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 precipitation. In this archipelago total annual precipitation varies Correspondenceto: M.I.P.deLima greatlywithgeographicallocation,andthis,togetherwiththe (iplima@esac.pt) 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 ≈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 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 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: =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.Lavalleéetal.,1991). The empirical codimension function c(γ) can be estimated 3 Theprecipitationdataandthestudyarea using the Probability Distribution/Multiple Scaling method (seee.g.LovejoyandSchertzer,1990;Lavalleéetal.,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 www.nonlin-processes-geophys.net/16/299/2009/ Nonlin. ProcessesGeophys.,16,299–311,2009 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 Nonlin. ProcessesGeophys.,16,299–311,2009 www.nonlin-processes-geophys.net/16/299/2009/ 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 characterized by a fractal dimension 0.56 (see Eq. 1); using the 10-mindata,precipitationoccurrencesontherangeofscales from10mintoroughly3.6daysarecharacterizedbyafractal www.nonlin-processes-geophys.net/16/299/2009/ Nonlin. ProcessesGeophys.,16,299–311,2009 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 Nonlin. ProcessesGeophys.,16,299–311,2009 www.nonlin-processes-geophys.net/16/299/2009/ 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)islinearforqC ,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 www.nonlin-processes-geophys.net/16/299/2009/ Nonlin. ProcessesGeophys.,16,299–311,2009 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, Nonlin. ProcessesGeophys.,16,299–311,2009 www.nonlin-processes-geophys.net/16/299/2009/

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