj,
(x,
c•)]
dP
(t]=n} j= 1
= Kx[us,
(xoo)]A(dus)/A(t)
e-a(t•{A(t)}'•
' n!
j=l
t' ,.t hn
=e-'•<t'
tjo'K•[u,
,x,
oo)]A,du)t,n,,
-•
Hencebyvirtueof (53) wehave
whichprovesthe assertion.
NOTATION
•'(s) streamflowhydrograph.
M(t) maximum of •'(s)in [0, t].
AFS annual flood series.
PDS partial duration series.
r• time of the maximum of the kth exceedance.
r/(t) number of r• in [0, t].
X• = t'(r•) - Xo.
X(t) = maxo_<•_<,t)X•.
ßt(x) distribution of M(t).
Ft(x) distribution of X(t).
A(t) = Eln(t)}.
H,•(x) = P{X,• < x}.
•[(0, x), s] = ?lx• <-xl• = s}.
N(t) randomindexn suchthat X,• = sup{X•; r• < t}.
p,•(t) = PIN(t)= n}.
T(t) time in [0, t] when X(t) is achieved.
V• volume of kth exceedance.
No(t) numberof upcrossings
of the levelXo.
V(t) = max,-•_<
t V•.
ßt(x) = PIV(t) < t}.
Acknowledgments.This researchwas supportedin part by the
National ResearchCouncil of Canada, grant CNR-A-8959. The sup-
port is gratefullyacknowledged.
REFERFNCES
Bartlett, M. S., The spectralanalysisof point processes,
J. Roy.
Statist. Soc., Ser. B, 25, 264-269, 1963.
Borgman,L. E., Risk criteria,J. Waterways
HarborsDie. Amer.Soc.
Civil Eng., 89(WW4), 1-35, 1963.

12. 356 TODOROVIC: STOCHASTIC FLOOD ANALYSIS
Chow, V. T., Discussion
of 'Annual floodsand the partial duration
floodseries'
byW. B, Langbein,EosTrans.AGU,31,939-941, 1950.
Chow,V. T., A generalformulafor hydrologic
frequency
analysis,
Eos
Trans. AGU, 32, 231-237, 1951.
Chow, V. T., Handbookof AppliedHydrology,McGraw-Hill, New
York, 1964.
Chung,K. L., A Course
in ProbabilityTheory,Academic,New York,
1974.
Cox, D. R., and P. A. W. Lewis,Multivariatepoint processes,
in
Proceedings
of the6th BerkeleySymposium,
pp. 401-448, 1972.
Cramer,H., andM. R. Leadbetter,
Stationary
andRelatedStochastic
Processes,
John Wiley, New York, 1967.
Dalrymple, T., Flood-frequency
analyses,U.S. Geol.Surv. Water
SupplyPap., 1543-A, 80 pp., 1960.
Dalrymple,T., Commenton 'Uniform flood-frequency
estimating
methods
for federalagencies'
by ManuelA. Benson,WaterResour.
Res., 6(3), 998, 1970.
Gupta,V. K., L. Duckstein,
andR. W. Peebles,
On thejoint distribu-
tion of the largestflood and its time of occurrence,Water Resour.
Res., 12(2), 295-304, 1976.
Karr, A., Two extreme
valueprocesses
arisingin hydrology,
J. Appl.
Probab., 13, 190-194, 1976.
Kartweliswilly,N. A., Stochastic
Hydrology(in Russian),Gidrome-
teoizdat, Moscow, 1975.
Kirby, W., On therandomoccurrences
ofmajorfloods,WaterResour.
Res., 5(4), 778-784, 1969.
Langbein,W. B., Annual floodsand the partial durationseries,
Eos
Trans. AGU, 30, 879-881, 1949.
Rousselle,
J., On someproblemsof flood analysis,Ph.D. thesis,226
pp., Colo. State Univ., Fort Collins, 1972.
Shane,R., and W. Lynn, Mhthematical model for flood risk evalua-
tion,J. Hydraul.Div. Amer.Soc.CivilEng.,90(HY6), 1-20, 1964.
Todorovic, P., On someproblemsinvolvingrandomnumberof ran-
domvariables,
Ann.Math. Statist.,41(3), 1059-1063,1970.
Todorovic,P.,On extremeproblems
in hydrology,paperpresented
at
Joint StatisticsMeeting, Amer. Statist. Ass. and Inst. of Math.
Statist., Colo. State Univ., Fort Collins, 1971.
Todorovic, P., and J. Rousselle,Some problemsof flood analysis,
Water Resour.Res., 7(5), 1144-1150, 1971.
Todorovic,P., andH. W. Shen,Someremarksonthestatistical
theory
of extremevalues,in Stochastic
Approaches
to WaterResources,
vol.
2, editedby H. W. Shen,UniversityofColoradoPress,
Fort Collins,
Colo., 1976.
Todorovic, P., and D. A. Woolhiser, On the time when the extreme
floodoccurs,WaterResour.Res.,8(6), 1433-1438,1972.
Todorovic,P., and V. Y evjevich,Stochasticprocess
of precipitation,
Hydrol. Pap. 35, p. 61, Colo. StateUniv., Fort Collins, 1969.
Todorovic,P., and E. Zelenhasic,
A stochastic
modelfor floodanaly-
sis,Water Resour..
R,es.,6(6), 1641-1648, 1970.
U.S. GeologicalSurvey,Magnitudeand frequencyof floodsin the
UnitedStates,U.S. Geol.Surv. WaterSupplyPap., 1963-1968.
Zelenhasic,E., Theoreticalprobability distributionfor flood peaks,
Hydrol. Pap. 42, Colo. StateUniv., Fort Collins, 1970.
(ReceivedFebruary 14, 1977;
revisedSeptember13, 1977;
acceptedSeptember15, 1977.)