Investigating the multifractality of point precipitation in the Madeira archipelago

13 Pages·2009·0.89 MB·English

Nonlin. ProcessesGeophys.,16,299–311,2009
Nonlinear Processes
www.nonlin-processes-geophys.net/16/299/2009/
©Author(s)2009. Thisworkisdistributedunder in Geophysics
theCreativeCommonsAttribution3.0License.
Investigating the multifractality of point precipitation
in the Madeira archipelago
M.I.P.deLima1,2andJ.L.M.P.deLima3,2
1DepartmentofForestry,ESAC/PolytechnicInstituteofCoimbra,3040-316Coimbra,Portugal
2InstituteofMarineResearch–CoimbraInterdisciplinaryCentre,Coimbra,Portugal
3DepartmentofCivilEngineering,FacultyofScienceandTechnology–Campus2,UniversityofCoimbra,
3030-788Coimbra,Portugal
Received: 1February2008–Revised: 30March2009–Accepted: 30March2009–Published: 15April2009
Abstract. The purpose of this work is to contribute to other processes, its temporal and spatial variability are im-
a better understanding of the variability of precipitation in portantissuesinmanystudiesandareasofresearch.
the Madeira archipelago. This archipelago is located in the
Althoughthescientificcommunityhasdedicatedmuchat-
Atlantic subtropical belt under the direct influence of the
tentiontostudyingprecipitationovertheyears,italsoagrees
Azores high pressure system. It is formed by Madeira Is-
thatmoreresearchisstillneededtogetabetterunderstanding
land(728km2)andPortoSantoIsland(42km2)andbytwo
ofthevariabilityinthisprocess.Therehasbeenacontinuous
other groups of very small inhabited islands. The complex
searchformethodsthatcanadequatelycharacterizeprecipi-
topography of the islands in the Madeira archipelago and
tationatdifferentscalesand,atthesametime,considerable
their small size play a crucial role in the local precipitation
efforthasbeenputintoanalyzingdatafromdifferentorigins.
regime,whichismarkedbyhighspatialvariability.
Butasyetthesetwotasksareincomplete.
This paper explores the invariance of properties mani-
Point precipitation data from some areas have not yet
festedacrossscalesanddeterminesthefractalandmultifrac-
been studied satisfactorily: for example, hardly any precip-
talbehaviourobservedinthetemporalstructureofprecipita-
itation studies have focused on the Portuguese and Spanish
tionusingdailyand10-mintimeseriesfromseverallocations
NorthAtlanticIslands(i.e.theAzores,MadeiraandCanary
scattered over the main islands. The period covered by the
archipelagos), and even fewer for scales of less than one
precipitationrecordsis34yearsforthedailydataandalmost
month. Themajorityofthefewstudiesthathavebeencon-
4monthsforthe10-mindata. Theresultsshowthatthetem-
ductedwerebasedonlargetemporalscales(seee.g.Garcia-
poral structure of precipitation in the Madeira Archipelago
Herrera et al., 2001, 2003; Pereira et al., 2006), and stud-
exhibitsscale-invariantandmultifractalproperties. Theem-
ies undertaken under research projects SIAM (e.g. Miranda
pirical exponent functions describing the scaling statistical
et al., 2006) and CLIMAAT (see e.g. de Azevedo et al.,
properties of the precipitation intensity were characterized
1999). ForstudiesofdailypointprecipitationseeMarzolet
using multifractal parameters; these parameters are increas-
al.(2006a,b). Typically,theseAtlanticislandshaveasmall
ing our awareness of the dynamics of this process in these
areaandstrongorographicamplitude,andsothespatialvari-
islands.
ation of climatic conditions is greater there than in flatter
regions. The diversity of climate singularities is explained
by the elevation, relief and advective processes, which are
1 Introduction highlysensitivetosmallvariationsinasynopticsituation.
Thisiswhatmotivatedthepresentstudyofprecipitationin
Precipitationisahighlynonlinearhydrologicalprocessthat
theMadeiraarchipelago.Itexplorestheinvarianceofproper-
exhibitslargevariabilityoverawiderangeoftimeandspace
ties manifested across scales and determines the fractal and
scales. Because precipitation is the driving agent of many
multifractal behaviour of the temporal structure of precipi-
tation. In this archipelago total annual precipitation varies
Correspondenceto: M.I.P.deLima greatlywithgeographicallocation,andthis,togetherwiththe
([email protected]) marked seasonal precipitation regime observed, has strong
PublishedbyCopernicusPublicationsonbehalfoftheEuropeanGeosciencesUnionandtheAmericanGeophysicalUnion.
300 M.I.P.deLimaandJ.L.M.P.deLima: MultifractalityofprecipitationinMadeira
impacts on society and the environment. The local pre- Fractal theory (e.g. Mandelbrot, 1977, 1982) deals with
cipitation regime affects the management of water and bio- simple scaling, which means that such behaviour is deter-
environmental resources, land use, agriculture and tourism, minedbyoneparameter,calledfractaldimension(e.g.Man-
andotherfactors. Itishopedthatthecharacterizationofthe delbrot, 1977, 1982; Feder, 1988; Falconer, 1990; Hastings
dynamicsofprecipitationwiththemultifractalapproachmay andSugihara,1993). Indefining(fractal)dimensiontheidea
improveunderstandingofthehydrologicalregimeintheis- ofmeasurementatscaleδ isfundamental. Foreachscaleδ,
lands, and therefore help to define measures supporting the irregularities (i.e. details, variability) on smaller scales (i.e.
sustainable development of the archipelago. Although the scalesofsizelessthanδ)areignored. Thenotionofdimen-
methodsusedheretostudyingrainarenotnovel,webelieve sion is associated with the (scaling) behaviour that is found
thatthisapplicationhasmorethanalocalinterest; thelarge inthemeasurementsforδ→0.
differencesinprecipitationtotalsobservedacrosstheislands, The number N of non-overlapping intervals of side
δ,A
overdistancesofonlyafewkilometres,areuncommonand δ necessary to cover an arbitrary (fractal) set A in a 1-
thisstudyhelpsclarifyingseveralfeaturesofthisprocessun- dimensionalspaceshouldsatisfythe(power-law)relation
dersuchconditions. N ≈λ−DA (1)
δ,A
meaning that N is proportional to δ−D in the limit δ→0.
δ
2 Someaspectsofscalingtheories In Eq. (1) parameter D is the (fractal) dimension of the
A
set A. This parameter can be obtained by the box-counting
The invariance of properties maintained across scales can method(seee.g.Feder, 1988; Falconer, 1990; Hastingsand
be mathematically investigated using fractal and multifrac- Sugihara, 1993). In practical applications of this method,
tal theories. These theories have been used to study many theD-dimensionalspaceoftheobservationsiscoveredwith
differentnaturalprocessesandsystems,includingprecipita- gradually-decreasing(itiscommonthatthesizeisdecreased
tion,andastheyarewelldocumentedintheliterature,weare gradually by a factor of 2), non-overlapping boxes of side
onlyreviewingtheintroductoryaspectofthesetheoriesand δ. For every grid-size, the number of boxes that contain at
analysis methods. There are presently different formalisms least one point of the set being analysed are “counted”. If
embracing multifractal theory. For a discussion of the rela- the set exhibits scale-invariance, plots of log(N ) against
δ,A
tionbetweentheformalismusedinthisworkandotherfor- log(δ)shouldyieldalinearrelationindicatingthescalingin
malisms, which is outside the scope of this work here, see, theformofthepowerlaw(Eq.1). ThefractaldimensionD
A
forexample,Marshaketal.(1997);Kantelhardtetal.(2006). can be estimated from the slope of the regression line fitted
Scale-invariantstudiesofthetemporalstructureofprecip- tothedata.
itation started by using fractal theory (e.g. Hubert and Car- For rainfall, the geometric structure that is the “support”
bonnel, 1989, 1991; Olsson et al., 1992) and evolved to- oftheprocesscanberegardedasafractalobjectembedded
wards the use of multifractal theory (see e.g. Lovejoy and in the 1-dimensional space of time and is defined as the set
Schertzer, 1990, 1995; Ladoy et al., 1991, 1993; Hubert, of rain periods observed in a particular location. Its frac-
1992, 1995; Tessier et al., 1992, 1993, 1996; Hubert et al., tal dimension, D, is between 0 and 1. Thus, this approach
1993;OlssonandNiemczynowicz,1994;deLimaandBog- deals only with the binary question of occurrence and non-
ardi,1995;Olsson,1995,1996;OlssonandNiemczynowicz, occurrenceoftheprecipitationprocess;inpracticethisdefi-
1996;BurlandoandRosso,1996;Harrisetal.,1996;Svens- nitionisassociatedwithanintensitythreshold.Veryroughly,
sonetal.,1996;Bendjoudietal.,1997;deLima,1998,1999; thedimensionofasettellshowdenselythesetoccupiesthe
deLimaandGrasman,1999;KielyandIvanova,1999;Dei- metricspaceinwhichitlies(e.g.Falconer,1990).
dda,2000;Sivakumar,2001;deLimaetal.,2002;Veneziano Multifractal theory (e.g. Mandelbrot, 1972, 1974;
and Furcolo, 2002; Nykanen and Harris, 2003; Veneziano HentschelandProcaccia,1983;Grassberger,1983;Schertzer
andLangousis,2005;Kantelhardtetal.,2006;Venezianoet and Lovejoy, 1983) deals instead with multiscaling. It can
al.,2006;Venugopaletal.,2006;LangousisandVeneziano, handlethedifferentintensitylevelsofprocesses,namelypre-
2007;Garc´ıa-Mar´ınetal.,2007;Royeretal.,2008). cipitation. Multifractalbehaviourisdeterminednotbyone,
Scale-invariance leads to a class of scaling rules (power butbyaninfinityofscalingexponents.
laws)characterizedbyscalingexponents. Thisallowsthere- The multifractal temporal structure of the precipitation
lationshipofvariabilitybetweendifferentscalestobequan- process can be investigated by studying the (multiple) scal-
tified: statisticalpropertiesofscale-invariantsystemsatdif- ing of the statistical moments of the precipitation intensity
ferent scales (i.e. on large and small scales) are related (SchertzerandLovejoy,1987). Thescalingofthemoments
by a scale-changing operation that involves only scale ra- of the precipitation intensity is described by the exponent
tios. Thusscalingtheoriesaredevelopedinnon-dimensional function K(q). The notion of moment can be generalized
frameworks. toanyrealvalueq. ThemomentsscalingfunctionK(q)sat-
isfies
<Rq>≈λK(q) (2)
λ
Nonlin. ProcessesGeophys.,16,299–311,2009 www.nonlin-processes-geophys.net/16/299/2009/
M.I.P.deLimaandJ.L.M.P.deLima: MultifractalityofprecipitationinMadeira 301
where <Rq> is the (ensemble) average qth moment of the for moments q in the nonlinear regime, q>0 and α6=1
λ
(non-dimensional)precipitationintensityR onascalespec- (Schertzer and Lovejoy, 1987). For universal multifractals,
λ
ifiedbyλ.Parameterλisintroducedasthescaleratioλ=L/δ the(dual)expressionforthefunctionc(γ),inEq.(3),isob-
used here instead of the length δ of the scale of homogene- tainedwiththehelpofLegendretransforms.
ity,definedforasetofsizeLembeddedina1-dimensional InEq.(5)theLe´vyindexα,alsocalledthedegreeofmul-
space. Tonon-dimensionalizetheprecipitationintensityR , tifractality, is defined in the interval [0,2] although for the
λ
theintensityonatimescaleofresolutionλcanbedividedby caseα=1anexpressiondifferentfromEq.(5)isusedtode-
the ensemble average intensity of the process. The scaling scribetheempiricalK(q)function. ParameterC describes
1
q
of the moments can be tested with log-log plots of <R > the sparseness or non-homogeneity of the mean of the pro-
λ
on different scales, against the scale ratio λ. The empiri- cess: itisthecodimensionofthemeansingularity. Themul-
calK(q)functionscanbeobtainedusingtheTraceMoment tifractal parameters C and α can be determined, for exam-
1
method; for a detailed discussion on this methodology see ple,usingtheDoubleTraceMoment(DTM)method(seee.g.
e.g.SchertzerandLovejoy(1987). LovejoyandSchertzer,1991).
Anotherequivalentwaytoinvestigatethemultifractaltem- For non-conserved processes a third parameter, H, is in-
poralstructureofprecipitationusestheprobabilitydistribu- troduced in the functional forms of the scaling functions,
tionsoftheprecipitationintensity.Theprecipitationintensity K(q)andc(γ),whichcharacterizesthedeviationfromcon-
threshold level is evaluated here with the order of singular- servation(thatis: <Rλ>=λ−H);forconservedprocessesis
ityγ oftheintensitiesR ∼λγ (e.g.FrischandParisi,1985; H=0. This parameter is sometimes known theoretically for
λ
Halsey et al., 1986; Schertzer and Lovejoy, 1987). Please different processes, although usually it must be determined
note the definition just given for singularity, since this term experimentally; for estimation methods of parameter H see
bearsseveralmeaningsintheliterature(seee.g.Marshaket e.g.Tessieretal.(1993)andLovejoyandSchertzer(1995).
al.,1997). Thescalingoftheprobabilitydistributionsofthe InterpretationsforthemeaningofH>0andH<0aregiven
precipitation intensity (e.g. Schertzer and Lovejoy, 1987) is intheseworkstogetherwithadiscussionofimplicationsfor
givenbytheexponentcodimensionfunctionc(γ): characterizingmultifractalprocesses.Foradiscussionofsta-
tisticalstationarityandstochasticcontinuityseee.g.Daviset
Pr(Rλ≥λγ)≈λ−c(γ) (3) al.(1994,1996)andKantelhardt(2008).
Discussions of the relation between the universal multi-
This statistical characterization of multifractals arises
fractalparametersandparametersusedinothermultifractal
directly from multiplicative cascade processes (see
formalisms,andoftheUniversalMultifractalModelincom-
e.g. Schertzer and Lovejoy, 1987, 1988). In this case
parison to multifractal cascade models that were developed
thescalingbehaviourcanbetestedwithlog-logplotsofthe
for applications in strange attractors and chaos are given in
probabilityofexceedingdifferentlevelsoftheprecipitation
e.g.Kantelhardtetal.(2006),LovejoyandSchertzer(2007)
intensity R , observed on scales of different levels of reso-
λ andKantelhardt(2008).
lutionλ,againstthescaleratioλ(e.g.Lavalle´eetal.,1991).
The empirical codimension function c(γ) can be estimated
3 Theprecipitationdataandthestudyarea
using the Probability Distribution/Multiple Scaling method
(seee.g.LovejoyandSchertzer,1990;Lavalle´eetal.,1991).
3.1 Briefdescriptionofthestudyarea
Thetwomultifractalscalingexponentfunctionsc(γ)and
K(q)arerelatedthroughtheformulas The precipitation data used in this work were recorded
in several meteorological stations located in the Madeira
K(q)=max{qγ−c(γ)} and c(γ)=max{γq−K(q)} (4)
γ q Archipelago (Portugal). This archipelago is situated in the
NorthAtlanticOcean,about900kmfromMainlandEurope
which are Legendre transform pairs that establish a one-to- (see Fig. 1). It is formed by Madeira Island (728km2) and
onecorrespondencebetweenordersofsingularityγ andsta- PortoSantoIsland(42km2)andbytwoothergroupsofvery
tisticalmomentsq (e.g.FrischandParisi,1985).
small inhabited islands. Porto Santo Island is 40km north-
Inmultifractals,thereisonlyaconvexityconstraintonthe
eastofthemainisland.
scalingexponentfunctionsc(γ)andK(q).Functionalforms
Theislands’mostfrequentcirculationpatterniscontrolled
for these scaling exponents are proposed by the Universal
bytheAzoreshighpressuresystemwithnortherlyflow,and
Multifractal Model (Schertzer and Lovejoy, 1987) that are
the second most important is the westerly flow, due to per-
theoretically derived on the basis of stable, attractive multi-
turbations associated with the normal track of Atlantic de-
fractal processes. This model has a multiplicative cascade
pressions. The climate in the archipelago is mild all year
structureandisbasedonLe´vystochasticvariables. Forthe
round, though there are exceptions at the highest altitudes,
momentsscalingfunctionofconservedprocessesitis:
wherethetemperatureislower.Meanannualairtemperature
is between 8◦C in the highest altitudes and 18–19◦C in the
K(q)= C1 (cid:0)qα−q(cid:1) (5) coastalregions.
α−1
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302 M.I.P.deLimaandJ.L.M.P.deLima: MultifractalityofprecipitationinMadeira

Fig.1.AltimetrymapoftheMadeiraarchipelagoandlocationofthe6precipitationmeasuringsitesusedinthisstudy.
The low pressure systems that cross the Atlantic in the thedatawererecordedinthemeteorologicalstationofFun-
winter and move down to Madeira’s latitude, or the ones chal/Observato´rio, from November 2001 to February 2003.
formedbetweenthearchipelagoandmainlandPortugal,can Theprecipitationinthisperiodwas555mm.
result in high precipitation. The northerly flows (which are Theselectionofthedailydatasetsusedinthisworkaimed
associatedwiththeeastbranchoftheAzoresHigh)arepre- atanalysingdatafromvariouslocationsintheislandsovera
dominantinsummer. However,intheMadeiraarchipelago, common period; this was considered crucial to understand-
thelocalscaleclimateisinfluencedbytheregionalscalecli- ingbetterthespatialvariationofrainfallinthearchipelago.
mateandbytheorographyandorientationofairmassmove- Acompromisewastoselectdailyprecipitationdatarecorded
mentovertheislands. ThereliefofMadeiraIslandisdom- at5locationsscatteredoverMadeiraIslandandatoneloca-
inated by a central peak (1862m) which divides the island tion on Porto Santo Island, for the period 1961–1994 (see
into a northern and a southern part. The highest point on Table 1). A daily time series from Funchal for the period
PortoSantoIslandis517m. 1961–2005 is also used; mean annual precipitation in this
period was 618.7mm. Figure 1 shows the location of the
3.2 Precipitationtimeseries measuring sites and also the altimetry of the islands. The
precipitationdatadonotprovideuniformspatialcoverageof
Thedataanalysedinthisworkaredailyand10-minprecip- the islands; in Madeira the measuring sites are located on
itation time series collected by the Portuguese Institute of the northern and southern coasts, and in the island’s central
Meteorology (IM) in the Madeira Archipelago. The distri- highlands.
bution of measuring sites with available long daily records
issparseinthearchipelago, andsomesectorsoftheislands 3.3 Annualandseasonalprecipitationregimes
are not covered (e.g. the western sector of Madeira Island).
Moreover, the time span of the records varies from station Annual precipitation differs not only between the two main
tostation. Althoughseveralautomaticstationsarepresently islandsofthearchipelagobutalsowithinMadeiraIslandit-
being operated on the islands, their continuous records of self. This was not confirmed for Porto Santo Island since
precipitation are quite short and there are a lot of gaps in thisstudyonlyusesonedatasetfromthisisland. Neverthe-
the time series records available. For this study we se- less, the small size of this island and a considerable (rela-
lected a 3.8 months long time series without missing data; tively) smoother relief than in the Madeira Island allow us
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M.I.P.deLimaandJ.L.M.P.deLima: MultifractalityofprecipitationinMadeira 303
Table1. Dailyprecipitationseries. Thestatisticsarefortheperiod1961–1994. Parametercvisthecoefficientofvariation,definedbythe
ratiobetweenthestandarddeviationandthemean.
Code Stationname Island Locationinthe Altitude Meanannual cvannual
island (m) prec.(mm)
524 PortoSanto/Aeroporto PortoSanto Central 78.0 379.4 0.27
522 Funchal/Observato´rio 49.3 631.9 0.30
Southernsector
385 LugardeBaixo 40.1 638.8 0.29
365 Santana Madeira Northernsector 400.5 1458.0 0.18
370 BicadaCana 1571.0 2906.3 0.25
Centralhighlands
373 Arieiro 1582.5 2923.5 0.21
to anticipate much smaller spatial precipitation variability aa)) 550000
than in the main island. In the Madeira Island there are
marked differences between the northern and southern re- m)m) 440000
mm
gions, and the more inland regions (see Table 1). In the n (n ( 330000
oo
period 1961–1994 the wettest site was Arieiro, in the cen- pitatipitati 220000
tral part of Madeira Island, with an average yearly precip- ecieci
prpr 110000
itation of 2924mm; the driest location was in Porto Santo
Island, with379mmperyear. Itshouldbestressedthatthe 00
SSeepp OOcctt NNoovv DDeecc JJaann FFeebb MMaarr AApprr MMaaii JJuunn JJuull AAuugg
mean annual precipitation in Madeira more than quadrupli-
cates over distances of roughly 10km (e.g. from Funchal, bb)) al al 2200 PPoorrttoo SSaannttoo
in the southern sector, to Arieiro, in the central highlands, nnunnu LLuuggaarr ddee BBaaiixxoo
wtiohAnerloethftohtuhegemhsuttoaltttiaipollnyp,irneitgcsipfmaitcaottonirothnilsys4tr.d6oi)sn.tgrilbyudtieopnenidssvoenrythseimloilcaar-. centage of acentage of acipitationcipitation 11110505 FBASFBASrruuiaiacciieennnnaaiiccttrr aaddhhoonnaaaaaall CCaannaa
The precipitation regime in the islands shows a strong sea- y pery perprepre 55
hh
sonality (Fig. 2): maximum monthly rainfall occurs during ntlntl
oo
autumnandwinterandtherearefrequentlyverydryoreven mm 00
SSeepp OOcctt NNoovv DDeecc JJaann FFeebb MMaarr AApprr MMaaii JJuunn JJuull AAuugg
rainlessmonthsfromApriltoSeptember. Althoughthereare
mmoonntthh
strongvariationsbetweenthedifferentlocationsintermsof
meanmonthlyprecipitation(Fig.2a),onaveragethemonthly Fig.2.Averagemonthlydistributionoftheannualprecipitation,for
percentagesofthecorrespondingannualprecipitationforthe sixlocationsintheMadeiraarchipelago,fortheperiod1961–1994:
different stations are almost identical (Fig. 2b). For the six (a)indepth(mm);(b)asapercentageofannualprecipitation.
stations studied, Fig. 2b shows the average percentage con-
tributionofthemonthlyprecipitationtotheannualprecipita-
tion.From1961to1994thewettestanddriestmonths,onav- 4.1 Distribution of rainfall occurrences characterized
erage,wereDecemberandJuly,respectively. Inthisperiod, byfractaltheory
approximately 44% of the annual precipitation felt between
NovemberandJanuary,whichwasthewettesttrimester. The Totestthescale-invarianttemporalstructureofprecipitation
driesttrimesterwasJunetoAugust,withlessthan5%ofthe we first applied the box-counting method (see Sect. 2). No
annualprecipitation. thresholding was considered to convert the time series into
setsofpoints. However, theminimumamountofprecipita-
tion recorded by conventional gauges is 0.1mm. The box-
4 Scalingandmultifractalanalysisofprecipitationdata countingplotsobtainedforFunchalareshowninFig.3aus-
ingdailyprecipitationdatafortheperiod1961–2005andin
This section presents results of the characterization of the
Fig. 3b using the 10-min time series for the 3.8 months pe-
fractal and multifractal behaviour of temporal precipitation
riod, November 2001–February 2002. Analysis of the box-
intheMadeiraarchipelago, studiedwithbothdailyand10-
countingplotsforthedailydatashowsthatprecipitationoc-
mintimeseries(seeSect.3).
currences on scales from 1 day to about 16 days are char-
acterized by a fractal dimension 0.56 (see Eq. 1); using the
10-mindata,precipitationoccurrencesontherangeofscales
from10mintoroughly3.6daysarecharacterizedbyafractal
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304 M.I.P.deLimaandJ.L.M.P.deLima: MultifractalityofprecipitationinMadeira
Table2.Summaryofresultsofthefractal(box-counting)analysisofprecipitationdatafromtheMadeiraarchipelago:scalingrange,fractal
dimension,D,andcorrespondingstandarderror.
Station Scalingrange D Stand. Island Locationin
error theisland
Fordailydata,1961–1994:
PortoSanto/Aeroporto 1day–16days 0.613 0.038 PortoSanto Central
Funchal/Observato´rio 1day–16days 0.565 0.022
Southernsector
LugardeBaixo 1day–16days 0.534 0.012
Santana 1day–16days 0.714 0.033 Madeira Northernsector
BicadaCana 1day–16days 0.720 0.022
Centralhighlands
Arieiro 1day–16days 0.757 0.019
Fordailydata,1961–2005:
Funchal/Observato´rio 1day–16days 0.562 0.020 Madeira Southernsector
For10mindata,November2001–February2002:
Funchal/Obervato´rio 10min–3.6days 0.562 0.005 Madeira Southernsector

ss 1100000000 isexpectedtobeinfluencedbythesmallersizeofthissam-
ee
oxox DD==00..5566 pleincomparisonwiththedailydatasample. Moreover,the
bb
y y 11000000 10-min sample covers a wet period, which leads to the ob-
ptpt
mm servation of “saturation” (i.e. trivially the slope is −1) for
ee 110000
n-n- scaleslargerthanroughlyoneweek. Alsoforthedailydata
oo
nn
ber ber 1100 ssllooppee==--11 plotstheanalyses oflargertimescalesare affectedby“sat-
mm aa)) uration”, which is a practical problem of applying the box-
nunu 11 countingmethodtoprecipitationoccurrences. Nevertheless,
11 1100 110000 11000000 1100000000 110000000000
resultsindicateascalingbehaviourspanningawiderangeof
bbooxx ssiizzee ((ddaayyss))
timescales,from10minuptoabout2weeks.
eses 1100000000 The results of the box-counting analyses for the remain-
xx
oo
bb ing stations aregiven in Table 2; they showthat the scaling
y y 11000000 DD==00..5566
ptpt range is the same for all locations. The fractal dimensions
mm
n-en-e 110000 giveninTable2provideanensemblecharacterizationofpre-
nono cipitationoccurrencesovertimeattheselocations. Theyre-
er er 1100 ssllooppee==--11 flect that in the northern sector and in the central highlands
bb
mm bb))
nunu 11 oftheMadeiraIslandthefractionofrainfalleventsislarger
(i.e.thefractaldimensionislarger)thaninthesouthernsec-
11 1100 110000 11000000 1100000000 110000000000
tor of the Island, an effect that is related to the effect of
bbooxx ssiizzee ((1100 mmiinn))
orography on the rain distribution in time and space. Oro-

Fig. 3. Box-counting plots obtained with precipitation from Fun- graphicprecipitationoccursonthewindwardsideofmoun-
chal, Madeira: (a) for daily time series from 1961 to 2005; (b) tains and is caused by the rising air motion of a large-scale
for 10-min time series, from November 2001 to February 2002. flowofmoistairacrossthemountainridge,resultinginadi-
Straight lines were fitted to the data: between 1 and 16 days, for abaticcoolingandcondensation. Consequently,theleeward
thedailydataplot; andbetween10minandroughly3.6days, for side is drier. Moisture is removed by orographic lift, leav-
the10-mindataplot. Theregressionlinesareusedtoestimatethe
ingdrieraironthedescending(generallywarming),leeward
correspondingfractaldimensionsD.
side. Thisphenomenonresultsinsubstantiallocalgradients
ofaveragerainfall. FortheMadeiraIsland(seeTable1)the
windwardsidecorrespondstothenorthernsectorandthelee-
sidetothesouthernsector.
dimension 0.56. These values are estimated from the abso-
lutevaluesoftheslopesoftheregression(heavy)linesfitted The scaling range and the fractal dimensions reported in
to the left-hand side sections of the plots. The smaller up- various studies of point precipitation suggest that they de-
per limit of the scaling range observed for the 10-min data pend on geographical location, while expressing different
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M.I.P.deLimaandJ.L.M.P.deLima: MultifractalityofprecipitationinMadeira 305

11
qq>>11
44
11..11 qq
00..55
33 11..44 00..11
22 2323 qq>)>)λλ 00 0000....3535
RR
44 << 00..77
11 og(og( --00..55 00..99
)) ll 11..11
>>
qqλλ 00 --11 11..33
RR
00
<<
g(g( --11..55
oo --00..11 00..0000 11..0000 22..0000 33..0000 44..0000
ll
--00..22 lloogg((λλ))

--00..33 qq<<11 Fig.5.Log-logplotoftheaverageqthmomentsoftheprecipitation
00..2255 intensityRλ onscalesbetween10minand3.8months,againstthe
--00..44
00..4455 scaleratioλ. Themomentsq plottedareindicatedinthelegend.
--00..55 00..6655 Data are 10-min precipitation from Funchal/Observato´rio, for the
00..8855 periodNovember2001–February2002.Straightlineswerefittedto
--00..66 thedatabetween10minand7.1days.
00..00 11..00 22..00 33..00 44..00
lloogg((λλ))
(1961–2005), Fig. 4 shows the log-log plot of the average
qthmomentsoftheprecipitationintensityR ontimescales
Fig.4. Log-logplotoftheaverageqthmomentsoftheprecipita- λ
from1day(λ=16384)upto16384days(44.9years, λ=1),
tionintensityRλ onscalesbetween1dayand16384days,against
the scale ratio λ: (top) for moments greater than 1; and (bottom) againstthescaleratioλ.Figure4(top)showsmomentslarger
formomentssmallerthan1.DataaredailyprecipitationfromFun- than1andFig.4(bottom)momentssmallerthan1. Themo-
chal/Observato´rio, for the period 1961–2005. Straight lines were ments q plotted in Fig. 4 are indicated in the legend. The
fittedtothedatabetween1and16days. scalingrangeseemstoextendfrom1daytoabout16days,
similarlytothebox-countingresult.
For the 10-min data, Fig. 5 shows the moments’ scaling
distributionsofprecipitationintime. Ascalingrangesimilar plot of the precipitation intensity R on time scales from
λ
to what was found for the daily data used in this work, be- 10min (λ=16384) up to roughly 3.8 months (163840min,
tween1and16days,wasobserved,forexample,byTessier λ=1), against the scale ratio λ. The moments q plotted in
etal.(1996)andRoyeretal.(2008)whenanalysingrainfall Fig. 5 are again indicated in the legend. In this case, the
daily time series over France. Studies focussing on data of scaling range seems to extend from 10min to 7.1 days (see
other origins and record lengths report different results for regression line fits in Fig. 5). Attention is drawn to the un-
the scaling range; see e.g. Fraedrich and Larnder (1993), certainty in the estimation of large moments, which can be
Hubert (1995), Olsson (1995), de Lima (1999), Lovejoy et attributedtothesmalllengthofthetimeseries. Evenso,re-
al.(2003),Garc´ıa-Mar´ınetal.(2007). Therearealsostudies sultssuggestthatthemultifractaltemporalbehaviourofpre-
reportingscalingrangesforsmallsamplesandevenforstorm cipitationisvaliddowntoscalesofatleast10min,thuscon-
events. Usuallysuchstudiesidentifymuchnarrowerscaling firmingthescalingrangefoundinthemono-fractalanalysis
ranges than the ones reported by studies that analyse large (see Fig. 3b). The multifractal structure observed down to
samples(whichaimatcharacterizingtheensemble). Analy- suchsmallscalesneedsfurtherinvestigationusingmoredata
sesofstormeventsamplesyieldupperlimitsforthescaling (i.e.longerseries,datafrommorelocations),whichwerenot
rangethataretypicallyoftheorderofafewhours(evenless availableforthisstudy.
thanonehour),scalescomparabletothemeanstormduration
(seee.g.Olssonetal.,1992;Onofetal.,1996;Harrisetal., 4.3 Multifractalityofprecipitation
1996; Purdy et al., 2001; Venugopal et al., 2006). Also the
precipitationstatisticsare,inthiscase,forparticularevents. The empirical functions K(q), in Eq. (2), that describe the
scaling of the moments of the precipitation intensity from
4.2 Multifractalscalingrange 1 day up to 16 days are plotted in Fig. 6 (left): the func-
tionsweredeterminedusingtheTraceMomentmethod(see
To test the temporal multifractal character of precipitation, Sect. 2) using daily data. For the daily data from Funchal,
thescalingofthemomentsoftheprecipitationintensitywere the magnitude of the standard error associated to the esti-
investigated. For the daily data from Funchal/Observato´rio mate of K(q) for the different moments q are in the range
www.nonlin-processes-geophys.net/16/299/2009/ Nonlin. ProcessesGeophys.,16,299–311,2009
306 M.I.P.deLimaandJ.L.M.P.deLima: MultifractalityofprecipitationinMadeira
33333333 66666666
2222....22552255....5555 FFFFuuuuFFnnFFnnuuccuuccnnhhhhnnccaaaacchhllllhhaaaallll 55555555 FFFFuuFFuunnuuFFnnccnncchhuucchhaannhhaallccaallhhllaall
22222222 PPPPooooPPrrPPrrttoottoooooorr ttrr SSoottSSoo aaaaSS nnSSnnaattttaaoonnoonnttoottoo 44444444 PPPPooPPoorroottPPrroorrttttoooo ooSSrr ttSSaaSSooaann aaSSttnnnnoottaattoooonnttoo
LLLLuuuuLLggggLLuuaaaauuggrrggrrBBaaBBaarraaaaBBrriiBBiixxaaxxaaooiiooxxiixxoooo LLLLuuLLuugguuggaaLLggaarruuaaBBrrggrrBBaaBBaaaaiirrxxaaBBiiooxxiixxaaooooiixxoo
K(q)K(q)K(q)K(q)K(q)K(q)K(q)K(q)01010101........01015155150101515155........515515515515 SABSABSABSABariariariariccSABSABiicciineneSABSABneneaaaaariarititicctitiiiariarirrCCaarrCCneneccaaiiooaaoonenennaatitiaannaarrCCaatitiaannoorrCCaaaannnnooaaaannaaaaaannaannaaaa K(q)/C1K(q)/C1K(q)/C1K(q)/C1K(q)/C1K(q)/C1K(q)/C1K(q)/C1123123123123123123123123 SABSABSABSABariariSABSABcciiariarineneccaaariariiiSABSABccnenetitiiiaarrCCneneaaaatitiooariarirrCCtiticcaanniiaarrCCooaaneneooaannaaaannnntitiaarrCCaaaannaaaaoonnaannaaaaaannaa
00000000 00000000
----0000--..--..00550055....555500000000 11111111 22222222 33333333 44444444 55555555 --11----1111--0011000000 11111111 22222222 33333333 44444444 555555 55
----1111----1111 --22----2222--22
qqqqqqqq qqqqqqqq
00000000 00000000
00000000 0000....00222200....2222 0000..00..444400..44..44 0000..00..666600..66..66 0000..00..888800..88..88 11111111 00000000 0000..0022....220022..22 0000..0044....440044..44 0000..0066....660066..66 0000..0088....888800..88 111111 11
----0000--..--..00110011....1111 --00----00..0022..--..220022..22
K(q)K(q)K(q)K(q)K(q)K(q)K(q)K(q)--------00000000----....----....0000323200003232........32323232 K(q)/C1K(q)/C1K(q)/C1K(q)/C1K(q)/C1K(q)/C1K(q)/C1K(q)/C1----0000--------0000....00006464....----....646400006464....6464
----0000--..--..00440044....4444 --00----00..0088..--..880088..88
----0000--..--..00550055....5555 --11----1111--11
qqqqqqqq qqqqqqqq

Fig.6. EmpiricalmomentsscalingfunctiondeterminedusingdailyprecipitationdatafromtheMadeiraarchipelago,fortheperiod1961–
1994;thescalingbehaviourisvalidfortherangeofscalesfrom1dayto16days.Theleft-handsideplotsshowtheusualK(q)functionsand
theright-handsideplotsshowversionsofthemomentsscalingfunctionsrescaledwithrespecttothecorrespondingmultifractalparameter,
C1.Theplotsatthebottomshowdetailsoftheempiricalscalingfunctionsformomentssmallerthan1.
from0.001and0.020. Forthisdatasetthehighestvaluesof tion c(γ) in Eq. (2), i.e. by conducting a systematic study
the standard error are observed for the smallest moments q oftheprobabilitydistributions, butthisisbeyondthescope
investigated(seeFig.6)andthelowestvaluesareassociated of this work. Multifractals contain singularities of extreme
tothemomentsaroundq=1(butdifferentfrom1); standard orders which are related to algebraic decays of the extreme
errorslargerthan0.01areassociatedtoestimatesofthescal- events (i.e. algebraic tails of the probability distributions);
ingfunctionforq<0.3andq>3. Fortheotherdatasetsthe theseareassociatedwithprocesses’“violent”character(see
behaviourissimilar. e.g.Hubertetal.,1993;Olsson,1995;deLima,1998;Purdy
Figure 6 shows that here the empirical functions K(q) etal.,2001;deLimaetal.,2002;DouglasandBarros,2003;
consist of both nonlinear and linear sections, and this sta- Garc´ıa-Mar´ınetal.,2007).
tisticalbehaviourisdiscussedbelow. Thevariousstatistical IntermsofK(q)andc(γ), thelowrainratesarecharac-
parametersestimatedintheanalysisarepresentedinTable3 terized by K(0)=−Cs=c(γmin), where Cs is the codimen-
forthedifferentdatasets. Forthelargerainratepartofthe sion of the support (i.e. the regions with R>0; the corre-
statistics, the empirical K(q) functions are linear for mo- spondingfractaldimensionofthesupportisD=1−Cs),and
mentsq exceedingacriticalvalueqcrit. Suchadiscontinuity γmin=min(K0(q)),whichisthelowestsingularityinthedata.
inthefirstorsecondderivativeofK(q)ariseseitherbecause For the cases analyzed in this study, where Cs>0, then the
of divergence of moments at qcrit (called qD in this first or- lowK(q)islinearforq<qmin sothatthemomentsq<qmin
der case) or it is simply due to the inadequate sample size aredeterminedbythelowestnonzerorainrates. Thevarious
(at q , in the second order case), so that all moments larger criticalexponentsarelistedinTable3. Theestimatesofthe
s
than q are determined by the largest value in the sample fractal dimension obtained using the box-counting method,
s
(see e.g. Schertzer and Lovejoy, 1991). The corresponding listed in Table 2, are consistent with the estimates of the
largest singularity present in the sample can be determined (fractal)codimensionCs obtainedviatheanalysisofthesta-
fromγ =max(K0(q));thissingularityγ dominatesthe tisticalmomentsoftherainrates.
max max
statistics. The intercept of the corresponding linear section
in the K(q) function is an estimate of c(γ ). A complete
max
and systematic characterization of the extremes is best un-
dertaken with the help of the complementary scaling func-
Nonlin. ProcessesGeophys.,16,299–311,2009 www.nonlin-processes-geophys.net/16/299/2009/
M.I.P.deLimaandJ.L.M.P.deLima: MultifractalityofprecipitationinMadeira 307
Table3. Summaryofestimatesofthecriticalstatisticalexponentsthatdescribethescalingoftheprecipitationintensitydeterminedfrom
daily data. The symbols are defined earlier in the text. Parameters α and C1 were estimated using two methods. Besides being esti-
mated using the DTM method, the α values were also estimated using the radius of curvature of the empirical K(q) functions at q=1:
(cid:20) (cid:21)
RK(q=1)= (1+KK000((qq)))3/2 q=1=(1+CC11α)3/2;wefurthermoreestimatedC1=K0(1).Scalingrangeisfrom1dayto16days.
Station γmin γmax γmax/C1 c(γmin) c(γmax) C1 α
DTM fromempirical DTM fromempirical
estimate K(q) estimate K(q)
PortoSanto/Aeroporto 0.35 0.80 1.62 0.39 0.55 0.49±0.01 0.49 0.55±0.03 0.39
Funchal/Observato´rio 0.45 0.68 1.44 0.43 0.48 0.47±0.01 0.47 0.48±0.03 0.18
LugardeBaixo 0.50 0.90 1.92 0.47 0.50 0.47±0.01 0.48 0.50±0.04 0.18
Santana 0.18 0.52 1.27 0.29 0.63 0.41±0.01 0.41 0.63±0.02 0.38
BicadaCana 0.25 0.54 1.58 0.28 0.63 0.34±0.01 0.34 0.63±0.02 0.30
Arieiro 0.20 0.52 1.53 0.24 0.71 0.34±0.01 0.35 0.71±0.02 0.38
ThefunctionK(q)forthevaluesnearthemean(q=1)can range of values already reported in the literature for other
becharacterizedusingC =K0(1)andtheradiusofcurvature raingaugeprecipitationstudies. Dependingongeographical
1
R atq=1bytheparameterαviatherelation location and characteristics of the time series, both positive
K
" # andnegativevaluesofH,aswellasvaluesroughlyequalto
(1+K0(q))3/2 (1+C )3/2
R (q=1)= = 1 . (6) zero,havebeendeterminedfromdataanalyses;e.g.Royeret
K K00(q) C1α al.(2008)forprecipitationinFrancereported−0.2<H<0.2.
q=1
To better compare the scaling behaviour observed for the
Thischaracterizationofthecurvaturebyαhastheadvantage
datarecordedatthedifferentlocations,ontheright-handside
thatinthecaseofuniversalmultifractalsthissingleparame-
of Fig. 6 the same empirical scaling functions that are on
ter,α,characterizestheentireK(q)function.
theleft-handsidearere-plotted,thistimeafterrescalingthe
For the data sets analysed there is a discrepancy between
functions with respect to C . These plots show that the be-
1
the estimates of parameter α using the DTM method (see
haviouroftheK(q)functionsaroundthemean(i.e.q=1),at
Sect.2)andthemethoddescribedabove(i.e.usingEq.6),in
thedifferentlocations, isverysimilar. Nevertheless, forthe
that the DTM estimates yielded larger values of α. For C ,
1 low intensities, which are highlighted by the low moments,
theestimatesgivenbythetwomethodologiesareveryclose.
the behaviours are quite different, leading to differences in
Asummaryofestimatesofstatisticalexponentsispresented
thegeometricsupportoftheprocessasexpressedbythedif-
inTable3.
ferent fractal dimensions D (see Table 2). The results in-
Results show that the degree of multifractality of rain, as
dicate the more persistent character of precipitation in the
expressedbyparameterα,islowerforthedriersouthernsec-
highlands and windward side of the Madeira Island. There
tor of the Madeira Island and for Porto Santo Island. The
are also differences in the high intensities, which are now
datafromthehighlandsincentralMadeiraIslandexhibitthe
highlightedbythehighmoments,withPortoSantoandLu-
lower parameters C which indicates the smaller intermit-
1 gardeBaixoexhibitingthelargest(relative)singularities: it
tencyandsparsenessofthemeanoftherainprocessatthese isγ =max(K0(q)).Thissuggestsamoreviolentbehaviour
max
locations.
of precipitation recorded in these locations. Note that the
The estimate of γ =0.50 for Lugar de Baixo (Table 3),
min Porto Santo/Aeroporto data set is the only sample analyzed
whichisgreaterthanC ,isinterpretedasresultingfromun-
1 fromtheislandofPortoSanto,andthatitisthedriestloca-
certainty in the estimation of statistical parameters; for this
tionstudiedinthearchipelago.
dataitisalsothecodimensionofthesupportC equaltoC
s 1 Results show that for the Madeira data, the low rain rate
(the codimension of the singularity of the mean). For the
cut-offisincertaincasesnearlyequaltothemean(climato-
othercasesinTable3itisC >C ,asexpected.
1 s logical)rainrate,ascanbeinferredfromthefactthatc(γ )
min
Parameter H can be estimated using the relation
is almost as large as C (see Table 3). For the daily data
H=(β−1+K(2))/2, where β is the spectral exponent. The 1
thislowrainratecut-offisimposedbythe0.1mmminimum
averagevalueestimatedfromthe6individualdailyseriesis
recordingdefinitionoftheconventionalraingauges.
H=−0.134±0.018, for values of K(2) estimated from the
empirical functions. The error estimate is the dispersion of
the corresponding parameters estimated from the individual
series. The values of H found in this work are within the
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308 M.I.P.deLimaandJ.L.M.P.deLima: MultifractalityofprecipitationinMadeira
5 Concludingremarks typeofdataacquisition)cannotbeignoredbecausethesecan
affectthestatisticalcharacteristicsofthesample,asitmaybe
The characterization of the scale-invariant structure of pre- expected.
cipitationcanhelptoimproveourknowledgeaboutthispro- The results of the analyses of the different data sets used
cess in the Madeira archipelago. At present there is al- in this work suggest and confirm that the different fractal
mostcompletelackofstudiesoflocalprecipitationatsmall andmultifractaldescriptionsofrainreportedintheliterature,
scales,whicharetherelevantscalesinthisterritoryregarding which are sometimes apparently inconsistent, can probably
rainfall-inducedhydrologicalprocesses. Theverysmallsize be explained in terms of the different characteristics of the
of the drainage basins in these islands, and the very scarce precipitationdata(forexample:dataresolution,lengthofthe
and limited high-resolution monitoring of the precipitation timeseries,localclimate). Wereferespeciallytothediffer-
processintheMadeiraarchipelagoarelikewisestressed. entscalingrangesintimeandtomultifractalparametersthat
This work shows that scale-invariant and multifractal havebeenreportedbyvariousstudies(seereferencesgivenin
properties are present in the temporal structure of precip- Sects.2and4).Inparticular,fororographicprecipitationsee
itation in the Madeira archipelago; scaling is maintained also e.g. Harris et al. (1996), Purdy et al. (2001), Nykanen
from at least 10min up to about two weeks. This result and Harris (2003) and Deidda et al. (2006). Two main rea-
can contribute to improving the estimation of the statistical sons were identified by Deidda et al. (2006) as contributing
properties of precipitation at scales of less than one day by to different outcomes in rainfall analyses: a physical factor
providing a theoretical support for extrapolating properties lies in the different orographic range covered by the data (a
acrossscales: fromlargerscales(thatcanbeinvestigatedus- limitedrangemaynotrepresentasignificantobstacleforper-
ing daily data) to smaller scales. This is an important issue turbations); a second explanation may be the differences in
becausemanyhydrologicalstudiesrequireknowledgeofthe thelengthoftheobservationperiods(relativelybriefobser-
statisticsofprecipitationatshortertimescales(i.e.lessthan vation period may not allow filtering the single events high
one day) although generally only daily data is available. It space-time variability from the possible spatial heterogene-
is also expected that multifractal theory and its application ity). These issues need to be further investigated but they
in models will offer tools to produce high-resolution syn- stressthatinnon-homogeneoushydrologicalregionsextrap-
thetic precipitation data. The synthetic precipitation data olation of results to different geographical areas should be
can be used in many hydrological applications and studies consideredcarefully.
(e.g.rainfall-runoff,soilerosion,spreadofpollutants,urban Future work will include a more complete and system-
drainage),thusmitigatingtheshortageofhistoricalhighres- aticanalysisofprecipitationextremes,includingtheanalysis
olutiondatathatposesaproblemformostconventionalsta- oftheprobabilitydistributionsofprecipitationintensitieson
tisticalapproachestothestudyofprecipitation. different scales. The exploration of the available (but lim-
The empirical multifractal exponent functions character- ited) data from continuously recording gauges installed in
izingthescalingstatisticalpropertiesoftheprecipitationin- the Madeira archipelago is worth being pursued, too, to al-
tensity were described using a multifractal model based on lowafirmerdefinitionoftheprecipitationscalingregimesin
Le´vyrandomvariableswhichiscalleduniversalmultifractal the region. This characterization should be correlated with
model. Theresultsshowthatthedifferentdriversofprecip- theprevailingweathertypesfortheterritory,inordertobet-
itation in the islands lead to several precipitation signatures terunderstandtheextentoftheinfluenceoforographyonthe
over time. The dynamics of precipitation characterized by precipitationobservedacrosstheislands.
themultifractalapproachisincreasingourunderstandingof
thehydrologicalregimeintheislands.Thisstudygivescom- Acknowledgements. This work was carried out under research
plementaryinsightnotonlyintothemeanbehaviourofrain, projects POCI/GEO/59712/2004 and PTDC/GEO/73114/2006,
fundedbythePortugueseFoundationforScienceandTechnology
which is explored more often, but also into the full dynam-
of the Portuguese Ministry of Science, Technology and Higher
ical range of the process. It shows that the wetter spots in
Education (Lisbon). The authors acknowledge the collaboration
the northern sector of the Madeira Island and in the central
in this study of M. F. E. S. Coelho, of the Portuguese Institute
highlandsarecharacterizedbylessintermittentrain,andthat
of Meteorology, within the framework of the above-mentioned
raintherehasahigherdegreeofmultifractality.
projects.
Thedatasetsusedinthisworkdonotprovidetheuniform
spatial coverage of the precipitation distribution over the Editedby:A.Tarquis
MadeiraIslandswhichcouldallowustofullyunderstandthe Reviewedby:twoanonymousreferees
spatialvariabilityofprecipitationoverthearchipelago. Also
the limited continuous records available impede the small-
scale characterization of rain in time. Nevertheless, the re-
sultsstressthatthedependencyofthemultifractalityofpre-
cipitationonclimatological,geographicalandevenotherfac-
tors(e.g.precipitation-generatingmechanisms, samplesize,
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