VOL.
4, NO. $
WATER
RESOURCES
RESEARCH
OCTOBER
1968
Noah,Ioseph,and Operational
Hydrology
BENOIT
B.
JAMES
International
I•fANDELBROT
R. WALLIS
Business Machines Research Center
Yorktown Heights, New York 10598
Dedicated
to Harold
Edwin
Hurst
ß . . were all the fountains of the great deep broken up, and the windows of heaven
were opened. And the rain was upon the earth forty days and forty nights. Genesis, 6,
11-12
ß .. there came seven years of great plenty throughout the land of Egypt. And there
shall arise after them seven years of famine ... Genesis,41, 29-30
Abstract. By 'Noah Effect' we designate the observation that extreme precipitation can be
very extreme indeed, and by 'Joseph Effect' the finding that a long period of unusual (high
or low) precipitation can be extremely long. Current models of statistical hydrology cannot
account for either effect and must be superseded.As a replacement, 'self-similar' models appear very promising. They account particularly well for the remarkable empirical observations of Harold Edwin Hurst. The present paper introduces and summarizes a series of
investigations on self-similar operational hydrology. (Key words: Statistics; synthesis; time
series)
INTRODUCTION
be of interest to note that they are instances
By 'NoahEffect'we designate
the fact that of a broadfamilyof 'self-similar
models.'
The
extremeprecipitation
canbe very extremein- conceptof 'self-similarity'
originatedin the
deed,and by 'JosephEffect'the fact that a theoryof turbulence,
to whichit waslongrelongperiodof highor lowprecipitation
canbe stricted,but it has recentlybecomeof value
extremely
long.In a seriesof papersto which in studyinga variety of naturalphenomena
the presentwork servesas Introduction
and (see for example,Mandelbrot[1963, 1966,
Summary,
we shalldescribe
in detaila family 1967a,1967b]).
of statisticalmodelsof hydrologywhichwe
A word of acknowledgment
beforewe probelieveadequately
accountfor the Noah and ceed.In investigations
of currentstatistical
Josepheffects.Differentpapersin the series modelsof hydrology,
oneof the mostactive
will be devoted,respectively,
to mathematicalgroupshas beenthat foundedby Professor
considerations,
to accounts
of computersimu- Harold A. Thomas,Jr., at Harvard.In view
lations,and to analyses
of empiricalrecords. of the criticaltoneof muchthat follows,the
Later paperswill studyvariousproblems
of authorshastento express
heretheir personal
water controlengineering
as problems
of op- indebtedness
to Harold Thomas.He directed
erationsresearch,
whichthey were long be- B.B.M.'s curiosityto Hurst's work and to
fore the term 'operations
research'
itself was hydrologyand later initiatedJ.R.W. into the
coined.
intricacies of 'synthetic hydrology' and simu-
The models to be described were advanced
lation.
and arguedin Mandelbrot [1965] and Mandel-
brot and Van Ness[1968].We havecarried PARTISAN
COMMEI•ITS
Ol•lCURREI•IT
STATISTICAL
outextensive
experiments
of everykindto test
tIYDROLOGY
and develop these models,and we have, in
our opinion,confirmedtheir soundness.
It may
909
Current modelsof hydrologyassumeprecipitation to be randomand Gaussian(i.e., fol-
910
I•IANDELBROT AND WALLIS
lowing the normal probability distribution, combininghydrologyand climatology,is far
with its 'Gaiton ogive') with successive
years' from being of Gauss-Markovform.
precipitationseither mutually independentor
Other approaches
to hydrologicalmodeling
with a short memory. 'Independence'implies also start with a Gauss-Markovprocessand
in particularthat a large precipitationin one then introducemodifications
that tend to be
year hasno 'aftereffect'on the followingyears; more extensivewhen recordsare long than
'short memory'meansthat all aftereffectsdie when they are short. This generalprocedure
out within a few years. The classicalshort can be illustratedusing two examples.The
memorymechanism,
the Gauss-Markovprocess, first involvesthe loosebut intuitive idea of
is a 'singlelag linear autoregressive
model.'In the duration of a drought,the secondthe
this ease, aftereffectsdie out in geometric morerigorousbut lessintuitiveconceptof the
progressionand decreaserapidly. More gen- Hurst range.
eral are the 'multiplelag linear autoregressive For droughtsthe point is that, if an indemodels.'One feature commonto all thesemo- pendent Gauss processor a Gauss-Markov
delsis their belonging
to 'the Browniandomain processis chosento fit best the other aspects
of attraction,' a term which we shall define of precipitation,it will greatly underestimate
later. It is our basic belief that modelsin the
the durations of the longest drought. There-
Browniandomaincannotaccountfor the Noah
and Josepheffects.These modelsunderestimate the complication
of hydrological
fluetuations and the difficultyof 'controllingthem
by establishing
reservesto make the future
less irregular' (to paraphrasethe title of
Mass• [1946]).
Disappointmentwith specificmodelsin the
Brownian domain is today very widespread
among hydrologists(for example,see Yevjevich [1968]). Therefore,our sweepingassettion can only be controversialin its blanket
condemnationof all modelsin the Brownian
domain.To try to minimizesuchcontroversy,
we shall now describevarious stop-gapsthat
have beenproposed.
We shall point out that,
in effect, such models end up outside the
Browniandomain.
fore, suchprocesses
must be modifiedby consideringmoredurableaftereffects
(for example,
through'multiplelag'models).Onewhoeonsiderssuchmodifications
asnuisance
corrections
to
a basicGauss-Markov
process
will naturallytry
to fit all availabledata with a 'minimal'modifled process,
havingas short a span of aftereffectsas possible.However,when the sample
durationis sufficientlyincreased,
'unexpectedly'
long droughtswill again be observed.This
shows,after the fact, that the 'minimal'model
had attributed a special significanceto the
longestsample T that was available when it
was constructed.
As the sampleincreases,
such
a model must be changed.(For example,the
numberof lagsmust be increased.)
'Drought' being,as we said, an elusiveeoneept, let us now proceedto the observedbe-
Some authors eventually conclude that a
descriptionof hydrologicalreality requires a
Gauss-Markovprocesswith time-varying parameters. Such models must, however, be
changedbefore their consequences
have had
time to develop fully. For example,Jbefore
the sample average of precipitationhas had
time to 'stabilize'near its expectedvalue, elimarie changeis assumedto modify that expectation.We believesuchmodelsto be rather
pointless,becausethe usefulness
of a statistical
modellies primarily in its large samplepredietions. Since a changingexpectedvalue easily
overwhelms Gauss-Markov fluctuations, a
Gauss-Markovhydrologicalmodel can be used
only in conjunctionwith some'mastermodel'
ruling climatic change.The over-all model,
havior of the 'Hurst range,'which is lessintuitire but easierto study.To defineit, onebegins
by evaluatingthe total capacityR(s) a reservoit musthave had, in orderto perform'ideally'
for s years.'Ideal performance'here means(a)
that the outflow is uniform; (b) that the reservoirendsthe periodas full as it began; (c) r
that the dam never overflows; (d) that the
capacityis the smallestcompatiblewith (a),
(b) and (c). The conceptof an ideal dam
is of coursepurely retrospective,since data
necessaryto design such a dam are only
known when it is too late. However, the past
dependence
of the ideal capacityupon s tells a
great deal about the long-run behavior of a
river on which an actual dam is to be built.
Postponingqualifications
to later papers,let us
OperationalHydrology
describea striking discoveryH. E. Hurst made
while examiningR(s) for the Nile and other
geophysicalrecords.Hurst divided the capacity
R(s) by the standard deviation $(s) of s successivedischarges.The empirical finding, then,
is that save perhaps [or small values o] s,
the rescaledrange R(s)/S(s) is proportionalto
s• with H a constant between 0.5 and 1. Hurst
judged H to be 'typically' near 0.7, but other
estimatesput H much higher, above 0.85. An
independentGaussmodel yields R(s)/S(s) •
0.5. Gauss-Markov models, 'multiple lag' models, and all other models in the Brownian domain give a more complex prediction' R(s)/
$(s) • sø'5for large s, but R(s)/S(s) grows
faster than sø'5for small or moderate s, which
we shall call the 'initial transient region.' In
this transient region a variety of different behaviors may be obtained. Moreover, many
modelsmay lead to the sametransientbehavior,
which makes them indistinguishablefrom the
viewpoint of predictionsconcerningR(s)/$(s).
Then, if one has only the values of R(s)/$(s)
for I _• s _• T (with T a finite duration),
many different modelsof the BrownJandomain
are likely to yield predictions undistinguishable from the data. When s exceedsT, however, the Hurst range of every one of these
processeswill soon merge into the classicalsø'*
pattern. So far such a convergenceto sø'• has
never been observedin hydrology.Thus, again,
those who consider Hurst's
effect to be a tran-
sient implicitly attach an undeserved importance to T, which is typically the currently
available sample sizes.These scholarscondemn
themselvesnever to witnessthe full asymptotic
developmentof the modelsthey postulate.
TOWARDS
A
CHANGE
HYDROLOGICAL
Or
DIRECTION
IN
MODELING
We have now sketched a few reasons,to be.
911
the main features of the problem, namely the
Josephand/or Noah Effects.
To characterizeour proposednon-Brownian
first approximations,the loose distinction between 'low-frequency'and 'high-frequoncy'phenomena is useful. Using a Gauss-Markov process implies fitting high frequency effects first
and worrying about low frequency effectslater.
We propose to invert this order of priorities.
Conveniently, we shall be able to use the term
'low frequency' in either of its main meanings,
to designateeither a rarely occurring phenomenon, or an oscillatingphenomenonwith a long
wavelength.
The conceptsof 'low' and 'high' frequency
are, of course, relative. Natural phenomena
cover a continuousspectrum,in which very low
frequenciesof turbulence theory and very high
frequenciesof hydrology overlap around one
cycle per day. This frequency, being fundamental in astronomy, may also separate zones
in which intrinsically different mechanismsrule
the fluctuations of precipitation. The same
holds for the wavelengthof one year. The third
important wavelength in hydrology is 50 to
100 years,which we shall refer to as a 'lifetime.'
This is roughly the horizon for which one designs water structures and also, coincidentally,
the length of most hydrological records. The
importance of this wavelength is of human, not
astronomical, origin; it is purely 'anthropocentric.' Whereas precipitation fluctuations of
wavelength near one day or one year may
participate in several physical mechanisms,
fluctuationsof precipitation of wavelengthnear
one lifetime are likely to participate in one
mechanism only. Thus, the latter are likely
to be simpler than the former. Now assume
that one wants an approximationvalid over a
wide band of frequencies.It may be convenient
to start by askingfor a goodfit in somenarrow
frequencyband, with the hope that the formula
so obtained will be applicable over the wide
band. Under these circumstances, the band
near one lifetime, although purely anthropocentricin its definition,constitutesin our opinion a better basis of extrapolation than the
band near one year, which has meaning in
fully developedlater, why we dislike hydrological models obtained by 'patching up' the
Gauss-Markovprocess.It shouldbe understood
that this criticism does not imply that we expect to be able, with some other model involving few parameters,to representfully the
tremendouslycomplicatedhydrologicalreality. astronomy.
A model having few parameterscan only be a
We realize that a stress on low frequencies
'first approximation,'and we believethat such emphasizesidiosyncrasies.But the purpose of
a first approximationmust endeavorto 'catch' hydrological engineering is to guard against
912
MANDELBROT AND WALLIS
the recurrence of such idiosyncrasies,and one
cannotafford to neglectany availableinforma-
double the variance
of each of them. It is un-
fortunate that this property doesnot yet have
a generally acceptedname. Let it be said imWe also realize that, modelsin the BrownJan mediatelythat this property fails to hold, either
domain having long been recognizedas ap- for fractional noises or for approximation
plicablein many fieldsof science(beginningof thereto used in Mandelbrot and Wallis [1968a].
course with the BrownJan motion of statistical
These processessatisfy, however, both the law
mechanics), their proponents among hydro- of large numbers and the .centrallimit theorem.
logists are often able to identify ready-made If a natural phenomenonobeys the conditions
answers to the standard problems. Our pro- of a validity of all three of these mathematical
posed approach requires more work, but the theorems,it will be called 'smooth' or 'in the
answers appear to be sufficiently better to Brownian domain of attraction.' There exist
phenomenathat fail to obey the conditionsof
make this work worthwhile. Moreover, the
conceptof self-similarity,to be discussedlater validity of the third theorem, or of the last two,
or even of all three. Such phenomenawill be
in the paper, will bring true simplicity.
called 'erratic.' For example, the average T-•
'SMOOTI-I' AND •ERRATIC' PROCESSES
Z•:•• X(t) may ]ail to tend to any limit. Or it
We finally come to the promised character- may tend to a limit, whereas its distribution
tion.
ization of the ']3rownian domain of attraction'
does not tend to the Gaussian. Or it may tend
and of related specificmeaningsfor the terms
to a Gaussianlimit, whereas'past' and 'future'
averages fail to become asymptotically independent. The importanceof this latter circumstancefor the hydroIogistlies in the coincidental
equality between the order of magnitude of
most past records and the horizon of most
designs(both equal one lifetime), and in the
fact that, true expectations being unknown,
planning requiresthe determinationof the differencebetweenthe expectedmean flow over a
future lifetime and the known past average.
It is readily verified that the Gaussianmodels with a limited memory all assumehydrological phenomenato be 'smooth.' The Noah
and Josepheffects,on the contrary, not only
suggestthat hydrologicaldata are 'erratic' but
also express the major two forms of erratic
behavior. We shall speak of 'Joseph.-erratic'
'smooth'
and
'erratic'
time
series. We
need
three resultsof probabilitytheory,two of which
are classical,and all three of which relate to
averagesof T successive
terms of a stationary
time seriesdesignatedby X(t).
One says that X(t) satisfiesa law of large
numberswhen its averagetends to a limit, the
expectationœX(t), when T -• o•. This law is
the theoretical justification of the common
practiceof taking sampleaveragesas estimates
of populationexpectations.
One says that X(t) satisfiesthe more demanding Central limit theorem in its original
form when, for large T, the distributionof the
averagebecomesapproximatelyGaussian,with
a variance tending to zero as T -• •. This is
the justificationof the commonbelief that the
sampleaverageis likely, if T is not small, to behavior when the wettest decade within a cenbe a goodestimateof the expectation.A corol- tury includes an extraordinary 'term' of wet
lary of this is that, for large T, eventhe largest years.We shall speakof 'Noah-erratic' behavior
of the T quantities T -• X(t) contributesneg- when a few of the years within the century wit-
ligiblyin relativevalueto the averageT-• Zt:••
ness 'floods' so major as to affect the average
precipitation for periodsof many years within
which the flood years occurred.Needlessto say,
important in applications.Let us call 'past a processcan be both Joseph-and Noah-erratic
average' the expressionT-• Y•t:-•2 X(t) and simultaneously, a complication that we shall
face much later. 'Pure Joseph-erratic'behavior
'future average'the expressionT-• Z•:•• X(t)
and consider the difference between these two.
will be said to apply when none of the yearly
The third basic result on the averages of precipitationsduring a 'wet term,' had it stood
random sequences
assertsthat, as T -• •, the alone, would have been interpreted as a flood.
Clearly, the word 'erratic' should not be
two averagescan be consideredindependent,
so that the variance of their difference is
construedto suggesta 'black-and-white' con-
X(t).
The final result is less well known but very
Operational Hydrology
trast. The three theoremsin questionare, indeed, asymptotic,but scientificapplicationsof
mathematicsalwaysdeal with T's in somefinite
horizon. Consider, for example, an infinite
(nonrandom)seriesa(t). For the mathemati-
913
distant 'outliers.' Also, high water levels, which
would
be considered 'millenium
floods' if one
extrapolatedthe tails of the histogram from its
body, occur much more frequently than they
shouldunder the Gaussianassumption.
cian, the basic distinction is whether the sum
Despite the importance of deviations from
Y.t=•
© a(t) is, respectively,finite or infinite. For the Gaussian,we believe it worthwhile to begin
the scientist,on the other hand, the ultimate our investigationof the JosephEffect by Gausconvergence
of Y.t=• a(t) is of little import, un- sian processesX(t), which are by definition
lessZ,• • a(t) is alreadycloseto its limit. There- such that the joint distribution of their values
fore, the conceptof 'erratic' must be considered at any finite number of instants is a multivarias comingin variousdegreesof intensity,rather ate Gaussianvariable. Such processeswill be
examined in the next several sections. In the
like 'grey.'
sectionnear the end of the paper, highly nonTHE
ISSUE
OF THE
MARGINAL
DISTRIBUTION
OF
Gaussianprocesseswith a Noah Effect will be
TI-IE
YEARLY
FLOW
mentioned. (Processesthat are only 'locally'
We shall now characterize more accurately Gaussian are studied in Mandelbrot [1968].)
the idea of a 'pure Joseph-erratic'process,
GAUSSIAN
PROCESSES
AND
TI--II•
COVARIANCE
using the concept of 'marginal distribution,'
which is defined as the distribution of the values
Gaussianprocesses
are well known to be fully
of a processirrespectiveof their chronological specifiedby their covariancefunction: If X (t)
order.We believeit reasonableto demandthat, is of zero mean and unit variance,the covariwhen the order of values of a pure Joseph- ance C(s) is the correlation between X(t) and
erratic sampleis scrambled,one shouldbe left X(t -[- s). (Of course,in the caseof Gauss;.an
with a smoothprocess.Thus, the marginal dis- variables zero correlation is identical to intribution of these values will draw a line bedependence.)Our problem is, then, to use the
tween, on the one hand, Noah-erraticprocesses behaviorof C(s) to classifya Gaussianprocess
and, on the other hand, processesthat are as smoothor Joseph-erratic.What we need here
either smoothor pure Joseph-erratic.
is a distinction between a form of high-freThe paragonof the pure Joseph-erraticis a quency effect--namely 'short lag' or 'short run'
processwith a Gaussianmarginal distribution. effects--and a form of low-frequency effect-We shouldtherefore,in every case,begin by namely 'long lag' or 'long run' effects.Short lag
checkingwhether the Gaussianmarginal dis- effectsdependupon the valuesof C(s) for a few
tribution applies.For this, in a first approxima- small valuesof s. Long lag effectsdependupon
tion, 'probabilitypaper' plots are goodenough. the other values of C(s). We shah now examine
They evidently showthat it is not exceptional this dichotomy on four examples.The first is
that the marginal distributionbe either nearly the processof independentincrements,whose
Gaussianor highly non-Gaussian.
To stay near covarianceC•(s) satisfiesC•(s) -- 0 for all s
the land of Joseph,an exampleof nearly Gaus- 0. The secondis the Gauss-Markov processof
sian marginal distribution is provided by the covariance
Ca(s) -- exp(--]sl/s). The last two
level of the Nile at the Rhoda Gauge near are the processesof covariancesrespectively
Cairo, an example of highly non-Gaussianby equalto Cs(s)-- (1 -]- Isl2/s•)
-• and C•(s) -the annual dischargefrom Lake Albert. To find (1 -[- Isl/s•)-ø'5,
wheres•.,s•ands•areconstants.
other examplesof either behavior,it sufficesto
The abovefour covariancesdiffer considerably
thumb through Boulos [1951]: Straight line from each other in the long run, but C2(s),
interpolations are quite acceptable in certain C3(s), and C•(s) are all three smoothand monocases,poor but perhapsbearablein someother tone in the short run. Therefore, if the sample
cases, and dreadful in still others. Much less duration is short, and the sample covariance
completedata are availablein other places,but correspondingly'noisy,' the graphs of C•(s),
the existenceof huge deviations from the Gaus- C•(s), Cs(s), and C•(s) may be undistinguishsian is very familiar' For example,runoffsdue able, not only to the eye but even from the
to major storms may appear on histogramsas viewpoint of many tests of statistical signifi-
914
MAAIDELBROT AND WALLIS
var lAX •] ~ s•'5,where• means'asymptoticallyproportionalto.' More generally,let C(u)
• u2u-•for large u, with 0.5 < H < 1. Then,
for large s, var [AX •] --• s2s.Incidentally,asC•(s), and C•(s), are not statisticallysignificant sumingimplicitlythat Z•:o•C(u) < oo, G.I.
cancethat examine each value of s singly. That
is, such statistical tests are liable to indicate
that the differencesbetweenthe samplecovariance and either of the functions C•(s), C.•(s),
for most s. The statistician could then conclude
Taylor suggested
this infinite sum as a measure
of the spanof temporaldependence
in a time
series.The Taylor span is thus infinite for
fore, he will advise the hydrologistthat there C•(s), finite and easyto estimatein caseslike
conis no evidencethat his data were not generated C•(s) or C•(s), wherethe seriesZ•C(u)
by an independentGaussprocess(C•), or a vergesrapidly, and, finally, finite but hard to
Markov-Gauss (C2), or perhaps some more estimate in caseslike C•(s), where the series
that all the data will be acceptablyfitted as
soon as short run data have been fitted. There-
involved short-memory process.
This would, however, be a very rash con-
clusion.For example,under the hypothesisthat
the true covarianceis C•(s), one would expect
the relative proportionof positiveto negative
samplecovariancesto be about one. This proportion would be larger under the hypothesis
C.o(s), still larger under C•(s), and larger yet
under C4(s). Thus, if statisticalcriteria geared
towardslow-frequencyeffectscan be developed,
it is reasonableto expect them to show the
Z•=o•C(u)
converges
slowly.
TI-IE RANGE, TI-IE JOSEPH EFFECT, AND
I-IURST'S LAWS
Curiously,empiricaldata aboutthe behavior
of vat [,AX•] in hydrologyhave been examined only after thoserelative to anothermeasure of over-all behavior of a process,namely
R(s)/S(s), where the sequentialrange R(s)
was defined earlier to be the capacity of a
reservoircapableof performing'ideally' for s
same data to be significantlycloserto C3(s), or yearsand S(s) to be the standarddeviationof
yearly flow. Among Gaussianprocesses,the
to C•(s), than to C2(s).
of R (s)/S(s) upon s sharply disOur need, then, is to enhancelong-run prop- dependence
erties of a processwhile eliminatingshort-run tinguishes smooth from Joseph-erratic proThis is alreadyobviousfor Joseph'sown
wiggles.To do so, the best procedure.is to in- cesses.
tegrate or to use moving averages (fancier exampleof the sevenyearsof highprecipitation
averages will not be consideredhere). Three followedby sevenyears of drought,for which
methods of dealing with long-run effects de- the ideal reservoir would have had to be enormous. If wet and dry years alternate at any
serveto be singledout.
TI-IE
Let
VARIAI•CE
OF
the 'accumulated
CUMULATED
pointof a record,thenidealreservoir
sizeis
FLOWS
flow' since time
decreased. It
1 be
definedas Z•q• X(u) and designatedas X•(t).
Then, G.I. Taylor's formula (see Friedlander
and Topper [1961]) can be used to evaluate
the variance of AX • X •(t + s) -- X •(t) -Z•:,,• • X(u) -- X(t + 1) ... • X(t +s). This
variance is given by sC(O) • 2 Z• • (s -- u)
C(u), which immediately introduces a basic
long-run dichotomy.
When Z• © C(u) • •, var lAX •] -- var
[X•(t + s) -- X•(t)] is found to be asymp-
toticallyproportional
to sZ•_•C(u),and X(t)
is found to be in the BrownJan
domain
of at-
traction.
When, on the contrary, Z•:o•C(u) diverges
sufficiently rapidly, var [.AX•] grows faster
and X is not in the Brownian domain. When,
for example, C(u) -- C,(u), one finds that
is the task of mathematics
expressthis reductionin numbers.
First considerthe casewhen X(t)
to
is an in-
dependent
Gaussprocess.
Then,whens is large,
bothR(s) andR(s)/S(s) equalN/s,multiplied
by some 'universal' random variable independentof s. The little that is knownabout
those random variables is due to Feller [1951].
For the .Gauss-Markovprocessand for other
models for which the memory Z•_-o'C(u) is
finite,the 'N/slaw' remainstrue, but the multi-
plyingrandomvariablesare changed.
The case of series exhibiting the Joseph
Effect is entirely different. For such series,
the N/s law fails, as was first notedby Harold
Edwin Hurst [1951, 1956, 1965]. For hydrologicalseries,as well as for many othernatural
time series,R(s)/S(s) increases
like Cs•. Here,
C and H are positiveconstants;H may range
Operational Hydrology
between 0 and 1 and is seldom near 0.5. We
shall call this empirical finding 'Hurst's law.'
Moreover, •/var[.AX•], consideredearlier, is
also proportionalto ss rather than to sø", as
suggested
by the usualsimplemodels.This will
be called 'Hurst's law for the standard devia-
tion,' or 'Langbein'scorollaryof I-Iurst'slaw,'
becauseit was first noted in Langbein's comments on Hurst [1956].
Strictly speaking,Hurst claimeda more demanding'one parameter ss law,' accordingto
whichR•s)/S(s) is asymptotically
equalto
(s/2)s. His reasonsfor claimingthat C • 2-s
are not too clear or convincing. Moreover,
915
RELATIOAl' BETWEEI•
THE
JOSEPIE EFFECT,
HURST'SLAW, AAI'DTHE LOl•G RUN
BEHAVIOR
OF
THE
COVARIAAI'CES
To accountfor the abovelisted findingsconcerningvar(AX*), R(s)/S(s), or the spectrum
has proved to be very hard. For example,
perusal of the discussionof Hurst [1951] and
Hurst [1951] demonstrates the kind of desperate expedient necessaryto fit his finding
within the universe of the simple statistical
models. Claiming (incorrectly, as we shall
demonstrate presently) that there exists no
stationary random processwith a range follow-
ing the ss law, severalamongthe discussants
have
suggestedeither giving up statistical staensure a better fit. It also yields a different
tionarity or invoking nonrandom 'climatic'
estimate of H. For example, Ven Te Chow
changes.
in his discussionof Hurst [1951] found a case
A more hopeful reaction, already mentioned
where H goesup from 0.72 to 0.87 when C
in
the partisan commentat the beginningof
is separatelyestimated.Also, we found cases
this
paper, is exemplified in such works as
where the best estimate of H is below 0.5, which
Ants and Lloyd [1953] and Fiering [1967].
contradicts Hurst's assertion that 0.5 ( H ( 1.
See Mandelbrot and Wallis [1968c] for revised These authors, and others, have constructed
values of H.
stationary stochasticprocessesof the usual
Note also that Hurst's original'(s/2) • law' kind (i.e., in the Browntan domain of attrac• •) for which
has provendangerous.
In somecasesit tempted tion, satisfyingZ•C(u)
him, as well as other authors,to estimateH both rangeand standarddeviationare proporfrom a single sample of natural or simulated tional to ss over a finite span of valuesof s.
values of X(t). Such estimatesshouldbe dis- But the usual•/s behaviorstill appliesbeyond
separateselectionof H and of C can obviously
carded.
The
revised
statement
we use means
that estimationof H requiresmany values of s
this span.Thus, the ss law is here a property
of what we have called a transient span. This
transient may be made arbitrarily long. But
and, for every value of s, a large number of
starting points t spread over the total sample long transientscan only be achievedwith comof length T.
plicated processes
with a long memory. For exOn the other hand, every specificmodel of
ample, Fiering [1967] (p. 85) had to use an
the JosephEffect, suchas the fractionalnoise
autoregressivemodel with 20 lags to ensure
to be describedin the sequel,will yield a relathat Hurst's law holds over the span I • s •
tion betweenC and H, whoseconformitywith 60.
experiencewill test the value of the model.
An alternative to this approach is based
SPECTRAL AI•ALYSIS:
PRIAI'CIPLE
AAI'D APPLICATI01•
upon the existenceof the self similar random
I1•
IEYDROLOGY
processes,pointed out by Mandelbrot [1965]
and examined below. For such processes,
In addition to the behavior of var (•iX •)
and of Hurst'srange,a third way of lookingat
low frequencyphenomena
is throughspectral
(or Fourier or harmonic) analysis.We only
mentionit here to say that the spectraldensity
of hydrological
recordspeakssharplyfor very
low frequencies,
as is alsothe casefor the socalled !:• noises[Mandelbrot, 1967a]. A full
discussion
of this topic is postponedto the
next paper.
Hurst's law holds for all values of s. Even more
important from our viewpoint, which emphasizes low frequency phenomena, is the existence of processes
for which Hurst's law holds
for the short as well as for the long run. For
the standarddeviation,this was already proved
whenwe notedin passing
that
%/var[AX*]•, s•
•1•
•/fANDELBROT AND •ALLIS
whenever
it easierto generalize'We claim that the independentGaussprocessX(t) is so convenient
C(s),'• s2•-•'
because
of the possibility
of interpolating
The sameasymptoticbehaviorcan be shown •:• X(u) to continuous
timeswith the help
to hold/or the rangeR(s) and the rescaled of a 'self-similar'
randomprocess
B(t), called
rangeR (s)/S(s).
Browntanmotion (also calledBachelierprocess,or Wiener process).To define self-simi-
Thisobservation
is central
to ourstudyof larity,onemustconsider
a portion
of B(t),
theJoseph
Effect.
Before
weexamine
it more witht varying
from0 to T, andrewrite
it as
closely,
let us showhowit .canexplain
the B(h T), withh varying
from0 to 1. 'Selfexistence
ofmodels
in which
Iturst's
lawholds similarity'
thenexpresses
the fact that the
in an initialtransient.
The key is that the rescaled
function
T-ø'SB(h
T) hasthe same
values
ofvar[X•(tq-s) -- X•(t)] fors • T distribution
forevery
value
ofT.
areaffected
onlyby thevalues
of C(s) for Fromthis,oneimmediately
deduces
that
s •
T. Hence, changesin the covariancefor
s • T willpass
unnoticed
whenonlythespan %/var[B(t+ s) -- B(t)]= sø'5 and
s • T is observable.Now supposethat, start-
ing from (say) the covariance
C(s) - (1 +
max B(t •- u) -- min B(t •- u) -- Csø'•,
10s)•-•, long lag covariances(s > T) are
decreasedsufiqcientlyto make Z•_-o•C(u) convergent. The modified processX(t) is thus
with C a constant.These statementsare forms
of the sø'5law, but they are valid uniformly
'brought back' into the Browntandomain of
attraction. It could even be made into a
'multiple lag' autoregressive
model, which is
the usualgeneralization
of the Markov model.
For suchmodifiedprocesses,
Iturst's law continuesto hold for s < T, and for sometime
beyonds = T. On the long run, however,it
(that is, for all s) rather than asymptotically
(that is, for highs).
By analogy,
whenstudyingthe lawsof Hurst,
it is good to know that more generalselfsimilar processes
exist. A Gaussianprocess
X(t), of integralBa(t) = j'o'X(u)du,is Joseph
self-similarif the rescaledfunctionT-•Bs(h T)
will be replaced
by the x/s law. For example,
the standarddeviationwill equal Cx/s. The
value of this constantdepends
uponthe tail
selectedfor the modifiedcovariance.
It is ad-
is independent
of T in distribution.
That the
ss lawsapplyto Ba(t) canbe seenby simple
inspection.
Ba(t)was called'fractionalBrowntan motion' by Mandelbrot and Van Ness
justable at will and quite arbitrary.
We considersuch models,in which T plays
[1968].
a central role, to be undesirable.
DEFINITION
OF
GAUSSIAN
SELF-SI•VIILAR
ERRATIC
PROCESSES
Unfortunately, the derivative But (t), called
'fractional Gaussian noise,' is too irregular to
be studied directly. As we interpolated the integral of the independent Gauss processby
Browntan motion, we must now replace X(t)
In criticizingthe usualstatisticalmodels,as by B•(i q- 1) --B•(t). The covariance
of this
appliedto hydrology,we don't underestimate functionB•(t •- 1) --B•(t) is givenfor s •_
their good features. In particular, wherever by
the independentGaussprocessis an acceptable
approximation, it is unbeatable. Where one
must amend this process,there are features
that one will want preserved.For example,all
the models of the Browntan domain preserve
the assumptionsthat the variance is finite, and
Taylor's scale [defined as Z•X(u)]
is also
finite. But they destroy another property that
makes the independentGaussprocessso convenient to manipulate. We shall now express
this property in an indirect way that will make
C,(s) = C[(s- 1)•"-+-(s-+- 1)•"
with 0 <: H <: 1. If % •
large, onefinds
H •
i and s is
C,s) • [2H(2H -- 1)C]s
This is preciselythe form we have proposedto
use to model phenomenaobeyingIturst's law.
Therefore, our models are approximationsto
fractionalGaussiannoise.
OperationalHydrology
APPLICATIONS
OF SELF-SIMILARITY
To apply self-similarity, one proceedsvery
much as in the classical'dimensional analysis'
917
sary to explain Hurst's findings, and in the
third paper that it is insufficient.
REFERENCES
of fluid and solid mechanics. This is no acci-
dent. Ifydrology can be consideredthe low Anis, A. A., and E. H. Lloyd, On the range of
frequencyapplicationof the theory of turbupartial sums of a finite number of independent
lence,in which self-similarityand dimensional normal variates, Biometrika, •0, 35, 1953.
analysiswere introducedby yon Karman and Boulos,Naguib, Probability in the Hydrology o[
The Nile, Cairo, Government Press, 1951.
Kolmogoroff (see Friedlander and Topper Feller, W., The asymptotic distribution of the
[1961].
rangeof sumsof independentrandomvariables,
An illustration of dimensionalanalysis is en-
Ann. Math. Star., 22, 427, 1951.
counteredwhen dealing with flows of water in Fiering, Myron, StreamflowSynthesis,Cambridge,
Harvard University Press, 1967.
vesselsof the same shape but of different diFriedlander,
S. K., and L. Topper, Turbulence'
mensionsand at proportionatelydifferentvelocClassicPapers on Statistical Theory, New York,
ities. Then the calculationsneednot be repeated.
Interscience, 1961.
It sufficesto solve all relevant problemsonce. Hurst, H. E., Long-term storage capacity of reservoirs, Trans. Am. Soc. Civil Eng., 116, 770,
Solutions to other cases will be obtained by
mere rescaling.
The application of self-similarity to hydrology is in the same spirit. Once one has performed the calculations relative to some 'refer-
1951.
Hurst, H. E., Methods of using long-term storage
in reservoirs, Proc. Inst. Civil Eng., Part 1,
519, 1956.
Hurst, H. E., R. P. Black, and Y. M. Simaika,
Long-Term Storage, an Experimental Study,
ence'horizon,answersrelative to other horizons
London, Constable, 1965.
can be obtainedby simple rescaling.This may
B., The variation of certain •eculamake it worthwhile,in the caseof the reference Mandelbrot,
rive prices, J. BusinessUniv. Chicago, 86, 394,
horizon, to perform some very lengthy and
1963; reprinted in The Random Character ol
involved calculations that would not otherwise
Stock Market Prices, Paul /5. Coother (ed.),
M.I.T. Press,Cambridge, 1964.
be economic. The new hydrology we propose
may demand readjustmentsof thought. But Mandelbrot, B., Une classede processusstochastiques homothetiques k sol; application k la
there is hopethat the ultimate outcomeof this
loi climatologiquede H. E. Hurst. Compt. Rend.
new hydrologywill be a set of new and better
Acad. Sci. Paris, 260, 3274, 1965.
'cookbookrecipes.'
Mandelbrot, B., Forecasts of future prices, unbiased markets and 'martingale' models, J.
Business Univ. Chicago, 89, 242, 1966.
FORETASTE
OF A DISCUSSION
OF THE
Mandelbrot, B., Some noises with 1/f spectrum,
NOAH
EFFECT
a bridge between direct current and white
noise, Inst. Electrical Electronic Eng., Tra.•s.
A discussion of the Noah Effect is several
Information Theory, 13, 289, 1967a.
papersremovedfrom the presentintroductory Mandelbrot,B., How long is the coas•of Britain?
article. Our approachwill resemblethe methods
Statistical self-similarity and fractional dimension, Science,155, 636, 1967b.
Mandelbrot [1963] usesto describethe variaMandelbrot,
B., Sporadic random functions and
tion of commodityprices. The above considconditional spectral analysis; self*similar exered function X•(t) -- Z•_•X(u), with X(u)
amples and limits. Proceedingso• the Fi•th
the annual flow for the year u, will be the
Berkeley Symposiumon Mathematical Statistics and Probability, edited by L. M. LeCam
counterpartof the price of a commodityat the
and J. Neyman, Vol. III, 155, 1967c.
instant t. The very rapid and large changes
B., Long-run linearity and locally
typical of the behaviorof priceswill be com- Mandelbrot,
Gaussianprocesses;roles of J-shaped spectra
paredto floods.Sporadicprocesses
(seeMandel-
and of infinite variance,Intern. Econ. Rev. (in
brot [1967c]) will also be needed.
The Noah Effect certainly raises important
press), 1968.
Mandelbrot,B., and J. W. Van Ness,Fractional
problemsin operationalhydrology.Suchproblems are, however,separatefrom thoseraised
by the JosephEffect. In the next paper we
experimentswith fractional noises,Water Re-
shall show that the Noah Effect is not neces-
BrownJan motions, fractional noises, and applications,SIAM Rev. (in press),1968.
Mandelbrot, B., and J. R. Wallis, Computer
sources Res. (in press), 1968.
918
MANDELBROT AI•D WALLIS
Mandelbrot, B., and J. R. Wallis, Robustnessof
the range statistic, as a measure of long-run
interdependence,Water ResourcesRes., to appear, 1959a.
Mandlebrot, B., and J. R. Wallis, Some long
run properties of geophysical records, Water
l'avenir dans la vie jconomique, 2 vols., Paris,
Hermann, 194õ.
Yevjevich, V. M., Misconceptionsin hydrology
and their consequences,Water Resources Res.
4, 225-232, 1968.
ResourcesRes., to appear,1969b.
Mass•, P., Les r•serves et la r•gularization de
(Manuscript receivedMay 17, 1968;
revised July 1, 1968.)