HIDROLOGÍA con modelos estocásticos

1. VOL.6, NO.3 WATER
RESOURCES
RESEARCH $UNE1970
Stochastic
Models
inHydrology
ADRIAN E. SCIIEIDEGGER
U.$. Geological
Survey,University
ofIllinois,Urbana,
Illinois61801
Abstract.The stochastic
models
that canbe used
to represent
growth
andsteady
state
phenomena
in hydro!ogy
are reviewed.
It is shown
that thereare essentially
two typesof
growthmodelspossible;
the cyclicgrowthmodeland the randomconfiguration
model.For
steady
statephenomena
(timeseries)
wearegenerally
restricted
to a Gaussian
typeof model
with or withoutautocorrelation.
Self-similarity
models(fractionalBrownfan
motion)leadto
physically
absurd
conditions
if theyareextrapolated
to highfrequencies.
INTRODUCTION
While investigating
the causes
of hydrologi-
cal processes,
a completelynovel approach
emergedduring the past five years. Whereas
suchcauses
were soughtin entirely determin-
isticphenomena
in the past,LeopoldandLa•g-
bein [1962] recognized
that while on a micro-
scopicscaleeach minute event affectingthe
evolutionof for examplea landscape,
a river
course, or • drainage basin is certainly de-
terministic, the combinedeffects of all the in-
dividualmicroscopic
eventscanbestbe treated
by modelsthat are analogous
to thoseusedin
statisticalthermodynamics.
The two main approaches
to statisticalhy-
drology,just as in statisticalthermodynamics,
are by two types of modelswhich might be
calledtime evolutionaryand ensemble
statisti'-
cal. In identical statistical populationsthe
ergodic hypothesisprovides a connectionbe-
tweenthe two approaches.
However,depending
on the generation
of the stochastic
models,
the
populationsin a particular evolutionarymodel
and a particular ensemble
statisticalmodel can-
not be assumed
a priori to be the same.Some
startlingdifferences
in basicpopulations
have
been noted in the statistical treatments of a
phenomenon,
althoughsuperficially
thesepop-
ulationswouldbe thoughtof asbeingidentical.
The reasonis that evolutionaryprocesses
nat-
urally suggest
the propertyof cyclicfry,
whereas
the ensemble
approachdoesnot. It is therefore
not the ergodichypothesisthat is infringed
upon,but the factthat the generation
of a pop-
ulationby a cyclicevolutionary
process
and the
750
populationobtainedby regarding
all possible
configurations
of systems
are generallyquite
different from one another.
The authorof this paper intendsto investi-
gate the stochasticmodels that can be used to
representgrowthand steadystate phenomena
in hydrology.
ENSEMBLES, OBSERVABLES,
AND
EXPECTATION VALUES
We begin by discussing
some fundamental
conceptsnecessaryfor the stochastictreatment
of geomorphology.
These concepts
are well
known from their applicationin statistical
physics.
For a generaldefinitionof the perti-
nent terms, the reader is referred to $ommer-
feld's[1964] standardtext.
We are concerned
with a system(landscape,
river net, etc.) in which certain observables
(elevation,bifurcationratio, etc.) are to. be
measured.The prob•.em
is to predict the nu-
merical values that will be obtained if these
observables
are measured
undercertainspeci-
fied conditions.
In deterministic
theory,thecorrectprediction
will simply state the value of the observable
that will be obtained.
In • stochastic
theory,
however,only an expectationvalue can be
specified.
This expectation
value is alwaysde-
fined as the average value of the observable
over an ensemble
of statesof the system.This
ensemble
of statesmustbe clearlyspecified
in
eachparticularcase.The reasons
why a prob-
abilistic approach rather than a deterministic

2. Stochastic Models 751
one may be indicated usually lies in the great
mechanicalcomplexityof the systemunder con-
sideration,
whichprecludes
the fo!lowing
up of
eachmicroscopic
eventin detail. Thus the single
processunder consideration
is replacedby an
ensembleof like processes
that are all equiva-
lent within the limits of one'sknowledge.
As an example,we consider
time evolutionary
processes.
A systemis subjectto influences
fluc-
tuating in time (they may actually be deter-
ministie,but causedby somany influences
that
they can be treated as fluctuating) according
to someprobabilistic
law, and we are interested
in the expectationvalue of some observable.
This expectation
valueisthenthe average
value
of the observab!e
in questionover the ensemble
of statesthat is obtainedif the process
isstarted
over and over againfrom the sameinitial con-
dition,but subjectto the fluctuating
influences
mentioned above.
Under certain specificcircumstancesthe time
average can be substituted for the ensemble
average (ergodieityof the system).In this case,
the ensemble over which the observable is av-
eragedare the statesof the systemat a series
of time instants.One of the necessary,
but by no
meanssufficient,conditionsfor this type of av-
eragingto be possibleis that the systembe in
a steadystate.
The general proceduresoutlined above are
standardpractice in statisticalphysicsand are
widely appliedin all fieldswherethe statements
are of a probabilistic
nature.Thus the scheme
doesnot only apply to statisticalphysicsas
such [Sommer]eid,1964], but also,to quantum
theory [Dirac, 1947].
It wouldbe conceivable,
givenan ensemble
of
states and an observable (which has a certain
value in each state), to define its expectation
value differentlyfrom the mannerin whichthis
was done above. For instance,we could de-
termine the most probable state of the system
and term that value of the observable which
it attains for this most probable state as its
expectationvalue [Shreve, 1966]. However this
procedure,while mathematically well defined,
is physicallyunsound.Measurementvaluesare
generally either time averageson a fluctuating
systemor the result of averagingvaluesfrom
repeated measurements
on similar systems.In
either ease,the averageof the observableover
the ensembleis the appropriate theoretical ex-
pression,and not someother mathematically
possible
expression.
CYCLIC MODELS
Let us considera time evolutionary process
as discussed
as an examplein the sectionon en-
sembles.Actually the time evolution of a sys-
tem represents
the most important application
of stochasticideasto hydrology.
Thus the growth of a system is followed
sequentiallythrough subsequent
stages.These
stagesdeve!opthroughthe actionof influences
that fluctuate accordingto some probabilistic
law. At any one time instant t, the ensemble
for calculatingexpectationvalues of a desired
observableis obtained by imagining that the
growthprocess
is startedmanytimesoveragain
from the initial condition to t, but subject to
differingrandom influences
and subjectto the
sameprobabilitylaw asmentionedabove.Evi-
dently this leads to an ensembleof possible
statesof the systemat eachstageof its develop-
ment specified
by the giventime t,. Expectation
valuesof any observable
at that time t, are then
calculated by the usual procedure of taking
averages
overthe ensemble.
In the procedure outlined above there is no
intrinsic specificationas to how the ensemble
at any particular time t, will look. This
dependsentirely on the probabilisticlaw re-
ferringto the fluctuatinginfluences.
However it is often difficult to specify this
law accurately,and the alternate path is then
to postulate dimefly (ab hypothesi) certain
propertiesfor the sequentialensembles
(at t•,
t2,..., t•,...).
The simplestsuchpostulateis that the stages
are cyclic regarding some time interval At,
(this may vary with i), and that each cycle
taken at its proper sea!eis similar to. the pre-
vious ones.Under such conditions,the growth
of the systemissaidto be eyelieandself-similar.
The notion of self-similarityis essentialfor the
definitionof a cycle.
It shouldbe emphasizedonce more that as-
suming a system's growth is eyelie and self-
similaris in lieu of, and thereforeequivalentto,
assuminga specificprobabilisticlaw regarding
the fluctuating influencesaffecting the growth
of the system.Such an assumption,therefore,
must be justified upon somephysicalor other
grounds.

3. 752 ADRIAN E. SCI-IEIDEGGER
The notionof cyclicityand self-similarityhas
bee
n tied up abovewith certainspecified,
but
not.necessarily constant, time intervals
The notionof cyclicitycanthen be generalized,
or rather extrapolated,to infinitely small or in-
finitely largeintervalsAt,. We shalldiscuss
such
extrapolations
laterin greater
detail,but let it
be statedhere that an observedcyclicityin
giventime intervalt• < t, < re,whereperhaps
self-similarityis empiricallyfound for a series
of finiteAt, -- a(t,) At,+• witha some
func-
tion, impliesnothingat all for t < t• or t > re.
Carrying self-similarity to infinitely small or
infinitelylargevaluesof t canleadto suchab-
surd
•notions
asa nonintegrable
length
ofa river
or to totally unjustifiedextrapolationso.ft.ime
series.
If:•a systemis cyclicallyself-similar,then it
isppSSible
to regard
each
cycle
asa compo-
nent:Cell.
If thequantitythat characterizes
each
cycleis reduced
by the correspo.nding
similarity
factorsto somecommonscale,then we end up
with a seriesof cells,in eachof whichthere is
(statistically) an equalamountof the quantity
mentionedabove. A systemo.f this type is
anMogous
in gasdynamics
to onein whichthere
is an equipartitionof energyin the individual
phase cells.Therefore there is an obviousanal-
ogyin suchsystems
with thetemperature
in gas
dynamics.The mean value o.fthe quantity in
questionis, therefore,canonicallydistributedin
the cells.The distributionof the quantity in
question
is exactly
canonical,
provided
it can
be further assumed that the deviations from the
mean are Gaussian. Since the statistical behavior
of the systemis generallydue to the action of
many linearly superposed
independentrandom
influences,
the central limit theoremapplicable
in this caseensures
that this is usuallythe case.
It is possible,
but not veryinstructive,
to put
the argumentsof this sectiongiven in words
into an abstract mathematical formalism. The
concepts
becomeclearwhen givensomespecific
examples.
River nets. It is well known that river nets
followHorton's
lawof stream
numbers;
i.e.,the
numbersn, of streamsegments
of (Strahler or
Horton) order i form (on the average) a geo-
metric sequence.
Horton [1945,p. 337] already
attempted a hydrophysicalexplanation of this
law in terms of a self-similarcyclic growth
model,an ideawhichwasformalizedby Wolden-
berg[1966].Accordingly,
ea.ch
newstreamorder
corresponds
to a cyclein the drainage
basinde-
velopment.In this modelit is easilypossible
to
definea temperature
analog;thisis simplythe
bifurcationratio for a givenchange(equalto
+ 1 in oneStrahlerorder for example) [Schei-
degger,
1968a].Unfortunately,
naturedoesnot
produce
river netsthat followtheabovegrowth
model[Ranalli (rndScheidegger,
1968].
Meanders. River meanders can also be re-
gardedin terms of a cyclicgrowthmodel.New
meanderloops are addedagain and again to
thealreadyexisting
ones.
In principle,a certainfixedlengthof river
segmentcan be regardedas generatingcells
(adjoining
segments,
numbered
by i) ofthesys-
tem; the stochasticvariable is the deviation
angle•, (change
of direction
angle)overeach
segment
i. Since• canbe positiveor negative,
it cannotdirectlybe relatedto a temperature
analog.
For this analog
we haveto take •,•'
[Scheidegger,
1967],andfor thisquantity•'
wehaveanequipartition
theorem
whichappears
to be satisfiedin nature [Thakur and Schei-
degger,
1968].
RANDON C0Nr•UR•ON •OD•rS
Aswasnotedin thelastsection,
the assump-
tion that the outcome
of a fluctuating
growth
processin a system is that the latter becomes
cyclically self-similar is in lieu of assum-
ing a specificprobabilitylaw for the fluctua-
tions.Thereis nothinginherentlysacred
about
makingsuchanhypothesis,
except
the apparent
simplicity of the resultingstructure of the
system.
Theoretically,
any otheroutcome
might
have been chosen.
In this view, an equallysimpleassumption
is that the probabilitylaw is suchthat the out-
comeis an ensemble
of systems
in whichevery
possible
internal configuration
within any one
systemis likely. We call sucha modela random
configuration
model. In gas dynamicsit cor-
respondsto assuminga microcanonical
proba-
bility distribution.
It is nowno longerpossible
to,splitthe sys-
temintocomponent
cells.
A temperature
analog
still exists,but this is with the microcanonical
temperaturein gasdynamics.
A slight generalizationof the abovemodelis
that the probabilitydistributionof the possible
configurations
of the systemis not constantbut

4. Stochastic Models 753
Gaussianregardingsomecharacterizing
param-
eters.It shouldbe noted that a constant
•proba-
bility distributionis a specialcaseof a Gaus-
sianone(infinitedispersion).
Thesebasicconcepts
areillustratedin the fol-
lowingexample.
We noted above that as far as river nets are
concerned,
nature doesnot supportthe cyclic
growth model.We will thereforetry to use a
randomconfiguration
modelof the type which
isthe subjectof the presentsection.
This model implies in the caseof river nets
that a particularriver networkis a realization
of a particular graph (arborescence)
in the en-
semble.
of all possiblegraphs (arborescences),
whereinthe probabilitydistribution
is suchthat
all possible
arborescences
(with a givennumber
of free ends) are equally likely. This corre-
spondsto. a microcanonical
probability distri-
bution of the individualpossible
statesof the
system.We can then set up a microcanonical
temperature
analogfor river nets[Scheidegger,
1968b].
OTHER MODELS
We have discussed
above growth modelsin
which the growth is a stochastic
process.
What
hasbeen investigatedis the probability of sys-
temsthat are the outcomeof the growthprocess.
Two typesof possible
outcomes
havebeencon-
sidered. The resulting systems can or cannot
be subdivided into. individual cells. The latter
casein somefashion can be t•ken as a special
case of the former. Only one single cell en-
compasses
the wholesystem.
The questionarisesas to whether there are
any other types of outcome that might rea-
sonablybe expected?Dependingon the prob-
abilisticlawsgoverningthe growth process,
the
outcomemight be anything; i.e., any arbitrary
probability distribution for the configurations
that the systemmight have at time t{ couldbe
the outcome.However, we have to assumethat
the growthprocess
is the outcomeof the effect
of many independent
individuallinearilysuper-
posedinfluences,
and under suchconditionsthe
central limit theorem imposessomelimitations.
Usually the particular individual possible
configurations
of a systemat a certain time t,,
which in their tota•lity representthe ensemble,
will be characterizedby certain parameters.Be-
causethe influencesproducingthese configura-
tions are usuallyassumed
to be great in num-
ber, additiveand independent,
it will be per-
missibleto apply the central limit theorem,
indicatingthat the distributionof probabilities
regarding
theseparameters
or configurations
is
Gaussian.Thus barring queer processes,
the
mostgeneraloutcome
for the probabilitydis-
tributions in each cell will be that the latter is
Gaussianaround somemean. A specialcaseof
a Gaussian
probabilitydistribution
isthat every
configuration
is equallylikely (infinitedisper-
sion).
Apart from queerprocesses,
the following
possibilities
existfor theoutcome
of the prob-
ability distributions:
1. Systemdivisibleinto cells (cycles).In
each cell there is a Gaussian (which includesa
constant)probabilitydistributionof the pos-
sibleconfigurations.
2. Systemnot divisibleinto cells(consisting
of a single
cell).Thedistribution
of probabilities
for the configurations
is Gaussian(which in-
cludesa constant).
These are in essencethe casesdiscussedabove.
Except for queerprocesses,
no othersare pos-
sible.
THE STEADY STATE
Up to now we have discussed
modelsof
stochasticgrowth processes.
However, an im-
portant class
of hydrologic
systems
is not grow-
ingbut is (presumably)
fluctuating
in a steady
state.
A systemis givenhere which,in a random
fluctuatingfashion,passes
in time through a
variety of states.The statesthat are formedat
timestl, t2, "', t{, "' form an ensemble.
The
expectation
value of any observable
is formed
by takingits average
valueoverthis ensemble
(time average).
Alternatively, we can consider all possible
states of the systemat a given time t,. These
statesalsoform an ensemble
with a givenprob-
ability distribution.Under generalassumptions
(ergodie hypothesis)averaging over this en-
semblecan be substitutedfor the time average.
In otherwords,the systempasses
in time (arbi-
trarily closely)througheachpossible
state,sub-
ject to a certain probability distribution.
In hydrologythe problem usually arisesto
predictthe stochastic
behaviorof someobserv-

5. 754 •DRX• r.
able probabilities,
suchas its mean,its dis-
persion,probabilitiesof high or low seriesof
values,
etc.Theproblem
canbesolved
by simple
averagingover the ensemble.
The di•culty is
that the ensemble
is only imperfectlyknown
from a short interval of measurements. There-
fore modelswill have to be set up for the
extrapolation from the measurementsto other
time ranges.
To establish such models we must look for
physicalreasons
for choosing
them. A randomly
fluctuatingsystemis subjectto certain influ-
ences.If theseinfluences
are independent,and
their effects
are additivewith regardto a certain
observableX(t), •hen the distribution of •he
latter will be Gaussian,and •he dispersionof
•he cumulativevariablex(t) • •X(t)dt will be
proportional •o the time t (BrownJan condi-
tion)
=
wherewe have assumed
that X(t) = 0 and
angiebracketsdenotethe average.
A modification of the above model is obtained
if it is assumed that there is a correlation be-
tween subsequent
influences.
This correlationis
givenby the correlationfunctionC(s)
C(s)
=•••••X(t)X(t
+s)dt
if the individual measurements X are taken at
the t•es t. The integralis to be taken over all
times; since,in practiceonly a finite numberof
measurements
can be made, the integral is an
approximationto reality. If there is • correla-
tion betweensubsequent
influences,
the auto-
correlation
timeL, is [Pai, 19'57]
Lt = C(8)ds
It is •plicitly assumed
that the integralcon-
verges.
If it doesconverge,
thenat leastasymp-
totically for longtimes (for t >>L•)
(x(t)) = x(t) at t
This is characteristic of Brownian behavior. For
shorttimes(• <<L•) wehave
t
SCYIEIDEGGER
sothat the behaviorof (x• (t)) is bracketed
be-
tween proportionalityto t"-and proportionality
to t.
The idea of a finite correlation time Lt is
basedon the assumption
that if we wait long
enough,
the influences
producing
the X(t) are
independent
and additive,so that in the long
run the processcan be consideredas Gaussian.
This reflectsitself in the integrabilitycondi-
tion for C(s). It is possible
to conceive
of
processes where the corre!ation time is not
finite,but infinite; i.e., C(s) is not integrable.
The behavior of (x2) must then lie between t
and t2 as demonstrated
above,but this is also
for infinitetimes(the limiting(x-ø)
• t isnever
reached).
It is evenpossible
to formally
setup
a mathematical
modelfor a stochastic
process
that always(for shortand longtime ranges)
hasa behaviorfor (x•) ~ t•, with 2H a con-
stant betweenI and 2, independently
of the
timerangeunderconsideration.
By postulating
a certain type of self-shnilarity[Mandelbrot,
1965], such mathematical modelscan be made
soasto bedetermined
by the soleparameter
H.
It shouldbe emphasized,
however,
that the
physicalreasons
for postulatingsuchself-simi-
lar models are not clear. In the modified Gauss
process,we have assumedan initial correlation
anda longrunindependence
of theeffects
giv-
ingriseto thetimeseries
X(t•) orx(t•). Physi-
callythisseems
to be • very satisfactory
pro-
cedure.
On the otherhand,whenpostulating
• self-similar
process
with H • • (fractional
Brownian motion), we really make a com-
pletely arbitrary extrapolationfrom short to
very long time ranges.Actuallypostulating
a certain
H -v•• simply
amounts
to postulating
• prioria certainasymptotic
behavior
for X(t)
for longtimes.Suchan assumption
is of course
also
madein theusualGauss-Markov
model(ul-
timate independence
of the events),but there
isat leasta physical
reason
for doingso.
Usingself-similarity
(withH v•:•) to ex-
trapolate the correlated behavior from a finite
time spanto.an asymptotically
infiniteoneis
physically
completely
unjustified.
Furthermore,
usingself-similarity
to intrapolate
to a very
short
timespan,
anultraviolet
catastrophe
arises.
Since
thecycles
aresupposed
tobeself-similar,
it canbe seen
that the phenomenon
becomes
so
highlyoscillatory
for smallfrequencies
asto be
nolonger
definable.
Thisisphysically
absurd.

6. Stochastic Models 755
Any physicaltime seriesconsistsof a finite
number of measurements that have been taken
over a finite total time range.Empiricallyit is
foundthat within this rangethe behaviorof the
mean-root square variance of the cumulative
series is describedby proportionalityto t
whereH is betweenx/• and 1. This is exactlyas
predictedby the usual Gauss-Markovcorrela-
tion model with a suitable choiceof Lt. To ex-
trapolate beyond the time range over which
measurements
havebeenmade,or to intrapolate
into the intervals between the individual meas-
urements,we haveto proceedfrom a reasonable
physicalassumption.
To simply postulateself-
similarity for the seriesand extrapolateto in-
finitelylongandto intrapo•ateto infinitelyshort
time intervalsis physically unsound,as evi-
denced
by the ultravioletcatastrophe
mentioned
above. There is therefore a physicalreasonto
reject the fractional Brownian motion model at
the high frequencyend.Why then shouldthere
be a reasonto acceptit for the low frequency
end? If the correlation between events is such
that the latter cannot be assumed to become in-
dependentand additive for the contemplated
extrapolation
time range,thereis no possibility
of predictingwhat this correlationwill be like.
This correlation has to be measured or some
physicalreason
mustbe givento makean edu-
cated guess.Self-similaritymay be a mathe-
maticallypretty idea•but unless
a physicalrea-
son is given for its realization in nature it is
not clear why the asymptoticbehaviorof the
correlationengendered
by it should
be any more
likely in realitythan any other.
The modelof hydrologicphenomenathat
physically most reasonableis therefore one of
additiverandomeventscausing
the fluctuation
of someobservable.
Theseeventsmay be cor-
relatedby a seriesof effects,eachwith a cer-
tain correlationintensity and time range
Unless
some
physical
reasons
aregiven,extrapo-
lationsin • steadystate can thereforebe made
only basedon empiricism.
The value of H (i.e.,
the correlationbehavior) for thetime rangeT is
found under investigation, and an educated
guessis madethat this H doesnot changevery
much for additional time intervals of the size
of maybe T/2.
REFERENCES
Dirac, P. A.M., The Principles o• Quantum Me-
chanics,309 pp., Oxford University Press,New
York, 1947.
Horton, R. E., Erosional development of streams
and their drainage basins; hydrophysical ap-
proach to quantitative mophology, Geol. $oc.
Amen.Bull., 56, 275-370,1945.
Leopold,L. B., and W. Langbein,The conceptof
entropy in landscapeevolution, U.S. Geol. Surv.
Pro•. Pap. 500A, A1-A20, 1962.
Mandelbrot, B., Une classede processus
stochas-
tiques homothetiquesa sol, Application a la loi
climatologiquede H. E. Hurst, Compt. Rend.,
260, 3274-3277, 1965.
Pai, S., ViscousFlow Theory, vol. 2, D. van Nos-
trand, New York, 1957.
Ranalli, G., and A. E. Scheidegger,
A test of the
topologicalstructureof river nets,Int. Ass.$ci.
Hydrol. Bull., 15(2), 142-153,1968.
Scheidegger,
A. E., A thermodynamicanalogyfor
meander systems, Water Resour. Res., 3(4),
1041-1046, 1967.
Scheidegger,A. E., Horton's law of stream order
numbersand a temperatureanalogin river nets,
Water Resour. Res., 4(1), 167-171, 1968a.
Scheidegger,A. E., Microcanonical ensemblesof
river nets, Bull. Int. Ass. $ci. Hydrol., 13(4),
87-90, 1968b.
Sommerfeld, A., Thermodynamics and Statistical
Mechanics,AcademicPress,New York, 1964.
Shreve, R., Statistical law of stream numbers,J.
Geol.,74(1), 17-37, 1966.
Thakur, T. R., and A. E. Scheidegger,
A test of
the statistical theory of meander formation,
Water Resour.Res.,4(2), 317-329,1968.
Woldenberg, M. J., Horton's laws justified in
terms of allometric growth and steady state in
open systems,Geol. Soc. Amer. Bull, 77, 431-
434, 1966.
(ManuscriptreceivedJanuary 13, 1970.)