WATER RESOURCES RESEARCH, VOL. 32, NO. 9, PAGES 2825-2839, SEPTEMBER
1996
A deterministic geometric representation of temporal
rainfall:
Results
for a storm
in Boston
Carlos E. Puente and Nelson Obreg6n
HydrologicSciences,Departmentof Land, Air and Water Resources,Universityof California,Davis
Abstract. The useof a deterministicfractal-multifractal(FM) representationto model
high-resolutionrainfall time seriesvia projectionsof fractal interpolatingfunctions
weighedby multifractalmeasuresis reported.It is shownthat the intrinsicshapeand
variabilityof an 8-hour Bostonstormrecordedevery 15 s on October 25, 1980, may be
encodedwholistically,
employingthe fractalgeometricmethodology.
It is illustratedthat
the FM methodologyprovidesvery faithful descriptionsof both major trendsand small
(noisy)fluctuationsfor this storm,resultingin preservationof not only classicalstatistical
characteristics
of the recordsbut alsornultifractaland chaoticpropertiespresentin them.
These results,and thosefor other storms,suggestthat a stochasticframeworkfor rainfall
maybe bypassed
in favor of a deterministicrepresentation
basedon projections.
the recordsandconsequently
do not captureall thevariability
observedat a fixedlocationwhen a stormpassesby. Typically,
thesemodelspreservesomestatisticsof the availabledata set
served in cluster models regarding their ability to have a
uniqueset of parametersirrespectiveof the data'sdegreeof
aggregation[Foufoula-Georgiou
and Guttorp,1986]. Despite
providing a theoreticalframework which includesbeautiful
ideasandmodels,theserepresentations
are typicallylimitedby
their analyticaltractabilitydue to the simplifyingassumptions
which they require, for example,stationarity,ergodicity,etc.
Also, the geometricoutcomesthey producedo not fully capture the very complexintermittencypatternsobservedin rain-
but do not keep the intermittentdetailspresentin the records.
fall.
1.
Introduction
Modeling the structureof temporal rainfall has attracted
muchattentionin the literaturefor the last30 years.Although
severalsophisticated(deterministicand stochastic)rainfall
modelsexist,they are built to preservesomecharacteristics
of
Sinceknowingdetailsmaybe importantin predictingthe rainfall process,an alternative approach to temporal rainfall
The possibilitythat rainfall couldbe understoodas a lowdimensionalchaotic processwas introducedby Rodriguezshould be considered.
Iturbeet al. [1989].Even thoughdeterminismbecomesan imExistingtemporalrainfall representations
may be classified portantcomponentof thisapproach,actualdynamicmodeling
dependcling
on theirbasicbuildingblocks,asfollows:(1) phys- (i.e., finding the equationsof motion from a time series)is
ically based,(2) point process,(3) chaotic,and (4) fractal quite difficult[e.g.,Crutchfield
andMcNamara,1987;Casdagli,
geometric.Physicallybasedmodelsapproximatethe physical 1989] and limits the practicalapplicabilityof the otherwise
processes
givingriseto the observedrainfall[e.g.,Georgakakos beautifulmethodology.The typicallengthof availablestorm
and Bras, 1984]. By employingbasicthermodynamics,
cloud records also precludesa complete analysisfollowing such
microphysics
principles,and integrationof the cloudon top of ideas.In any event,whetheror not there is a climaticattractor
a recordingstation,theserepresentations
produceintermittent remainsa relevanttopic of research[e.g., Tsonisand Elsner,
outcomeswhich nonethelessdo not capturethe inherentde- 1989].
tails presentin the records.This happens,in part, becauseof
A stochastic
fractalrepresentationof rainfallwasintroduced
approximatedparameterizations
which reflectour lack of un- byLovejoyand Schertzer
[1985,1990]andSchertzer
andLovejoy
derstandingof all the processes
takingplacewithin the atmo- [1987]via the notionof universalmultifractals.Their idea is to
sphere.
represent
rainfa!lasa realization
of a Levyprocess
andpa-
Typicalstochastic
point processmodelsapproximate
the rameterizeit via its codimensionfunction (basicallythe left
irregularandcomplexrain patterns(of a fractaland/ormultifractalnature)by superimposing
randomlyarrivingEuclidean
objects(e.g., rectangularpulses).Models with Poissonand
cluster-basedarrival processeshave been defined such that
somestatisticalfeaturesof the rainfalltime series(e.g.,mean,
variance,autocorrelations,
e.tc.)are preserved(see,for instance,Eagleson[1978];Kavvasand Delleur[1981];Smithand
Karr [1983];and Rodriguez-Iturbe
[1986],amongothers).Althoughit has been found that cluster-based
proceduresare
better than Poisson-basedones [e.g., •aldes et al., 1985;
Rodriguez-Iturbe
et al., 1987], inconsistencies
have been obCopyright1996by the AmericanGeophysicalUnion.
Paper number96WR01466.
0043-1397/96/96WR-01466509.00
portion of the multifractalspectrum[e.g.,Feder, 1988; C. E.
Puenteet al., Deterministicmultifractalswith negativedimensions?,submittedto PhysicalReviewE, 1996].Success
hasbeen
attained in characterizingsucha functionfor alternativedata
sets[e.g.,LovejoyandSchertzer,
1990].Eventhoughreasonable
lookingsimulations,havingintermittencyas found in rainfall,
may be obtained,it is difficultto find conditionalsimulations
with suchan approach.It is also pertinent to note that the
codimension
function(evenwhen compactlyparameterized),
as well as other classicalstatistical characteristicslike mean,
variance, and autocorrelation,do not really characterizea
givendata set. In fact, havingthe codimensionfunctionpreserveddoesnot imply the capturingof the actuallocationsof
the detailspresentin the records.
2825
2826
PUENTE AND OBREGON: GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
dy
lows.Beginwith a uniform distributionover an intervalI, say,
[0, 1], andselectan intermittency(redistribution)parameterp,
0 < p < 1. Then, redistributethe mass such that in the
interval [0, 1/2],p % of the massis uniformlydistributedand
likewisethe remaining(1 - p)% in the interval [1/2, 1].
Repeat this same processon each of the uniform pieces,ad
infinitum, to arrive at a deterministic binomial multifractal
f
y
measurewith parameterp. A sketchof sucha measure,dx, is
shownin the bottom of Figure 1 forp -- 0.3 and for 13 stages
in the cascade.As is seen, the measure dx is multifractal, i.e.,
it is singularand contains(in the limit) infinitelymanylayers
which correspondto intertwined Cantor sets in x. General
deterministicmultifractalsmay be constructedby splittingthe
massinto more than two piecesand by usingdifferentlength
scales[Mandelbrot,1989].
The set of transformations
dx
Figure 1. The fractal-multifractalframework in two dimen-
sions;dy = dx o f- •.
used in the context
of derived
distributionsare the fractalinterpolationfunctionsintroduced
byBarnsley[1986,1988].Theseare continuousfunctions
f that
interpolatea givensetof N + 1 pointsin the plane { (Xo,Yo),
(xi, Yi), "', (XN, YN); XO < Xi < "' < Xn) and whose
graphsmay be fractal. They are obtainediteratingN contracN
tile afiSnemappingsWn suchthat G = U n= 1tOn
(G), where
G = {(x, f(x)): x • [x o, XN]} is the graph off. The
followingare the mappingsusedand the conditionsthat guaranteethe existence
of a unique(andhencedeterministic)
setG:
x
In this work, a new procedurefor the quantificationof hydrologic(geophysical)
phenomenais reviewedanda particular
applicationto temporal rainfall for a high-resolutionstormis
given. The procedure to be used is the fractal-multifractal
representation(FM), as introducedby Puente[1992, 1994].
The basisfor developingsuchan approachis the fact that latest
developments
in physicsrecognize(1) the relevanceof details
in our ability to predict and (2) the possibilityof describing
apparentlyrandomsetsby meansof simpledeterministicrules
[e.g.,Moon, 1987].
The ideabehindthe FM approachis to think of the complex,
jagged,and intricatehydrologicpatternsasprojectionsof fractal functionswhich are "illuminated"via simple multifractal
measures.An important trait of the FM approachis that it is
entirely deterministic.Also, it doesnot require any statistical
assumptionssuch as stationarityor ergodicityor a minimal
lengthfor the recordsunder study.As will be illustrated,the
FM representation results in "random-looking" outcomes
which are not random
Wn(y)
---(an•n)(;)
-It(•n
n)
(1)
Wn(;:)
=(Xn-ll
Wn(•;)
• (•)
(2)
Cn
'
such that
Yn-1/ '
and0 -< Idnl < 1, for n = 1,...,
N.
These previousequationsallow solvingfor the parameters
an, Cn, en, andfn in terms of the interpolatingpoints' coordinatesand the "free" scalingparametersdn. The fractal dimensionD of the graphof an interpolatingfunctionis (1) D >
1, the solutionof X;Idnla
n - 1, if Eldnl> 1, and(2) 1 if
D--1
•ldnl -< 1 [Barnsley,
1988].Figure1 includes
an exampleof a
dyz
at all and which resemble actual rainfall
(geophysical)
records,not onlyin their appearancebut alsoin
z
z
{heirstatistical,
multifractal,
andchaotic
properties.
Thislast
point will be shownfindinga FM approximationof an 8-hour
storm recordedin Bostonevery 15 s.
dz
y
2.
x
The Fractal-Multifractal Approach
A large numberof deterministicmeasuresmay be obtained
usingthe FM methodology[Puente,1992, 1994]. These meady
suresdy are definedfollowinga classicalderiveddistributions
approach,usinga genericmultifractaldx as the parent distributionand a fractalinterpolatingfunctionf asthe transforma-
dx
tion(dy = dx o f- •). Thetwocomponents
thatmakeupthe
9onstruction,dx and f, and how they are combinedare reviewed next.
Figure 2. Three-dimensional
fractalinterpolatorprojections
and derivedmeasures:{(0, 0.1, 0), (0.5, 1, 0.4), (1, 0, 0.2)}.
The classicalcascademodel in turbulencegivesrise to the Herer? ) = r? ) = -0.6, 07 ) = 07) = 45, r? ) = 0.6,
most genericmultifractalmeasures.The binomial multiplica- r(22)= -0.6, 07 ) = 0(22)
= 45, p• = 0.3,P2 = 0.7. Angles
tive processresultsin a binomialmultifractalmeasureas fol- are in degrees.
PUENTE AND OBREG(SN: GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
2827
3O
2O
0.0
0.5
.0
0.5
.0
0.5
1.o
30
0
0.0
30
2O
lO
o
o.o
3ø1
I
20
•
10
0.0
0.5
30
1.0
'
20
0.0
1.0
0.5
30
'
20
0.0
0.5
Figure 3. Projectionsfrom two-dimensionalfractai interpoiatorsand a storm in Boston.
fractal interpolatingfunctionf that passesby the three data
points((0, 0), (0.5, -0.35), (1, -0.2)) shownby the soliddot
and which hasdx = -0.8 and d2 - -0.6. Observethat the
graph G is a self-a•ne fractal set with dimensionD = 1.48.
For detailson how to generatethe fractal interpolatingfunctionsin practice,seeBarnsley[1988]andPuente[1994].
Figure 1 also illustrateshow the derived distributionsapproachis usedto transforma binomialmultifractalmeasuredx
via a fractal interpolatingfunctionf, in order to determinea
deriveddistributiondy. Mathematically,dy is definedconsideringall relevantparenteventsin x and addingtheir measures,
tifractal) and (2) not strictlyself-similarnor self-atfine.As D
growsfrom oneto two,the measures
(1) progressively
become
absolutelycontinuous
(i.e., havea density)and (2) in the limit
become Gaussian.
Given the relevanceof multinomial multifractalsto represent intermittent natural phenomena, [e.g., Meneveauand
Sreenivasan,1987; Sreenivasan,1991], the derived measures
have a physicalinterpretation:They could be thought of as
imagesor "projections"of turbulence.The seeminglyrandom
appearanceandthe complexintermittencyof the measuresdy
(e.g.,seeFigure1) represents
the basisfor tryingto usethe FM
i.e.,dy(B) = dx(f-•(B)) = dx{x: f(x) • B), for a Borel approachto modelcomplexseries,as the onesgivenby rainsubsetB. The measuredy can be interpretedas a weighted fall. It is worth emphasizingthat havingdy not strictlyselfprojectionof the functionf, with the weightsgivenby dx.
similaror self-atfineis a welcomedproperty,sincea common
A greatvarietyof derivedmeasuresare obtainedby varying objectionagainstthe use of fractal geometryhas been that
the parametersoff anddx [Puente,1992,1994].Dependingon natural objectsdo not exhibitsuchgeometricbehaviorad inthe nature of the fractal interpolatingfunction,the following finitum.
overall behavior is found. When the fractal dimension D is
It is importantto stresssomemeritsthat the FM approach
closeto one, the derivedmeasuresare (1) singular(i.e., mul- may have over other methodscurrentlyin use. First, the FM
2828
PUENTE AND OBREG6N: GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
50
'
20
0.0
0.5
1.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
20
0.0
50
20
10
0
50
'
20
10
0.0
0.5
1.0
0.5
1.0
30
20
0.0
Figure 4. Projectionsfrom three-dimensional
fractal interpolatorsand a stormin Boston.
approachis entirelydeterministic:
Both the parentmultifractal
measureand the transformingmappingmay be uniquelyobtainedvia simplerecursiveprocedures[Puente,1994].The data
at hand are interpretedas a normalizeddistribution,a probabilitymeasure,whichis encodedvia a parentmultifractalmeasure and a unique fractal interpolatingfunction.Second,instead of concentrating on the statistics of the actual
realization(s),the FM approachfocuseson a wholisticdescription of geophysicalpatterns.Rather than concentratingon
describingor characterizingthe distributionof the data, the
FM approachusesderiveddistributions
to describethe data.It
appearsthat the FM procedure,or othersbasedon a similar
idea, may providea very parsimoniousrepresentationof natural data sets.
Table 1. SurrogateParametersfor "Storms"in Figure 3
Localization
Storma Xo
Xl
X2
X3
0.324
0.320
0.418
0.312
0.753
0.750
0.750
0.757
X4
1
2
4
5
-0.055
-0.050
-0.047
-0.062
0.158
0.160
0.149
0.162
6
-0.060
0.161 0.339 0.771 0.903
Y0
Regularity
Yl
Y2
Y3
Y4
dl
d2
d3
Intermittency
d4
0.936
0.77!
0.077
0.241 -0.337
-1.142
-0.747
-0.082
0.482
0.744
0.940
0.770 -0.500
-0.500
0.500 -0.800
0.750 -0.080
0.480 -0.500
0.949 -0.853
0.566
0.598 -0.530
0.004
0.745
0.058 -0.172 -0.475
0.940
0.780 -0.021
0.129 -0.355
0.883 -0.739
0.073 -0.500
0.598
0.768
0.324
0.416 -0.338
Numberscorrelatewith panelsin Figure3, startingat the top.
-1.108
-0.759
-0.062
0.500
Pl
P2
P3
P4
0.331
0.330
0.308
0.322
0.160
0.160
0.173
0.191
0.135
0.140
0.168
0.142
0.374
0.370
0.351
0.345
0.723 0.313 0.149 0.114 0.424
PUENTEAND :OB
'REG6N:GEOMETRICREPRESENTATION
OF •MPORAL RAINFALL
2829
Alternative
deriveddistributions.
maybe obtained
by con- Table 3. RelevantStatisticsfor Real and Fractalsideringfractalinterpolatingfunctionsoverhigherdimensions Multifractal Fitted Stormsin Boston
[Barnsley,1988;Barnsleyet al., 1989].Analogousto the two- Statistic
Order
Real value
Fitted value
dimensional
case,one mayconsiderN + 1 datapoints,{ (Xn,
1
0.445
0.439
Yn, Zn): XO< '" < XN; n = 0, 1, "', N}, and a set of N
M(y)
2
0.237
0.215
contractilemapsof the form
Wn
-cndnhn
kn In
+fn,
mn
(3)
M(y) *
an
such that
M(ey)
An= dn
hn r?
0(n
1)r(2)
.(2)
,.l•.
ta(2)\
In mn
r?COS
sin0(n
1>
,',,•
a(2>
] (4)
3
4
1
2
3
4
1
2
3
4
)
hasnormlessthan1 (i.e.,square
rootofmaximum
eigenvalue 13
ofA nrA
n < 1) and
,(q)
Wn YO =
Yn
\ ZO/
Zn
for n = 1, ...,
,
Wn YN
ZN
=
Yn ,
(5)
Zn
N. Now a three-dimensional "wire" G =
w• (G) t_J... t_JwN(G) appears,the graphof a continuous
deterministicfunctionf from [Xo, x•] to the yz plane. This
functionperfectlyinterpolatestheN + 1 datapoints,provided D(1)
-1.0
-0.2
0.6
1.4
2.2
3.0
3.8
4.6
5.4
6.2
0.265
2.502
0.547
0.258
-0.854
2.272
0.075
0.074
3.643
28.76
23
2.711
2.034
1.204
0.396
-0.391
-1.161
- 1.908
-2.629
-3.330
-4.014
-4.688
0.982
0.211
2.412
0.535
0.235
-0.925
2.273
0.077
0.072
3.169
24.65
16
2.686
2.139
1.202
0.396
-0.392
-1.153
- 1.873
-2.545
-3.184
-3.803
-4.411
0.984
thatall scaling
parameters
r(n
i) havemagnitudes
lessthan1.
The constructionfor a three-dimensional
fractal interpolating
functionleavesnowfour free parametersper map,withan, Cn,
(interen,fn, an, andkn all determined
in termsof r(n
•), r(n
2), 07), eters),and (3) the parentmultifractalredistributions
mittencyparameters).
0(n
2), andthe N + 1 datapoints.As in two dimensions,
G
becomes
fractalasthe magnitude
of r(n
i) increases
toward1
3.
Preservation
of Overall Features
[Barnsley,
1988;Puenteand Klebanoff,1994].
Figure2 is analogous
to Figure1 for the three-dimensional It is shownin thissectionthat it ispossibleto find parameter
case.In additionto a parent measurein x, a fractal interpo- combinations, both for two- and three-dimensional fractal inlating functionfromx to y, and a derivedmeasurein y, now terpolatingfunctions,suchthat the deriveddistributions
they
there are alsoan interpolatingfunctionfromx to z (a projec- produceresembleactualrainfallrecords.In orderto illustrate
tion of the uniquewire in thex - z plane),a derivedmeasure
thispoint,a varietyof derivedmeasures
are shownin Figures
in z, anda joint derivedmeasureinyz. The advantagein going 3 and 4 for two- and three-dimensionalfractal interpolating
to higher dimensionsis that "hidden"relationshipsbetween functions,respectively.So that comparisons
could readilybe
processes
in y and in z can now be incorporated.As may be made,thesemeasureswere generatedhavingthe samesizeof
seen,rainfallin time maybe modeledby considering
eitherdy an October25, !980, stormgatheredin Bostonevery15 s (i.e.,
or dz. Notice that as in the two-dimensional case, these mea-
1990 data points)and were scaledso that they all have the
sameintensityrangeasthe actualstorm(in 100 x millimeters
nature.
per 15 s). In both figures,made of six series,the data set
In summary,the FM approachrelieson the descriptionof depictingthe Bostonstormis includedin the third panelfrom
intermittent(normal•ed) data setsasweightedprojectionsof the top.
fractal interpolatingfunctions.The relevantparametersthat
As maybe seen,all "storms"havefeatureswhichare similar
needto be specifiedfor a givendataset are (1) the pointsby to those of the Boston data: They have a large peak, few
which the fractal interpolationfunctionpasses(localization intermediateones,and low intensitieswhichappearto contain
parameters);
(2) thescalings
(androtations)(regularity
param- noise.Observethat the deterministicFM representations
result in measureswhichresemblethe detailspresenton actual
recordsat a wide range of scales.In fact, it is not easy to
Table 2. RelevantSurrogateParametersfor "Storms"in
discriminatebetweenthe real seriesand thosethat were genFigure 4
erated,as theselatter representations
do sharesimilarstatisLocalization
tical and multifractalcharacteristics.
A completecomparison
betweenthe first deterministicstorm in Figure 3, which capStorma
Y2
Z1
Z2
tureswell the timingof the majorpeak, and the actualrecords
from the Bostonstormwill be givenin the next section.
1
0.0
0.90
-0.20
2
0.1
0.40
0.20
All projectionsin Figure 3 were obtained from a two4
0.0
1.00
0.00
dimensionalfractal interpolatingfunctionthat passedby five
5
0.1
0.50
0.00
datapoints.Table 1 includesall the FM parametersthat were
6
0.1
0.20
0.40
used.The projectionsin Figure 4, on the other hand, came
from three-dimensional
fractal interpolatingfunctionswhich
Numbers
correlat
e withpanelsin Figure4, startingat the top.
surespossessintermittencypropertiestypicallyobservedin
2830
PUENTE
ANDOBREGON:
GEOMETRIC
REPRESENTATION
OFTEMPORAL
RAINFALL
1.0
0.5
0.0
-0.5
-1.0
0.0
0.2
0.4
0.6
0.8
0.01 $
o.ooo
1
1ooo
Figure
5. Fitted
andrealrainfall
records
forthestorm
inBoston.
TheFMparameters
arethose
ofthefirst
storm in Table 1.
passed
bythreedatapoints.
Table2 includes
thevarying
FM thata cataloging
exercise
is a mustbefore
attempting
any
parameters
forthese
cases,
astheyalways
share
xo = 0, x• = sophisticated
searchalgorithm.
0.5,X2 = 1,yl = 1,Y2= 0, zo = 0; r(••) = r(•2) = -0.6,
Besides
interacting
witha catalog,
it isclearthata multidi0(11)
-- 012)
= 45,1'(21
) -=0.6,r(22)
----0.6, 07) = 0(22)
=- mensional
optimization
procedure
anda properly
defined
ob45;andp•= 0.3,P2= 0.7.Theangles
above
areindegrees.jective
function
are
required
to
fine-tune
the
FM
parameters
A pictorial
representation
of thebuilding
blocks
of thesecto a dataset.Thissection
reports
theresults
ondgraph
fromthetopinFigure
4wasalready
given
inFigure corresponding
obtained
with
a
heuristic
two-step
optimization
procedure
2. In fact,thesecond
stormcorresponds
to themeasure
dz in
which
wasfound
reliable
afterextensive
experimentation.
The
Figure2 (beawareof properorientation).
method
relied
onthemultidimensional
simplex
method
[Press
et al., 1989]to obtainpreliminary
FM parameters
andon
4. A Fitted Storm for the BostonRecords
simulated
annealing
[Otten
etal.,1989]
andsequential
quaIt iseasytoverifythattheFM approach
iscontinuous
with dratic
programming
[Zhou
andTits,1993]
forfinetuning.
The multidimensional
simplexmethodwasusedfroman
respect
toitsparameters,
i.e.,small
changes
in eitherdxorf
yielda small
change
intheoutcome
dy.Unfortunately,
there initialsimplex
defined
around
a setofFMparameters
fromthe
isnosimple
analytical
formula
thatgives
thederived
measurecatalog,
which
resulted
in "reasonable"
visual
agreement
with
dy,oritsmost
common
statistics,
interms
ofthese
parameters.
therecords
at hand.In orderto specify
theprocedure,
an
Thisimplies
thattheinverse
problem
offinding
theFMpa- objective
function
wasdefined
accounting
forweighted
sums
of
rameters
fora given
datasetcannot
beobtained
analytically
squareddifferences
betweenqualifters
of the real and FM
butratherrequires
a numerical
solution.
At theend,thisin- records.Theseattributesincludedclassical
statistical
indicaverse
problem
becomes
nontrivial
dueto(1)thelargenumber torsandmultifractal
characteristics:
(1) thefirsttenmoments
ofcombinations
ofsurrogate
parameters
(even
withfewinter- aroundthemeanin thetimeaxis,(2) thefirsttenmoments
polating
points)and(2) thepractically
infinitenumber
of de- aroundthemodein thetimeaxis,(3) thefirsttenmoments
rivedmeasures
thatmaybegenerated,
many
sharing
commonaround
themean
intheintensity
axis,
and(4)themass
expostatistical
andmultifractal
features.
It hasbeenourexperience
nentsr(q) (-2 <= q <= 7, at 0.2increments),
which
PUENTE AND OBREG6N: GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
0.0
0.5'
2831
1.0
0.0064
1.0
i
0.0032
0.0000
0.5
p
1
0.0
996
1990
1.000
10.0
0.719
1.0
0.438
0.1
0.157
0.0
-o.
125
100
0
0.0
0.•8
0.2.
....
200
0.4
l
•,,.•
O.
f4
0.00
•
2.14x10-0
0.6
' ' ' ' ' ' ' ' ....
300
0.8
400
500
0.01
o. 1o
1.oo
I 0
' ' '
1.05
!(01 ) = 0.446
a(01) = 0.2.32'
=
ß
,,if'4) =
D1
0.98
0.97
0.87
u
= o.o?
o(o ) = o.o?
1,n,(3,)= 3.644
=
6.4,?.x
10-3
I
0.68
I
I
I
0.50
0.70
I
i
0.97
i
1.25
1.50
Figure 6. Statistics
for the observedstormin Boston:autocorrelation
(p(,)), powerspectrum(S(•o)), data
histogram(f(dy)), andmultifractalspectrum(f(a)).
accountfor the sca.
ling lawsof the qth-order mo_ments
of the
data sets(t•):
the optimizationprocedure.The bestparametervaluesfound
via the multidimensionalsimplexmethod were used to redefine suchquantities.
• [/J•i(/5)]q
• /5-*(q),
(6)
Clearly, the procedureused for assigningweightsis by no
i
meansunique,asit maybe modifiedin a numberof ways.It is
where t•i(/5) is the total measure(mass)in the ith piece of worthremarking,however,that the objectivefunctionselected
resolution/5.
did not accountexplicitlyfor the autocorrelationor power
Each one of the entries on the four sets of attributes above
spectraof the real and FM records.In regardsto usageof the
had a componentin the objectivefunctionhavingas general optimizationalgorithms,it shouldbe addedthat no particular
structure
wi(1 - qi/Oi)2, wherewi is theweightof the ith care mustbe exercisedother than demandingthat the matheattribute, with qi and Oi representingthe FM and real at- matical structureof the model be preserved,for example,
tributes,respectively.The weightson each of the setswere Idnl < 1, •Pn = I, etc.
determinedas follows:(1) For the momentsthey decrease
The "optimal" fit obtained for the storm in Boston was
linearlyas the order of the momentincreasesand (2) for the already
included
asthetopstormin Figure'3.
Figure5 shows
massexponentstheyare constant.In additionto the individual the whole FM construction and the Boston records, for the
weightsjust explained,a setof four extraweightswasdefined bestfitted parametersas reportedin the first row of Table 1.
in order to properlygive emphasisto the four classesof at- Table 3 providesrelevant statisticalinformationfor the obtributesconsidered.These were determinedusingthe initial served and fitted data sets. These statistics include the followFM parametersaroundwhich the initial simplexwasbuilt, in ing: central momentsfor the recordsseenfrom the time axis,
sucha waythat at that pointin space,all four setsof attributes M(y); momentsaroundthe modefor recordsseenfrom the
contributeto the objectivefunctionequally.In orderto havea time axis,M(y) *; centralmomentscomputedfrom the rainfall
dynamicprocedurethat builds on current knowledge,these intensityaxis,M(dy); time lag where first local minimumof
four weightswere modifiedbeforestartingthe secondstagein autocorrelationfunctionhappens,,(tim); the scalingexpo-
2832
PUENTE AND OBREGON: GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
1.0
0.5
6.47x10
-•'0
1.0
-3
--0.5
• 3.24x10
0.0
996
1990
t•
1.00
10. oo
O. 72
1. oo
0.44
O. 1o
0.17
O, o1
-o.
11
100
o
o.o
0.266
f .....
200
300
o.s
o.a
' ' ' ' ' ' '
400
o. ol
500
D1
0.98
0.68
=
0.50
O. 0065
0.98
0.87
u
•(01) = 0.0?8
= o.
0.000
o. oooo
1.oo
1.05
'
p,(01) = 0.440
a(01)-- 0.2,15
ß
re,O) = o.ze
=
•;• 0.f 33
o. 1 o
0.70
,
0.97
•
1.23
1.50
Figure 7. Statistics
for the FM fittedstormin Boston:autocorrelation
(p(•')), powerspectrum(S(•o)), data
histogram( f(d y ) ), andmultifractalspectrum( f( a ) ).
nent/3 for the powerspectrumat largefrequencies,
i.e., S(
spectrumscaling(alsonot accountedfor duringthe optimiza-
--• to-s; massexponents
•'(q) for theindicated
ordersq; and tion) wasalsofound,as evidencedby a FM fitted exponent/3
the informationdimensionfor the data,D(1) = -d•'(q)/dq
at q = 1, whichmeasuresthe dimensionof a Cantorsetwhere
the measureconcentrates[Feder,1988].The massexponents
abovewere computedfitting the best regressionto equation
(6) usingfour consecutive
resolutions& in powersof 2.
As maybe seen,the FM description
(having17 parameters:
10 coordinates,
4 scalings,
and 3 independentintermittencies)
not onlymatcheswell the actualrecordspictorially,but it also
preservesthe optimizedstatisticaland multifracta!characteristics of the Boston storm. As observed in Table 3, excellent
agreementis foundfor all momentsof orderslessthan 4, with
modal momentson the time axisbeing sl!ghtlybetter than
those found around the mean. This fact is a clear indication
that the FM representationnicely capturesthe timing and
locationof the largestpeak observed.Even thoughno autocorrelationfunction attributeswere accountedfor in the optimization exercise,the first local minimum of the autocorrela-
tion function(oftentimesusedto define a scalefor chaotic
analYSiS;
seenextsection)wasproperlypreserved,
i.e., 16versus 23 lags are very close,consideringthat the time series
studiedare made of 1990values.A closeagreementon power
(computedfor frequenciesgreaterthan 0.4) whichdiffersby
only 1% from the exponentof the real records.As is seen,the
massexponents
functionz(q) is nicelycaptured,especially
for
valuesof the order q between -1 and 3. The information
dimensionD (1) is alsofitted accurately.
In orderto furtherverifyhowthe FM fit behaves,Figures6,
7, and 8 comparerelevantfunctionsof the observedand FM
fitted recordsfrom Boston.Observethat despitethe duration
of 1990datapointsfor the recordsin Boston,the FM projection providesreasonablyclosefittingsof both the autocorrelation functionand the powerspectrumof the actualdata.
Notice the similarityin the shapesof thesefunctions,with the
autocorrelation
of the actualdata beingrougherthan the one
predictedby the FM approach,which qualitativelymaintains
the delayswherethe autocorrelation
equalse-• and0, and
with the powerspectrumof the predictedrecordsexhibitinga
lessstablepowerlawbehaviorthanthe actualrecords(in terms
of regression
fit).
For the data histogram(f(dy)), there is indeed a good
visualagreementandexcellentfit of the momentsaspreviously
reportedin Table 3. Overall, the momentsin both time'and
PUENTE AND OBREGON: GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
2833
10000
10000
IO0
1 oo
....
!
....
i
5
....
10
....
i
i
15
i
....
10
15
ORDER
ORDER
10
10
•
....
5
05
1
-2
10o
-4
5
10
i
i
i
-2
0
2
4
6
ORDER
Figure 8. Momentsfor observedand FM fitted (*-*) stormin Boston:centralmomentsin time (M(y)),
modalmomentsin time (M(y)*), centralmomentsin intensityaxis(M(dy)), and massexponents(r(q)).
intensityaxesareverywell preservedfor ordersnot considered
during the optimizationexercise.As seen in Figure 8, this
happensfor time momentsof ordersup to 15.
As previouslyincludedin Table 3 and as depictedin Figure
8, the massexponents
functionsof the actualrecordsarenicely
preservedby the FM representation.
This impliesthat the
multifractalspectra,
f(a) versusa, for the observedandfitted
storms(found via Legendretransforms:a = - d,/dq and
f(c•) = aq + r(q)) shouldbe close,especiallyfor valuesof q
between-1 and3. Indeed,the bottomright-handcomersof
5.
Are Rainfall
Time
Series Chaotic?
Whether rainfall may be viewed as a deterministicchaotic
processhas attracted attention in the literature [e.g., Rodriguez-Iturbe
et al., 1989]. Given that it is possibleto model
complexdata setsby meansof the FM approach,it is relevant
to askif the outcomesof the deterministicprocedurelead to
deterministic
chaos.This sectionstudiesthisquestionby analyzingthe actualrecordsandthe "best"representation
for the
Boston
storm.
Figures6 and7 exhibitgoodagreementbetweenreal and FM
fitted multifractalspectra,especiallyfor its left-handportion.
The specificchaoticanalysiscarriedfor both seriesrelieson
standardtests as definedin the literature. The actual steps
function Of the data, i.e., the scalingstructurewhich corre-
the time seriesusinga time delay[e.g.,Packardetal., 1980];(2)
are asfollows:(1) phase-space
reconstruction
of
Thismeansthat the FM approach
preserves
the codimension undertaken
sponds
to largesingularities.
As realandfittedmassexponents determinationof minimal embeddingdimensionvia correla[e.g.,Grassberger
and :Pro'caccia,
deviatefor largemagnitudesof the exponentq, the multifrac- tion dimensionstabilization
Grassberger,
1990];
(3)verification
ofchaotic
behavior
via
tal spectraof both seriesdifferin boththeir tails.It is clearthat 1983;
K2
entropy
[e.g.,
Provenzale
et
al.,
1991];
(4)
verification
of
the right-handportionof the predictedmultifractalspectrumis
not pres6rvingthe scalingon the very smallvaluespresentin minimalembeddingdimensiohvia the false neighborsalgoetal., 1992];and(5) verification
of sensitivthe data, evenwhile changingthe numberof resolutionscon- rithm[e.g.,Kennel
via a positiveLyapunovexponent[e.g.,
sideredto define the number of regressionpointsnr used in ity to initial conditions
equation(6). This fact and the observationthat manymulti- Wolf et al:, 1985].
The resultsof the analysis
for bothof theBoston"records"
fractal models exhibit instabilities on the right-hand side
branchof the spectrumsuggestthat negativevaluesof q may are includedin Figures9-12. Figures9 and 11 includethe
analysis
and K2 entropycalculations
havebeenomittedfrom the optimizationexercise.Overall,the correlationdimension
to the first localminimumof
entropydimension,a stablequalifterof the spectrum,is well usinga delayr that corresponds
preservedby the FM measure.This is particularlytrue for the autocorrelationfunction of the real records (i.e., 23).
Thesefiguresincludethe followinginformation.
regressions
madewith four and sevenresolutions.
2834
PUENTE AND OBREGON: GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
Ioo
0.50
'• 0.25
1o-5
0.00
o.ool
1 .ooo
6
I
0
5
I0
!
1o
o)
,
o. ooi
1.ooo
i
5
1o
1o
8
(0
6
4
2
0
0
2
4
6
8
I0
N
Figure 9. Observedstormin Bostoncorrelationdimensionand K2(N ) entropy.Figure 9a showsphasespacecorrelationfunctions;Figure9b showslocalslopesof correlationfunctions;Figures9c,9d, and 9e show
correlationdimensions
for five large,five intermediate,and five smalldistances
r, respectively,
as indicated
fromrightto left on theuppermost
correlationfunctionin Figure9a bytheplussigns,asterisks,
opentriangles,
opensquaresand crosses;
Figure9f showsK 2 meanentropyand one standarddeviationband;and Figure9g
shows1/K 2 and one standarddeviationband.
Figures9a and 11a showthe phase-space
correlationfunctionsCs(r), which"count"the numberof points,in a phase
spacewith N coordinates,whose distancesare less than r.
Figures9b and lib showthe local slopeof Cs(r) (computed
via lag-onedifferences).Figures9c and 11c,9d and lid, and9e
and lie showthe correlationdimensionsV•v,definedfrom the
scalingrelation Cs(r) "• r vN,computedfor five large, five
intermediate,and five smallvaluesof the distancer, respectively.Suchvaluesare indicatedon the uppercorrelationfunction curve(i.e., for two coordinates)and are foundvia a regressionof three successive
pointsendingwith the symbolin
question;for example,the valuesreportedfor V•vand the plus
signare obtainedemployingCs(r) for distancesindicatedby
the opentriangle,the asterisk,andthe plussign.Figures9f and
11f showthe averageK2(N ) entropy,definedasthe logarithm
of the ratio of two consecutivephase-space
correlationfunctions,foundfor the "stable"regionin the correlationfunction
slopeand the band of plusand minusone standarddeviation,
andFigures9g and 11gshowthe averagevalueof 1/K2(N ) and
its corresponding
band of one standarddeviation.
As may be seenin Figures9a and 1l a, the computedcorrelation functions for both observed and fitted records, consid-
ering up to ten coordinates,are not perfect straightlines in
log-log scale.This leads to stable and unstablelocal slopes
(foundfrom two successive
points),asindicatedin Figures9b
and 1lb. While the actualdata leadsto a slopevalleybetween
two mounds,the local slopeson the FM fitted recordsshow
onlyone moundat smalldistances.
Noticefrom Figures9c,9d,
and 9e that the real data exhibit stable correlation
dimension
estimates(foundvia a regression
of threeconsecutive
points)
in the intermediatezone,yieldinga correlationdimensionof
3.68; see the graph with the crossesin Figure 9d. The FM
records,on the other hand, also lead to stable behavior and a
close correlation dimension of 3.44, but now for the asterisk
PUENTE AND OBREG(•N: GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
'T=
'7'=8
'
1oo
c,
.•
ß
ß
i
'
'
ß
!
ß
'
'
i
'
ß
'
!
'
ß
'
80
•
6o
!
ß
,
!
!
,
!
,
80
.• 60
• •o
2, 40
•
16
.........
1oo
2835
•
20
0
, , ,
0
i , , , i , , , i , ,
2
4
6
20
0
, i , , ,
8
, , , I , , , i , , , i , , , i , . ,
0
lO
2
4
6
ßr=
8
10
32
lOO
1oo
c,
80
c,
80
•
60
.•
6o
2, 40
2, 40
•
•
,80
20
o
o
,
o
o
• =
.....
16
Aro=
,
.
i
,
2
,
.
i
,
,
,
4
i
6
,
,
,
i
8
,
,
10
4
i , ' '[ ' i, ......
0.002
O. OO2
0.000
......................................................................................
,•
0.000
..........................................................
-0.002
-0.002
500
1000
1500
0
500
1000
1500
Figure 10. Observedstorm in Boston:false neighborsfor alternativedelays(r) and largestLyapunov
exponentfor alternativedimensions(No).
corresponding
to a smalldistance,i.e., Figure 11e. Thesecorrelation dimensionsare quite close to the value of 3.78 reported for this stormby Rodriguez-Iturbe
et al. [1989].
The K2(N ) entropyanalysisyieldsinconclusive
resultsfor
both the real and FM fitted storms.As may be seenin Figures
9f and 9g and 11f and 11g,K 2 remainspositivefor all embedding dimensionsN, but no apparent stabilizationof either
K2(N ) or its inverseis attained.The shortlengthof the time
seriesprecludesa more completeanalysisregardingthis at-
boundingthe optimal embeddingdimensionusingthe method
of false neighbors[Kennelet al., 1992]. By followingthe evolution of pointsin phase-space
andkeepingtrackof pointsthat
remain closedynamically,a verificationof a true attractor is
made. These figuresalso include the largestLyapunovexponents under alternativeconditions,i.e., the growth in phase
spaceof the largestprincipal axisin which an initial sphere
progressively
deforms[Wolfet al., 1985].Notice that the percentageof falseneighborsbehavesvery similarlyfor both real
tribute. Notice, however, that the behaviors of real and FM
and FM fitted records. As is seen, there is a marked decrease
fitted K2(N ) entropiesare quite similar.
Figures10 and 12verifythe correlationdimensionresultsby
in false neighborsas the embeddingdimensionis increased
from 2 to 4 for delaysr that rangefrom 8 to 32. At this stage,
2836
PUENTE AND OBREG6N: GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
ßr
=
23
N
=
100
10
-5
0.00'1
o
1.000
5
N
0
I
0.001
1.000
0
5
,
I0
10
8
6
i
2
4
6
8
i
i
,
10
N
Figure 11. FM fitted stormin BostoncorrelationdimensionandK2(N) entropy.Figure11ashowsphasespacecorrelationfunctions;Figure11bshowslocalslopesof correlationfunctions;Figures11c,11d,and 11e
showcorrelationdimensionsfor five large, five intermediate,and five small distancesr, respectively,as
indicatedfrom right to left on the uppermostcorrelationfunctionin Figure11aby the plussigns,asterisks,
open triangles,open squares,and crosses;
Figure 11f showsK 2 meanentropyand one standarddeviation
band;and Figure 11gshowsl/K2 and one standarddeviationband.
the percentageof falseneighborsfalls in all casesbelow 10%,
and only a mild rise is observedthereafter. These results
clearlysupportthe validityof the correlationdimensionvalues
givenbefore.
The largestLyapunovexponent(h•) for both real and FM
fitted seriesis givenin Figures10 and 12 for an optimalembeddingof No = 4 andfor a coupleof plausibledelays•. The
resultsfor the real data are clear.A positiveLyapunovexponent is found for both delaysconsidered.For the predicted
recordsthe Lyapunovexponents
varyin character.For mostof
the time, h• givespositivevalueswhichsignifychaoticbehavior, but there are seriesdurationsfor whichnegativevaluesare
found.Overall,the behaviorof bothreal andpredictedrecords
is similarin relationto the largestLyapunovexponent:Notice
that the scaleson both casesare quite closeand that, indeed,
positivevalues are found. Compare in particular the cases
when •-=
16.
In order to avoidissuesrelated to the small lengthof the
time series,the FM procedurewasusedto obtaina refinement
of theBostonrecords
over2TM
points.A complete
analysis
of
these"records"confirmedthe chaoticnatureof the FM projectionwhichnicelyapproximates
the Bostondataat the lower
resolution. This time, a correlation dimension close to 3.2 was
found, and the percentageof falseneighborsalsocamebelow
10% for four embeddingdimensions.
Thistime a positivelargestLyapunovexponentwasfound,evenfor the durationof the
series.
6.
Summary and Conclusions
A deterministic
geometricframeworkfor the description
of
complexhydrologic(geophysical)time serieshas been reviewed.The approachrelies on the use of fractal-geometric
objectsand in particularon the combinationof fractalinter-
PUENTE AND OBREG6N: GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
'7'=8
.7-=16
ß
ß
ß
i'
'
ß
ß
i
ß
'
'
i
ß
ß
''1
ß
,
,
i
,
,
,
i
,
.
,
i
,
,
,
100
1oo
•
2837
80
'
•
ß
,
,
.
• 80
ß• 60
ß• 60
•
40
•
20
40
•
20
ß
ß. . ! . . , i . , , i . , . i . . . •
0
2
0
4
6
-r=
•n
ß
ß
!
,
.
ß
i
o
10
2
4
6
!
8
10
24
,
ß
100
8
o
ß
ß
ß
i
ß
ß
'
i
'
ß
ß
1oo
80
•
80
.•
6o
,..9, 40
•
4o
•
R
20
½
20
0
. ß . i ß , ß ! . . ß !
0
2
T =
4
o
i . ß ß
6
o
8
l•
•vo=
o
4
T=
•4
•v•=
4
0.00œ
o. ooto
,.•
...............................................................
0.000
,.E
0.0000
.............................................
-o. ooto
-0.002
o
i
i
i
,i
I
i
,
ß
.
i
,
,
,ooo
,
i
.
,
-0.0020
................
o
500
lOOO
15oo
Figure 12. FM fitted stormin Boston:false neighborsfor alternativedelays(,) and largestLyapunov
exponentfor alternativedimensions
(No).
polationfunctionsand multinomialmultifractalmeasures.It
hasbeenillustrated,by meansof examples,
that somederived
measures
producedbythe fractal-multifractal
(FM) procedure
resemblehigh-resolution
rainfall records,both for two- and
three-dimensional
fractal interpolatingfunctions.
A faithfulFM representation
of a high-resolution
stormin
Bostonhasbeenpresentedin thiswork.A detailedcomparison
of the real and FM fitted time seriesrevealsthat the geometric
procedurenot onlycapturesthe timingandsizeof the largest
peakbut alsopreserves
the overallappearance
of the actual
records,includingsecondarypeaksand small noisyfluctuations. The FM representationwas reached minimizing a
weightedsumof squareddifferences
betweenattributesof the
real and FM outcomesof equal length.The obtained"optimal" solutionindeedpreserves
the followingattributes,explicitlyaccounted
for in theobjective
function:
(1) thefirsttencentral
and modal momentsof the recordswhen seenfrom the time axis;
(2) thefirstten centralmoments
of therecords
asseenfromthe
intensityaxis;and (3) the massexponents
functionof the data.
The FM representation
alsocaptured(1) higher-order
moments
alongboththetimeandintensity
axes;(2) theoverallshapeof the
autocorrelation
andhistogram
functions
for the records;(3) the
scaling
properties
presentin thepowerspectrum
of thedata;and
(4) the chaoticnatureof the setof observations.
2838
PUENTE AND OBREG(SN:GEOMETRIC REPRESENTATION OF TEMPORAL RAINFALL
Acknowledgments.The researchleading to this article was supSincesimilarlyfaithful FM representations
(to be reported
elsewhere)have also been found for a coupleof the high- portedby the Universityof CaliforniaWater ResourcesCenter,aspart
of Water ResourcesCenter Project UCAL-WRC-W-804. Valuable
resolutionstorms(one havingtwo peaks)gatheredin Iowa commentsby anonymousreviewersare acknowledged.
City every5 s by Georgakakos
et al. [1994],theseresultsimply
that there maybe no need in separatingtrendsand (arguably References
unimportant)small fluctuationswhen dealingwith rainfall Barnsley,M. F., Fractalfunctionsandinterpolation,Constr.Approx.,2,
records.In fact, thiswork suggests
a newglobalperspectivefor
303-329, 1986.
understandingrainfall, a perspectivein which major features Barnsley,M. F., FractalsEverywhere,Academic,San Diego, Calif.,
1988.
and noisydetailsare capturedjointly. Clearly,the presented Barnsley,M. F., J. Elton, D. Hardin, and P. Massopust,Hidden varianalysishintsthat a stochastic
frameworkfor rainfallmodeling
able fractal interpolationfunctions,SIAM J. Math. Anal., 20(5),
1218-1242, 1989.
maynot be necessary
and revealsthat the notionof projections
may providethe proper alternative.The FM procedureis in- Casdagli,M., Nonlinear predictionof chaotictime series,PhysicaD,
35, 335-356, 1989.
deed quite general, as it may generatepatternsof arbitrary Crutchfield,J.P., and B. S. McNamara, Equationsof motion from a
lengthswith a multitude of peaksand recordswhich contain
data series,ComplexSyst.,1, 417- 452, 1987.
periodsof no rain. This last point is easilymadeusinga Can- Eagleson,P.S., Climate, soil, and vegetation,2, The distributionof
annualprecipitationderivedfrom observedstormsequences,
Water
torian measureto computeweightedprojections.
Resour.Res.,14(5), 713-721, 1978.
There are of courseunansweredquestionsregardingthe FM
Feder, J., Fractals, Plenum, New York, 1988.
methodology,which need to be studiedin the future. These Foufoula-Georgiou,E., and P. Guttorp, Compatibilityof continuous
rainfall occurrencemodels with discrete rainfall observations,Water
include(1) the completeunderstanding
of the kindsof meaResour.Res.,22(8), 1316-1322, 1986.
suresthat may be generatedby the FM procedure;(2) the Georgakakos,K. P., and R. L. Bras, A hydrologicallyusefulstation
searchfor efficientalgorithms(includingthe proper objective
precipitationmodel, 1, Formulation, Water Resour.Res., 20(11),
function)to properlydescribereal hydrologic(geophysical) 1585-1596, 1984.
data setslike the one analyzedhere; (3) the identificationof Georgakakos,K. P., A. A. Carsteanu,P. L. Sturdevant,and J. A.
Cramer, Observationand analysisof midwesternrain rates,J. Appl.
the mostparsimonious
representation
(with the leastpossible Meteorol.,33(12), 1433-1444,1994.
numberof interpolatingpoints)of a givendataset;and (4) the Grassberger,P., An optimizedbox-assistedalgorithmfor fractal dimensions,Phys.Lett.A, 148, 63-68, 1990.
determinationof the physicalmeaningthat the FM parameters
Grassberger,
P., andI. Procaccia,Measuringthe strangeness
of strange
attractors,PhysicaD, 9, 189-208, 1983.
Clearly,the merit of the FM methodology
to representdata Kavvas,M. L., and J. W. Delleur, A stochasticclustermodel of daily
rainfall sequences,
WaterResour.Res.,17(4), 1151-1160,1981.
setswill reston our abilityto solvethe properinverseproblem
in the least possibleamount of time. As argued by Puente Kennel, M. B., R. Brown, and H. D. I. Abarbanel,Determiningembeddingdimensionfor phase-space
reconstructionusinga geomet[1996],rainfall(geophysical)
data setsare unique"signatures" rical construction,
Phys.Rev.A, 45(6), 3403-3411,1992.
of the physical,chemical,andbiologicalprocesses
takingplace Lovejoy, S., and D. Schertzer,Generalizedscale invariancein the
atmosphereand fractal modelsof rain, WaterResour.Res.,21(8),
within the atmosphere,and understandingthe geometryof
may possess.
theserecordsmay be very valuablefor rainfall modelingand
prediction.It is envisionedthat onceFM parametersare available for subsequent
data sets,suchsurrogategeometricinformation may be useful to study the dynamicsof rainfall. If
trendsin FM parameterspacemaybe elucidated,thismaylead
to predictionsof rainfall by chunks.Insteadof allowingpredictionsfew time stepsaheadat a time (e.g.,minutes,hours,or
daysahead),the FM approachmay resultin wholisticrepresentationsof rainfallrecordsat the sametimescaleasthe input
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(ReceivedOctober 13, 1995;revisedMay 3, 1996;
acceptedMay 9, 1996.)