different methods for spatial interpolation of rainfall

BASE Biotechnol. Agron. Soc. Environ.201317(2),392-406 Focus on:

Differentmethodsforspatialinterpolationofrainfalldataforoperationalhydrologyandhydrologicalmodelingatwatershedscale.AreviewSarannLy(1,2),CatherineCharles(3),AuroreDegré(1)(1)Univ.Liege-GemblouxAgro-BioTech.Soil-WaterSystems.PassagedesDéportés,2.B-5030Gembloux(Belgium).E-mail:[email protected](2)InstituteofTechnologyofCambodia.DepartmentofRuralEngineering.RussianFederationBoulevards.POBox86.PhnomPenh(Cambodia).E-mail:[email protected](3)Univ.Liege-GemblouxAgro-BioTech.AppliedStatistics,ComputerScienceandMathematics.AvenuedelaFacultéd’Agronomie,8.B-5030Gembloux(Belgium).

ReceivedonNovember3,2011;acceptedonDecember6,2012.

Watershedmanagementandhydrologicalmodelingrequiredatarelatedtotheveryimportantmatterofprecipitation,oftenmeasured using rain gauges orweather stations.Hydrologicalmodels often require a preliminary spatial interpolation aspartofthemodelingprocess.Thesuccessofspatialinterpolationvariesaccordingtothetypeofmodelchosen,itsmodeofgeographicalmanagementandtheresolutionused.Thequalityofaresultisdeterminedbythequalityofthecontinuousspatialrainfall,whichensuesfromtheinterpolationmethodused.Theobjectiveofthisarticleistoreviewtheexistingmethodsforinterpolationofrainfalldatathatareusuallyrequiredinhydrologicalmodeling.Wereviewthebasisfortheapplicationofcertaincommonmethodsandgeostatisticalapproachesusedininterpolationofrainfall.Previousstudieshavehighlightedtheneedfornewresearchtoinvestigatewaysofimprovingthequalityofrainfalldataandultimately,thequalityofhydrologicalmodeling.Keywords.Rain,spatialdistribution,geostatistics,kriging,Thiessenpolygon,InverseDistanceWeighting(IDW),computerapplications,simulationmodels,hydrology.

Méthodes de spatialisation de données pluviométriques dédiées à l’hydrologie opérationnelle et à la modélisation hydrologique à l’échelle du bassin versant (synthèse bibliographique). Lagestionhydrologiquedesbassinsversantsetlamodélisationhydrologiqueexigentdesdonnéesrelativesauxprécipitations,variabletrèsimportante,leplussouventmesuréepar des pluviomètres ou des stationsmétéorologiques. Lesmodèles hydrologiques demandent souvent une spatialisationpréalableàlamodélisation,laspatialisationestdépendantedutypedemodèleetdesonmodedegestiongéographiqueetdelarésolutionutilisée.Laqualitéd’unrésultatestconditionnéeàlaqualitédelapluiespatialecontinuequidécouledelaméthoded’interpolationutilisée.L’objectifdecetarticleestdefournirunerevuesurlesméthodesdespatialisationdelapluieexistantesquisonthabituellementexigéesparlamodélisationhydrologique.Nouspassonsenrevuelesméthodesdebasegénéralementutiliséesetdesapprochesgéostatistiques.Lesétudesprécédentesmettentenlumièreunbesoindenouvellerecherchesurlesmoyensnécessairespouraméliorerladonnéedepluieetin fine,laqualitédelamodélisationhydrologique.Mots-clés. Pluie, distribution spatiale, géostatistique, krigeage, polygone de Thiessen, distance inverse, application desordinateurs,modèledesimulation,hydrologie.

1. INTRODUCTION

Computerhydrologicalmodelsthatsimulatemostofthehydrologicalcycleareanessentialtoolforhydrologistsand engineers in understanding and describing thehydrologicalsystem.Whenthesemodelsaresuccessfulin attaining accurate results, they can forecast whatwilloccurwithinthehydrologicalsystem.Thiscanbe

usefulforclimatestudies(e.g.intermsofprecipitationorevaporation),optimizedwatermanagementandlanduse changes: so-called scenario analysis. In the last30years,notonlythenumberbutalsothecomplexityof hydrological computer models have increasedenormously,due to theavailabilityofmorepowerfulcomputersandGeographicInformationSystems(GIS)(Singh,1995).Watershedmodelscanbecategorizedas

Spatialinterpolationofrainfall 393

physically-basedorconceptual,accordingtothedegreeof complexity and physical completeness present intheformulationofthestructure(Refsgaard,1996).Inaddition, hydrologicalmodels are classified as either“lumped”or“distributed”,dependingonthedegreeofdiscretizationwhendescribingtheterraininthebasin.The physically-based models describe the naturalsystem using the basic mathematical representationsoftheflowofmass,momentumandvariousformsofenergyatlocalscale(Refsgaard,1996).Thesemodelsarethereforealsodescribedas“distributed”andtheycan explicitly account for spatial variability withina watershed. Physically-based distributed modelsaregenerallybelieved tobepreferable to conceptualmodelsbecausetheybetterrepresentacertainrealityofthehydrologicalcycle(Ruellandetal.,2008).

Fully distributed and physically-based modelsalwaysrequireasinputsthemainspatiallydistributeddataset for theDigitalElevationModel (DEM), landuseanditsmanagement,soil,andclimate.Thequalityof these inputshasasignificant impactonthemodelformulationprocessandontheresults.Climaticdata,air temperature, solar radiation, and precipitation allprovideessentialcontrolsonsurfaceenergybalanceandecosystemprocesses.Among these climaticdata, theamount of precipitation, traditionally collected usingrain gauges or weather stations, is a very importantparameter, which has a direct impact on runoff orwatersheddischarge (Obledetal.,1994).Fora largewatershedscale,thespatialvariabilityofrainfallneedstobetakenintoaccountinsteadofusingarealaveragerainfall as the input for themodel. In this context, itisnecessarytogaininsightintotheday-to-dayspatialvariabilityofwatersheddischarge,groundwater levelandsoilmoisturecontent(Schuurmansetal.,2007a).In order to gain insight into the general behavior ofthehydrologicalsystem,itissufficienttouseaccuratepredictionsofarealaveragerainfalloverthewatershed.

The spatial variability of rainfall represents thedominant effect in the production of runoff; as thespatial variability increases, so does the significanceof appropriate rainfall characterization (Segondet al., 2007). Averaging of the rainfall input limitsthe accuracy of the model’s results. Under suchcircumstances,catchmentresponseishighlynonlinear,whichmeansthattheresponsetoanaveragedinputwilldiffermuchmore from the response to a distributedinput (Shahet al.,1996b).Whena single raingaugeis used tomodel the catchment response, the resultsbecome less accurate at both the sub-catchment andcatchmentscalesandthisalsoaffectsthereproductionofthehydrograph(Segondetal.,2007).Whenspatialhomogeneity of rainfall is assumed to be used in ahydrologicalmodel,rainfallvariabilitycausescertaineffects to occur. Spatial variability in rainfall affectsthecatchmentresponse(Shahetal.,1996a;Shahetal.,

1996b), the timingofpeak runoff (Singh, 1997), theestimationofmodelparameters(Chaubeyetal.,1999)andthehydrologicalmodeloutputs(Belletal.,2000;Segondetal.,2007).

Thedistributedmodelhasbeenshowntobesensitivetothelocationsoftheraingaugeswithinthecatchmentandhencetothespatialvariabilityoftherainfalloverthe catchment (Bell et al., 2000).Failing to consideradequately the spatial variabilityof rainfallwill leadtoerrorsinthevaluesofthemodelparameters,whichwillbewronglyadjusted tocompensate forerrors intherainfallinputdata(Schuurmansetal.,2007a).Thisisproblematicsincetherequireddensityofraingaugestocapturethespatialvariabilityexceedsthatnormallyavailable from routinemonitoring networks (Segondet al., 2007). Furthermore, rain gauge density overthe forecast catchments isoneof themain factors inattainingforecastaccuracyduringanextremeeventthatresultsinsignificantfloodinginamajormetropolitanarea(Looperetal.,2011).Therefore,theprecipitationinput,aswithotherclimaticdata,shouldbepreparedas spatially distributed data before being forced intothe hydrological modeling. However, measuring ateverypointwheredataareneededisprohibitedbytheassociatedhighcosts.

Spatially distributed rainfall can be interpolatedby a range of different methods but the complexityliesinchoosingtheonethatbestreproducesthemostaccurate data (Caruso et al., 1998).One approach isto measure associated ancillary data, which havebeenavailable since the late1960svia ground-basedmeteorological radarsandby remotesensingdeviceslocated on satellite platforms. The accuracy andconsistencyoftheseindirectprocessesforhydrologicalpurposesstillremaintobedetermined.Techniquesforinterpolatingrainfallmustbecalibratedandvalidatedby means of historical information (Lanza et al.,2001).From2000sonwards,standardrange-correctedradar products proved to be sufficiently informativeto capture the spatial variability of rainfall to beused inhydrologicalapplications(Schuurmansetal.,2007a). In particular, the use of radar products incombinationwithmultivariate geostatisticalmethodsproved tobebeneficial for spatial rainfall estimation(Velasco-Foreroetal.,2009;Schiemannetal.,2011;Verworn et al., 2011). However for regions withoutthese sophisticated instruments, direct ground-basedmeasurement deserves to be considered for spatialinterpolationprocesses.

The major problem, prior to the choice of themost suitable interpolation method, is related to theavailability of rainfall data. Sometimes, data arecontinuously recorded but the rain gauges are tooscattered. This is particularly true in mountainousareas,whereamountsofprecipitationaremoredifficultto forecast due to complex topography, distance to

394 Biotechnol. Agron. Soc. Environ. 201317(2),392-406 LyS.,CharlesC.&DegréA.

the sea and the presence of largewater bodies suchas lakes(Johnsonetal.,1995;Buytaertetal.,2006).Withinacomplextopography,thespatialscalefeaturesof rainfall are characteristically difficult to captureevenbymeansofamoderatelydensenetworkofraingauges. Topography impacts rainfall pattern throughso-called orographic effects, which refer to the riseinprecipitation rates induced throughaltitudedue touplift,adiabaticcoolingandresultingcondensationofhumidairmassesonwindwardmountainsides(Chowetal.,1988).Observationoftheseorographiceffectsandweather patterns has promptedongoing investigationintowhether precipitation, in general, increaseswithaltitude(Groismanetal.,1994;Sevruk,1997;Sinclairetal.,1997).Theseauthorsfoundthattherelationshipbetweenprecipitation and elevationdependedon theregion’s exposure to wind and synoptic conditions.Depending on the predominant wind direction, rainshadowsmayappearwhenmorerainfalloccursatornearthemountainpeakandmuchlessrainfalloccursatloweraltitudes(Sinclairetal.,1997).Eveninflatterareas, raingaugesneed tobe correctlydistributed inordertodetectairflowinfluences,thermalinversionsandotherphenomenathatcouldaffectclimaticpatterns.This difficulty of accurately reproducing continuousspatialrainfallhasledtonotablefailuresintheresultinghydrological responsemodels,whichare sensitive toinput volume at thewatershed scale (Nicotina et al.,2008).Atasmallerscale,rainfallvariabilityalsohasagreaterimpactonpeakflows(Mandapakaetal.,2009).Asthescaleincreases,theimportanceofspatialrainfalldecreasesanddistributionofcatchmentresponsetime,ratherthanspatialvariabilityofrainfall,becomesthedominantfactorgoverningrunoffgeneration(Segondetal.,2007).

Theobjectiveofthispaperistoprovideareviewofexistingspatialinterpolationmethodsofrainfall,whicharerequiredforhydrologicalmodeling.Wereviewthebasisfortheapplicationofsomecommonlyusedspatialinterpolation methods and geostatistical approachesandprovideanoverviewof thecharacteristicsof themethods.

2. SPATIAL INTERPOLATION METHODS FOR CALCULATING RAINFALL

A number of interpolation techniques have beendescribed in the literature, which reproduce thespatial continuity of rainfall fields based on raingaugemeasurement.Thesemethodscanbegenerallyclassified into two main groups: deterministicmethodsandgeostatisticalmethods.Somecommonlyused methods are briefly introduced here. Spatialinterpolation is generally carried out by estimatinga regionalized value at unsampled points based on a

weight of observed regionalized values. The generalformulaforspatialinterpolationisasfollows:

where Zg is the interpolated value at the requiredpoints,Zsiistheobservedvalueatpointi,nsisthetotalnumberofobservedpoints andλ=(λi) is theweightcontributingtotheinterpolation.

The problem lies in calculating the weights, λ,whichwillbeusedintheinterpolation.Thedifferentmethodsforcomputingtheweightswillbepresentedinthefollowingsections.

2.1. Deterministic interpolation methods

Regarding the first group of spatial interpolationmethods for measuring rainfall, the most frequentlyuseddeterministicmethodsare theThiessenpolygon(THI)andInverseDistanceWeighting(IDW),whicharebasedonthelocationofthemeasuredstationsandonmeasuredvalues.Inageneralway,theforecastoftheregionalizedvaluetakesintoaccounttheweightedaverageoftheobservedregionalizedvalues.– Thesimplestandmostcommonspatialinterpolation method,particularlyinrelativelyflatzones,istouse the simple average of the number of stations. However,useofthismethodhasdecreasedbecause itdoesnotproviderepresentativemeasurementsof rainfallinmostcases(Chow,1964).– TheThiessenpolygon (THI)method assumes that theestimatedvaluescantakeontheobservedvalues of the closest station. The THI method is also knownasthenearestneighbor(NN)method(Nalder etal.,1998).Themethodrequirestheconstruction of a Thiessen polygon network. These polygons areformedbythemediatorsofsegmentsjoiningthe nearbystationstootherrelatedstations.Thesurface ofeachpolygonisdeterminedandusedtobalancethe rain quantity of the station at the center of the polygon.Thepolygonmustbechangedeverytime astationisaddedordeletedfromthenetwork(Chow, 1964). The deletion of a station is referred to as “missing rainfall”. This method, although more popular than taking the simple average of the numberofstations,isnotsuitableformountainous regions,becauseoftheorographicinfluenceofthe rain(Goovaerts,1999).– The Inverse Distance Weighting (IDW) method isbasedonthefunctionsoftheinversedistancesin whichtheweightsaredefinedbytheoppositeofthe distanceandnormalizedsothattheirsumequalsone. Theweightsdecreaseasthedistanceincreases.This methodismorecomplexthanthepreviousmethods,

ns Zg=∑λiZsi(1)

i=1


since the power of the inverse distance function must be selected before the interpolation is performed.A lowpower leads to a greaterweight towardsagridpointvalueof rainfall fromremote rain gauges.As the power tends toward zero, the interpolatedvalueswillapproximatetheareal-mean method,whileforhigherlevelsofpower,themethod approximates the Thiessen method (Dirks et al., 1998). There is a possibility of including in this method elevation weighting along with distance weighting,InverseDistanceandElevationWeighting (IDEW). IDEWprovidesmore suitable results for mountainousregionswheretopographicimpactson precipitationareimportant(Masihetal.,2011).– In the polynomial interpolation (PI) method, a globalequationisfittedtothestudyareaofinterest using either an algebraic or a trigonometric polynomialfunction(Tabiosetal.,1985).Inorderto express the polynomial equation in the form of equation (1), the least squares and Lagrange approaches can be used. For more details on this method,seeTabiosetal.(1985).– The spline interpolation method is based on a mathematical model for surface estimation that fitsaminimum-curvaturesurfacethroughtheinput points.Themethodfitsamathematicalfunctionto aspecifiednumberofthenearestinputpoints,while passing through thesamplepoints.Thismethod is not appropriate if there are large changes in the surface within a short distance, because it can overshootestimatedvalues(Ruellandetal.,2008).– TheMovingWindowRegression(MWR)methodis agenerallinearregression,whichisconductedonly inareaswherearelationshipbetweentheprimaryand secondaryvariablesisthoughttoexist(Lloyd,2005). Forexample,inapplyingtheMWRmethodtorainfall, rainfallrepresentstheprimaryvariableandelevation the secondary variable. The rainfall estimation is based on the modeled relationship between the rainfallandelevationdataclosesttotheestimation location.

2.2. Geostatistical interpolation methods

The second group of spatial interpolation methodsfor measuring rainfall, geostatistical methods,constitutes a discipline connecting mathematics andearth sciences. Kriging is an example of a groupof geostatistical techniques used to interpolate thevalueofarandomfield.Matheron(1971)namedandformalizedthismethodinhonorofDanielG.Krige,aSouthAfricanminingengineerwhopioneeredthefieldofgeostatistics.Krigingisbasedonstatisticalmodelsinvolving autocorrelation. Autocorrelation refers tothe statistical relationshipsbetweenmeasuredpoints.Notonlydogeostatisticalmethodshavethecapability

of producing a prediction surface, but they can alsoprovidesomemeasuresof thecertaintyandaccuracyofthepredictions.

In kriging, the value of the interest variable isestimatedforaparticularpointusingaweightedsumoftheavailablepointobservations.Theweightsofthedataarechosensothattheinterpolationisunbiasedandthevarianceisminimized.Ingeneral,thekrigingsystemmust be Linear, Authorized, Unbiased and Optimal(LAUO).Kriging is thefirstmethodof interpolationto take into account the spatial dependence structureofthedata.Thereareseveraltypesofkriging,whichdifferaccordingtotheformappliedtothemeanoftheinterestvariable:– when it is assumed that themean is constant and known,simplekriging(SK)isapplied;– wherethemeanisconstantbutunknown,ordinary kriging(ORK)isapplied;– finally, universal kriging (UNK) is applied where themeanisassumedtoshowapolynomialfunction of spatial coordinates. So, in contrast to the other twotypes,thislasttypeofkrigingisnotstationary withregardtothemean.

Stationarity defines itself here by the constancyof themean,butalsobythecovariancebetweentwoobservationsthatdependonlyonthedistancebetweenthese observations.All the different types of krigingapply the stationarity of the covariance, or, moregenerally, the semi-variogram. This function, whichrepresents the spatial dependence structure of thedata,must be estimated andmodeled beforemakingthe interpolation. First of all, the experimental semi-variogramcanbecalculatedasbeinghalfthesquareddifference between paired values to the distance bywhichtheyareseparated:

whereN(h)isthenumberofpairsofdatalocationsatdistancehapart.

In practice, the average squared distance can beobtainedforallpairsseparatedbyarangeofdistancesand theseaverage squareddifferencescanbeplottedagainst theaverage separationdistance.A theoreticalmodelmight thenbefitted to theexperimentalsemi-variogram(Figure 1)andthecoefficientofthismodel(nuggeteffect,sillandrange)canbeusedforakrigingequationsystem.

Thekrigingmethodencompassesseveralwaysofintegratingauxiliaryvariables:– ifthemeanisnotconstant,butwecanestimatethe meanatlocationsinthedomainofinterest,thenthis

^N(h)γ(h)=1∑(Zsi-Z(si+h))

2(2)2N(h)i=1


locallyvaryingmeancanbeusedtoinformestimation usingSK;thisisreferredtoasSimpleKrigingwith aLocallyvaryingmean(SKL)(Goovaerts,2000);– Kriging with External Drift (KED) assumes that the mean of the interest variable depends on auxiliaryvariables;thetheorybehindKEDisinfact the same as the theory behind universal kriging, whichalsocontainsanon-constantmean.Thedrift

is defined externally through certain auxiliary variables(Hengletal.,2003);– inordertobettermeettheassumptionsofstationarity, linear regression may be carried out against secondaryvariablestoremovefirstordertrends.The residualscanbeusedtogenerateanewvariogram and then ordinary kriging can be applied to these residuals.The resulting estimates can be added to

Figure 1. Example of an experimental semi-variogram with different permissible models fitted—Exemple d’un semi-variogramme expérimental sur lequel différents modèles possibles sont ajustés.

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thetrendtogivetheestimatedvalues.Thistechnique hasbeentermedResidualKriging(RK)orDetrended Kriging(DK).– the other type of kriging, Ordinary Cokriging (OCK),involvesestimatingthevariableofinterest bytheweightedlinearcombinationofitsobservations andtheobservationsoftheauxiliaryvariables.This techniquerequiresthestudyofthespatialdependence betweenvariablesinadditiontothestudyofsimple spatialdependences.

A detailed presentation of geostatistical theoriescan be found in Cressie (1991); Goovaerts (1997);Chilèsetal.(1999)andWebsteretal.(2007).

3. APPLICATIONS AND PERFORMANCE OF DIFFERENT METHODS FOR SPATIAL INTERPOLATION OF RAINFALL DATA

Studies relating to the interpolation of precipitationofteninvolveacomparisonofmethods.Whenalargenumber of data are available, these comparisons aremadebydividingthedatasetintotwo:onesetofdatafor interpolation and the other for validation. Thismethod is idealbecause thevalidation is completelyindependentoftheformulationofthemodel.Oftenthedataareveryfewand,insuchcases,thecomparisonofmethodsisinsteadmadebycross-validation(Isaakset al., 1990). However, whether the validation isindependent or crossed, it allows the identificationof errors of prediction. Another way to compareinterpolationmethodsistouseahydrologicalmodel.Here,theinterpolatedrainfalldatacanbeusedasaninput into thehydrologicalmodel.Theobservedandsimulateddischargecanbecomparedandtheerrorofpredictioncanbefound.

Spatial interpolation techniques differ in theirassumption,deterministicorstatistical(geostatistical)nature,andlocalorglobalperspective.Deterministictechniques,suchasIDW,havebeenusedinnumerousstudies.EventhoughIDWisafairlystraightforwarddeterministic interpolation technique, which offersadaptable weights, the selection of the weightingfunction is subjective and no measure of erroris provided. Therefore, the literature has soughtto address questions regarding the bases for theapplication and further development of multivariategeostatistical techniques, such asKEDor cokriging,usingvariousco-variables.Itisoftenrecognizedthatthe statistical approach, geostatistical techniques orkriging,presentseveraladvantagesoverdeterministicmethods. Kriging presents an important advantagein its ability to give unbiased predictions withminimumvarianceandtotakeintoaccountthespatialcorrelationbetweenthedatarecordedatdifferentrain

gauges orweather stations. In addition to providinga measure of prediction error (kriging variance),another major advantage of kriging over simplermethods is that its geostatistical framework is alsoabletoaccommodatesecondaryinformationinorderto improve the interpolation results. For countrieswithaccesstosatellites,radar,microwavelinks,etc.,thedataobtainedvia these instrumentsaregenerallyusedtoimproveprecipitationinterpolation.However,in countries where these modern instruments arenot available, measurements of altitude, especiallyas extracted from a digital elevation model (DEM),form an extensively accessible data source, whichcan be incorporated into multivariate geostatisticalinterpolation of rainfall. Nevertheless, some studieshave shown that deterministic interpolationmethodsperformbetterthangeostatisticalmethodsandthattheresultsdependon the samplingdensity (Dirkset al.,1998).Dirksetal.(1998)comparedtheperformanceofIDW,THIandkrigingininterpolatingrainfalldatafrom a network of thirteen rain gauges on NorfolkIslandinallmultipletimesteps:hour,day,monthandyear.Theresults led theauthors torecommendIDWforinterpolationsforspatiallydensenetworksofraingauges.Most studies have used only daily,monthlyor annual time steps for precipitation interpolation.Moreover, someotherstudieshaveusedonlyhourlytime steps for large-scale extreme rainfall events.Validationsinthesestudieshaveoftenbeenperformedusingcross-validationmethods,althoughafewotherstudies have been based on results obtained throughhydrological modeling. However, no single methodhasbeen shown tobeoptimal for all time steps andconditions.

3.1. Studies investigating the performance of spatial interpolation methods for annual and monthly rainfall

Some studies have tested both deterministic andgeostatisticalmethods for interpolating rainfall data.Mostofthesestudieshaveusedonlymonthlyorannualtimestepsforprecipitationinterpolationandmapping.There have been many comparative assessments ofcommoninterpolationtechniques.

In their study of monthly totals in a large scalenetwork from a 30-year dataset of annual rainfall at29stations located in the North Central continentalUnited States, Tabios et al. (1985) found that thestatisticalmethodsofkrigingandoptimalinterpolationwere superior to other methods. The comparisonwas based on the error of estimates obtained at fiveselected sites.The authors found thatTHI and IDWgave fairly satisfactory results, while PI did notproduce good results. In a separate study, Phillips


et al. (1992) evaluated three geostatistical methodsfor making mean annual precipitation estimates ona regular grid of points in the mountainous terrainof theWillamette River basin. Results showed that,compared with ORK, DK and OCK both exhibitedbetter precision and accuracy. In that study, contourdiagramsforORKandDKexhibitedsmoothzonationfollowinggeneralelevationtrends,whileOCKshowedapatchierpatternmorecloselycorrespondingtolocaltopographic features. In another study,Abtew et al.(1993) applied six methods of spatial interpolationover a 4,000 squaremile area in South Florida andthe results validated historical observations. Resultsindicated that themultiquadric, kriging, and optimalinterpolationschemeswerethebestthreemethodsforinterpolationofmonthlyrainfallwithinthestudyarea.Theoptimalandkrigingmethodshavetheadvantageof providing the error of interpolation.Nalder et al.(1998) later used four types of kriging and threesimple alternatives to estimate 30-year averages ofmonthly precipitation at specific sites in westernCanada.Oneofthealternativeswasanewtechnique,termed “gradient-plus-inverse distance squared”(GIDS), which combines multiple linear regressionanddistance-weighting.Basedon themeanabsoluteerrors fromcross-validation tests, theauthors rankedthemethodsinorderofeffectivenessforinterpolatingmonthlyprecipitationasfollows:GIDS,OCK,IDW,NN,ORK,DKandUNK.TheyconcludedthatGIDSwasasimple,robustandaccurateinterpolationmethodforuseintheirregionofstudy,andthat itshouldbeapplicable elsewhere, subject to careful comparisonwithothermethods.Theauthorsalsoconcluded thatitwas unfair to use localmultiple linear regressionsfortherelevantstochasticprocedurefordeterministicmethods,butnotforgeostatisticalmethods(withthesameancillaryvariable).

Basistha et al. (2008) used data from 44stationsto generate a normal annual rainfall map in theHimalayanregionofIndialyinginUttarakhandstateata1-kmspatialresolution.Theauthorscarriedoutacomparativeanalysisbycross-validatinginterpolationtechniquesandfoundthatUNKincombinationwiththehole-effectmodel(thismodelrelatestotheexistenceoftwohighvaluedrainfallfieldsinthestudyarea)andnaturallogarithmictransformationwithconstanttrendand the smallest Root Mean Square Error (RMSE)constitutedthebestchoice.ThatwasfollowedbyORK,spline,IDWandPI.Goovaerts(2000)employedthreemultivariate geostatistical algorithms (SKL, KED,and OCK) incorporating a digital elevation modelforthespatialpredictionofrainfallusingannualandmonthlyrainfallobservationsmeasuredat36climaticstations in a 5,000km² region of Portugal. Duringcross-validation, the author found that these threemultivariate geostatistical algorithms outperformed

other interpolators, in particular linear regression, atechniquewhichstressestheimportanceofaccountingforspatiallydependentrainfallobservationsinadditionto theco-locatedelevation.Lastly,Goovaerts (2000)foundthatORKyieldedmoreaccuratepredictionsthanlinearregressionwhenthecorrelationbetweenrainfallandelevationwasmoderate. InGreatBritain,Lloyd(2005) applied monthly precipitation from sparsepointdatatoarangeofinterpolationmethods:MWR,IDW, ORK, SKL and KED. The MWR, SKL andKEDmethods relied on elevation data as secondaryinformation. Based on his examination of mappedestimates of precipitation and cross-validation, theauthor found that KED provided the most accurateestimatesofprecipitationforallmonthsfromMarchtoDecember,whereasforJanuaryandFebruary,ORKprovided the most accurate estimates. However, thedata for these few months cannot be used to drawaccurateconclusionsregardingthebetterperformanceofaparticulartechnique.ThereasonforLloyd(2005)finding KED to be the most accurate precipitationinterpolation method from March to December isthat, during these months, more neighborhood datawereusedforinterpolation.KEDestimatesbasedona larger neighborhood tend to be more accurate. Inanotherstudy,Diodato(2005)studiedtheinfluenceoftopographic co-variableson the spatial variabilityofprecipitation using rainfall observationsmeasured at51climaticstationsinacomplexmountainousregionof southern Italy (Benevento province). In additiontoemployingtheORKmethod, theauthoraddedforOCK twoauxiliaryvariablesof annual and seasonalprecipitation:terrainelevationdataandatopographicindex. Cross-validation indicated that ORK yieldedthe largest prediction errors.The smallest predictionerrorswereproducedbyamultivariategeostatisticalmethod.Diodato(2005)concludedthatOCKisaveryflexible and robust interpolation method because itis capable of taking into account several propertiesof the landscape. More recently, Moral (2010)applied a wide range of geostatistical methods tomonthly and annual precipitation data measured at136meteorologicalstationsinaregionofsouthwesternSpain (Extremadura). Cross-validation revealed thatthe smallest prediction errors were obtained for thethreemultivariatealgorithms. Inparticular,SKLandORKwerefoundtooutperformOCK,whichrequiresamoredemandingvariogramanalysis.

Inthestudiesdescribedinthissection,geostatisticalmethods were generally found to outperformdeterministic methods for spatial interpolation andmapping of monthly and annual precipitation. Inparticular, the use of multivariate geostatisticalmethods in combination with elevation data as thesecondary variablewas generally found to yield themostaccuratepredictions.


3.2. Studies investigating the performance of spatial interpolation methods for daily rainfall

Otherstudieshavefocusedontheuseofgeostatisticalandnon-geostatisticalapproachesfortheinterpolationof daily rainfall in different sizes of area. Therehave also been several comparative studies of theperformance of interpolationmethods used for dailyrainfall.

Employingageostatisticalapproach,Creutinetal.(1988)usedaraingauge-radarcombinationtomeasureelevendailyeventsofarealrainfallintheParisregion.AnexternalindependentvalidationindicatedthatOCKimprovedslightlytheperformanceoftherawradardataandthatthetechniqueexceededtheperformanceoftheclassicaluniformcalibrationmethod.Inanotherstudy,Beeketal.(1992)selectedfourdaysin1984inwhichto investigate the spatialvariability in theamountofdailyprecipitationinnorth-westernEuropeinrelationtometeorological conditions.Datawere interpolatedusingkriging.Cross-validationshowedtheoccurrenceof large differences in the spatial structure of theamount of daily precipitation as a result of differentmeteorological conditions. Stratification of the studyarea intoacoast, amountainandan interior stratumproved to be successful, reducing theMeanSquaredErrorofpredictiontoalevelof55%.

Kyriakidis et al. (2001) mapped the seasonalaverage of daily precipitation for the period from1November 1981 to 31January 1982 over a regioninnorthernCaliforniaata1-kmresolution.Thestudydemonstrated the feasibility of constructing realisticanalysesofprecipitation.Theauthorsintegratedreadilyavailable and physically relevant predictors, such asatmosphericandterraincharacteristics,whichcontrolthe spatial distribution of precipitation at regionalscale.Differentinterpolationmethodswerecomparedin terms of cross-validation statistics and the spatialcharacteristics of cross-validation errors. Interactionsbetween lower-atmosphere state variables (humidityand horizontal wind components) and terrain (bothelevation and its local gradients) provide valuableinformation for mapping the spatial distribution oforographic precipitation. A geostatistical frameworkusing themaximum amount of relevant atmosphericand terrain information could lead to more accuraterepresentations of the spatial distribution of rainfallthanthosefoundintraditionalanalysesbasedonlyonraingaugedata.Themagnitudeof this improvementin accuracy, however, would depend on the densityoftheraingaugestations,onthespatialvariabilityoftheprecipitationfield,andonthedegreeofcorrelationbetweenrainfallanditspredictors.Classicalobjectiveanalysisschemesignoreimportantrelevantinformationsuch as humidity and vertical wind, and theyconsequently produce over-smooth representations

ofthespatialdistributionofrainfall;suchanadverseeffect is intensified when the rain gauge network issparse.

Buytaert et al. (2006) studied the variability ofspatial and temporal rainfall in the southEcuadorianAndesusingtheTHImethodandkrigingwith14raingaugesinthewesternmountainrangeoftheEcuadorianAndes.However,thenumberofraingaugesinthatstudywastoosmalltoallowtheproductionofaninformativevariogram using standard estimation (means of thedifference between each data pair; see equation2).Therefore, when data series were available at eachpoint,theexperimentalsemi-variogramproducedwascalculatedinanotherway,resemblingmorecloselythedefinitionofsemi-variance:

Cross-validation undertaken by Buytaert et al.(2006) showed that spatial interpolationwithkrigingprovided a better result than the one with THI, andthat the accuracy of both methods improved whenexternalvariableswereincluded.TheexternalvariableintegratedintoTHIreferredtodatanormalizationbasedonthecorrelationbetweenthemeandailyrainfallandtheexternalparameters.TheexternalvariableincludedinthekrigingmethodreferredtotheUNKprocess.

Schuurmansetal.(2007b)performedpredictionsofpoint rainfall usingORK and investigated the addedvalue of operational radar for KED and OCK withrespect to rain gauges in obtaining a high-resolutiondaily rainfall field. The spatial variability of dailyrainfallwasinvestigatedatthreespatialextents:225,10,000and82,875km²,withselectedrainfallevents.Cross-validationundertakeninthestudyshowedthatmethodsusingboth radar and raingaugedata (KEDandOCK)provedtobemoreaccuratethanusingraingaugedataalone(ORK),inparticular,forlargerspatialextents. Inaseparatestudy,Carrera-Hernandezetal.(2007) used various forms of geostatistical methodto analyze daily climatic data from approximately200stations located in the Basin of Mexico for themonths of June 1978 and June 1985. The results ofcross-validationshowedthattheinterpolationofdailyevents was improved by the use of elevation as asecondaryvariableevenwhenthatvariableshowedalowcorrelation.

In the studies described in this section, cross-validation of results showed that kriging methodsoutperformeddeterministicmethodsforthecalculationofdailyprecipitation.However,bothtypesofmethodwerefoundtobecomparableintermsofhydrological

γ(h)=

Var(Zsi-Z(si+h))=2

Var(Zsi)+Var(Z(si+h))-2Cov(Zsi,Z(si+h)).(3)


modelingresults.Inthestudiesdescribed,bothelevationandradardatawereusedinmultivariategeostatisticalcalculationsforthedailytimestep.However,veryfewmethodswerecomparedwithinanyonestudy.

3.3. Studies investigating the performance of spatial interpolation methods for hourly rainfall

A small number of studies have considered usinghourly time steps for large-scale extreme rainfallevents. With specific reference to flood events,Haberlandt (2007) used the combined techniques ofKEDandindicatorkrigingwithexternaldrift(IKED)forthespatialinterpolationofhourlyrainfallfromraingauges using secondary variables from radar, dailyprecipitationwithin a dense network of rain gauges,and elevation. The methods were performed usingdatafromthestormperiodof10thto13thAugust2002,whichledtoaseverefloodeventintheElberiverbasininGermany.Cross-validationcarriedoutinthestudyshowed that the multivariate geostatistical methodsKEDandIKEDobviouslyoutperformedtheunivariatemethods,withthemostsignificantaddedvaluebeingradar,followedbyrainfallfromthedailynetworkandelevation.TheauthorsfoundthatthebestperformancewasachievedwhenallsecondaryinformationwasusedatthesametimewithKED.Insomecases,indicator-based kriging techniques give a smaller RMSE thanunivariatetechniques,whichuseonlyasinglesourceofinformation,butthisisattheexpenseofaconsiderablelossofvariance.

Also focusing on hourly precipitation, Velasco-Forero et al. (2009) compared three geostatisticalmethods, all incorporating radar data as auxiliaryvariables in combination with a non-parametrictechnique to automatically compute correlograms.Cross-validation and spatial pattern analysis showedthat KED produced the most accurate results.Schiemann et al. (2011) also used a geostatisticalradar-rain gauge combination with non-parametriccorrelograms and parametric semi-variogrammodelsfor the construction of hourly precipitation grids forSwitzerland, based on data from a sparse real-timenetworkofraingaugesandfromaspatiallycompleteradarcomposite.Inthatstudy,cross-validationshowedtheKEDformulation tobebeneficial,particularly inmountainous regions where the quality of the Swissradar composite was comparatively low. Recently,Verwornetal.(2011)usedamultivariategeostatisticalapproach(KED)forthespatialinterpolationofhourlyrainfall, using auxiliary topographic data, rainfalldata from dense daily networks and weather radardata. The study analyzed certain inundation eventsoccurringbetween2000and2005causedbydiversemeteorological conditions in northern Germany.Through cross-validation, the authors found that

weather radar data were the most useful secondarydataforKEDforconvectivesummerevents,whilefortheinterpolationofstratiformwinterevents,theuseofdailyprecipitationassecondarydatawassatisfactory.Generally, the density of rain gauges is usually notsufficient toproduceuseful variogramson anhourlybasis;so theadvantageofusinga radarestimate liesnotonlyinitsabilitytogivetheapproximateexternalvariable,butalsointhecluesitprovidesregardingthespatialstructureofrainfall.Thevariogramisnotgivenbythereferencedata(raingauges)butbytheancillarydata.

Intheirinvestigationofhourlyrainfall,thestudiesdescribed in this section focused on large-scaleextreme rainfall events.A multivariate geostatisticalmethod(KED)wastheonemostcommonlyemployed,typically with the incorporation of radar data asthe secondary data source. Generally, multivariategeostatistical methods were shown to outperformunivariatemethodsinthesestudies.

3.4. Validation of interpolation methods using hydrological modeling

Another way to compare the alternative spatialinterpolation methods is to produce and comparevarious time-series of daily areal precipitationdistributions using not only an internal precipitationvalidation, but also an objective verification basedonstreamflowsimulations(Haberlandtetal.,1998).Haberlandt et al. (1998) used the Mackenzie RiverBasin in north-western Canada as their study area,carryingouthydrologicalsimulationsusingtheSemi-distributedLandUse-basedRunoffProcesses(SLURP)model. The authors found that better interpolationtechniquesandtheuseofcombinedprecipitationdatawereabletoimprovehydrologicalsimulationsandthattheseimprovementswererelatedtotherelativesizeofthesimulationunitsused.Inaseparatestudy,Ruellandetal.(2008)analyzedthesensitivityofalumpedandsemi-distributed hydrological model (Hydrostrahler)to several spatial interpolations of rainfall data. Thestudywas carriedoutwithin a contextof scarcityofdataoveralargeWestAfricanwatershed.

PointbypointassessmentshowsthatinterpolationusingIDWandkrigingmethodsismoreefficientthanthesimpleThiessentechnique,andspline,particularlywhen a monthly time step is used. In fact, splineinterpolationcanbeshowntoperformequallyaswellasthekrigingmethodiftheappropriatecovarianceisused. Inotherwords,asplineestimatewitharbitraryparameterswillperforminthesamewayasakrigingtechniquewithanarbitraryvariogram.InthestudyofRuellandetal.(2008),theuseofthesedatainadailylumped modeling showed a different ranking of thevarious interpolationmethodswith regard to various


hydrological assessments. That model seems to beparticularlysensitivetothedifferencesinthevolumeof rainfall input produced by each interpolationmethod.Infact,themodeliscalibratedforeachrainfallinput.Compensationmaybebuilt into themodel forinadequaterainfalldata.

Recently, Masih et al. (2011) used a semi-distributedhydrologicalSoil&WaterAssessmentTool(SWAT)model tocompare thatmodel’sperformanceunder standard precipitation input and modifiedareal precipitation input obtained through the spatialinterpolationInverseDistanceandElevationWeighting(IDEW)method.The authors concluded that the useof areal precipitation, obtained through interpolationof the available stationdata, improvedSWATmodelsimulated stream flows. The results were stronglyinfluencedbythespatialextentoftheinvestigationsaswellasbythestationdensityandspatialdistributionoftheavailableraingaugedatausedintheinterpolation.Moreover, theauthors strongly recommended furthertesting of the SWATmodel using areal precipitationasan inputobtained through theapplicationofotherinterpolation methods to rain gauge records. Theyhighlighted the fact that the SWAT model addedvaluetostreamflowsimulationsandotherprocesses.Theyalsosuggested thatdevelopmentofanoptionalcomponentfortheinterpolationofclimaticdatawithinthe existing SWAT model would benefit multipleSWATusers. In a separate study,Tobin et al. (2011)presented a comparative analysis of the performanceof IDW, ORK, and KED for hourly precipitationfieldsincomplexAlpineterrain.Therelevanceofthestudyhingedonthemajorimpactmadebyinadequateinterpolationsofmeteorologicalforcingontheaccuracyofhydrologicalpredictionsregardlessofthespecificsof the semi-distributed GSM-SOCONT models,during threefloodevents.Thegeostatisticalmethodsusedreliedona robustanisotropicvariogramfor thedefinitionofthespatialrainfallstructure.ResultsfromcumulativeprecipitationanalysisinthestudyindicatedthatIDWtendedtosignificantlyunderestimaterainfallvolumes, whereas ORK andKEDmethods capturedspatialpatternsandrainfallvolumesinducedbystormadvection.Theuseofnumericalweatherforecastsandelevationdataascovariatesforprecipitationprovidedevidence forKED outperforming the othermethods.Hydrological simulations run with KED-interpolatedinputfieldssignificantlyimprovedresults intermsofspecificrunoffvolume.Themodelwasnotre-calibratedwitheachtechnique.

Aswehaveseen in thissection,veryfewstudieshave focused on comparing the performance ofdifferent interpolation methods as evaluated byhydrological modeling. Based on these few studies,the performance of the IDWmethod can be said tobecomparable to thatof theORKmethod(Ruelland

etal.,2008;Masihetal.,2011).KEDwasnotincludedin these twostudies,but this techniquedemonstratedthe best performance in the study of Tobin et al.(2011).

3.5. Use of variogram models and negative weight in geostatistical methods for interpolating rainfall

Previousstudiesinvestigatinggeostatisticalapproacheshavegenerallyappliedthesametheoreticalvariogrammodelforalltimesteps.Thesphericalmodelhasbeentheonemostcommonlychosentointerpolaterainfall.Recently, Van De Beek et al. (2011) applied thesphericalmodelinexaminingtheseasonalvariogramparameters of daily rainfall in The Netherlands.Similarly,Verwornetal.(2011)appliedthesphericalmodeltointerpolatehourlyrainfallinthenorthernpartofGermany.However,negativeinterpolatedvaluescanoccurinkriging(Deutsch,1996).Onceasampleclosetothelocationbeinginterpolatedexhibitsanoutlyingvalue,negativeweightsinORKoccur.Thesenegativeweights can be large, depending on the variogrammodelandthedatavalues.Moreover,negativeweightsmay produce negative and nonphysical interpolatedvalueswhen applied to high data values. In general,twoapproachescanbeusedtoavoidanegativevalue:a posteriori correction as recommended by Deutsch(1996) or replacing all negative interpolated valueswith a zero value. Both are realistic solutions, butneitherisperfect.Despitethefactthat,insomecases,thenegativevaluesarelarge,nostudyhasyetexaminedthisproblem.

4. DISCUSSION

In the literature, the results of the comparison ofinterpolationmethodsdifferfromonestudytoanother.The successful performance of themethods dependson several factors, inparticular, temporal and spatialresolutions of the data, and the parameters of themodels, such as the semi-variogram in the case ofkriging. The studies discussed here focused on theanalysis of annual, monthly, daily, hourly or totalrainfall for precipitation events of some durationwithdifferentdensitiesofobservationnetworks. It isthus difficult to draw a general conclusion. No oneinterpolationmethod stands out as being universallythebest.Someauthorsrecommendaparticularmethodasbeingthebestaccordingtotheirjudgmentastowhatisthemostpractical(Tabiosetal.,1985;Abtewetal.,1993; Syed et al., 2003). These authors note all therelatively equivalent levels of performance betweentheORK technique and themultiquadratic functions(splinetype).However,bothTabiosetal.(1985)andAbtewet al. (1993) recommend theuseof theORK


techniquebecauseitallowsthecalculationoferrorsofprediction.Ontheotherhand,Syedetal. (2003)chosetoemploythemultiquadraticfunctionsbecausethey consider them easier to use. Also, within thecontext of a dense network of stations,Dirks et al.(1998) did not obtain significant improvements inresults by using ORK rather than IDW. They thusrecommend the simpler IDW method. In fact, thisobservation of the better performance of IDW hasbeen extended to its use with other types of data.For example, in an analysis of synthetic data froma computational experiment, Zimmerman et al.(1999) obtained a better interpolationwithORKorwithUNKthanwiththeIDWmethodonlywhenthesampling point was regular, the noise low and thespatialcorrelationstrong.

Several studies have examined methodsof multivariate interpolation. In some studies,radar-rainfall data have been used in combinationwith measurement at weather stations for spatialinterpolation of precipitation (Creutin et al., 1988;Haberlandt, 2007; Schuurmans et al., 2007b;Velasco-Forero et al., 2009). However, the bulk ofstudieshavemadeuseofacheaper,widelyavailabledata source, the Digital Elevation Model (DEM),takingadvantageoftherelationshipbetweenamountofprecipitationandelevation.Inparticular,Phillipsetal.(1992);Nalderetal.(1998);Goovaerts(2000)and Lloyd (2005) incorporated elevation into theinterpolation of precipitation.These authorsmainlyusedspline,SKL,RKorDK,KEDandOCK.Thesemultivariatemethodsseemtogivebetterresultsthanunivariate methods in mountainous regions for ascale of about 10,000km² (Phillips et al., 1992) orwhen the correlation between the rainfall data andtheelevation ishigher than0.75(Goovaerts,2000).It is important to note that all these studies wereconductedusingannualormonthlyprecipitation.Forfinertemporalresolutions,suchasadailyresolution,a strong relationship between elevation andprecipitationisquestionable,accordingtoHaberlandtet al. (1998). Even though these authors observedan average correlation of 0.52 between elevationand the annual accumulation of precipitation, thiscorrelation fell to0.06 fordailyobservations.Theythusreliedmoreonintegrationintotheinterpolationofanotherauxiliaryvariable:precipitationsimulatedby an atmospheric model. Furthermore, theseauthors studied the interpolation of precipitationwithin a context of hydrological modeling andused hydrological simulations in addition to cross-validation to compare their tested methods. Theonly multivariate method that they examined wasKED. They applied this method either to all thetime steps of their test period, or only when thecorrelationbetweentheobservationsofstationsand

theauxiliarydataexceededacertainthreshold(0.5or0.3dependingontheauxiliaryvariableinquestion).Wherethecorrelationwastoolow,theyusedORK.The authors obtained better results by applying theKEDmethodconditionallyratherthanbyusingitforevery time step.On the other hand, the indicationsobtainedviacross-validationfortheconditionalKEDwere only very slightly better than those forORK.Furthermore,forthehydrologicalsimulations,itwasORKthatgavethebestresults.Multivariatemethodswere found to bring an improvement to the qualityoftheinterpolationonlywhentheywereusedattheright time, but this improvement did not seem tohaveagreatimpactonthequalityofthehydrologicalmodeling.

Incontrast,Ruellandetal.(2008)foundadifferentranking of the various interpolation methods usedbetweenpointbypointassessmentandhydrologicalsimulation.They found that accurate assessment ofthe rainfall input volume was more important thanthe rainfallpattern itself for simulating streamflowhydrographs.They reached this conclusion throughthe use of a lumped model. This model does notaccount for the spatial variability of precipitationinput with the basin. Masih et al. (2011) foundIDEW to be a good method for rainfall input in asemi-distributedSWATmodel.However,theauthorsdidnotmakeacomparisonbetweenIDEWandanygeostatisticalmethods,whichareusuallyfoundtobesuperiortosuchasimplemethod.Toovercomesuchlimitations,moretypesofgeostatisticalmethodsarecurrentlybeingtestedtopreparehourlyrainfallinputforhydrologicalmodelingduringfloodevents(Tobinet al., 2011).The use of improved rainfall input inKEDprovidesevidenceforincreasedaccuracyinthepredictionofdischargevolumeandpeaks.

For annual and monthly rainfall, geostatisticalmethods appear preferable particularly,multivariategeostatistical methods which can be beneficialwhen using elevation data as a secondary variable.On the other hand, for daily rainfall, multivariategeostatisticalmethods and IDWare in competition.ThisisprobablyduetothefactthatstudiesindicatingthebetterperformanceofIDWwereconductedusingonly one geostatistical method (ORK) and/or othersimplemethodssuchasTHIandspline,whilestudiesindicating the better performance of multivariategeostatisticalmethodsonlymadeacomparisonwithinthe family of geostatistical methods. Some authorshaveusedradardataasasecondaryvariable,whichis normally well correlated with rainfall from raingauges thanks to the similar nature of the variable.A comparison between common deterministicmethods(suchasTHIandIDW)anddifferenttypesofgeostatisticalmethodshasnotyetbeenmadefordailyrainfall.


More recent studies have focused on hourlyrainfall rather than on calculating rainfall in othertime steps, as in previous studies. These morerecent studies have been conducted in developedcountrieswheremodern instruments, such as radar,are available.As a result, radar rainfall has alwaysbeenusedinthesestudiesasasecondaryvariableinmultivariategeostatisticalanalysis(KEDandOCK).Thesetwomethodshavealwaysdemonstratedbetterperformance in comparison with other types ofgeostatistical methods. However, the transpositionofthesemethods(usingradarrainfall)todevelopingcountriescannotbemadeunlessmoderninstrumentsare installed. This is very costly. Therefore, othercheap data such as elevation should be used as asecondaryvariableforincorporationintomultivariategeostatisticalmethods.

Radar rainfall andelevationhavegenerallybeenused as the secondary variables for integration intomultivariategeostatisticalmethods.Radarrainfallhasbeen found to be beneficialwhen incorporated intomultivariategeostatisticalmethodsintheinterpolationofdailyandhourlyrainfall.Elevationhasalsoprovidedamajoradvantageinimprovingtheuseofmultivariategeostatisticalmethodsforinterpolatingmonthlyandannual rainfall. However, very few studies havefocusedon incorporatingelevation intomultivariategeostatisticalmethodsfordailyrainfallinterpolation.Thereliabilityofpredictionsmayvaryifadifferenttime step is chosen. The stochastic nature of dailyrainfall, in particular, differs from that of monthlyorannualrainfall.Therefore,itwouldbeinterestingto discover whether integration of elevation as asecondary variable improves interpolation accuracy,because rainfalldata aremostly available at adailytimestep incountrywideor regionalmeasurements.Daily rainfall is the most important meteorologicalinputintowaterresourcesandagriculturalmodelingsystems.Thequestioniswhetherwhatconstitutesthebest technique when applied to monthly or annualrainfall is also appropriate to apply todaily rainfallwhenprecipitationpatterndifferencesexistbetweendailyandmonthlytimescales.

Veryfewanalyseshavebeenmadeoftheimpactofraingaugedensityoninterpolationmethods.Somestudies have focusedprimarily on the effect of raingaugedensityusingonlyonemethod(ORK,UNKorKED)ortwomethods(multiquadraticsurfacefittingandkriging)(Borgaetal.,1997).Thisuseofonlyoneortwomethodsmayhavebeenduetothecumbersomenatureoftheanalysis,intermsofcomputationtime.However, given that computational facilities arenow better developed and more widely available,it would be interesting to nowmake a comparisonbetween a wider range of techniques. This mightprovidesomeinsightsintermsofparticularstrengths,

weaknessesandapplicabilityofavarietyofmethods.Such analyses related to rain gauge density wouldbe valuable for engineers, hydrologists or decisionmakersworkingwithsparseraingaugedata.

Solutions to the problem of negative weightsin kriging are extremely limited in the method’sapplication to rainfall. We recommend furtherinvestigation into how negative results can beeliminatedthroughusingkriging.Forexample,sincenegativekrigedvaluesmaybegeneratedastheresultof a chosen variogram model, several variogrammodelscouldbeusedtominimizetheriskofnegativeresults appearing. Using a variety of variogrammodelsmightavoidnegativerainfallcalculations.

5. CONCLUSION

This article has presented and discussed previousstudiesrelatedtospatialinterpolationofrainfall.Themainconclusionsdrawnherecanbesummarizedasfollows:– for annual and monthly rainfall, geostatistical interpolation methods seem preferable to deterministic methods. In particular, the use of multivariategeostatisticalmethodsincombination with elevation data has generally yielded more accurateinterpolations;– fordailyrainfall,geostatisticalmethodsandIDW have proved to be comparable approaches, in particular for hydrological modeling. However, veryfewstudieshavefocusedonincorporatingthe variableofelevationintomultivariategeostatistical interpolation of daily rainfall.Moreover, the use ofdifferently interpolated rainfallasan input for hydrologicalmodelshasbeenverylittlestudied;– most authors have applied radar data as the secondaryvariablewhenanalyzinghourlyrainfall. Studiesfollowingthistrendhavebeencarriedout mostlywithmultivariategeostatistics(KED)anda fewotherunivariatemethods;– limitedcomparisonhasbeenmadewithinastudy between the use of common deterministic and different types of geostatistical interpolation methods,inparticularfordailyrainfall;– the impactof raingaugedensityon interpolation methodshasbeenverylittlestudied.

Thestudies reportedherehavemadeusstronglyaware of the need for further research in order todiscoverthewaysandmeanstoimprovetheaccuracyof rainfall input for hydrological modeling. Theinvestigationsundertakensofarhavebeenrestrictedin numerous aspects, thereby stressing the need forfurtherresearch.Theyhave,however,providedveryusefulstepsinthatdirection.


Quantificationandawarenessoftheuncertaintiesassociatedwith hydrological data are thus essentialfor the correct interpretation of the results of themodeling.Thepreciseevaluationofthespatiotemporalvariabilityofrainfallonthewatershedscalepresentsacomplexproblembecauseofthesmallnumberofraingaugesinmostcasesandbecauserainfallisextremelyvariedinspaceandtime.Thechoiceofinterpolationmethodformeasuringrainfalldependsonthequantityofvalidmeasures,thenatureoftherainintheregionsunder study and the quality of the observations.Thechoice of method is therefore crucial. Furthermore,a sensitivity analysis of a hydrological model canbe a complementary indicator of the quality of theinterpolation of rainfall and of other meteorologicalparameters. Thus, strategies for the acquisition andthepretreatmentofdatacanbebetterrealizedsoastoachieveamoreefficienthydrologicalmodeling.

Acknowledgements

Sarann Ly is financially supported by a scholarship fromthe cooperation project CUI-ITC01 funded by CUD(Commission Universitaire pour le Développement),Belgium.

Bibliography

AbtewW.,ObeysekeraJ.&ShihG.,1993.Spatialanalysisfor monthly rainfall in South Florida. J. Am. Water Resour. Assoc.,29(2),179-188.

BasisthaA., AryaD.S. & GoelN.K., 2008. SpatialdistributionofrainfallinIndianHimalayas-acasestudyofUttarakhandregion.Water Resour. Manage.,22(10),1325-1346.

BeekE.G., SteinA. & JanssenL.L.F., 1992. Spatialvariability and interpolation of daily precipitationamount.Stochastic Hydrol. Hydraul.,6(4),304-320.

BellV.A.&MooreR.J.,2000.Thesensitivityofcatchmentrunoffmodelstorainfalldataatdifferentspatialscales.Hydrol. Earth Syst. Sci.,4(4),653-667.

BorgaM. & VizzaccaroA., 1997. On the interpolationof hydrologic variables: formal equivalence ofmultiquadratic surface fitting and kriging. J. Hydrol.,195(1-4),160-171.

BuytaertW. et al., 2006. Spatial and temporal rainfallvariabilityinmountainousareas:acasestudyfromtheSouth Ecuadorian Andes. J. Hydrol., 329(3-4), 413-421.

Carrera-HernandezJ.J. & GaskinS.J., 2007. Spatiotemporalanalysisofdailyprecipitationandtemperaturein the Basin of Mexico. J. Hydrol., 336(3-4), 231-249.

CarusoC. & QuartaF., 1998. Interpolation methodscomparison.Comput. Math. Appl.,35(12),109-126.

ChaubeyI., HaanC.T., GrunwaldS. & SalisburyJ.M.,1999.Uncertaintyinthemodelparametersduetospatialvariabilityofrainfall.J. Hydrol.,220(1-2),48-61.

ChilèsJ.P. & DelfinerP., 1999. Geostatistics: modelling spatial uncertainty. 1st ed. New York, USA: Wiley-Interscience.

ChowV.T., 1964. Handbook of applied hydrology: a compendium of water-resources technology. 1st ed.NewYork,USA:McGraw-Hill,Inc.

ChowV.T., MaidmentD.R. & MaysL.W., 1988. Applied hydrology. International ed. Singapore: McGraw-Hill,Inc.

CressieN.,1991.Statistics for spatial data.1sted.NewYork,USA:Wiley.

CreutinJ.D.,DelrieuG.&LebelT.,1988.Rainmeasurementbyraingage-radarcombination:ageostatisticalapproach.J. Atmos. Oceanic Technol.,5(1),102-115.

DeutschC.V., 1996. Correcting for negative weights inordinary kriging. Comput. Geosci.-Uk, 22(7), 765-773.

DiodatoN.,2005.Theinfluenceoftopographicco-variableson the spatial variability of precipitation over smallregionsofcomplexterrain.Int. J. Climatol.,25(3),351-363.

DirksK.N.,HayJ.E.,StowC.D.&HarrisD.,1998.High-resolutionstudiesof rainfallonNorfolk Island.Part2:Interpolationofrainfalldata.J. Hydrol.,208(3-4),187-193.

GoovaertsP., 1997. Geostatistics for natural resources evaluation.NewYork,USA:OxfordUniversityPress.

GoovaertsP.,1999.Usingelevationtoaidthegeostatisticalmappingofrainfallerosivity.Catena,34(3-4),227-242.

GoovaertsP., 2000. Geostatistical approaches forincorporatingelevation into thespatial interpolationofrainfall.J. Hydrol.,228(1-2),113-129.

GroismanP.Y. & LegatesD.R., 1994. The accuracy ofUnited States precipitation data. Bull. Am. Meteorol. Soc.,75(2),215-227.

HaberlandtU.,2007.Geostatistical interpolationofhourlyprecipitationfromraingaugesandradarforalarge-scaleextremerainfallevent.J. Hydrol.,332(1-2),144-157.

HaberlandtU.&KiteG.W.,1998.Estimationofdailyspace-time precipitation series for macroscale hydrologicalmodelling.Hydrol. Process,12(9),1419-1432.

HenglT.,GeuvelinkG.B.M.&SteinA.,2003.Comparison of kriging with external drift and regression kriging.Enschede,TheNetherlands:ITC.

IsaaksE.H. & SrivastavaR.M., 1990. An introduction to applied geostatistics. New York, USA: OxfordUniversityPress.

JohnsonG.L. & HansonC.L., 1995. Topographic andatmosphericinfluencesonprecipitationvariabilityoveramountainouswatershed.J. Appl. Meteorol.,34(1),68-87.

KyriakidisP.C.,KimJ.&MillerN.L.,2001.Geostatisticalmapping of precipitation from rain gauge data using


atmospheric and terrain characteristics. J. Appl. Meteorol.,40(11),1855-1877.

LanzaL.G., RamirezJ.A. & TodiniE., 2001. Stochasticrainfall interpolation and downscaling. Hydrol. Earth Syst. Sci.,5(2),139-143.

LloydC.D., 2005. Assessing the effect of integratingelevationdataintotheestimationofmonthlyprecipitationinGreatBritain.J. Hydrol.,308(1-4),128-150.

LooperJ.P.&VieuxB.E.,2011.Anassessmentofdistributedflash flood forecasting accuracy using radar and raingauge input for aphysics-baseddistributedhydrologicmodel.J. Hydrol.,412-413,114-132.

MandapakaP.V.,KrajewskiW.F.,MantillaR.&GuptaV.K.,2009.Dissectingtheeffectofrainfallvariabilityonthestatistical structure of peak flows.Adv. Water Resour.,32(10),1508-1525.

MasihI.,MaskeyS.,UhlenbrookS.&SmakhtinV., 2011.Assessing the impact of areal precipitation input onstreamflowsimulationsusing theSWATmodel.J. Am. Water Resour. Assoc.,47(1),179-195.

MatheronG., 1971. The theory of regionalized variables and its applications.Paris:ÉcoleNationaleSupérieuredesMinesdeParis.

MoralF.J., 2010. Comparison of different geostatisticalapproaches to map climate variables: application toprecipitation.Int. J. Climatol.,30(4),620-631.

NalderI.A. & WeinR.W., 1998. Spatial interpolation ofclimaticnormals:testofanewmethodintheCanadianboreal forest. Agric. For. Meteorol., 92(4), 211-225.

NicotinaL.,CelegonE.A.,RinaldoA.&MaraniM.,2008.On the impact of rainfall patterns on the hydrologicresponse.Water Resour. Res.,44(12).

ObledC.,WendlingJ. & BevenK., 1994. The sensitivityof hydrological models to spatial rainfall patterns: anevaluation using observed data. J. Hydrol., 159(1-4),305-333.

PhillipsD.L., DolphJ. & MarksD., 1992. A comparisonof geostatistical procedures for spatial analysis ofprecipitation in mountainous terrain. Agric. For. Meteorol.,58(1-2),119-141.

RefsgaardJ.C., 1996. Terminology, modelling protocoland classification of hydrological model codes. In:AbbottM.B. & RefsgaardJ.C., eds. Distributed hydrological modelling. Dordrecht, The Netherlands:KluwerAcademicPublishers,17-39.

RuellandD.,Ardoin-BardinS.,BillenG.&ServatE.,2008.Sensitivityofalumpedandsemi-distributedhydrologicalmodel to several methods of rainfall interpolation ona large basin inWestAfrica.J. Hydrol.,361(1-2), 96-117.

SchiemannR. et al., 2011. Geostatistical radar-raingaugecombination with nonparametric correlograms:methodological considerations and application inSwitzerland. Hydrol. Earth Syst. Sci., 15(5), 1515-1536.

SchuurmansJ.M. & BierkensM.F.P., 2007a. Effect ofspatialdistributionofdailyrainfalloninteriorcatchmentresponse of a distributed hydrological model.Hydrol. Earth Syst. Sci.,11(2),677-693.

SchuurmansJ.M., BierkensM.F.P., PebesmaE.J. &UijlenhoetR., 2007b. Automatic prediction of high-resolutiondailyrainfallfieldsformultipleextents:thepotential of operational radar.J.Hydrometeorol.,8(6),1204-1224.

SegondM.L., WheaterH.S. & OnofC., 2007. Thesignificance of spatial rainfall representation for floodrunoffestimation:anumericalevaluationbasedon theLeecatchment,UK.J. Hydrol.,347(1-2),116-131.

SevrukB., 1997. Regional dependency of precipitation-altitude relationship in the SwissAlps.Clim.Change,36(3-4),355-369.

ShahS.M.S., O’ConnellP.E. & HoskingJ.R.M., 1996a.Modellingtheeffectsofspatialvariabilityinrainfalloncatchment response. 1. Formulation and calibration ofa stochastic rainfall field model. J.Hydrol., 175(1-4),67-88.

ShahS.M.S., O’ConnellP.E. & HoskingJ.R.M., 1996b.Modellingtheeffectsofspatialvariabilityinrainfalloncatchmentresponse.2.Experimentswithdistributedandlumpedmodels.J. Hydrol.,175(1-4),89-111.

SinclairM.R., WrattD.S., HendersonR.D. & GrayW.R.,1997.Factorsaffectingthedistributionandspilloverofprecipitationin theSouthernAlpsofNewZealand–acasestudy.J. Appl. Meteorol.,36(5),428-442.

SinghV.P., 1995.Watershedmodeling. In: SinghV.P., ed.Computer models of watershed hydrology. Colorado,USA:WaterResourcesPublications,1-22.

SinghV.P.,1997.Effectofspatialandtemporalvariabilityinrainfallandwatershedcharacteristicsonstreamflowhydrograph.Hydrol. Process,11(12),1649-1669.

SyedK., GoodrichD., MyersD. & SorooshianS., 2003.Spatialcharacteristicsofthunderstormrainfallfieldsandtheirrelationtorunoff.J. Hydrol.,271(1-4),1-21.

TabiosG.Q. & SalasJ.D., 1985. A comparative analysisof techniques for spatial interpolation of precipitation.Water Resour. Bull.,21,265-380.

TobinC. et al., 2011. Improved interpolation ofmeteorologicalforcingsforhydrologicapplicationsinaSwissalpineregion.J. Hydrol.,401(1-2),77-89.

VanDeBeekC.Z.,LeijnseH.,TorfsP.J.J.F.&UijlenhoetR.,2011.Climatologyofdailyrainfallsemi-varianceinTheNetherlands.Hydrol. Earth Syst. Sci.,15(1),171-183.

Velasco-ForeroC.A., Sempere-TorresD., CassiragaE.F.& Gomez-HernandezJ.J., 2009. A non-parametricautomatic blending methodology to estimate rainfallfieldsfromraingaugeandradardata.Adv. Water Resour.,32(7),986-1002.

VerwornA. & HaberlandtU., 2011. Spatial interpolationof hourly rainfall - effect of additional information,variograminferenceandstormproperties.Hydrol. Earth Syst. Sci.,15(2),569-584.


WebsterR. & OliverM.A., 2007. Geostatistics for environmental scientists. 2nd ed. Chichester, UK: JohnWiley&Sons.

ZimmermanD., PavlikC., RugglesA. & ArmstrongM.,1999. An experimental comparison of ordinary and

universalkrigingandinversedistanceweighting.Math. Geol.,31(4),375-390.

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